Using the Cosmic Microwave Background to Probe Structure at Intermediate Redshift

ILLUMINATING THE UNIVERSE: USING THE COSMIC MICROWAVE BACKGROUND TO PROBE STRUCTURE AT INTERMEDIATE REDSHIFT

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF PHYSICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Stephen John Osborne June 2013

This dissertation is online at: http://purl.stanford.edu/sz245wk1516

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Sarah Church, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Steven Allen

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Chao-Lin Kuo

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii iv Abstract

The cosmic microwave background (CMB) provides a backlight that allows us to probe structure out to the last scattering surface. We exploit observations of the sky at microwave and sub-mm wavelengths to measure properties of galaxies and galaxy clusters, as well as to search for possible pre-inflationary signals. In Chapters 2–4 we measure the correlation between the dark matter distribution and the microwave and sub-mm emission from galaxies to probe the connection between dark and luminous matter at redshifts 1 3. ⇠ CMB photons are gravitationally deflected by dark matter overdensities, with the majority of the 3 arcminute RMS deflection occurring between redshift 2 and 3. The dark matter ⇠ structures that lens the CMB are traced by dusty star-forming galaxies that emit strongly in the infrared, and have a redshift distribution that peaks between redshift 1 and 3. We use observations of the CMB from the Planck satellite to reconstruct the deflection angles with statistical estimators, and we correlate the deflections with observations of the infrared background light at 100-850 GHz. We find that the two signals are strongly correlated, with a correlation coecient of approximately 0.8, and we use the measured cross spectrum to estimate the minimum mass scale at which dark matter halos host a CIB source, as well as the star formation rate density in three redshift bins between redshift 1 and 7. In Chapter 5 we use the Doppler shift of CMB light scattered by moving galaxy clusters, known as the kinetic Sunyaev-Zeldovich (kSZ) e↵ect, to put a limit on the large-scale velocity distribution of a sample of galaxy clusters observed in WMAP CMB data. On 100 Mpc scales cluster velocities relative to the CMB are expected to be small, originating from gravitational instabilities. Larger motions could be generated by pre-inflationary inhomogeneities that leave a “tilt” across our horizon, resulting in a uniform matter flow across the horizon. The kSZ e↵ect is sensitive to such a flow, and we use it to constrain the radial and dipole

v velocity of a sample of 736 clusters with mean redshift 0.12, finding no evidence for either. In Chapters 6 and 7 we search for a possible pre-inflationary signal in CMB data. Mod- els of inflation suggest that our current patch of the universe could have been created as a nucleation bubble from a phase of false vacuum eternal inflation. If additional bubbles are produced, then it is possible that one of them intersected our past lightcone at the time of decoupling, imprinting a disk-shaped signal in the CMB. We have searched for this signal in the WMAP data using optimal algorithms that evaluate the exact posterior likelihood in an ecient and computationally fast way. We find no evidence for the signal, and place limits on the curvature perturbation generated by a collision intersecting the last scattering surface.

vi Preface

Each chapter in this thesis, with the exception of the introduction and conclusion, is a complete paper, and was written in collaboration with others. Some of the papers are in the process of journal submission, and the final versions may di↵er from those presented here. The formatting has been changed for consistency between chapters, and additional sections have been added to Chapter 2 to provide background information. The work was done in collaboration, and I will now outline the contributions that I made. Several of the results use data from the Planck satellite, which is a European Space Agency experiment, with a significant NASA contribution, designed to measure CMB temperature and polarization anisotropies. The project grew from two proposed missions, the cosmic background radiation anisotropy satellite (COBRAS) and the satellite for measurement of background anisotropies (SAMBA), which were combined and selected as the 3rd Medium-sized mission in the Horizon 2000 Scientific Programme. The success of the mission and the high quality of the data are due to the combined e↵ort of a large team of scientists and engineers. I am a member of the Core Team of scientists on the High Fre- quency Instrument (HFI), one of two instruments onboard Planck (the other being the Low Frequency Instrument) and I joined the project after the satellite had been constructed. I was involved in the pre-flight testing of the instrument, analyzing data from tests done at Centre Spatial de Liege` with Jean-Michel Lamarre. I worked with Andrew Lange’s group at Caltech and the Planck group at JPL on several aspects of the low-level data processing. My main contribution was to better determine the HFI detector time response, by analyzing data from tests where the detector bias current was stepped, and by comparing scans of the Galaxy made six months apart. These projects were done in collaboration with Brendan Crill and Guillaume Patanchon.

vii The analysis presented in Chapter 2 was developed over many iterations between my- self and Duncan Hanson, with Olivier Dore´ providing valuable comments, suggestions, and guidance. The paper presented in Chapter 3 is one of the Planck Collaboration papers, and as such represents the work of many people. I made significant contributions to all aspects of the analysis directly related to the work in Chapter 3, including the measurement, null tests, and modeling sections. The project was initiated, proposed, and lead by Olivier, who contributed to all aspects of the paper. Ultimately, the lens reconstructions used in the analysis were produced by Duncan; although I made my own reconstructions, we used Duncan’s to be consistent with the Planck lensing power spectrum paper. In addition, Duncan generated the simulations necessary to calculate the estimator normalization 1 and mean-field, and developed and implemented the C pipeline. I performed the calculations in the modeling section, however the code to calculate the CIB auto and cross power spectra with the halo model was given to me by Olivier. The work in Chapter 4 is an extension of the modeling work in Chapter 3 and was done with Olivier. Chapter 5 is largely my own work. I wrote the code and performed the analysis, while Sarah Church, Elena Pierpaoli, and Daisy Mak provided valuable comments and suggestions. Many of the results were checked by Daisy who developed an independent analysis pipeline. Chapters 6 and 7 were written in collaboration with Leonardo Senatore and Kendrick Smith. The algorithms were developed in collaboration, and it is dicult to separate individual contributions. The theoretical background was largely written by Leonardo, and the frequentist analysis was primarily developed by Kendrick. I wrote the code, with the 1 exception of the C function and the code to calculate the bubble profiles smoothed by the CMB transfer function, both of which were written by Kendrick.

viii Acknowledgments

Without the help, advice, and encouragement of many people this work would not have been successfully completed. The main person I have to thank is my advisor, Sarah Church, who has supported me throughout, providing advice and assistance, and ensuring that I always had funding to pursue interesting research ideas. Thanks to Sarah, I was able to participate in several observing runs, first to Mauna Kea with the SuZIE experiment, and later to Chile with the QUIET telescope. I am grateful to Charles Lawrence for letting me join the Planck Collaboration and allowing me to work at JPL for extended periods of time, and to Andrew Lange who let me work with his group at Caltech when I first joined the Planck team. I owe a huge amount to Olivier Dore,´ who advised me on all aspects of research, helping me to fully participate in the Planck scientific analysis and introducing me to several of his collaborators. It was a pleasure writing papers, proposals, and presentations with Olivier, and I learned a lot about galaxies, star formation, and the infrared background. I visited Caltech and JPL many times to meet with Olivier, and every trip was engaging. I enjoyed working with Duncan Hanson on the Planck lensing analysis and learning about statistical estimators, and am grateful for the significant amount of time he has spent answering my— often trivial—questions. It was always entertaining visiting Duncan at Caltech and taking his dog on walks. After I started working with Duncan and Olivier, analyzing Planck data became much easier and more fun! I enjoyed collaborating with Elena Pierpaoli and Daisy Mak on the bulk flow project. They provided invaluable help and guidance, especially when trying to get the paper published. I would like to thank both Leonardo Senatore and Kendrick Smith for making the bubbles project so much fun. I learned a lot from Leonardo about how to approach

ix problems, and I am in awe of Kendrick’s knowledge of statistics. In addition, Leonardo, Kendrick, and Olivier organized a superb conference in Aspen! In my ﬁrst three years at Stanford I shared an oce with Ed Wu, and I want to thank him for teaching me about computer science and data analysis—I learned a lot discussing research ideas with Ed. I have had the privilege of working with a talented group of ex- perimentalists in Sarah’s group, and I have beneﬁted from working in the lab with Matt Sieth, Patricia Voll, Kiruthika Devaraj, and Judy Lau, learning how to design experiments and build instruments. I want to thank Keith Thompson for working with me to design and build the QUIET star-tracker, and for teaching me about telescope design, as well as how to use the telescopes at the student observatory. Dana Volponi and Maria Frank were friendly and supportive, and ensured that I successfully navigated the Stanford bureaucracy. Finally, I want to thank my beautiful and brilliant girlfriend Becky Du↵ett, who has put up with many late nights working, and many months travelling. In addition, I am grateful to Becky’s family. The Du↵etts welcomed me for many Christmases and holidays in Portland when I couldn’t make it home.

x Contents

Abstract v

Preface vii

Acknowledgments ix

1 Introduction 1 1.1 Our Cosmology ...... 1 1.1.1 The Friedmann Equations ...... 5 1.2 The Early Universe ...... 6 1.2.1 Inﬂation ...... 6 1.2.2 Eternal Inﬂation ...... 8 1.3 Growth of Structure ...... 10 1.4 The Cosmic Microwave Background ...... 14 1.4.1 The Sunyaev Zeldovich E↵ect ...... 15 1.5 The Cosmic Infrared Background ...... 16 1.6 Lensing of the CMB ...... 19 1.7 The Planck Mission ...... 20

2 Foreground Contamination in CMB Lens Reconstruction 39 2.1 Introduction ...... 40 2.2 Lens Reconstruction ...... 43 2.2.1 Lensing Potential ...... 43 2.2.2 Lensing Power Spectrum ...... 45

xi 2.3 Source Trispectrum ...... 46 2.4 Source Estimators ...... 49 2.5 Foreground Simulations ...... 52 2.6 Source Modeling ...... 54 2.6.1 Radio Point Sources ...... 58 2.6.2 SZ ...... 62 2.6.3 Dusty Galaxies ...... 65 2.7 Additional Bias Terms ...... 67 2.7.1 Inhomogeneous Flux Sensitivity ...... 67 2.7.2 Source Masking ...... 69 2.8 Results and Discussion ...... 71 2 A Flat Sky Expressions ...... 83 2 B Supplementary Plots ...... 85 2 B 1 Estimator Sensitivity ...... 85 2 B 2 Frequency Dependence of the Foreground Bias ...... 85 2 B 3 Bispectrum- and Trispectrum-Related Spectra ...... 87 2 B 4 Results on Simulations ...... 88

3 Gravitational Lensing by Dusty Star-Forming Galaxy Halos 93 3.1 Introduction ...... 94 3.2 Data Sets ...... 96 3.2.1 Planck Maps ...... 96 3.2.2 External Data Sets ...... 99 3.3 Cross-Correlation Formalism and Implementation ...... 101 3.3.1 Reconstructing the CMB Lensing Potential ...... 101 3.3.2 Decreasing the Foreground Noise ...... 102 3.3.3 Measuring Cross-Correlations ...... 105 3.3.4 Simulating Cross-Correlations ...... 106 3.4 A Strong Signal Using Planck HFI Data ...... 107 3.5 Statistical and Systematic Error Budget ...... 109 3.5.1 Statistical Error Budget ...... 109

xii 3.5.2 Instrumental and Observational Systematic E↵ects ...... 111 3.5.3 Astrophysical Contamination ...... 115 3.5.4 Final Statistical and Systematic Error Budget ...... 129 3.6 Interpretation and Discussion ...... 129 3.6.1 Model Comparison Methodology ...... 130 3.6.2 Two Modelling Approaches ...... 131 3.6.3 Interpreting the Reconstructed Emissivities ...... 140 3.6.4 Discussion and Outlook ...... 141 3 A Statistical Errors ...... 163

4 Reconstructing The Redshift Distribution Of The CIB Emissivity 167 4.1 Introduction ...... 168 4.2 Reconstructing the CIB Emissivity ...... 169 4.3 Fisher Matrix Analysis ...... 172 4.3.1 Noise Covariance Matrix ...... 173 4.3.2 Experimental Model ...... 174 4.3.3 Principal Components ...... 176 4.4 Results ...... 177

5 Measuring The Galaxy Cluster Bulk Flow From WMAP Data 184 5.1 Introduction ...... 185 5.2 Theory ...... 188 5.2.1 SZ Clusters as Tracers of the Velocity Field ...... 188 5.2.2 Expected Signal for a Generic Velocity Tracer ...... 190 5.3 Data ...... 193 5.3.1 The X-ray Selected Cluster Sample ...... 194 5.3.2 External Simulations ...... 195 5.4 Method ...... 195 5.4.1 Outline of the Method ...... 195 5.4.2 Cluster Modeling ...... 196 5.4.3 Filters ...... 199 5.4.4 Dipole Fitting Procedure ...... 210

xiii 5.4.5 Conversion to Velocity Dipole ...... 212 5.4.6 Tests of the Method ...... 213 5.5 Systematic E↵ects ...... 214 5.5.1 Thermal SZ ...... 214 5.5.2 Radio Point Sources ...... 219 5.5.3 Galactic Emission ...... 220 5.6 Results ...... 224 5.6.1 Dipole ...... 224 5.6.2 Monopole and Higher Moments ...... 227 5.6.3 Comparison with SuZIE Measurements ...... 228 5.7 Conclusions ...... 228 5 A Expected Cluster Velocity Dipole ...... 241 5 B Comparison with KAKE Results ...... 242 5 B 1 Filter Pipeline Comparison ...... 242 5 B 2 Cluster Dipole Comparison ...... 244

6 Optimal Analysis of Azimuthal Features in the CMB 248 6.1 Introduction ...... 249 6.2 Mini-Summary of the Theory of Bubble Collisions ...... 252 6.3 Bubble Signal ...... 255 6.3.1 Ramp and Step Models ...... 255 6.3.2 CMB Temperature Proﬁles ...... 256 6.3.3 Can CMB Transfer Functions be Neglected? ...... 258 6.3.4 Discretizing the Bubble Radius ...... 260 6.4 Data Analysis ...... 262 6.4.1 Deﬁnitions and Notation ...... 262 6.4.2 An Algorithm for Fast Calculation of 2 ...... 264 6.4.3 Bayesian Analysis ...... 267 6.4.4 Frequentist Analysis ...... 270 6.4.5 Computational Cost ...... 274 6.5 Large Numbers of Bubbles ...... 275

xiv 6.6 Conclusions ...... 278 1 6 A Details of the C Filter ...... 284 6 B Likelihood Model ...... 286

7 Collisions with Other Universes: the Optimal Analysis of the WMAP Data 290 7.1 Introduction ...... 291 7.2 Bubble Signal ...... 292 7.3 Method ...... 293 7.3.1 Bayesian Analysis ...... 294 7.3.2 Frequentist Analysis ...... 297 7.4 Many Bubbles ...... 299 7.5 Conclusions ...... 299

8 Summary 302 8.1 Lensing and the Cosmic Infrared Background ...... 302 8.2 Kinetic SZ Velocity Constraints ...... 304 8.3 Cosmic Bubble Collisions ...... 305

xv List of Tables

1.1 Evolution of a single mode of the potential , baryon density b, dark mat-

ter density c, and radiation temperature ⇥r, based on Ma [19]. Each mode begins evolving when it enters the horizon, which may not occur before decoupling. D(a) is the growth function...... 13

2.1 Primary source terms discussed in Sec. 2.3. The first row gives the trispectrum due to lensing for comparison purposes...... 48 2.2 One-point amplitudes calculated at 150 GHz, with a 200 mJy (6 mJy in brackets) maximum flux density. For radio point sources S is not shown h i since Eq. (2.45) gives an unphysical negative flux...... 62

3.1 Point source estimator. The measured quantity S 3 , as deﬁned in Eq. 9, is given as a function of frequency...... D . .E ...... 121 3.2 Cross-spectrum detection band-powers. All values are in units of µK.sr. fore The extragalactic foreground contribution, C` has been removed from

C`. Both statistical and systematic errors are given (see Sect. 3.5 for details). 128 3.3 Reconstructed emissivity as a function of redshift and associated star formation rate. At each frequency and for each of the three redshift bins the ﬁrst quantity corresponds to the mean emissivity in the corresponding red- 1 1 shift bin, ¯j(z), in Jy Mpc sr , while the second corresponds to the SFR 3 1 density, ⇢SFR, in M Mpc yr ...... 143

5.1 Simulated thermal SZ monopole and measured monopole...... 200 5.2 Monopole and dipole in the PSM simulations for the z = 0 1 shell. . . . . 223 xvi 5.7 95% Conﬁdence Limits on the Dipole...... 227

5.3 Results from the KAKE ﬁlter.⇤ ...... 239

5.4 Results from our Wiener filter.⇤ ...... 239 5.5 Results from the matched filter...... 240 5.6 Results from the tSZ bias removing filter...... 240 5.8 Number of clusters satisfying the luminosity and redshift cut in our sample and the KAKE sample...... 245 5.9 Comparison of the cluster monopole and dipole in our cluster sample with the KAKE sample...... 245 5.10 Same as table 5.9 for the W band channels only...... 246 5.11 Result from Kashlinsky et al. [54]...... 246 5.12 Mean optical depth of our cluster sample...... 247 5.13 Result from our filter pipeline with the luminosity cut...... 247

xvii List of Figures

1.1 Simulation of the spatial distribution of bubbles at a late time in an eternally inﬂating false vacuum. From [16]...... 9

1.2 Map of CMB temperature anisotropies measured by the WMAP satellite (WMAP Collaboration)...... 15

1.3 Power spectrum of CMB temperature anisotropies measured by the WMAP satellite (WMAP Collaboration)...... 16

2 1.4 Maps of the 26 deg N1 ﬁeld (galactic coordinates l, b = 85.33, 44.28)

at 217, 353, 545, and 857 GHz, smoothed with a 100 Gaussian, with the CMB, bright point sources, and galactic cirrus subtracted. From Planck Collaboration et al. [39]...... 18

1.5 Power spectrum of the CMB lensing potential in the ⇤CDM model. The solid line is the linear theory model, with the dashed line also including non-linear corrections from HALOFIT [44]. From [43]...... 20

1.6 Theoretical temperature and polarization power spectra. The primordial B-mode signal is shown in blue with two di↵erent values for the tensor to scalar ratio. The dashed curves are expected noise levels at 143 GHz (red) and 217 GHz (blue) for two sky surveys...... 22

xviii 2.1 Trispectrum-related power spectra biases âb calculated for an L = CL cd max 1500 experiment. The amplitudes of the four trispectrum shapes are arbitrary (such that âb 1 at its peak.), but are used consistently in each of CL ab ⇡ the four plots. Note that the relative size of the bias terms in each plot has no physical significance. The colored solid lines give the total contributions to âb from the indicted trispectra. Corresponding dashed lines give only CL the contribution from the primary contraction of the trispectrum (which in the upper left panel is equal to the lensing potential spectrum)...... 52

2.2 Same as Fig. 2.1, but for an experiment which is cosmic variance (C.V.)

limited to Lmax = 3000...... 53

2.3 Temperature power spectra of the simulated sky components for the Lmax = 1500 and 3000 experimental models, including unlensed and lensed CMB (black and orange), radio sources (red), infrared sources (blue), thermal SZ (green), and kinetic SZ (purple) at 148 GHz. Bright sources have been removed before taking the spectra (see Sec. 2.5 for details)...... 54

2.4 Power spectra calculated using our foreground model. The solid lines show the CMB (unlensed in black, lensed in orange), radio point source (red), SZ (green), and CIB (blue) spectra. Radio and CIB point sources with ﬂux density S > 200 mJy are assumed to be masked. The dashed lines show the point source signals for a 6 mJy ﬂux density cut. For comparison the blue dotted lines show the CIB one and two-halo terms calculated using the Ade et al. [27] CIB model...... 59

2.5 Redshift distribution of power in the radio point source (red), SZ (green), and CIB (blue) spectra calculated using our foreground model at 150 GHz for ` = 50 (solid) and ` = 2000 (dotted). The power in the lensing potential spectrum is shown in black. The curves are normalized by area...... 61

2.6 As in Fig. 2.5, except that the foreground spectra show the redshift distri- S 2S 2 bution of power in C` ...... 61

xix 2.7 Power spectrum of S 2 from simulation (colored lines) and from our model (solid black lines) for radio point sources (upper panel), infrared sources (middle panel), and SZ (lower panel). The infrared model has been scaled by a factor of 30 to better ﬁt the simulation results. The additional dashed and dotted lines are results from models with di↵erent parameters, discussed in the text...... 63

2.8 Cross power spectrum of S 2 and calculated from the simulated S and maps (colored boxes) and the model predictions (black lines). To better ﬁt the simulations the CIB model spectrum has been multiplied by 25 for the B12 model and by 10 for the B10 model. The SZ spectrum has been multiplied by 2. The empirical model from Sec 2.4 is almost identical to the radio point source model up to an amplitude...... 64

2.9 Contribution to the reconstructed lensing spectrum from simulations for

each foreground component when Lmax = 1500 (red line). The blue line gives the result after subtracting the disconnected term, which is the green line. The residual foreground contribution calculated using the results from Sec. 2.3 is the orange line. The blue bars give the lensing spectrum reconstructed from the full-sky simulated CMB map...... 69

2.10 Same as Fig. 2.9, but for Lmax = 3000. The di↵erence between lensing the reconstructed lensing potential and the simulated lensing potential is due to the N(1) noise bias [17], which has not been subtracted...... 70

2.11 Detection signiﬁcance as a function of the minimum (black) and maximum (blue) multipole used in the estimator. The black curve is calculated with

`max = 3000 and the blue curve with `min = 2...... 86

ˆ 2.12 Bias to C` calculated using the Lmax = 1500 (upper panel) or Lmax = 3000 (lower panel) estimator and masking scheme from the simulated radio point source (red), infrared point source (blue), thermal SZ (green), and kinetic SZ (purple) maps. The gray bars show the error bars from a Fisher matrix calculation...... 90

xx 2.13 The bispectrum and trispectrum-related spectra calculated from the 148 GHz simulated map. The red line shows the best-ﬁt spectrum, which depends on a single parameter: S 3 for the bispectrum, and S 4 for the trispectrum. The noise bias has beenD E subtracted from the trispectrumD E estimate, and the bias terms caused by CMB lensing are shown in blue...... 91

2.14 The green line shows the bias from the unresolved foreground components in the masked 148 GHz simulated map. The solid blue line is the measured bias, the dashed blue line is the disconnected term, and the red line is the sum of the blue lines. The gray error bars are from a Fisher matrix

calculation for the Lmax = 1500 estimator...... 92

3.1 Redshift- and mass- integrand for the CIB and CMB lensing potential power spectra at ` = 500, calculated using the CIB halo model of [17], evaluated at 217 GHz. The good match between the redshift and halo mass distributions leads to an expected correlation up to 80 %. The sharper features in the CIB kernel are artefacts from the [27] model. We note that the low mass, high z behavior of our calculation is limited by the accuracy of the 5 mass function we use [28]. All of our mass integrals use Mmin = 10 M .. . 97

3.2 Combined Galactic, point-source and Hi mask with sky fractions 16, 30 and 43 %...... 100

3.3 Angular cross-spectra between the reconstructed lensing map and the temperature map at the six HFI frequencies. The error bars correspond to the scatter within each band. The solid line is the expected result based on the PER model and is not a ﬁt to these data (see Fig. 3.16 for an adjusted model), although it is already a satisfying model. In each panel we also show the correlation between the lens reconstruction at 143 GHz and the 143 GHz temperature map in grey. This is a simple illustration of the frequency scaling of our measured signal and also the strength of our signal as compared to possible intra-frequency systematic errors...... 104

xxi 3.4 Temperature maps of size 1 deg2 at 545 and 857 GHz stacked on the 20,000 brightest peaks (left column), troughs (centre column) and random map locations (right column). The stacked (averaged) temperature maps is in K. The arrows indicate the lensing deﬂection angle deduced from the gradient of the band-pass ﬁltered lensing potential map stacked on the same peaks.

The longest arrow corresponds to a deflection of 6.300, which is only a fraction of the total deflection angle because of our filtering. This stacking allows us to visualize in real space the lensing of the CMB by the galaxies

that generate the CIB. A small and expected o↵set ( 10) was corrected by ' hand when displaying the deﬂection ﬁeld...... 107

T 3.5 Naive analytical estimates of the contribution to the C` variance as a function of multipole and frequency as given in Eq. 4. We assume the same bin sizes as in Fig. 3.3. The di↵erent lines are the contribution to the ana- 2 CIB CIB lytical error from the signal only: C` C` + C` (green), noise only: ˆ,N ˆCIB,N ˆCIB,N C` C` (blue), and the mixed signal and⇣ noise terms:⌘ C` C` (yel- ˆ,N CIB low) and C` C` (orange). The total contribution is the solid black line, CIB 2 and the theory spectrum, C` , is the dashed line...... 111 ⇣ ⌘

3.6 Null tests at 545 GHz. Left: di↵erence spectra obtained by nulling the signal in the HR temperature map before correlating it with our nominal reconstruction. Middle: temperature signal nulled using di↵erent detectors at 545 GHz. Right: temperature signal nulled using the ﬁrst and second survey maps. The black error bars correspond to the scatter measured within an `-bin, while the coloured bands correspond to the analytical estimate. Except for the survey null test (see text for details), these tests are passed 2 satisfactorily except, as illustrated by the quoted and Ndof, thus strengthening conﬁdence in our signal...... 115

xxii 3.7 Left: di↵erence between the cross-spectra measured using the 20 % Galac- tic mask (20 % is the unmasked sky fraction) from that measured with our default 40 % Galactic mask. Middle: spectra obtained when di↵erencing the 60 % and 40 % Galaxy mask measurements. For both left and middle panels and all Galactic masks, the same point source and Hi masks are used, which removes an additional fraction of the sky. Right: di↵erence between the cross-spectra calculated with the Hi cleaned temperature maps from those with no Hi cleaning. This cross-spectrum is thus the correlation between the Hi template and the reconstruction. The error bars are calculated in the same way as in Fig. 3.6. Again, the null tests are passed with an acceptable 2...... 116

3.8 Left: di↵erence between cross-spectra calculated using the lens reconstruction at 100 GHz with the nominal 143 GHz reconstruction. We see an overall shift, which leads to a high reduced 2. This shift can be explained by the expected overall normalization uncertainties of the 100 GHz and 143 GHz reconstructions. While this uncertainty is not included in the 2 or the solid bars, it is included later in our analysis in Sect. 3.6.1. Middle: same as the left panel, but the 217 GHz reconstruction is used instead of the 100 GHz reconstruction. Right: di↵erence between cross-spectra when we consider the 143 GHz lens reconstruction calculated with a less restrictive Galaxy mask (that excludes only 20 % of the sky) and the nominal reconstruction mask that excludes 40 % of the sky...... 117

3.9 Results from the point source contamination estimator of Eq. (5). The best- ﬁt cross-spectra associated with shot noise are plotted in black. We do not show the best-ﬁt at 545 and 857 GHz since the signal-to-noise ratio is low. The grey line is a prediction for the bias from the CMB lensing - infrared correlation, and has been subtracted from the spectra (plotted as black points). We see that with the subtraction of the bias from CMB lensing, the measured bispectrum-related spectrum is generally consistent either with zero, or with the shape expected for shot noise...... 120

xxiii 3.10 Frequency spectrum of our cross-spectra averaged within `-bins (black points with associated error bars). The light shaded regions correspond to the HFI frequency bands. The solid black curve corresponds to the best- fit CIB assuming a [64] spectrum, while the dot-dashed line assumes a [9] spectrum. The dashed black line corresponds to the best-fit model when allowing for an SZ component in addition to the [64] CIB shape. The blue dots correspond to the associated absolute value of the best-fit SZ component. We conclude from this plot that the SZ e↵ect is not an important contaminant...... 122

3.11 Foreground components at each frequency. The data points and error bars show our results. The dashed line is an estimated upper limit on the magnitude of the SZ contamination derived in Sect. 3.5.3. We show the absolute value of this contribution, which is negative at frequencies less than 217 GHz. The dot-dashed line is the extragalactic point source contribution, with an amplitude measured from our data as derived in Sect. 3.5.3. Again we show the absolute value, with the signal being negative below ` 1200. The oscillating solid line corresponds to the calculated ISW ⇠ contamination...... 125

3.12 Cross-spectrum of the 545 GHz lens reconstruction correlated with the 545 GHz temperature map with di↵erent Galactic masks. The legend gives the visible sky fractions. The solid line represents the analytic unclustered shot-

noise contribution ﬁt to the fsky = 0.09 points above ` = 1300...... 126

3.13 Cross-correlation coecients calculated from the model spectrum and best-ﬁt halo model at each frequency. The CIB is a spectacular tracer of CMB lensing, and vice-versa. The data points represent the measured cross-correlation divided by the best-ﬁt auto power spectra models at 545 GHz.132

3.14 Marginalized 2-D distributions of zc and z for the linear bias model, ﬁt to all frequencies simultaneously. The orange dots indicate the parameter values at the minimum 2...... 133

xxiv 3.15 Measured cross-spectra with the best-fit j reconstruction model fit to both the CIB auto- and CIB-lensing cross-spectra (solid coloured), and the best- fit linear bias model (dashed coloured). The 2 values quoted in each panel are the contribution to the global 2 from the data in the panel for the halo model, and loosely indicate the goodness of fit (see text for details). The one and two-halo contributions are shown as the dashed and solid black lines, respectively. A light dashed black horizontal line is indicating the zero level...... 136 3.16 PIR auto-spectra with the best-fit mean emissivity j reconstruction model fit for the CIB auto and CIB-lensing cross-spectra (solid coloured). The 2 values are defined as in Fig. 3.15. The one and two-halo contributions are shown as the dashed and solid black lines, respectively, while shot noise is the dot-dashed black line...... 137

3.17 Marginalized 2-D distribution of log10 (Mmin/M ) and log10 (Msat/M ) for our overall HOD model when the CIB-lensing cross-spectra are combined with the CIB auto-spectra and FIRAS measurements. The orange dot shows the best-fit value. The contours correspond to 68%, 95% and 99.7% confidence intervals...... 139 3.18 Reconstructed mean emissivity, ¯j, for each frequency as a function of redshift. The solid line at low z and the dashed line at higher z correspond to the B11 model. The B11 emissivity model at z > 1 is not used, and is shown only for reference. The black error bars correspond to the 68% C.L. while the color shading display the full posterior distribution...... 142 3.19 Marginalized 1-D distribution of the emissivity in the high redshift bin at 353 GHz with (black line) or without (blue line) including the CIB-lensing correlation. Its inclusion helps to constrain the emissivity at high redshift, transforming an upper limit into a detection...... 143 3.20 Correlation between the lensing potential and the IRIS map at 100 µm using our nominal lens reconstruction. We clearly see a correlation and estimate the significance to be 9 , ignoring possible systematic e↵ects. The solid line represents a simple reasonnable prediction for this signal...... 144

xxv 3.21 Ratio of various error estimation procedures to the errors obtained with the data-based analytical estimate. At each frequency the numerator is given by: (i) the scatter within an `-bin in simulations (solid black line); (ii) the scatter within an `-bin in the data (solid dashed black line); (iii) the scatter of bins across simulated realizations (solid coloured line); (iv) the analytical errors calculated from the simulations (dashed coloured line); (v) the scatter across realizations for the cross-correlation between the simulated temperature map and the lensing potential reconstructed from the data (coloured dot-dashed line). The grey envelope is the precision of the simulated errors expected from 100 simulations (shown as a spread around unity)...... 163

4.1 Derivatives of the CIB auto spectrum (top and middle panels) and CIB lensing potential cross spectrum (bottom panel) with respect to the amplitudes,

ji, of the binned emissivity...... 171

4.2 CIB auto and CIB lensing potential cross spectra relative to our ﬁducial model at 217 GHz assuming 10 µK-arcmin noise, 5’ beam FWHM and

Lmax = 3000 (dark gray bars), and 100 µK-arcmin noise, 5’ beam FWHM

and Lmax = 1500 (light gray bars). The colored lines show the spectra if the emissivity in a single redshift bin is doubled while holding the mean CIB intensity constant...... 174

4.3 The ﬁve (normalized) principal components of the CIB emissivity with the largest eigenvalues, obtained with experimental model 1. In the upper panel we assume no knowledge of the frequency dependence of j, while in the lower panel we assume that the frequency dependence is known exactly. Note the di↵erent y-scale in the two panels...... 175

xxvi 4.4 Reconstructed emissivity relative to the true emissivity for experimental model 1 (solid) and model 2 (dashed), assuming that the frequency dependence of j is precisely known. The colored lines give the reconstruction calculated using only the principal components measured with signal-to- noise ratio greater than 3, as determined from a Fisher matrix calculation, using only the CIB auto spectra, auto and cross frequency spectra, or CIB lensing potential cross spectra...... 177

5.1 Redshift distribution for the cluster sample used in this paper...... 191 5.2 Expected bulk flow velocity in ⇤CDM cosmology with a selection function calculated from our cluster sample in redshift shells extending from z = 0 to the specified x-axis value. The dashed line shows the result with a uniform selection function. The dark and light shaded areas are the 68% and 95% confidence limits from cosmic variance...... 192 5.3 Multi-frequency matched filters for the WMAP channels...... 202 5.4 Unbiased multi-frequency matched filters. A discussion of the filter shape is given in the text...... 202 5.5 Thermal SZ bias removing filters for an experiment with four frequencies:

100 GHz, 143 GHz, 217 GHz and 353 GHz, beam FWHM of 100,70,50

and 50, and white noise levels 25, 15, 25 and 75 µK/K arcmin...... 204 5.6 Spectra of a simulated kSZ map (with signal only at the locations of our cluster sample) convolved with the WMAP beams. The simulations were performed both with (top 3 spectra) and without (bottom 3 spectra) a bulk flow component of amplitude 1000 km/s. The dashed lines are the WMAP beam functions, scaled to the signal amplitude...... 205 5.7 Sum of the eight WMAP maps in the Q, V and W bands filtered by the matched filters in Fig. 5.3. The map is noise dominated...... 206 5.8 Sum of the eight WMAP maps in the Q, V and W bands filtered by the tSZ removing filters in Fig. 5.4. The values in this map are larger than in Fig. 5.7 since the map is noise dominated and the unbiased filters give a lower signal to noise measurement of the kSZ signal...... 206

xxvii 5.9 Wiener ﬁlter used by KAKE...... 208

5.10 Wiener ﬁlter for each of the WMAP channels...... 208

5.11 Suppression of the kSZ signal by the beam and ﬁlter in the V band...... 209

5.12 Left: Error in the recovered dipole direction from simulated kSZ maps with a bulk ﬂow velocity of 0 km/s (top), 500 km/s (middle) and 1000 km/s

(bottom) in the directions: (latitude, longitude) = (0,0) black lines, (0,

90) blue lines and (90,0) red lines, using the MF. Right: Same for the UF. The arrows are the 95% conﬁdence limits...... 215

5.13 Recovered bulk flow velocity in simulated maps containing CMB, noise and kSZ using the MF (black) and the UF (blue). All clusters are included in the fit. The x-axis is the bulk velocity input into the simulated maps, the y-axis is the recovered velocity. The red line indicates perfect recovery. When the input bulk flow is small, a dipole is not detected and the data points provide an estimate of the scatter in the recovered amplitude. . . . . 216

5.14 Error in the direction of the bulk ﬂow velocity in simulated maps containing CMB, noise and kSZ with the MF (black) and UF (blue), when all clusters are included in the ﬁt...... 217

5.15 Error in the recovered monopole velocity in simulated maps containing CMB, noise and kSZ with the MF (black) and UF (blue)...... 218

5.16 Sum of the PSM simulated maps convolved with the WMAP beams and ﬁltered with the MF. This map shows galactic emission around the edges of the mask that is absent from the WMAP foreground reduced maps due to WMAP’s foreground reduction method [6]. The color scale is altered to better show the galactic emission by mimicking the e↵ect of replacing the 1 data by sinh (data) [29]...... 221

5.17 Sum of the PSM simulated maps convolved with the WMAP beams and 1 ﬁltered with the UF. The sinh color scale that was used in Fig. 5.16 is used here...... 221

xxviii 5.18 Left: Cluster dipole amplitude in the maps filtered by the KAKE filter. Points with a plus sign have redshift shells with minimum redshift of 0, those with a cross have shells with minimum redshift of 0.05 and 0.12. The green line is the noise bias, the red line is the 95% confidence limit that there is no bulk flow and the blue line is the 99.7% confidence limit. Right: Cluster dipole amplitude in km/s...... 225

5.19 Cluster dipole amplitude in the maps ﬁltered by the WF...... 225

5.20 Cluster dipole amplitude in the maps ﬁltered by the MF...... 226

5.21 Cluster dipole amplitude in the maps ﬁltered by the UF...... 226

5.22 95% confidence upper limit to the bulk flow in the redshift 0 1 shell. We find a bias in simulated thermal SZ maps equivalent to a bulk velocity of 2500 km/s...... 227 ⇠ 5.23 Spectrum of the W1 channel map filtered by the KAKE filter (black) and the di↵erence between the filtered W band map spectra from our KAKE filter pipeline and the spectra of the publicly available maps used in the Kash- linsky et al. [54] analysis...... 243

5.24 Filtered W1 channel map spectrum from the KAKE pipeline and our pipeline.243

5.25 Integrated signal from a beam smoothed and ﬁltered cluster in our simulated maps...... 244

6.1 A spacetime diagram showing the causal structure of a single bubble collision, based on Fig. 3 from [15]. Coordinates are chosen so that light

propagating in the plane of the diagram moves along 45 lines...... 253

6.2 The Earth’s last scattering surface at the time of decoupling, based on Fig. 4 from [15]. The color of the shaded region indicates the magnitude of the curvature perturbation, assumed to be of the “ramp” form given below in Eq. (4)...... 253

xxix 6.3 Angular bubble profiles b(✓) defined in 6.3.2, for bubbles at comoving dis- § tance r = 13886.6 Mpc (corresponding to angular size ✓bubble = 11.39) and arbitrary normalization. The ramp and step profiles are obtained by evolving an initial feature in the adiabatic curvature perturbation (Eqs. (4), (5)) forward to obtain a CMB temperature profile, using the CMB transfer function. The cosine and disc profiles are approximations to the ramp and step profiles in which the CMB transfer function is omitted (Eq. (14)). Qual- itatively, the e↵ect of including the CMB transfer function is to smooth

sharp features in the proﬁle, with smoothing length 1 given by the CMB ⇡ acoustic scale...... 261

6.4 The WMAP 7-year best fit ⇤CDM CMB spectrum (black line), the spectrum of bubbles with the ramp profile normalized to the CMB spectrum at ` = 2 (blue), and the spectrum of bubbles with the step profile normalized in the same way (red)...... 275

6.5 Left panel: Histogrammed maximum likelihood bubble amplitude aML, for

many Monte Carlo simulations of the data d. Counterintuitively, aML is within 0.001 of zero 11% of the time. Right panel: Posterior likelihoods (d a) for three randomly simulated data realizations d. It is seen that the L | derivative (@ /@a) is always zero at a = 0, but this point can be either the L global maximum likelihood (blue, solid curve), a local maximum which is not the global maximum (magneta, dotted curve), or a local minimum (red, dashed curve). Likelihoods in this ﬁgure were generated using a toy model

with 1000 µK-arcmin isotropic noise, 5 Gaussian beam, no sky cut, and

assuming ﬁxed bubble size ✓bubble = 30...... 286

xxx 7.1 Bayesian analysis of the bubble parameter space, assuming the “ramp model” for the bubble proﬁle. Top panel: Posterior likelihood (aramp d) for the L | amplitude parameter aramp, deﬁned to be the slope of the initial curva- 1 ture perturbation in Mpc , given the WMAP data d (solid black), after marginalizing the bubble radius. As explained in the text, the tails of the likelihood are slow to decay, due to a poorly constrained region of parameter space with small bubble radius. We illustrate this by showing the likeli-

hood calculated using bubbles with a subset of angular sizes: ✓bubble < 20

(blue short-dashed), and ✓bubble > 20 (red long-dashed). Bottom panel: Posterior likelihood (↵ramp d), obtained from the top panel by changing L | ramp ramp ramp 4/3 variables from a to ↵ = a (sin ✓bubble) . After this change of variables, the likelihood is narrower and less sensitive to marginalization over the bubble radius. Vertical lines are 95% conﬁdence limits on the amplitude parameter ↵ramp. The likelihood is consistent with no bubbles (↵ramp = 0)...... 295 7.2 Bayesian analysis of the “step” bubble model. We show the posterior likelihood (↵step d) after changing variables from astep to ↵step = astep(sin ✓ )1/3 L | bubble to remove degeneracies. The dashed coloured lines have the same meaning as in Fig. 7.1. The likelihood is consistent with no detection of the signal. . 297 7.3 Distribution of the likelihood ratio from simulations. An x-axis value of zero means that the likelihood peaks at a = 0. The red up-shaded distribution is consistent with the likelihood peaking at a = 0 within numerical precision. The WMAP value is consistent with a peak at a = 0, which occurs in 51.4% of the simulations...... 298

8.1 Comparison of the Planck and SPT-SPIRE CIB-CMB lensing potential cross spectrum. The SPT-SPIRE data is from Holder et al. [2]...... 304

xxxi xxxii Chapter 1

Introduction

“I dont know if you have had the same experience, but the snag I always come up against when I’m telling a story is this dashed dicult problem of where to begin it.”

Right Ho, Jeeves P.G. Wodehouse

1.1 Our Cosmology

Observations over the past few decades have allowed us to build a relatively simple cosmological model of the universe that can account for many of the observed structures we see. The ⇤ Cold Dark Matter (⇤CDM) model allows us to explain the structure that we see in cosmic microwave background (CMB) images as well as in galaxy surveys, and it enables us to calculate the age and size of the universe. The model assumes that our place in the universe is not special (the Copernican principle) and that we observe a sample of the universe that is representative of the whole, with the same physical laws at all locations. The following properties are assumed:

Isotropy — The large scale properties of the universe look the same in all directions. • This means that global properties of the large scale structure of the universe can be inferred from observations made in one direction. In addition, isotropy implies that a

1 2 CHAPTER 1. INTRODUCTION

preferred direction or axis of the universe cannot to be chosen, for example on large enough scales objects do not preferentially move towards us in one direction and away from us in another.

Homogeneity — Any property of the universe when averaged over large scales is the • same at all locations. This is observed to be true even for regions of the universe that have, based on the ⇤CDM model, never been in causal contact. The explanation for this is currently unknown, with one possibility being a period of rapid expansion early in the history of the universe.

These assumptions are found to be in excellent agreement with all measurements, and we present one such test in Chapter 5, where we measure the bulk motion of a sample of galaxy clusters. As well as these assumptions, there are several other important features of the universe that must be accounted for by any cosmological model:

Expansion of space — Evidence for the expansion of space was obtained in 1929 • when Hubble observed that galaxies are receding from us with a velocity proportional to their distance from us. This can be accounted for in General Relativ- ity by including a cosmological constant in the Einstein equations (which can either be interpreted as a constant energy density of space, or equivalently as being a new type of matter/energy with negative pressure). The expansion of space implies that the universe starts in a singularity, the Big Bang, and allows us to define an age for the universe. The current expansion rate of the universe is measured to be H = 74.2 3.6 km/s [1] and the dark energy density (defined relative to the critical 0 ± +0.015 density) is ⌦⇤ = 0.728 0.016 [2]. Presence of dark matter — First postulated in 1932 by Jan Oort to account for higher • than expected velocities of stars in the Milky Way given the amount of visible matter present in the Galaxy, and in 1933 by Fritz Zwicky to account for the discrepancy between the Coma cluster mass inferred from its luminosity and the mass inferred from the velocities of galaxies within it. Both observations can be explained by the presence of matter which does not interact electromagnetically with baryonic matter. Any explanation using modified gravity theories must explain the spatial separation 1.1. OUR COSMOLOGY 3

of visible and dark matter observed in colliding galaxy clusters (for example, the Bullet Cluster, Clowe et al. 3). To form structures with the density contrasts observed today, N-body simulations suggest that the dark matter is non-relativistic (or “cold”). The dark matter density has been measured to be ⌦ h2 = 0.1123 0.0035 [2, using c ± the WMAP7+BAO+H0 combination].

Spatial ﬂatness — The evolution of the universe is described by the Friedmann equa- • tions (which we brieﬂy describe below in Section 1.1.1), which derive from Ein-

stein’s equations. They allow us to deﬁne a critical density, ⇢c, which is the energy

density of a ﬂat universe. If the energy density is greater than ⇢c, then the universe

is closed, if it is below ⇢c, then the universe is open. Observations of the CMB combined with measurements of the Hubble Constant and Type Ia supernovae show that the curvature is ﬂat to within experimental error, with ⌦ ⌦ + ⌦ = tot ⌘ i i ⇤ 1.002 0.011 [4] where ⌦ is the energy density relative to the critical density. A ± P major focus of this work is to measure gravitational lensing of the CMB, which will ultimately allow more precise measurements of curvature using the CMB alone [e.g. 5].

Combining the Standard Model of particle physics with the ⇤CDM model allows the universe to be described by 31 free parameters [6], with all current cosmological observations being explained with 5 parameters, although more are likely to be needed to accom- modate more precise observations. In addition to the dark matter and dark energy densities above, the additional parameters are the baryonic matter density, ⌦ h2 = 0.02260 0.00053, b ± +0.088 9 the amplitude of scalar fluctuations, As = 2.441 0.092 10 , and the spectral index of ⇥ scalars n = 0.963 0.012 [2]. Additional parameters may be required, such as the tensor s ± ⇣ ⌘ to scalar ratio, r < 0.72 (95% confidence) [7], the tensor spectral index, and the neutrino 2 density, ⌦⌫h = ⌫ m⌫/93 eV with ⌫ m⌫ < 0.17 eV (95% confidence) [8], with a constraint obtained from the CMB, large scale structure, Lyman ↵ forest, and SNIa data, and P P a lower limit of ⌫ m⌫ & 0.05 from neutrino oscillation experiments, as well as parameters needed for moreP complex models. In Chapter 2 we will present methodology designed to aid measurements of the sum of neutrino masses. 4 CHAPTER 1. INTRODUCTION

While this model has been hugely successful in allowing us to explain observations of structure in the universe it does not tell us the nature of dark energy or dark matter. Several unsolved problems remain:

Horizon problem — The universe appears statistically homogeneous and isotropic on • large scales, and there is no observational evidence to the contrary. We observe the CMB to have the same temperature on opposite sides of the sky, despite the fact that the regions are separated by a distance large enough that, according to the ⇤CDM model, they have never been in causal contact.

Flatness problem — The energy density of the universe is (to within observational • limits) equal to the critical density required to make the universe flat, despite there being no clear reason why this should be the case. The problem is even more serious given that a flat universe is in an unstable equilibrium if the energy density is slightly over the critical value in the early universe, then the universe will become denser relative to the critical density as it evolves. We will show in Section 1.1.1 that the e↵ective curvature density has a (1 + z)2 dependence, whereas the matter density has a (1 + z)3 dependence and the radiation density has a (1 + z)4 dependence. In the past the curvature contribution therefore dominates, and for the universe to be flat today the curvature contribution must have been exponentially small in the early universe.

Absence of defects — If the early universe is hot, then a large number of stable • particles, such as magnetic monopoles, should be produced [e.g. 9]. If they survived until the present day, then they would be the primary constituent of the universe, however, searches for them have not been successful1

We now overview the equations that underpin the ⇤CDM model before discussing a possible solution to the above problems, known as inflation. This will motivate work done to search for evidence of eternal inflation, which we describe in Chapter 7. We will then describe how density perturbations grow in the universe and discuss specific astrophysical signals that we can use to learn about structure growth, namely the CMB, the Sunyaev- Zeldovich e↵ect, and the cosmic infrared background. We will end this chapter with an 1When assessing the importance of these objects in the universe it should be remembered that their existence is not motivated by any observational evidence. 1.1. OUR COSMOLOGY 5

overview of the Planck experiment and brieﬂy mention work we have done to mitigate systematic uncertainties in the Planck data.

1.1.1 The Friedmann Equations

Starting from the Einstein equations of general relativity we can derive the Friedmann equations governing the evolution of the scale factor. The Einstein equations are:

1 Rµ⌫ gµ⌫R + gµ⌫⇤ = 8⇡GTµ⌫ , (1.1) 2 where Tµ⌫ is the stress-energy tensor, Rµ⌫ is the Ricci curvature tensor, ⇤ is the cosmological constant, gµ⌫ is the metric tensor, and we are using units with c = 1. On large scales the universe is found to be homogeneous and isotropic, and the most general metric satisfying these properties is the Friedmann-Robertson-Walker (FRW) metric:

2 2 µ ⌫ 2 2 dr 2 2 2 2 ds = gµ⌫dx dx = dt + a (t) + r (d✓ + sin ✓d ) , (1.2) 1 kr2 " # where a is the scale factor, r is a distance coordinate, and k is the spatial curvature with values of -1, 0, or +1 for an open, ﬂat, or closed universe. To proceed further we require information about the energy density of the universe, which can be encoded in a stress- energy tensor. The di↵erent components of the stress-energy tensor are the energy density

T00, energy ﬂux density T0i, momentum density Ti0, and the spatial stress tensor Tij. For a perfect ﬂuid with no shear stresses, viscosity, or heat conduction, that is in thermodynamic equilibrium the stress-energy tensor is:

Tµ⌫ = (⇢ + p) uµu⌫ + pgµ⌫ , (1.3)

where ⇢ is the energy density, p is the hydrostatic pressure, and uµ is the velocity of the ﬂuid, with gµ⌫u u = 1. If the ﬂuid is at rest then the only non-zero entries are T = ⇢, µ ⌫ 00 and Tii = p. Solving the Einstein equations with the FRW metric gives the Friedmann Equations: a¨ 4⇡G ⇤ = (⇢ + 3p) + (1.4) a 3 3 6 CHAPTER 1. INTRODUCTION

a˙ 2 8⇡G⇢ + ⇤ k = . (1.5) a 3 a2 ✓ ◆ The dimensionless density parameters are deﬁned by:

8⇡G⇢ ⇢ ⌦ = = , 2 (1.6) 3H ⇢c where H = a˙/a is the Hubble constant, and ⇢c is the critical density.

1.2 The Early Universe

1.2.1 Inﬂation

Inflation provides a solution to all three of the horizon, flatness, and defect problems. In the 1970s Guth discovered inflation while trying to understand why magnetic monopoles are not observed today [10]. He realized that an expansion of space resulting from a constant energy density would also resolve the horizon and flatness problems [11]. Inflation would cause the universe to rapidly expand making regions causally connected before expansion appear causally disconnected today and, depending on the number of e-folds (multiplication by e), driving the universe to flatness. This expansion is thought to occur between 36 32 10 and 10 seconds after the Big Bang. After this period of rapid expansion ⇠ ⇠ the universe continues expanding but at a slower rate. Since the universe is known to be expanding now such a period of expansion is attractive theoretically since the two mecha- nisms could be related [e.g. 12]. In addition, inflation can provide a mechanism for generating the observed density fluctuations in the universe today. Any particle that causes inflation (known as the inflaton) would undergo quantum fluctuations in density, which would be magnified in size during expansion. Since space expands rapidly the fluctuations cannot be “undone”, as would be the case in a non-inflating universe, and they become the seeds for structure to grow. In a similar way quantum fluctuations in the graviton (the hypothetical particle that transmits the gravitational force) would lead to gravitational waves in the universe today that can potentially be detected. We note that inflation is only a hypothesis, and that models of the universe can be created to overcome the horizon problem without invoking inflation. 1.2. THE EARLY UNIVERSE 7

From equation 1.6 we see that the e↵ective curvature density is ⌦ = ⌦ 1 = /H2a2. k tot The ﬂatness problem can therefore be solved if:

d 1 < 0 , (1.7) dt aH ! and the curvature density would decrease with time until it is close enough to zero that deviations from ﬂatness after inﬂation would appear at much later times. Equation 1.7 also solves the horizon problem. The size of the comoving horizon (the radius of the region causally connected to an observer2) is:

a da0 1 ⌘ = . (1.8) a a H(a ) Z0 0 0 0 If 1/aH decreases with time then the size of the comoving horizon will shrink and regions previously in causal contact will become disconnected. If the Hubble constant is positive and approximately constant during inflation, then since H = a˙/a we find a(t) = a exp (H(t t )), where a and t are the values at the end of inflation. 0 0 0 0 One hypothesis is that inflation is caused by a scalar field, (t). For inflation to generate accelerating expansion d2a/dt2 must be positive, and so from the Friedmann equations p < ⇢/3. Since the energy density is positive the pressure is negative. Following Dodelson [14] we can find the pressure and energy density of the scalar field from the stress-energy tensor:

1 2 ⇢ = ˙ + V() 2 (1.9) 1 p = ˙ V() , 2 where V() is the potential for the field. From these equations we see how the pressure can be negative: if the field has more potential energy than kinetic energy. If the scalar field is in a local minimum of the potential but not a global minimum (known as a false vacuum) then it will have little kinetic energy but non-zero potential energy, and hence negative pressure.

2The number of “universe sized” patches with the same volume as ours must be at least 21 (95% conﬁ- dence) based on current ﬂatness limits Scott and Zibin [13]. 8 CHAPTER 1. INTRODUCTION

The Friedmann equations for the scalar ﬁeld are:

dV ¨ + 3H˙ = d (1.10) 8⇡G 1 H2 = ˙2 + V() . 3 2 ! Inflation ends when the scalar field rolls down the potential and the potential energy of the field is converted into radiation and particles in a process known as “reheating”. The universe then enters a period where the energy density is dominated by radiation.

1.2.2 Eternal Inﬂation

Quantum fluctuations in the inflaton field give some regions of space a larger inflaton potential. From equation 1.10 we see that regions with larger potential have greater expansion rates, and so at a later time they will occupy more space than regions with a lower potential. Over time these regions dominate the volume, and so if the decay rate to non-inflating universes is suciently low then the universe will end up “eternally inflating”. Regions with lower vacuum energy density appear as bubble universes. It is therefore possible that our universe is a bubble with a relatively low vacuum energy density, and hence low expansion rate, compared with the surrounding universe. An alternative possibility is that the inflaton potential could contain many false vacuum states. In standard inflation expansion occurs while the universe is in a false vacuum state and ends when there is a phase transition to the true vacuum. If there are many false vacua then the inflaton in each region of space will be in one of many possible false vacuum states and can undergo a phase transition to a region with a lower vacuum energy density and di↵erent physical constants. The universe would then have a fractal like structure with bubbles in one vacuum state containing bubbles with lower vacuum energy densities. That the universe could contain a large number of false vacua is a prediction of string theory [e.g. 15]. In Fig. 1.1 we show a simulation of the spatial distribution of bubbles at a late time in an eternally inflating false vacuum from Kleban [16]. At late times the physical volume of the false vacuum is larger, resulting in a larger number of small bubbles. It is dicult to test this hypothesis experimentally since we can only observe events 1.2. THE EARLY UNIVERSE 9

Figure 1.1 Simulation of the spatial distribution of bubbles at a late time in an eternally inﬂating false vacuum. From [16]. within our horizon. However, if other bubble universes have collided with our own universe, and the collision boundary is within our horizon (or has passed entirely through it) then there would be observational e↵ects that we could detect. There are several ways to search for a collision:

A collision leaves an imprint on the temperature and polarization pattern of the CMB • if the collision intersects the last scattering surface. We discuss this possibility further in Chapter 7, where we present results from a search for this signal.

If the collision wall has passed entirely through the last scattering surface then there • is a dipole induced in the CMB. A dipole asymmetry would be dicult to detect 10 CHAPTER 1. INTRODUCTION

above the motion of the Earth and Solar System, and depending on its amplitude may be impossible to measure.

In principle any measurement of large scale density perturbations is sensitive to the • e↵ects of a bubble collision. For example a collision would introduce a coherent ﬂow of objects.

In addition to the observational challenges associated with providing evidence for eternal inﬂation, falsifying the theory would be dicult since a detection of any of these signatures could be explained by other phenomena. However, the polarization pattern introduced during a collision has a signature that would be dicult to explain in other models [17].

1.3 Growth of Structure

After inflation density perturbations begin to grow due to gravity. We now overview the equations governing the linear growth structure so that we can connect our observations with underlying physical processes. While we do not know the details of the evolution of the universe immediately after inflation (or even if inflation occurred) we know that at some point the universe had cooled suciently to form an ionized plasma of protons and electrons. Coulomb scattering strongly couples the electrons and protons in the plasma and Compton scattering couples the electrons and photons, with all three components a↵ected by gravity. Over-dense regions of the plasma therefore collapse under gravity, resisted by the pressure of the fluid. The relative strength of these two forces at a given time determines whether the fluid collapses or expands, and it undergoes acoustic oscillations with the compressed regions becoming hotter and the rarefied regions colder. The universe had cooled suciently 380,000 years after the Big Bang for electrons and protons to condense into atoms, and as the number of free electrons becomes smaller the Coulomb length gets larger, decoupling the photons from the electrons and protons. Once the Coulomb length is greater than the Hubble length photons travel through the universe without scattering, and the baryons fall into the dark matter potential wells. Dark matter and neutrinos do not interact electromagnetically (with neutrino decoupling occurring 1 second after the Big ⇠ Bang), and their dynamics are only a↵ected by gravity. 1.3. GROWTH OF STRUCTURE 11

For small density ﬂuctuations, = ⇢/⇢ 1, the evolution of each component is accu- ⌧ rately described by perturbing the Einstein equations and neglecting second order terms. At 1 the present time, linear perturbation theory applies on scales larger than k & 10 Mpc. On smaller scales gravitational collapse must be treated using a non-linear theory, for example with the halo model, which we use in Chapters 3 and 4. We will only state the main results here since a full description requires lengthy treatment. We closely follow Dodelson [14], which should be referred to for more details.

For each component of the ﬂuid a Boltzmann equation can be written that describes its evolution from the initial inhomogeneities and accounts for its interaction with all of the other components. The Boltzmann equation describes the time evolution of the phase space density ⇢(x, p, t), where x and p are position and momentum. It formalizes the statement that the rate of change of the number of particles is equal to the di↵erence between the rate of production and the rate of destruction. While analytical solutions to the equations can be found in limiting cases when either a single component dominates or on large or small scales, on intermediate scales when the matter and radiation energy densities are similar numerical solutions must be found. This can be done using a numerical code such as CAMB [18].

We start by writing down the perturbed metric. While the unperturbed metric depends only on the scale factor a(t), the perturbed metric depends on two additional functions: and :

g00(x, t) = 1 2 (x, t) g0i(x, t) = 0 (1.11) 2 gij(x, t) = a ij (1 + 2(x, t)) , where is the perturbation to the spatial curvature and is the Newtonian potential. For linear theory it is easiest to work in Fourier space where the linearity ensures that each Fourier mode of the density and velocity fields evolves independently. Solving the Boltz- mann equations to first order in and v and assuming that the velocity field is irrotational 12 CHAPTER 1. INTRODUCTION

we ﬁnd:

˙ + ikv = 3˙ v˙ + Hv = ik (1.12) ⇥˙ + ikµ⇥ + ˙ + ikµ = ⌧˙ ⇥ ⇥ + µv , 0 b ⇥ ⇤ where ⇥ = T/T, over-dots represent derivatives with respect to conformal time (d⌘ = dt/a), (k, a) and v(k, a) depend on the Fourier mode and scale factor (which acts like a time variable), ⇥0 is the monopole, and µ = kˆ p is the photon direction. The ﬁrst equa- · tion is analogous to the continuity equation in Newtonian physics and the second equation is analogous to the Euler equation. Equivalent equations exist for the baryons, with an additional term due to Compton scattering on the right hand side of the Euler equation: ⌧˙[v + 3i⇥ ]/R, where R = 3⇢ /4⇢ is the ratio of baryon to photon density,⌧ ˙ = n a b 1 b e T is the optical depth to Compton scattering, and ⇥1 is the dipole in the photon distribution. There is an equivalent equation for the neutrinos. While the dark matter and baryons can be described by their density and velocity, the photons additionally have a quadrupole, octupole and higher moments, as well as polarization, which is an important observational probe of structure. We will not discuss polarization since most of this work is concerned with temperature anisotropies.

The evolution of the and potentials in equation 1.11 is described by the perturbed Einstein equations. From equation 1.1:

Gµ⌫ = 8⇡GTµ⌫ + ⇤g⌫µ . (1.13)

Of the ten equations we only need two, with the other equations either redundant or describing vector and tensor perturbations. We omit details of the calculation and quote the result:

2 2 2 k + 3H ˙ H = 4⇡Ga ⇢mm + 4⇢r⇥r,0 (1.14) k⇣2 ( + ⌘) = 32⇡Ga2⇢ ⇥ , ⇥ r r,2 ⇤ where quantities with the subscript r describe both photons and neutrinos. From these 1.3. GROWTH OF STRUCTURE 13

Radiation Matter ⇤ Component t < teq teq < t < tdec tdec < t < t⇤ t > t⇤ oscillation oscillation constant constant b oscillation oscillation D(a) constant c logarithmic D(a) D(a) constant ⇥r oscillation oscillation constant constant

Table 1.1 Evolution of a single mode of the potential , baryon density b, dark matter density c, and radiation temperature ⇥r, based on Ma [19]. Each mode begins evolving when it enters the horizon, which may not occur before decoupling. D(a) is the growth function. equations it can be determined that there are three time periods where a di↵erent component dominates the dynamics:

1. Radiation domination - The radiation density has a (1 + z)4 dependence and so dominates at early times. As modes in the baryon and radiation ﬁelds enter the horizon they begin to oscillate, while dark matter modes grow logarithmically.

2. Matter domination - The matter density has a (1 + z)3 dependence and dominates at intermediate times. Baryon and radiation modes that enter the horizon before recombination oscillate, whereas those that enter the horizon later do not. Since the baryons interact both gravitationally and electromagnetically they damp the oscillations. Af- ter recombination the baryons are no longer supported by radiation pressure and fall into the dark matter potential wells, with both densities then growing proportionally to the growth function.

3. Dark energy domination - The dark energy density is constant and so will dominate at late times. Once dark energy dominates, the growth of density perturbations slows down due to the rapid expansion and eventually the modes freeze out.

The general behavior of each component is given in table 1.1. 14 CHAPTER 1. INTRODUCTION

1.4 The Cosmic Microwave Background

The CMB was first observed in 1964 by Penzias and Wilson who noticed excess noise in their receiver while working on an antenna to detect radio waves bounced o↵ metallic bal- loons in the atmosphere. It was soon realized that they were observing the 2.725 K CMB and the measurement provided an important test of the Big Bang model. As detector technology improved it became possible to search for anisotropies in the light, which were first measured in 1992 by the Cosmic Background Explorer (COBE) [20] and are now known to have an 18 µK RMS, corresponding to fluctuations of a part in 105. The low angular resolution of COBE prevented a measurement of the first acoustic peak in the spatial spectrum of the radiation temperature, which was first measured by the Toco experiment [21] and confirmed by the BOOMERanG [22] and MAXIMA [23] experiments. Many experiments, including the WMAP and Planck satellites, have now measured the smaller scale peaks in the spectrum, and the polarization of the light has been measured with high signal to noise ratio. The CMB light was emitted as blackbody radiation with peak emission at infrared wavelengths, which has been redshifted to microwave wavelengths today. In Fig. 1.2 we show the WMAP Internal Linear Combination CMB map (created from a combination of frequency channels to minimize galactic emission). The CMB temperature is close to a Gaussian random field, and so the statistical properties of the temperature map can be described by its power spectrum (to the extent that it is Gaussian), shown in Fig. 1.3. On scales larger than the horizon, modes are out of causal contact and so evolve adiabat- ically. At the time of CMB recombination this corresponds to regions on the last scattering surface separated by ✓ 1, and modes larger than this have approximately constant power ⇠ in the spectrum. The power in these modes is modified by the redshifting of CMB photons as they climb out of gravitational potential wells, known as the Sachs-Wolfe e↵ect [24]. Small-scale modes (` & 2000) lose power during recombination as photons have time to di↵use from small-scale hot spots to cold spots over the timescale of recombination, an e↵ect known as Silk damping [25]. The peaks and troughs in the spectrum are a signature of oscillations in the plasma, with multiple peaks indicating that the plasma has undergone multiple compressions and rarefactions. The peaks come from regions at an extrema of 1.4. THE COSMIC MICROWAVE BACKGROUND 15

Figure 1.2 Map of CMB temperature anisotropies measured by the WMAP satellite (WMAP Collaboration). their oscillation, either maximally compressed or rareﬁed, with troughs occurring between the extrema. Although the CMB we observe today is close to Gaussian, non-Gaussianity is introduced as the light travels to us, for example by gravitational lensing. In addition, it is possible that there is primordial non-Gaussianity, which can be introduced during inﬂation.

1.4.1 The Sunyaev Zeldovich Effect

Since the CMB is the most distant object that we can observe with visible light it provides a backlight to the universe, allowing intervening structure to be observed at microwave frequencies. Matter falling into the gravitational potentials of galaxy clusters is heated, and this energy can be transferred to CMB photons scattering o↵ electrons within the cluster. Approximately 1% of CMB photons passing through a massive cluster are scattered, leaving a characteristic decrease in CMB brightness at frequencies less than 220 GHz ⇠ and an increase in brightness at higher frequencies. This is the thermal Sunyaev Zeldovich (SZ) e↵ect and has been observed by many experiments. An additional signal is generated if electrons within a cluster have a non-thermal velocity relative to the CMB, which causes the scattered light to be Doppler shifted. This is the kinetic SZ e↵ect and gives a 16 CHAPTER 1. INTRODUCTION

Figure 1.3 Power spectrum of CMB temperature anisotropies measured by the WMAP satellite (WMAP Collaboration). temperature change of: T ~v ~l = d⌧ e · , (1.15) T c Z ~ where ⌧ is the optical depth to Compton scattering, ~ve is the electron velocity and l is a unit vector in the direction of the cluster. The kinetic SZ e↵ect has recently been measured in a large sample of clusters by ACT [26]. Observations of the kinetic SZ e↵ect allow the velocity of matter on large scales to be probed, and we place limits on the motion of galaxy clusters in Chapter 5.

1.5 The Cosmic Infrared Background

The cosmic infrared background (CIB) is the combined infrared emission from galaxies, with a large fraction of the light emitted by galaxies at redshifts 1 3 undergoing rapid ⇠ 1.5. THE COSMIC INFRARED BACKGROUND 17

star formation. The light is emitted with a graybody spectrum from dust grains heated by young UV-bright stars, and is redshifted to infrared and far-infrared wavelengths today. The peak star formation rate of the universe occurred around redshift 1 2 [e.g. 27], ⇠ partly because of a higher number of galaxy collisions and mergers in the past. The history of star-forming events and starburst activity is encoded in the CIB, and measurements of the CIB brightness can be used to estimate the star formation rate. Resolving the objects that contribute to the CIB at high redshift is dicult due to their low brightness and small angular size (but will be possible with ALMA and JWST). How- ever, variations in the number density of galaxies results in spatial fluctuations in the unresolved CIB emission. On large scales, the angular power is determined by the clustering of dark matter halos hosting the galaxies, as well as the redshift distribution of CIB brightness. The dark matter halos hosting dusty galaxies gravitationally lens the microwave background light, correlating the spatial variations in the infrared background with the CMB lensing potential. Early attempts to measure the integrated starlight from galaxies were made in the 1950s and 60s [28–30], but it was not until the mid 1970s that it was realized that there would be a significant contribution from extragalactic dust emission at infrared wavelengths [e.g. 31, 32]. The emission was estimated to be fainter than the galactic foreground emission in Partridge and Peebles [33], but by the 1980s only upper limits had been obtained. Limits were placed on the absolute brightness of the emission using sounding rockets, and with satellite observations in the late 1980s and 90s the signal became apparent, with some evidence from the Infrared Astronomy Satellite [IRAS, 34] and then a conclusive detection from the Di↵use Infrared Background Experiment (DIRBE) aboard COBE [35]. Later measurements were made by the Infrared Space Observatory (ISO), and angular fluctuations in the unresolved signal were first detected by Spitzer [36]. More recently, the Her- schel Space Observatory and Planck satellite, launched together in 2009, have made maps of the infrared background at 250, 350, and 500 µm (Herschel), and 350, 550, 850, and 1380 µm (Planck). We show maps of the CIB measured by Planck in Fig. 1.4. The emission is largely unresolved but is measured at high signal to noise ratio. We will present a measurement of the correlation between the infrared emission and CMB lensing potential in Chapter 3. For 18 CHAPTER 1. INTRODUCTION

(a) 217 GHz (b) 353 GHz

2 Figure 1.4 Maps of the 26 deg N1 ﬁeld (galactic coordinates l, b = 85.33, 44.28) at 217, 353, 545, and 857 GHz, smoothed with a 100 Gaussian, with the CMB, bright point sources, and galactic cirrus subtracted. From Planck Collaboration et al. [39].

a review of the CIB see Hauser and Dwek [37] and Kashlinsky [38]. 1.6. LENSING OF THE CMB 19

1.6 Lensing of the CMB

The CMB is gravitationally lensed by structure between ourselves and the surface of last scattering, with an RMS deflection angle of 2.70 [40, 41] and deflections coherent over ⇠ scales of a few degrees. The majority of the deflection is from structures at redshift 1 3 ⇠ 2 1 with sizes k few 10 Mpc [42]. The lensing deflection primarily comes from matter ⇠ ⇥ on scales where linear theory provides a good description of the growth of structure, and so the lensing structure is close to a Gaussian random field. We write the deflection angle as the gradient of a lensing potential, ↵~ = . In the linear regime the power spectrum r of can be calculated by solving the geodesic equations of the metric in equation 1.11. Since the deflection angles are small the Born approximation can be used, which allows lensing deflections to be evaluated on the null-geodesics of unlensed photons. With this approximation the deflection angle is [see, for example, the review by 43, and references therein]: ⇤ ↵(nˆ) = nˆ (nˆ) = 2 d ⇤ nˆ w(nˆ, ⌘0 ) , (1.16) r 0 r Z ⇤ where w ( )/2, is the comoving distance to the last scattering surface, ⌘0 is ⌘ ⇤ the conformal time when the photon is at position nˆ, and we have assumed a flat universe. Under the Limber approximation the linear theory lensing power spectrum is:

4 2 2 2 9H0⌦0 ⇤ ⇤ = `/, , C` 4 d P (k ) (1.17) ⇡ ` 0 a Z ✓ ◆ ⇤ ! where P is the matter power spectrum. In Fig. 1.5 we show the spectrum calculated from equation 1.17. The CMB lensing signal has been measured through cross-correlation with galaxy surveys [45–47] as well as directly in CMB maps by the ACT, SPT, and Planck experiments [48–50]. Measurements of the lensing potential spectrum allow useful constraints to be placed on cosmological parameters such as the sum of neutrino masses and dark energy equation of state [e.g. 51–53]. In addition, degeneracies between some parameters obtained from measurements of the CMB power spectrum, such as the dark energy and curvature densities both of which depend on the angular diameter distance, can be broken using measurements of the lensing potential, which is sensitive to structure at a di↵erent 20 CHAPTER 1. INTRODUCTION

−6 10 2

/ −7 10 l C 2 +1)] l

( −8 l

[ 10

−9 10 1 10 100 1000 l

Fig. 2. The powerFigure 1.5 spectrum Power spectrum of the of deﬂection the CMB lensing angle potential (given in in the terms⇤CDM of themodel. lensing The solid potential by )foraconcordanceline is the linearCDM theory model. model, The with linear the dashed theory line spectrum also including (solid) non-linear is compared corrections with the same model includingfrom non-linearHALOFIT [44]. corrections From [43]. (dashed) from halofit (65).

The lensingredshift potential range can [52, be 54] expanded as has been in demonstrated spherical in harmonic van Engelensas et al. [49], Ade et al. [50], Sherwin et al. [55]. In Section 1.7 we will discuss the estimator we use to reconstruct (nˆ)= lmYlm(nˆ), (3.9) the CMB lensing potential, and in Chapter 3 we will present a measurement of the cross lm correlation between the CIB and lensing potential measured by Planck. and for a statistically isotropic ﬁeld the angular power spectrum Cl is deﬁned by

lm = ll mm C . (3.10) 1.7 The Planck Mission lm l

Taking the sphericalPlanck is a Europeanharmonic Space components Agency mission, of Eq. with (3.8) a significant we the NASArefore contribution, get that was launched in May 2009 and designed to measure the temperature and polarization dk Cl =16anisotropies ofd the CMB.d It contains (k; two0 instruments:, 0 the)j Lowl(k Frequency)jl(k) Instrument (LFI) . k 0 0 P with detectors at 30, 44, and 70 GHz, and the High Frequency Instrument (HFI) with de- (3.11) tectors at 100, 143, 217, 353, 545, and 857 GHz. HFI has 52 spider web and polarization In linear theorysensitive we bolometers can define cooled a to transfer 100 mK, withfunction light fedT by(k three; )sothat corrugated horns,(k; two)= at 4T K (k; ) (k) R where (k)and is theone at primordial 100 mK, and comoving a set of filters curvature and lenses cooled perturbation to cryogenic (or temperatures other variable [56]. for isocur- vatureR modes). We then have

2 dk Cl =16 (k) d T(k; 0 )jl(k) , (3.12) k PR 0 where the primordial power spectrum is (k). Given some primordial power spectrum this R can be computed easily numerically usingP Boltzmann codes such as camb 8 (66). Since it is deﬂection angles that are physically relevant, it is usual toplotthepowerspectrumofthe deﬂection angle given by l(l +1)C,asshownforatypicalmodelinFig.2.Notethatfor l 8 http://camb.info

27 1.7. THE PLANCK MISSION 21

One of the goals of Planck is to measure the polarization signal from primordial gravitational waves as well as from gravitational lensing of the CMB. The polarization signal can be decomposed into E and B-modes, with the primordial signal contributing only B- modes, and the gravitational lensing signal contributing E-modes and B-modes generated from E-modes by lensing. The theoretical CMB temperature and polarization power spectra are shown in Fig. 1.6. The dashed red and blue lines show polarization noise levels at

143 and 217 GHz assuming beam FWHM of 70 and 50, white noise standard deviations of T/T = 4.2 µK/K and 9.8 µK/K in a pixel of size equal to the beam FWHM [57], and multipole bins of width d` = 50. This is approximately equal to the sensitivity obtained from two full-sky surveys (the HFI instrument aboard Planck has completed over ﬁve surveys). Additional information about the instrument can be found in [58–60]. 22 CHAPTER 1. INTRODUCTION

104

Temperature

2 10 E−modes ) 2 K

µ 0 ( 10 π /2 l Lensed B−modes +1)C l

( −2 l 10

B−modes r=0.1 10−4 r=0.01

500 1000 1500 2000 Multipole number l

Figure 1.6 Theoretical temperature and polarization power spectra. The primordial B-mode signal is shown in blue with two di↵erent values for the tensor to scalar ratio. The dashed curves are expected noise levels at 143 GHz (red) and 217 GHz (blue) for two sky surveys. Bibliography

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Cosmic Microwave Background (CMB) light is gravitationally lensed by large-scale structure, with RMS deflection angles of 2.70. The majority of the deflection occurs between ⇠ 2 1 redshift 1 3 by lenses of size k few 10 Mpc [1–4], and traces the large-scale ⇠ ⇠ ⇥ dark matter structure. Measurements of the lens deflection angles are useful for several reasons: 1) They allow useful constraints to be placed on cosmological parameters such as the sum of neutrino masses and dark energy equation of state [5–7], as well as enabling parameter degeneracies arising in CMB power spectrum analyses to be broken. 2) They can be used to “delens” the CMB, which can lead to higher signal to noise ratio measurements of primordial B-modes. 3) Lensing of the CMB is a source of bias in primordial non-Gaussianity measurements, and so the lens deflection angles can aid in bias removal. 4) Cross-correlating the deflection angles with tracers of large scale structure allows us to measure properties of the tracer over the redshift range where the lensing signal originates.

Before presenting an analysis of the extragalactic foreground contamination in CMB lens reconstruction, we will give a brief overview of the estimator we use to measure the lens deﬂection angles in CMB maps. We focus on the temperature data, since for Planck the polarization information does not signiﬁcantly increase the lens reconstruction signal to noise ratio.

The lensed CMB temperature ﬁeld is a remapping of the unlensed ﬁeld, which depends

33 34 BIBLIOGRAPHY

on the lensing potential :

T(nˆ) = T˜(nˆ + (nˆ)) r (1.18) = T˜(nˆ) + (nˆ) T˜(nˆ) + (2) , r · r O where T(nˆ) is the lensed CMB temperature at position nˆ, T˜(nˆ) is the unlensed CMB temperature, is the lensing potential, and is the photon deﬂection angle. The lensing r potential can be calculated from the gravitational potential :

⇤ (nˆ) = 2 d ⇤ (nˆ, ) , (1.19) 0 Z ⇤ where is the comoving distance and is the distance to last scattering. The convergence, ⇤  = 1 2, is a measure of the projected density perturbation [see e.g. 4, for details]. 2 r Since depends linearly on the gravitational potential it is, to a good approximation, a Gaussian random field with power peaking on angular scales of a few degrees. If we think of the lenses as being fixed and average over CMB realizations, then gravitational lensing introduces a source of statistical anisotropy into the CMB map. Using the flat sky approximation for simplicity, and working in Fourier space, the CMB covariance is:

2 d `0 T(~` )T(~` )⇤ = T˜(~` )T˜(~` )⇤ T˜(~` )T˜(~`0)⇤ (~` ~`0)⇤ (~` ~`0) ~`0 1 2 1 2 (2⇡)2 1 2 2 · 2 Z D E D d ` E D E (1.20) T˜(~`)T˜(~` )⇤ (~` ~`)(~` ~`) ~` (2⇡)2 2 1 1 · Z D E = C˜TT(~` ~` ) + C˜TT(~L) ~L ~` C˜TT(~L) ~L ~` + (2) , `1 1 2 `1 · 1 `2 · 2 O where ~L = ~` ~` . Ignoring second order terms in , the dependence of the covariance on 1 2 a single mode of the lensing potential is:

@ TT TT T(~` )T(~` )⇤ = ~L ~` C˜ ~L ~` C˜ (~L ~L0) . (1.21) ~ 1 2 1 `1 2 `2 @(L0) · · D E ⇣ ⌘ The covariance of modes T(~` ) and T(~` ) is therefore proportional to (~` ~` ), which 1 2 1 2 allows us to estimate by measuring the CMB covariance. Although the CMB power BIBLIOGRAPHY 35

spectrum is sensitive to lensing, which smooths the peaks and troughs and shifts power from large to small scales, we will not consider the extraction of lensing information from the power spectrum. See Benoit-Levy et al. [8] for more details.

Quadratic Estimators

The quadratic estimator that we use can be derived from the maximum likelihood estimator by making the approximation that the CMB covariance is linear in , ignoring terms of (2) in Eq. (1.20). In this section we brieﬂy outline the derivation of the estimator, fol- O lowing Hirata and Seljak [9] (but also see [3, 10–12] and references therein). The negative logarithm of the lensing potential likelihood, given the observed CMB map Tˆ, is:

1 1 1 ln L[ Tˆ] [ Tˆ] = Tˆ t CˆTT[] Tˆ + ln det CˆTT[] , (1.22) | ⌘ L | 2 2 ⇣ ⌘ where CˆTT[] = CTT[] + CN is the covariance of the data, and CN is the noise covariance. We wish to maximize the likelihood with respect to the lensing potential, which is equivalent to maximizing the log likelihood [9]:

[ Tˆ] 1 1 CˆTT[] 1 1 1 CˆTT[] L | = Tˆ t CˆTT[] CˆTT[] Tˆ + Tr CˆTT[] . (1.23) 2 2 ⇣ ⌘ ⇣ ⌘ "⇣ ⌘ # The ﬁrst term can be written as [9]:

TT 1 TT 1 TT 1 Tˆ Cˆ [] ⇤[] C˜ ⇤[] Cˆ [] Tˆ , (1.24) r · r  ⇣ ⌘ ⇣ ⌘ where we have introduced the operator ⇤[]T˜(nˆ) = T˜(nˆ + (nˆ)). We calculate the second r term using simulations of the observed CMB, which includes a model of the instrumental noise and beam, as well as data processing e↵ects such as the pixelization and foreground mask (see [13] for details). From the simulations we calculate a lens “mean-ﬁeld”, which is the mean of the reconstructions in the simulations, and we subtract this from Eq. (1.24). The likelihood peak can be found by solving @ /@ = 0 iteratively, for example using L 36 BIBLIOGRAPHY

the Newton-Raphson method. After i + 1 iterations the solution is:

1 2 ˆ = ˆ L L . (1.25) i+1 i " i# i To ﬁrst order in it suces to use a single iteration, starting from ˆ = 0:

ˆ = F L , (1.26) =0 1 where the inverse second derivative of the likelihood, 2 / , is the Fisher matrix L F, which acts as the estimator normalization. For simplicityh we replacei F by its average over realizations of Tˆ. For full-sky data it is natural to use the spherical harmonic basis. Derivatives on the sphere can then be evaluated using spin-1 spherical harmonics, with

Y`m(nˆ) = `(` + 1) 1Y` (nˆ) . (1.27) r m p This allows us to write the likelihood derivative at = 0 (equivalently the unnormalized estimator, ¯, in the approximation that we neglect the mean-ﬁeld):

TT TT 1 TT 1 ¯ = ⇤ ˆ ˆ ˆ ˆ ˆ . `m dnˆ 1Y`m(nˆ) 1Y`1m1 (nˆ) C`1 C T 0Y`2m2 (nˆ) C T 2 `1m1 3 2 `2m2 3 Z `1m1 ✓ ◆ `2m2 ✓ ◆ 6X ⇣ ⌘ 7 6X ⇣ ⌘ (1.28)7 6 7 6 7 46 57 46 57 Writing the normalization as A` , and including the mean-ﬁeld gives:

ˆ = A¯ MF . (1.29) `m ` `m `m D E 1 We calculate CˆTT Tˆ using the conjugate gradient descent method [13, 14]. On oc- 1 TT casion we approximate⇣ ⌘ the noise as being isotropic, in which case C T T`m/C`, `m ⇡ and the estimator reduces to the Okamoto and Hu [10] estimator with✓⇣ a⌘ mean-ﬁeld◆ correction. For more information about our estimator see [3, 9, 10, 12, 13]. Bibliography

[1] R. D. Blandford and M. Jaroszynski. Gravitational distortion of the images of distant radio sources in an inhomogeneous universe. ApJ, 246:1–12, May 1981. doi: 10. 1086/158892.

[2] A. Blanchard and J. Schneider. Gravitational lensing e↵ect on the ﬂuctuations of the cosmic background radiation. A&A, 184:1–6, October 1987.

[3] M. Zaldarriaga and U. Seljak. Reconstructing projected matter density power spectrum from cosmic microwave background. Phys. Rev. D, 59(12):123507, June 1999. doi: 10.1103/PhysRevD.59.123507.

[4] A. Lewis and A. Challinor. Weak gravitational lensing of the CMB. Phys. Rep., 429: 1–65, June 2006. doi: 10.1016/j.physrep.2006.03.002.

[5] J. Lesgourgues, L. Perotto, S. Pastor, and M. Piat. Probing neutrino masses with CMB lensing extraction. Phys. Rev. D, 73(4):045021–+, February 2006. doi: 10. 1103/PhysRevD.73.045021.

[6] R. de Putter, O. Zahn, and E. V. Linder. CMB lensing constraints on neutrinos and dark energy. Phys. Rev. D, 79(6):065033–+, March 2009. doi: 10.1103/PhysRevD. 79.065033.

[7] K. M. Smith, A. Cooray, S. Das, O. Dore,´ D. Hanson, C. Hirata, M. Kaplinghat, B. Keating, M. LoVerde, N. Miller, G. Rocha, M. Shimon, and O. Zahn. CMBPol Mission Concept Study: Gravitational Lensing. ArXiv e-prints, November 2008.

37 38 BIBLIOGRAPHY

[8] Aurelien Benoit-Levy, Kendrick M. Smith, and Wayne Hu. Non-Gaussian structure of the lensed CMB power spectra covariance matrix. Phys.Rev., D86:123008, 2012. doi: 10.1103/PhysRevD.86.123008.

[9] C. M. Hirata and U. Seljak. Analyzing weak lensing of the cosmic microwave background using the likelihood function. Phys. Rev. D, 67(4):043001–+, February 2003. doi: 10.1103/PhysRevD.67.043001.

[10] T. Okamoto and W. Hu. Cosmic microwave background lensing reconstruction on the full sky. Phys. Rev. D, 67(8):083002–+, April 2003. doi: 10.1103/PhysRevD.67. 083002.

[11] Martin Bucher, Carla Soﬁa Carvalho, Kavilan Moodley, and Mathieu Remazeilles. CMB Lensing Reconstruction in Real Space. Phys.Rev., D85:043016, 2012. doi: 10.1103/PhysRevD.85.043016.

[12] Duncan Hanson, Anthony Challinor, George Efstathiou, and Pawel Bielewicz. CMB temperature lensing power reconstruction. Phys.Rev., D83:043005, 2011. doi: 10. 1103/PhysRevD.83.043005.

[13] P.A.R. Ade et al. Planck 2013 results. XVII. Gravitational lensing by large-scale structure. 2013.

[14] Kendrick M. Smith, Oliver Zahn, and Olivier Dore. Detection of Gravitational Lens- ing in the Cosmic Microwave Background. Phys.Rev., D76:043510, 2007. doi: 10.1103/PhysRevD.76.043510. Chapter 2

Foreground Contamination in CMB Lens Reconstruction

“Reeling and Writhing, of course, to begin with,” the Mock Turtle replied; “and then the di↵erent branches of Arithmetic–Ambition, Distraction, Ugliﬁcation, and Derision.”

Alice in Wonderland Lewis Carroll

We have investigated the e↵ect of unpolarized and unresolved point source and Sunyaev-Zel’dovich emission on CMB lens reconstruction, and on trispectrum estimators in general. Using quadratic estimators we show that the foregrounds bias the lensing potential power spectrum, but that the di↵erent contributions to the bias can each be measured in the data. For a lensing estimator with

resolution Lmax = 1500 (for example Planck) the dominant contribution to the bias can be calculated by assuming that the foreground sources are unclustered, but as foreground sources are removed down to lower ﬂux density limits the clustering terms become more important. We describe an optimal estimator to measure the amplitude of the unclustered foreground component, as well as estimators to measure the amplitude of the clustering terms. We test

39 40 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

our estimators on simulations, and ﬁnd that the unclustered component can be measured to a precision of 6% with an L = 1500 estimator (assuming ⇠ max 60% usable sky fraction) and 3% for an L = 3000 estimator (for example ⇠ max SPT of ACT, assuming 0.06% sky fraction). The di↵erent bias terms can be subtracted from the lensing potential spectrum giving a signiﬁcant reduction in the foreground contamination.

2.1 Introduction

Gravitational lensing of the cosmic microwave background (CMB) by large-scale structure is a long predicted e↵ect [1], which has only recently become detectable [2, 3]. Modern high-resolution CMB experiments such as ACT, SPT, and Planck are now able to measure the power spectrum of the CMB lensing potential at high (4 25 ) signiﬁcance, and have ! successfully used it for cosmological parameter ﬁtting [4–6]. The power spectrum of the CMB lensing potential is not directly measurable, but must be inferred from the non-Gaussian statistical properties that it induces in the observed CMB. The trispectrum, or connected 4-point function, of the CMB is very sensitive to lensing [7] and is the basis for all current estimators of the lensing potential power spectrum [e.g. 8–10]. A potential source of error in these measurements can be caused by additional sources of non-Gaussian signal in the observed sky, such as extragalactic foreground contamination from radio/infrared point sources and Sunyaev-Zel’dovich e↵ects from clusters, that may be misinterpreted as a lensing signal and provide a source of bias for the lensing estimates. In this paper, we will address this problem using a twofold approach:

1. We model the contribution of unresolved sources semi-analytically, to develop an intuitive understanding of the dominant biases to lensing and the signature of their e↵ects on the lensing potential.

2. We develop modiﬁed estimators which may be used either to measure the amplitude of these contaminating terms, or to project their contribution out of the standard lensing estimates, testing our approach using the simulations of [11]. 2.1. INTRODUCTION 41

We can split the bias to the lensing measurement from point sources into two pieces: (1) Detected (and therefore masked) sources can be tracers of the same large-scale structures which lens the CMB. Masking of these sources preferentially removes regions of the sky where the lensing potential is large. (2) Unresolved, un-detected sources have non- Gaussian statistics which may mimic lensing. From an analysis point of view, the bias caused by masking resolved sources is easier to measure than the bias from the unresolved source population. The fraction of the sky masked by sources is adjustable, and so biases from the resolved (or nearly resolved) source population can be investigated by varying the cut in flux density that defines the mask. For this reason most of our focus will be on char- acterizing unresolved source contamination, although in Sec. 2.7.2 we will briefly consider masking e↵ects.

The bias to lens reconstruction from extragalactic foregrounds has been studied before. The most detailed analyses have focussed on the contamination of the cross-correlation of the CMB lensing potential with external tracers. Unresolved source contamination was found to provide the largest source of systematic error for cross-correlation of the CMB lensing potential with NVSS quasars [2, 3], and the SPT lensing potential-Herschel galaxy cross-correlation was estimated to have a foreground contribution of a few percent based on contamination of the lensing potential auto spectrum [12]. The additional noise expected in the Planck lensing potential-Herschel galaxy cross-correlation caused by infrared point source emission is expected to dominate the signal on scales L & 1000 [13]. Although useful as a guide to the magnitude of lensing bias e↵ects, these bispectrum (3-point) calculations are not directly relevant to the trispectrum (4-point) terms of concern for lensing auto-spectrum measurements, which are the subject of this paper. The trispectrum has received relatively less attention. In the case of polarization lens reconstruction, the complete set of trispectrum terms can be calculated under the assumption that polarization angles from di↵erent sources are uncorrelated [14]. Under this assumption the contamination from unresolved radio point sources in polarization was found to be small, however this situation does not necessarily carry over to temperature, where radio source emission is larger relative to the CMB ﬂuctuations, and Sunyaev-Zel’dovich (SZ) and infrared source emission are also a larger concern. In the case of temperature, published analyses have used the simulations of Sehgal et al. [11] to investigate numerically the magnitude of 42 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

source contamination, finding percent-level biases [4, 5], although the wealth of new sub- mm data from Herschel, Planck, ACT, and SPT have lead to improvements in our modeling of extragalactic CMB foregrounds, which in some cases conflict with the simulation inputs, therefore making interpretation of the simulation results dicult. This paper presents a detailed analytical framework for estimating point-source contamination to the temperature trispectrum, as well as estimators which can be used to produce lensing estimators with reduced sensitivity to the trispectrum “shapes” generated by extragalatic foregrounds. We will ultimately use the simulations of Sehgal et al. [11] to test several aspects of our results, with the caveats mentioned above. Throughout this work, the following simple model of point sources will provide useful intuition. Consider laying down a field of density contrast 1 + (~x) throughout the Universe. Then proceed to populate this Universe with point sources such that the source number density traces the density perturbations. On suciently large scales, the distribution function from which these sources are drawn can be approximated as a linear function of the density, so that e.g. a region of space where (~x) is twice as large will have (on average) twice as many sources. If we consider the density field (~x) to be fixed, and look at multiple realizations of the source distribution, then the noise has statistical properties that are similar to inhomogeneous instrumental noise, but with the “shot noise” modulated by the large-scale density field rather than by instrumental e↵ects. This modulation introduces a source of statistical anisotropy, with properties that can be studied in an analogous way to the statistical anisotropy that lensing introduces. Of course, in reality we do not have multiple realizations of the Universe with the same density contrast. Both point sources and lensing ultimately appear in CMB maps as a source of trispectrum non-Gaussianity, rather than statistical anisotropy. Nevertheless, we will find that this picture of point sources as a source of inhomogeneous noise is a useful intuitive picture. The remainder of this paper is as follows. In Sec. 2.2 we review the procedure of lens reconstruction. In Sec. 2.3 we derive templates for the shape of extragalactic foreground contributions to the trispectrum, and calculate their overlap with the lensing trispectrum. In Sec. 2.4 we discuss how to construct lensing estimators which are less sensitive to these foreground shapes. In Sec. 2.5 we test our predictions of the lensing contamination using the simulations of Sehgal et al. [11]. In Sec. 2.6 we use simple models to estimate the 2.2. LENS RECONSTRUCTION 43

amplitude of each template shape for various source populations. In Sec. 2.7 we estimate the bias introduced by masking resolved sources. We conclude in Sec. 2.8.

2.2 Lens Reconstruction

2.2.1 Lensing Potential

In this section, we brieﬂy review the process we use to estimate the lensing potential power spectrum CL from temperature maps of the microwave background, using a formalism which will make the discussion of the preceeding sections more compact. Lensing is a remapping operation, described in real space by

T(ˆn) = T˜(ˆn + (ˆn)), (2.1) r where T˜ is the primordial CMB temperature, and is the CMB lensing potential. We begin by considering the lensing potential to be fixed, and consider the statistics of the CMB after averaging over realizations of the Gaussian, unlensed T˜(ˆn). Lensing is a linear operation, and so if the lensing potential is fixed the observed sky remains Gaussian, however it becomes statistically anisotropic and its covariance acquires o↵-diagonal elements. At first order in the lensing potential we have

`1 `2 L T` T` = W , (2.2) 1m1 2m2 `1`2L LM LM 0 m1 m2 M 1 X B C ⌦ ↵ B C @B AC where the ensemble average is taken over realizations of T˜lm and the lensing “weight function” is given by

(2` + 1)(2` + 1)(2L + 1) W = 1 2 CTT ` (` + 1) `1`2L `1 1 1 r 4⇡ p ` +` +L 1 + ( 1) 1 2 `1 `2 L + (`1 `2). (2.3) ⇥ 2 0 10 1 1 $ ! B C B C @B AC 44 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

The statistical anisotropy introduced by lensing can be probed with quadratic estimators x¯, derived from maximum likelihood estimators [e.g. 15, and references therein], and constructed as 1 ` ` L = 1 2 x ¯ ¯ , x¯LM W`1`2LT`1m1 T`2m2 (2.4) 2 ` ,` 0 m1 m2 M 1 1mX1 2m2 B C B C where T¯ are inverse-variance filteredB CMB multipolesC and W x is a weight function lm @ A `1`2L for the quadratic estimator. The weight function is usually taken to be a matched filter for lensing with W x = W and the estimator denoted as ¯, however it can also be advantageous to use other weight functions with reduced sensitivity to certain systematic e↵ects [16]. We will find this is also the case with point source contamination. Throughout this paper we will assume homogeneous instrumental noise, and that the inverse-variance filter can TT TT 1 TT be calculated on the full sky with the filter function F` = [C` + N` ] , where C` is TT a fiducial CMB power spectrum and N` is the (beam-deconvolved) instrumental noise power spectrum. This is applied to an observed (beam-convolved) data map T obs(ˆn) as

¯ 1 obs Tlm = F` B` dnYˆ lm⇤ T (ˆn), (2.5) Z where B` is the instrumental beam transfer function. The estimatorx ¯ responds to such that averaging over CMB realizations with a ﬁxed realization of gives x¯ = x , h LMi RL LM where the response function is given by R

x 1 1 x = W W F` F` . (2.6) L + `1`2L `1`2L 1 2 R 2L 1 ` ` 2 X1 2

x x 1 Estimates of the lensing potential may therefore be formed as ˆ = ( ) x¯ . The Okamoto LM RL LM 1 1 and Hu [8] lensing estimator, for example, is ˆ = ( ) ¯ , with ( ) acting as the LM RL LM RL estimator normalization. We will discuss this formalism further in Sec. 2.4, giving the response of additional estimators. 2.2. LENS RECONSTRUCTION 45

2.2.2 Lensing Power Spectrum

Estimates of the lensing potential power spectrum may be constructed straightforwardly from the potential estimates above. The ensemble average (over both realizations of the primordial CMB and the lensing potential) is given explicitly by

2 1 M x x x¯LM = ( 1) W W | | 4 `1`2L `3`3L ` m ` m ` m ` m D E X1 1 X2 2 X3 3 X4 4 ` ` L ` ` L 1 2 3 4 ¯ ¯ ¯ ¯ T`1m1 T`2m2 T`3m3 T`4m4 . (2.7) ⇥ 0 m1 m2 M 1 0 m3 m4 M 1 B C B C D E B C B C The ensemble-average of the@ 4-point functionA @ can be brokenA into connected (C) and disconnected (D) parts as

T¯` m T¯` m T¯` m T¯` m = T¯` m T¯` m T¯` m T¯` m + T¯` m T¯` m T¯` m T¯` m . (2.8) 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 C 1 1 2 2 3 3 4 4 D D E D E D E The disconnected part is that which can be formed from the three Wick contractions of the four multipoles, and is given by

T¯` m T¯` m T¯` m T¯` m = C¯` m ,` m C¯` m ,` m + C¯` m ,` m C¯` m ,` m + C¯` m ,` m C¯` m ,` m , 1 1 2 2 3 3 4 4 D 1 1 2 2 3 3 4 4 1 1 3 3 2 2 4 4 1 1 4 4 2 2 3 3 D E (2.9) ¯ ¯ ¯ where C`1m1,`2m2 = T`1m1 T`2m2 is the covariance matrix of T. We denote the contribution xx of the disconnectedD part to theE ensemble average of Eq. (2.7) as NL .

The connected part of the 4-point function is zero for purely Gaussian ﬂuctuations, but in the presence of lensing it becomes non-zero. Following [7], for a statistically isotropic signal such as lensing the connected 4-point function must take the form

` ` ` ` M `1`2 1 2 L 3 4 L T` m T` m T` m T` m = ( 1) T (L) , (2.10) 1 1 2 2 3 3 4 4 C `3`4 LM 0 m1 m2 M 1 0 m3 m4 M 1 X B C B C ⌦ ↵ B C B C @B AC @B AC 46 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

where T `1`2 (L) is known as the trispectrum. Symmetry of the four multipoles further re- `3`4 quires that the trispectrum may be encoded as

T `1`2 (L) = P`1`2 (L) + (2L + 1) `3`4 `3`4 XL0 ` ` ` ` ` +` 1 2 L `1`3 L+L 1 2 L `1`4 ( 1) 2 3 P (L0) + ( 1) 0 P (L0) . (2.11) `2`4 `3`2 " 8 `4 `3 L0 9 8 `4 `3 L0 9 # <> => <> => ` ` > > > > The ﬁrst term P 1 2 (L): is referred to; as the primary contraction: of the trispectrum,; while the `3`4 ﬁnal two terms are known as secondary contractions. The primary contraction introduced by lensing is given by P`1`2 (L) = CW W . (2.12) `3`4 L `1`2L `3`4L Given the discussion above, for an observed sky consisting of lensed CMB + Gaussian noise and using the standard lensing estimator with W x = W,

2 L0 1 1 2 C + M C = ¯ LM N , (2.13) L L L0 2L + 1 | | L R ! 2 M 3 6 X 7 6 7 4 L0 5 where the L0 index is summed over. The mixing matrix ML would be zero if there were only primary contractions of the trispectrum, however it has small o↵-diagonal contributions due to the secondary contractions of the trispectrum [17]. The mixing matrix may be inverted and used to obtain an estimate of CL using Eq. (2.13). Alternatively, if a good approximation of the true power spectrum CL is available, the o↵-diagonal contributions can simply be subtracted from the LHS of Eq. (2.13).

2.3 Source Trispectrum

In this section we outline the trispectrum conﬁgurations that can be sourced by point sources. We will focus in particular on the subset of these trispectra which we expect to project most strongly onto the conﬁgurations probed by the lensing estimator described in the previous section. To simplify this discussion, we will assume discrete point sources 2.3. SOURCE TRISPECTRUM 47

with temperature S i, such that the observed sky temperature in direction nˆ is given by

obs T (nˆ) = T(nˆ) + S i(nˆ) + N(nˆ), (2.14) Xi where T is the lensed CMB temperature, and N is an instrumental noise contribution. We will assume that only multpoles L > 100 are used, so that we may ignore CMB temperature anisotropies generated by the ISW e↵ect. Discarding multipoles at L < 100 has negligible impact on the signal to noise ratio of the lensing estimator. Ignoring the ISW e↵ect, T is linear in the primordial, unlensed CMB temperature and therefore every non-zero n- point function must have an even number of multipoles associated with T. We may then group point source terms of the trispectrum by the number of individual sources which they contain. Up to permutation symmetries of the trispectrum and source indices, there are 7 types of source term, which we describe in the following subsections.

4 1 source terms: There are two types of source term containing a single source: S i and 2 4 S i TT. The S i term is simply the kurtosis of the unresolved source population, and 2 is analogous to the shot-noise term in the power spectrum of the sources. The S i TT term probes the correlation between the sources and the lensing potential.

2 2 3 2 source terms: There are three types of term containing two sources: S i S j , S i S j, and 2 2 3 S iS jTT. The S i S j and S i S j terms probe the clustering of the sources, while the the

S iS jTT term probes the source-lensing bispectrum.

2 3 source terms: There is only one non-zero term containing three sources: S iS jS k , which probes the bispectrum of the sources.

4 source terms: There is only one non-zero term containing four sources: S iS jS kS m. This term probes directly the trispectrum of the sources; each of the four multipoles in the trispectrum is sourced by a separate point.

In all of the expressions it should be understood that the point source indices are disjoint (i.e. i , j , k , m). For several of the terms above, statistical isotropy implies that the 48 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

Term Primary Contraction P`1`2 (L) `3`4 TTTT CW W L `1`2L `3`4L S 4 1 S 4 WS 2 WS 2 i 3 `1`2L `3`4L 2 S 2TT 1CS (WSD2 WE + W WS 2 ) i 2 L `1`2L `3`4L `1`2L `3`4L S 2S 2 CS 2S 2 WS 2 WS 2 i j L `1`2L `3`4L S 3S 1 (CS 3S + CS 3S + CS 3S + CS 3S )WS 2 WS 2 i j 3 `1 `2 `3 `4 `1`2L `3`4L S S TT 1 (bSS WS 2 W + bSS W WS 2 ) i j 2 `1`2L `1`2L `3`4L `3`4L `1`2L `3`4L S S S 2 1 (bSSS2 + bSSS2 )WS 2 WS 2 i j k 2 `1`2L `3`4L `1`2L `3`4L S S S S 1 sT `1`2 (L) i j k m 3 `3`4 Table 2.1 Primary source terms discussed in Sec. 2.3. The ﬁrst row gives the trispectrum due to lensing for comparison purposes.

details of the point source model enter only through the following power spectra:

2 S 2 dnˆ YLM(nˆ)S i (nˆ)L0 M0 = CL LL0 MM0 *Xi Z + 2 2 S 2S 2 dnˆ YLM(nˆ)S i (nˆ)(S j )L0 M0 = CL LL0 MM0 *Xij Z + 3 S 3S dnˆ YLM(nˆ)S i (nˆ)(S j)L0 M0 = CL LL0 MM0 . (2.15) *Xij Z + However, for two of the terms the point source model enters through reduced bispectra b`1`2L of the sources/lensing:

m1m2 M SS (S i)` m (S j)` m LM = b (2.16) 1 1 2 2 G`1`2L `1`2L *Xij + 2 2 m1m2 M SSS (S i)` m (S j)` m (S )LM = b , (2.17) 1 1 2 2 k G`1`2L `1`2L *Xi jk + 2.4. SOURCE ESTIMATORS 49

where the Gaunt integral is given by

(2` + 1)(2` + 1)(2L + 1) `1 `2 L `1 `2 L m1m2 M 1 2 . (2.18) `1`2L ⇡ G ⌘ r 4 0 0001 0 m1 m2 M 1 B C B C B C B C We will calculate approximate values for several@ of these spectraA @ in Sec. 2.6.A We present the primary contractions of the trispectra for the seven di↵erent source terms in Table 2.1. To simplify our presentation, we ﬁnd it useful to introduce the “point source weight function” given by

2 (2` + 1)(2` + 1)(2L + 1) ` ` L S = 1 2 1 2 . W`1`2L (2.19) r 4⇡ 0 0001 B C B C Most of the trispectrum terms in Table 2.1 include one or more@ factorsA of W s . This will `1`2L prove useful in the next section, when we discuss estimators for the source trispectra.

2.4 Source Estimators

One approach to mitigating point source biases in the lensing spectrum is to construct physical models for the various source trispectra presented in the previous section, and then to propagate these to biases in the lens reconstruction, which may then be subtracted to obtain unbiased estimates of C` . However, this is subject to considerable uncertainty in modeling the source populations, as well as issues with precisely determining, for example, the map flux density cut. A potentially more robust approach is to jointly estimate both the lensing and point- source trispectra, or alternatively to construct lensing estimators which probe trispectrum configurations orthogonal to those generated by point sources. To elaborate on this approach, we consider first constructing a quadratic “point source estimator” S¯ 2 following Eq. (2.4), with weight function WS 2 . As one might expect, in LM `1`2L real space this estimator corresponds to squaring the inverse-variance filtered sky map

2 S 2S 2 1 2 S 2S 2 1 1 2 Sˆ = ( ) S¯ = ( ) dnYˆ ⇤ (ˆn) T¯ (ˆn). (2.20) LM RL LM RL 2 LM Z 50 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

S 2S 2 S 2 Estimates for CL and CL may be formed intuitively by taking the auto-spectrum of ˆ 2 ˆ S LM, or its cross-spectrum with the lensing potential estimate . To remove the point source bias to the lensing estimator we can use a weight funtion W which projects out `1`2L any contributions associated with S 2. This is given by

S 2 S 2S 2 1 S 2 W = W ( ) W . (2.21) `1`2L `1`2L RL RL `1`2L

S 2S 2 1 S 2 S 2 In Eq. (2.21), ( ) is the normalization of the point source estimator W , and RL `1`2L RL is the response of the lensing estimator to the point source signal. Of course, the point source estimator in Eq. (2.20) is biased by lensing, and the estimator in Eq. (2.21) is biased by secondary contractions of the trispectra themselves. An alternative approach, following [16], is to construct a “bias-hardened” lensing estimator. This is formed by noting that, from Eq. (2.6) and Eq. (2.20), both estimators pick up unwanted contributions:

2 ¯ = ˆ + S Sˆ 2 LM RL LM RL LM 2 2 2 (2.22) S¯ 2 = S S Sˆ 2 + S ˆ . LM RL LM RL LM The lensing estimator can then be constructed to project out the contribution associated with S 2, and vice-versa:

1 ˆ S 2 ¯ LM L L LM = R 2 R . (2.23) 2 S S 2S 2 ¯2 2Sˆ 3 2 3 2S LM3 6 LM7 6RL RL 7 6 7 6 7 6 7 6 7 46 57 46 57 46 57

To formalize our understanding of the bias contributions, it is useful to generalize the estimator for the lensing power spectrum of Eq. (2.13). Consider a trispectrum with the form 1 abP`1`2 (L) = Cab(Wa Wb + Wb Wa ), (2.24) `3`4 2 L `1`2L `3`4L `1`2L `3`4L where (a, b) denote a pair of weight functions. All of the primary contractions in Table 2.1 except (possibly) for the ﬁnal trispectrum term can be written in this form, or as a sum over a small number of terms with this form. We denote the response of a pair of estimators 2.4. SOURCE ESTIMATORS 51

(x, z) to the (a, b) trispectrum as

xz 2 1 xz ˆ = x¯ z¯⇤ N . (2.25) CL,ab xa zb + xb za 2L + 1 LM LM L R R R R 2 M 3 6 X 7 6 7 Eq. (2.25) is an estimator for the x-z cross-spectrum4 obtained from maps5 x and z, where x has an underlying non-Gaussian signal caused by a, and z has a non-Gaussian signal caused by b. The estimators x and z are arbitrary, and may or may not be bias-hardened. In the case that (x, z) = (a, b) we will use the shorthand âb. If the data contains non-Gaussianity CL with an (a, b) trispectrum, then averaging over CMB temperature realizations with fixed a and b we have ˆxz = Cab + ML0Cab, where the mixing matrix M depends on the CL,ab L L L0 estimator being considered,D E and as before L0 is summed over. We refer to Eq. (2.25) as the trispectrum-related spectrum, of which the lensing estimator in Eq. (2.13) is a specific case. If the data contains non-Gaussianity of some other type (c, d), then ˆ may be biased. C We denote this bias as ˆxz . For example, part of the point source bias to the lensing CL,ab cd estimator is ˆ . Depending on the (x, z) and (c, d) weight functions, the secondary L, 2 2 C S S contributions to this bias can be costly to evaluate using the harmonic space expressions, and we will usually evaluate these terms using analogous flat-sky expressions, presented in App. 2 A.

In Fig. 2.1 we show the interactions between the trispectrum shapes that we will pri- S 2 S 2S 2 4 marily consider in this paper: those related to C` , C` , C` , and S . We consider experiments with two di↵erent resolutions, which we incorporate intoD theE estimator by restricting the maximum multipole in the data that is used in the reconstruction. The two estimators have Lmax = 1500, for a Planck-like estimator, and Lmax = 3000 for an SPT/ACT-like estimator. To calculate the foreground spectra we use simple empirical mod- S 2 4.25 S 2S 2 2.5 els: C` = (` + 50) and C` = (` + 300) , which we will justify in Sec. 2.6. As is S 2S 2 4 4 clear from Table 2.1, the CL and S trispectra are inextricably linked, because S is S 2S 2 degenerate with a constant contributionD E to C` . D E 52 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

Figure 2.1 Trispectrum-related power spectra biases âb calculated for an L = 1500 CL cd max experiment. The amplitudes of the four trispectrum shapes are arbitrary (such that âb CL ab ⇡ 1 at its peak.), but are used consistently in each of the four plots. Note that the relative size of the bias terms in each plot has no physical significance. The colored solid lines give the total contributions to âb from the indicted trispectra. Corresponding dashed lines give CL only the contribution from the primary contraction of the trispectrum (which in the upper left panel is equal to the lensing potential spectrum).

2.5 Foreground Simulations

To test our foreground model, as well as the performance of the estimators derived in Sec. 2.4, we use the Sehgal et al. [11] simulations of the microwave sky. These consist of a single realization at typical CMB observing frequencies, with separate maps for the radio and infrared point source emission, and the thermal and kinetic SZ signal. The large scale structure in the simulations was created from an N-body simulation to which the foreground components were added, thus attempting to generate a realistic correlation structure between the foregrounds and lensing structure. Although there have been several new foreground models developed since these simulations were produced [e.g. 18], the simulations are accurate enough to test the performance of our estimators. We will primarily use the 148 GHz map to test our analysis, but we also calculate 2.5. FOREGROUND SIMULATIONS 53

Figure 2.2 Same as Fig. 2.1, but for an experiment which is cosmic variance (C.V.) limited to Lmax = 3000. the lensing bias in the 90, 219, and 350 GHz maps for comparison. When using the

Lmax = 1500 estimator we remove sources with S > 200 mJy from the radio and infrared point source maps, comparable to the sensitivity of Planck [19]. For simplicity we do not mask the SZ emission, and discuss this case separately in Sec. 2.6.2. For the Lmax = 3000 estimator we mask sources with S > 6 mJy, comparable to the SPT and ACT sensitivities [20, 21]. We mask sources by zeroing the pixel containing the source in the map pixelized with Nside = 2048 in the HEALPix scheme [22]. In Fig. 2.3 we show the power spectra of the masked foreground components from the simulation.

For simplicity, we only use full-sky estimators, but calculate the foreground terms on one-eighth of the sky to remove spurious signal that might arise from symmetries in the map, which are present because the large scale structure was generated in an eighth of the sky and mirrored to create the full-sky map [11]. For the full-sky, Lmax = 3000 estimator, the sensitivity to is comparable to what has been achieved with SPT/ACT on smaller sky areas.

To test the analysis of the preceding sections we take the following approach: 54 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

Figure 2.3 Temperature power spectra of the simulated sky components for the Lmax = 1500 and 3000 experimental models, including unlensed and lensed CMB (black and orange), radio sources (red), infrared sources (blue), thermal SZ (green), and kinetic SZ (purple) at 148 GHz. Bright sources have been removed before taking the spectra (see Sec. 2.5 for details).

4 S 2S 2 S 2 1. We compare the foreground model of S , C` , and C` with the simulations. D E 2. We compare an analytical estimate of two of the lensing bias terms, calculated using 4 S 2S 2 S and C` measured from the simulations. D E 3. We compare the lensing bias obtained from the estimators calculated in Sec. 2.4 with the lens reconstruction from the combined CMB and foreground simulations.

By taking this three stage approach we aim to better understand the origin of the bias terms.

2.6 Source Modeling

S 2S 2 S 2 4 We now describe how foreground models can be used to calculate C` , C` , and S , which are needed to calculate the lensing bias terms. As explained in Sec. 2.4 our ultimateD E goal is to measure these quantities from the data, however by constructing a foreground model we can better understand the origin of the bias terms. In particular, we wish to assess the relative importance of the di↵erent foreground components, as well as the importance 2.6. SOURCE MODELING 55

of the di↵erent bias terms. Our model will include radio point sources, infrared point sources, and SZ emission, and we will distribute the sources such that their number density is correctly correlated with the structure lensing the CMB.

We ﬁrst describe our model in general, and will then discuss each foreground in detail. We assume that the signal from a single source projected onto the sky is azimuthally symmetric with a proﬁle y(✓). For a source located at the north pole, the harmonic-space signal is then: 4⇡ y` y` = 2⇡ d(cos ✓) P`(cos ✓) y(✓) , (2.26) ⌘ 2` + 1 0 r Z with the corresponding inverse relation:

2` + 1 y(✓) = P (cos ✓) y . (2.27) ⇡ ` ` ` 4 X ! For a source that is unresolved, y is constant for ` L . Rotating the source to location `  max nˆ on the sky gives the signal:

R(nˆ)y `m = Y`⇤m(nˆ) y` . (2.28) ⇥ ⇤ The total foreground signal S is the sum of the signal S i from each source:

S `m = [R(nˆ i)S i]`m = Y`⇤m(nˆ i) yi,` , (2.29) = Xi 0 Xi

S 2S 2 S 2 2 where NS is the number of sources. To calculate C` and C` we instead rotate y(✓) , which has the (normalized) spectrum:

2 1 1 `1 `2 ` (2` + 1)(2` + 1) y y , (2.30) ` ⇡ 1 2 `1 `2 Y ⌘ 4 N2 ` ` 0 0001 X1 2 B C B C @B AC where we have introduced the normalization factor N2, since it will appear with this term in trispectrum calculations. The normalization factor is equal to (2` + 1) calculated ` Y` P 56 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

S 2S 2 with y` = 1. The C` spectrum is given by

2 2 2 2 S S ⇤ = dnˆ dnˆ 0 Y⇤ (nˆ)Y` m (nˆ 0) S (nˆ)S (nˆ 0) , (2.31) `m `0m0 C `m 0 0 D E Z Z D E where NS NS 2 S (nˆ) = S i(nˆ) S j(nˆ) . (2.32) = = Xi 0 Xj 0

We assume that the properties of individual sources, such as brightness or angular pro- ﬁle, are independent of the source location on the sky. Consider ﬁrst the case where the source density does not depend on sky position. In this case the foreground signal has the statistical properties of shot noise and so

NS 2 2 2 2 S S ⇤ = dnˆ dnˆ 0 Y⇤ (nˆ)Y` m (nˆ 0) S (nˆ)S (nˆ 0) . (2.33) `m `0m0 C `m 0 0 i i i=0 D E Z Z X D E 2 Using S = Y⇤ (nˆ i) i,` we ﬁnd: i,`m `m Y

NS 2 2 1 2 S S ⇤ = Y⇤ (nˆ i)Y` m (nˆ i) i,` i,` = . (2.34) `m `0m0 C `m 0 0 Y Y 0 4⇡ Y` i=0 D E D E X

The single-source signal y` depends on redshift, as well as additional parameters such as the source ﬂux S or the source host-halo mass M. In the limit that the number of sources

NS 1, NS N S S 1 max dN dN = dz dS , (2.35) = ! 0 0 0 dS dz Xi 0 Z Z Z where S max is a cut in ﬂux density above which sources are resolved and masked, and dN/dS dz encapsulates the distribution of source number counts. We use an equivalent relation for sources described by M instead of S .

Combining Eqs. (2.34) and (2.35) the shot noise part of the spectrum is

2 2 dN CS S = dz dS 2(S, z) . (2.36) ` dS dz Y` Z Z 2.6. SOURCE MODELING 57

For the case of point sources with ﬂux S this reduces to:

2 2 dN CS S = dS S 4 . (2.37) ` dS Z

Fluctuations in the dark matter over-density = (⇢ ⇢¯)/⇢¯, where ⇢ is the dark matter density, lead to ﬂuctuations in the source number density. In our simplest model we account for this by multiplying the mean source number density by , and including a redshift dependent bias b(z). For example, the source variance is given by:

S max dN S 2(nˆ) = dS dz S 2 1 + b(z) (nˆ, z) , (2.38) S =0 dS dz Z Z ⇣ ⌘ In this model, dN/dS dz describes the single-source PDF, which can be arbitrarily non- Gaussian, while correlations between sources are governed by large-scale structure . For this model we assume that we are working on scales suciently large that (nˆ, z) is Gaus- 3 sian and therefore described by the matter power spectrum: (k, z)(k0, z)⇤ = P(k, z) (2⇡) (k h i k0). The source terms of Table 2.1 which involve the bispectrum/trispectrum of : S iS jTT, 2 S iS jS k and S iS jS kS m, are then zero. S 2 Consider C` , which from Eq. (2.38) is equal to:

2 dN 2 S ` m = dnˆ Y`m(nˆ) dz b(z) dS S (nˆ, z) ` m . (2.39) `m 0 0 dS dz h 0 0 i D E Z Z Z On small scales ` & 100 we use the Limber approximation [e.g. 23]. This gives the general form for the linear model foreground spectra:

S 2 S 2 C` = d K ()K ()P(k = `/, ), (2.40) Z where P(k, ) is the matter power spectrum, and the lensing kernel is [e.g. 23]

3 ( ) K () = ⌦ H2 ⇤ , (2.41) `2 m 0 a ⇤ where is the comoving distance to last scattering, ⌦m is the present day matter density ⇤ as a fraction of the critical density, H0 is the present day Hubble constant, and a is the scale 58 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

factor. To account for signal power on small scales where the dark matter density is determined by non-linear collapse, we use a halo model which approximates the density ﬁeld as a set of discrete halos that are populated with galaxies and galaxy clusters. On small scales the power spectrum of the foregrounds is determined by the correlation structure of sources within the same halo, and the cross-spectrum of the foregrounds with the lensing potential is determined by the correlation of the lensing signal from the halo and the sources within it. For a review of the halo model see, for example, Cooray and Sheth [24]. S 2 S We use the one-halo term to calculate C` for SZ, replacing y(✓) in the C` spectrum with y(✓)2 to give [e.g. 25, 26]

S 2 dn M C = dK () dM u(k, M) `(M, z) , (2.42) ` dM ⇢¯ Y Z Z where n is the source number density, u(k, M) is the profile of the halo density normalized to one on large scales, and k = `/. We now discuss each foreground in detail and calculate the dominant spectra using either a linear bias or halo model. For simplicity, in this section we will ignore correlations between the di↵erent source populations, since these will give an (1) increase in the O lensing bias and will not a↵ect our conclusions. We will consider two di↵erent point source masking schemes. The first assumes that all sources with flux density greater than S max =

200 mJy are masked, which we use when Lmax = 1500, and the second uses S max = 6 mJy for Lmax = 3000. We will perform all calculations at 150 GHz.

2.6.1 Radio Point Sources

Radio point sources are an important CMB contaminant at ⌫ < 150 GHz. At the sensitivities that we consider the unresolved radio emission is dominated by unclustered sources [e.g. 28–32] and so we expect the dominant contribution to the lensing bias to come from the S 4 term. D EWe use the model of Babich and Pierpaoli [31, hereafter BP], which splits the radio 2.6. SOURCE MODELING 59

Figure 2.4 Power spectra calculated using our foreground model. The solid lines show the CMB (unlensed in black, lensed in orange), radio point source (red), SZ (green), and CIB (blue) spectra. Radio and CIB point sources with flux density S > 200 mJy are assumed to be masked. The dashed lines show the point source signals for a 6 mJy flux density cut. For comparison the blue dotted lines show the CIB one and two-halo terms calculated using the Ade et al. [27] CIB model. source number counts into redshift-independent and flux density-independent pieces:

dN dN = (S ) n(z) , (2.43) dS dz dS with a redshift distribution that contains two populations:

(z z )2/22 3z n(z) = 0.75 e 0 + 0.25 10 , (2.44) ⇥ where = 0.4 at z < 1 and 0.9 at z 1, z = 0.95 and the number density is normalized to 0 one below z = 3.1. The ﬁrst term describes the quasar population at high redshift and the second term describes the radio loud galaxy population, with both terms consistent with the observations of Sadler et al. [33] and Gonzalez-Nuevo´ et al. [34]. The radio loud population 60 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

lies at z . 0.1 and accounts for 10 15% of the total population (BP, and references therein). As in BP we adopt a constant bias of 1.7, which is consistent with measurements of NVSS radio galaxies [e.g. 2, 35, 36]. The one-point functions that characterize the statistical properties of the radio emission are calculated from Eq. (2.43):

S max dN S i = dS S i (S ) , (2.45) S =0 dS D E Z with dN = N S . (2.46) dS 0 The parameters are extrapolated to 150 GHz from the Waldram et al. [37] model, equal to 1.15 1 N0 = 9 Jy sr and = 2.15. This model was designed for use at WMAP frequencies (23-90 GHz), and so the extrapolation to higher frequency CMB channels is somewhat uncertain. However, we find good agreement with the flux density distribution of SPT [20] 3 and ACT [21] radio sources in the range 5 10 . S [Jy] . 1, and with the de Zotti ⇥ et al. [38] point source model within the same range. We calculate S 4 using Eqs. (2.45) and (2.46) with i = 4, giving the result in Table 2.2. D E From Eq. (2.38) we find the radio point source kernel defined in Eq. (2.40):

i 1 dz KS () = S i n(z)b(z) . (2.47) d D E SS S 2S 2 S 2 Using Eqs. (2.40) and (2.47) we calculate C` , C` , and C` . In Fig. 2.4 we show SS S 2S 2 C` , which has the redshift distribution of power shown in Fig. 2.5. We compare C` and S 2 C` to the simulated spectra in the upper panel of Figs. 2.7 and 2.8 for Lmax = 1500. The black line gives the model result and the colored lines and boxes give the spectra obtained S 2S 2 from the simulation. The redshift distribution of signal power in C` is shown in Fig. 2.6. S 2S 2 We find good agreement between our shot noise model and the simulated C` , as well S 2 S 2S 2 as between the linear bias model and C` . We find the clustering part of C` to be approximately three orders of magnitude smaller than the shot noise, which we find to be slightly higher than in the simulation. The di↵erence between the shot noise amplitudes in the model and simulation can be removed with only a small change in parameters, and the 2.6. SOURCE MODELING 61

Figure 2.5 Redshift distribution of power in Figure 2.6 As in Fig. 2.5, except that the the radio point source (red), SZ (green), and foreground spectra show the redshift distri- S 2S 2 CIB (blue) spectra calculated using our fore- bution of power in C` . ground model at 150 GHz for ` = 50 (solid) and ` = 2000 (dotted). The power in the lensing potential spectrum is shown in black. The curves are normalized by area.

1.15 1 S 2S 2 dashed line shows the model calculated with N = 5.4 Jy sr . Above ` 2900 C 0 ⇠ ` becomes large in the simulation because of the Lmax = 1500 bandlimit, which causes the spectrum to be poorly measured near ` = 2Lmax.

As seen in Table 2.2 there is a large reduction in S 4 when the ﬂux density cut is lowered from S max = 200 mJy to S max = 6 mJy, which isD becauseE of the steep dependence: 2.15 dN/dS S . The one-point amplitudes, and hence the radio point source lensing bias, ⇠ can therefore be signiﬁcantly lowered by masking only a small number of additional radio sources.

There is an additional source of anisotropy in the point source shot-noise caused by lensing of the sources by intervening matter. Since the sources are lensed by the same matter distribution that modulates the point source number density, we do not expect this e↵ect to lead to an order of magnitude larger lensing bias, and we do not consider it further. For details of how source lensing can be included in the model see, for example, BP. 62 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

Foreground S [µK] S 2 105 [µK2] S 3 109 [µK3] S 4 1012 [µK4] h i ⇥ ⇥ ⇥ D E D E D E 5 Radio sources - 1.70 (0.086) 3.92 (0.0060) 1.28 (5.8 10 ) ⇥ 5 CIB (B10) 25.0 (25.0) 1.02 (0.91) 0.173 (0.015) 0.044 (7.0 10 ) ⇥ 5 CIB (B12) 21.7 (21.7) 0.89 (0.84) 0.044 (0.0013) 0.0042 (5.8 10 ) ⇥ Table 2.2 One-point amplitudes calculated at 150 GHz, with a 200 mJy (6 mJy in brackets) maximum ﬂux density. For radio point sources S is not shown since Eq. (2.45) gives an h i unphysical negative ﬂux.

2.6.2 SZ

Thermal SZ emission can be a significant lensing contaminant at 100-150 GHz. The ⇠ thermal SZ signal is caused by CMB photons scattering o↵ high-temperature electrons trapped in the gravitational potential of galaxy clusters, which increases the CMB temperature along the line-of-sight to the cluster [39]. The signal has a frequency dependence of ex + 1 g⌫ = x 4 , (2.48) ex 1 where x = h⌫/k T , with a null at 220 GHz, a temperature decrement at lower fre- B CMB ⇠ quency and an increment at higher frequency. In addition to the thermal signal there is a kinetic signal caused by the Doppler shifting of light scattered from electrons within a moving cluster, which is subdominant to the thermal signal [39–42]. Since the kinetic signal is subdominant we will not consider it further in this section. The significance of the thermal SZ signal as a contaminant to CMB measurements is well known, and the contribution to the lensing noise bias is briefly discussed in Cooray and Kesden [43].

We use the Komatsu and Seljak [25] model to calculate the SZ signal, g⌫ y`(M, z), from a cluster in a halo of mass M at redshift z, and use the Tinker et al. [44] halo mass function to calculate the foreground spectra using Eqs. (2.36) and (2.42). For the dark matter density profile u(k, M) we use an NFW profile [45]. For the mass integral we use a lower limit of 10 1 15 1 5 10 h M and an upper limit of 5 10 h M . We find the redshift distribution of ⇥ ⇥ SS S 2S 2 thermal SZ power in C` and C` in Figs. 2.5 and 2.6, with most of the power at z < 1. The foreground spectra are shown in the middle panel of Figs. 2.7 and 2.8. There is good 2.6. SOURCE MODELING 63

Figure 2.7 Power spectrum of S 2 from simulation (colored lines) and from our model (solid black lines) for radio point sources (upper panel), infrared sources (middle panel), and SZ (lower panel). The infrared model has been scaled by a factor of 30 to better ﬁt the simulation results. The additional dashed and dotted lines are results from models with di↵erent parameters, discussed in the text.

S 2S 2 agreement between the model and simulations for C` below ` 1500, with a small S 2 ⇠ discrepancy at higher `. We ﬁnd worse agreement for C` , with the model a factor of two lower than the simulations (note that the model spectrum is scaled by two in Fig. 2.8 to more easily compare the shape of the spectra). Masking bright clusters decreases the spectra in Figs. 2.7 and 2.8, lowering the SZ lensing bias. The e↵ect on the spectra can be modeled by excluding bright clusters from the mass and redshift integral in Eq. (2.34) by limiting the M and z range. To demonstrate 4 15 1 this we calculate S with a lower upper mass limit of 10 h M , showing the result as 15 1 the dashed line inD Fig.E 2.7, which is a factor of 4.5 lower than for Mmax = 5 10 h M ⇡ ⇥ at ` . 1500. As a simple test of the impact of masking in the simulations we calculate S 4 from the D E 64 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

Figure 2.8 Cross power spectrum of S 2 and calculated from the simulated S and maps (colored boxes) and the model predictions (black lines). To better ﬁt the simulations the CIB model spectrum has been multiplied by 25 for the B12 model and by 10 for the B10 model. The SZ spectrum has been multiplied by 2. The empirical model from Sec 2.4 is almost identical to the radio point source model up to an amplitude.

simulated SZ map with di↵erent fractions of the sky masked. We use a simple thresholding procedure, masking all pixels above a flux density threshold S max in the SZ map. While this clearly over-simplifies the process of finding and masking clusters, it is a simple way to probe the SZ flux density distribution. We degrade the SZ map to resolution Nside =

2048 and create a mask by selecting all pixels with flux density greater than S max. This procedure is equivalent to assuming that the clusters are point sources up to ` 2N . ⇡ side Although this is a simplification, by choosing an e↵ective S max appropriate for the particular 4 experiment, it is still informative. We bandlimit the map to Lmax and calculate S in the unmasked pixels. We find that the following function provides a good approximationD E to 2.6. SOURCE MODELING 65

4 S at di↵erent S max: D E 4 S exp ( q) = + , log10 12 4 [mx c] (2.49) [10D µEK ] 1 + exp ( q) 0 1 B C B C where @B AC S x S x = log max , q = 0 . (2.50) 10 [mJy] ! For L = 1500 we find the parameters: m = 1.5, c = 1.1, S = log (40), and = max 0 10 log10(2.75). For S max = 200 mJy we find that no clusters are masked, whereas for S max = 6 mJy we find that S 4 is reduced by a factor of four. More sophisticated cluster finding algorithms that areD optimizedE to the cluster SZ profile and make use of multiple frequency channels can be incorporated in this simple scheme by lowering S max. We will discuss this further in Sec. 2.7.1.

2.6.3 Dusty Galaxies

Above 300 GHz emission from dusty galaxies is the dominant extragalactic CMB fore- ⇠ ground. The light is emitted as graybody radiation from warm dust surrounding young UV-bright stars, and is redshifted to infrared wavelengths today. With experiment resolutions Lmax . 3000 the galaxies are largely unresolved, and the observed signal is the integrated light from many faint sources [46], which is known as the cosmic infrared background (CIB). The mean CIB signal and its anisotropies have been observed by many experiments [47–55] and the signal is known to be correlated with the lensing potential [12, 26, 27]. S 2S 2 In the C` spectrum we expect the CIB shot noise to be dominant at the angular scales and frequencies important for lensing measurements. In the CIB temperature power spectrum the shot noise is the largest contribution above ` 1000 at 217 GHz [e.g. 27], and ⇠ we expect it to be more important at 150 GHz based on the Ade et al. [56] CIB model. Since we expect the shot noise to dominate over the one and two-halo terms, at least on small scales, for simplicity we will not use the halo model and instead use a similar approach to our radio point source model, but with a di↵erent source distribution, dN/dS dz. 66 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

For the source number counts we use the Bethermin et al. [57, hereafter B12] model, finding the one-point amplitudes in Table 2.2. Masking sources from 200 mJy to 6 mJy reduces the amplitudes by a smaller amount than for radio sources. We find that the model S 2S 2 underestimates the simulated C` in Fig. 2.7 (note that the model in the figure is scaled by a factor of 30). However, we find that the Bethermin et al. [58, hereafter B10] model gives a larger shot noise level, closer to the value found in the simulation, and is the dashed line in Fig. 2.7 (this line is not scaled). We find that the clustering term in Fig. 2.7 is over three orders of magnitude smaller in the multipole range shown.

S 2S 2 The C` spectrum is not flat as would be expected for a point source shot noise signal. One explanation for this is the way the dark matter halos are populated in the simulations, with galaxies added to dark matter halos so that the one-halo temperature power spectrum is correctly reproduced [11]. The shot noise therefore has two contributions: one from the sources populating the halos, and one from the shot noise of the halos. The shot noise of the halos will have reduced power on angular scales smaller than the size of halos contributing S 2S 2 the majority of the emission, which could be the cause of the roll-o↵ of power in C` at ` & 2000. In this explanation ` = 2000 is the scale of halos at z 2 3 that contribute ⇠ a majority of the signal. We model this e↵ect by assuming that the halo profile is a 90 S 2S 2 Gaussian. This is the dotted line in Fig. 2.7, fit to the simulated C` .

Although this explanation provides a reasonable fit to the spectrum, another possibility is spurious large-scale signal in the CIB map at ` . 600. On these scales the simulated CIB temperature power spectrum is larger than the Planck Collaboration et al. [53] CIB model (which is based on the measured spectra at 217-857 GHz). If we bandlimit the CIB map with Lmin = 600 we find that the spectrum flattens but still shows a roll-o↵ in power, and so spurious low-` power does not appear to fully explain the shape of the spectrum. 2.7. ADDITIONAL BIAS TERMS 67

2.7 Additional Bias Terms

2.7.1 Inhomogeneous Flux Sensitivity

Point sources in the CMB map can be masked down to a limiting ﬂux density that is determined by the noise level (or di↵use foreground emission). We write the detection signiﬁ- cance of a point source in the CMB data as [14]:

S 2 Lmax 2` + 1 2 = max , (2.51) ⇡ 4 ` C` X=0 where C` is the CMB and instrument noise spectrum. If we ignore instrument noise, but 2 restrict Lmax, then with = 9, S = 204 mJy for Lmax = 1500, approximately the value found for Planck in Ade et al. [19], and 13 mJy for Lmax = 3000. If multiple frequency channels can be used to subtract the CMB, then sources can be masked to a lower ﬂux density. For example, for 30 µK-arcminute noise S max = 25 mJy for Lmax = 1500, and

12 mJy for Lmax = 3000.

The noise level in the data will often be inhomogeneous, for example if the experiment does not spend an equal amount of time observing all parts of the sky. The peak brightness of sources that remain in the map after masking then depends on nˆ, and so S max 4 becomes S max(nˆ), which introduces an nˆ dependence to S . This statistical anisotropy in the data will be mistaken for a lensing signal, in an analogousD E way to the bias introduced by inhomogeneous noise in Hanson et al. [59].

The inhomogeneity couples di↵erent modes of :

4 LML M dnˆ Y⇤ (nˆ)YL M (nˆ) S (nˆ) . (2.52) h 0 0 i / LM 0 0 Z D E Calculating 2 /(2L + 1) we see that the amplitude of the S 4 lensing bias is M | LM| modiﬁed as: D E D E P 1 S 4 dnˆ S 4 (nˆ) . (2.53) ! 4⇡ D E Z D E 68 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

In addition, the inhomogeneous ﬂux sensitivity generates a mean-ﬁeld in the lens reconstruction:

/ p ⇡ 2 1 4 `1`2L LM = S W (2`1 + 1)(2`2 + 1) , (2.54) LM + m1m2 M h i 2L 1 ` ` D E X1 2 p where S 2 is the harmonic transform of S 2 (nˆ). If this is not subtracted then ˆ is bi- LM ased, withD E the bias equal to the mean-ﬁeld spectrumD E plus additional terms if the mean-ﬁeld is correlated with the lensing potential (which is not the case if the point source inhomogeneity is caused by the instrument noise).

To estimate the size of this e↵ect we use the radio point source model. With this model the ratio of the inhomogeneous one-point amplitude to the homogeneous amplitude is:

S =S (nˆ) (1/4⇡) dnˆ max dS (dN/dS ) S 4 = S =0 r S =S (2.55) R maxR / 4 S =0 dS (dN dS ) S R Using Eq. (2.46) and setting S max to the average of S max(nˆ) over the sky gives:

5 (1/4⇡) dnˆ S max(nˆ) = r 5 (2.56) (1/4⇡) R dnˆ S max(nˆ) h R i As an example, consider the inhomogeneous sensitivity introduced by the Planck scanning strategy [60]. This can be approximated as a noise variance that depends on ecliptic latitude 2 ✓ as sin ✓. If we neglect the CMB noise for simplicity, then S max can be calculated from 2 Eq. (2.51) with the denominator C` sin ✓, giving S max(nˆ) sin ✓. Setting = 2 gives a / / ratio r = 12/⇡2 1.22. We ﬁnd a similar increase in the lensing bias in reconstructions on ⇡ point source maps masked using a sensitivity map created directly from the Planck detector pointing information, with no obvious sign of the mean-ﬁeld term. 2.7. ADDITIONAL BIAS TERMS 69

Figure 2.9 Contribution to the reconstructed lensing spectrum from simulations for each foreground component when Lmax = 1500 (red line). The blue line gives the result after subtracting the disconnected term, which is the green line. The residual foreground contribution calculated using the results from Sec. 2.3 is the orange line. The blue bars give the lensing spectrum reconstructed from the full-sky simulated CMB map.

2.7.2 Source Masking

To reduce foreground contamination in the lens reconstruction bright foregrounds are masked before calculating ˆ. Since the unresolved foreground emission is correlated with the lensing potential any mask that covers the bright foreground sources will preferentially mask the peaks of , potentially biasing the spectrum. It is dicult to gain insight into this e↵ect analytically since it will depend on the size of the regions excised around sources, as well as the number of sources masked. 70 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

Figure 2.10 Same as Fig. 2.9, but for Lmax = 3000. The di↵erence between lensing the reconstructed lensing potential and the simulated lensing potential is due to the N(1) noise bias [17], which has not been subtracted.

In the ﬂat-sky approximation the covariance of the masked data is:

2 2 d ` d `0 T(~` )T(~` ) = D⇤(~` ~` )D⇤(~`0 ~` ) 1 2 (2⇡)2 (2⇡)2 1 2 Z Z D E 2 (~` ~`0) C˜ + (~` + ~`0)(~` + ~`0) ~`0 C˜ + (~` + ~`0)(~` + ~`0) ~` C˜ + ( ) , (2.57) ` · `0 · ` O h i where the average is over data realizations with ﬁxed , and D is the harmonic transform of the mask. If there is no mask then D(~` ~`0) = (~` ~`0). The e↵ect of the mask is to couple modes of , with a coupling distance determined by the sharpness of features in the mask. Splitting the mask into a correlated piece and an uncorrelated piece: D(~`) ⇠ 2.8. RESULTS AND DISCUSSION 71

~ ~ ~ ~ D1(`) + D2(`)(`) where D1(`) is the uncorrelated part and D2(`)(`) is the correlated part. If the mask- correlation is weak, then the largest bias to the spectrum will be D3D C ⇠ 1 2 L and so will depend linearly on the amplitude of the correlation. We test how large this bias can be by generating a realization of the lensed CMB and reconstructing the lensing potential using masks that cover the peaks of . We create masks with seven di↵erent masked sky fractions between 0.0001 and 0.4 (the first SPT lens reconstruction masks 1% of the sky due to point sources and galaxy clusters; for Planck ⇠ the point source and cluster mask covers 2% of the sky). In order to isolate biases caused ⇠ by the correlation of the mask with the lensing potential from other mask related biases we also rotate the masks so that they no longer cover the peaks of and calculate the lens reconstruction. Any bias caused by the mask itself, and not due to its correlation with , will be the same in the rotated and unrotated masks. We rotate each of the seven masks by multiples of 36 degrees to give a total of 70 masks, and reconstruct the lensing potential from the lensed CMB map using each one. We find no significant di↵erence in the spectrum of when any of the masks are used. We find that for all seven sky fractions the bias in the spectrum is less than 1% at all multipoles, and less than 0.2% at ` > 10. Since this bias is small we do not consider this e↵ect further, and leave a more detailed study for the future.

2.8 Results and Discussion

4 S 2S 2 We calculate the lensing bias analytically from S and C` obtained from the foreground maps, showing the results in Figs. 2.9 andD 2.10E for the Lmax = 1500 and 3000 estimators respectively, with the corresponding 200 mJy and 6 mJy point source ﬂux density cuts. The orange line is the analytical estimate, which should be compared with the reconstructed lensing potential spectrum estimated from the foreground maps with the disconnected noise bias subtracted, which is the blue line. Figs. 2.9 and 2.10 do not include the bias terms arising from correlations between the foregrounds and the lensing potential, which are more dicult to extract from the simulated maps. Before discussing the analytical results we make some general remarks about the lensing bias in the 148 GHz maps. For the Lmax = 1500 estimator with the 200 mJy mask, the 72 CHAPTER 2. LENSING FOREGROUND CONTAMINATION

thermal SZ and radio point source signals give the largest bias, which is larger than C` at ` & 1200. The relative importance of the two foregrounds depends on the SZ mask used, as discussed in Sec. 2.6.2. For the Lmax = 3000 estimator all of the biases are larger because the estimator uses higher-` modes in the data, where the foreground emission is a greater contribution relative to the CMB, as can be seen in Fig. 2.3. For this reason lowering Lmax in the estimator can be used to reduce the foreground bias at the expense of a lower signal to noise ratio lensing measurement. The thermal SZ and CIB are the largest biases, with a large contribution at all multipoles. The kinetic SZ signal is small for both estimators, justifying our decision to ignore it in our modeling calculations. For radio point sources we find good agreement between the analytical calculation and the simulations, as we would expect since at these sensitivities it is the simplest foreground (in a statistical sense), being shot noise dominated. For the other foregrounds the analytical bias generally over-estimates the measured bias at high ` by a factor of 2. The discrep- ⇠ ancy can be reduced by scaling the analytical calculation, but for CIB this does not give a good fit at all multipoles. At the null in the analytical spectrum at ` 700 and ` 1800 ⇠ ⇠ for the Lmax = 1500 and 3000 estimators respectively, we would expect the reconstruction to have no bias. However, because the reconstruction has statistical noise, when the spectra are smoothed the result is non-zero. The purple line in the upper left panel is the bias expected from a 90 Gaussian-smoothed shot noise map, which models a possible roll-o↵ S 2S 2 in C` power at high-` due to the halo occupation model used in the simulations, as discussed in Sec. 2.6.3. For the Lmax = 1500 estimator this provides a good fit at ` > 1000, but a poor fit at lower-`. A possible explanation for the discrepancy between the analytical calculation and the measured lensing bias is the presence of other terms from Table 2.1 that we have not calculated, although terms containing cannot be responsible. In the 90 GHz simulation we find that radio point sources contribute the largest bias for the Lmax = 1500 estimator, and thermal SZ for the Lmax = 3000 estimator, depending on how the SZ is masked. At 219, 277, and 350 GHz the CIB is the largest bias, with a negligible thermal SZ bias at 219 GHz, as we would expect from the thermal SZ frequency dependence. The kinetic SZ signal is essentially the same at all frequencies due to its flat frequency dependence, and is negligible for the estimators we use. We measure the bias in the simulated maps using the estimators developed in Sec. 2.4. 2.8. RESULTS AND DISCUSSION 73

On the combined 148 GHz CMB and foreground map, we ﬁnd that we can measure the shot noise bias to a precision of 6% for the L = 1500 estimator with a 60% unmasked ⇠ max sky area, and to 3% precision for the Lmax = 3000 estimator with 0.06% unmasked sky area. By measuring the shot noise bias, we e↵ectively remove the discrepancy in the overall amplitude of the analytical measurements in Figs. 2.9 and 2.10, allowing a signiﬁcant reduction in the foreground contamination. Our methodology is generally applicable to other trispectrum estimators, and can be used to remove foreground contamination from them. The overlap with the foreground trispectrum shapes will be di↵erent, but can be calculated using the results from Secs. 2.3 and 2.4 by replacing the lensing weight functions with the weight functions for the particular trispectrum being investigated. In addition, the methodology can be used to remove the polarized foreground bias from polarization sensitive lensing estimators, which is a simpler case to analyze because of the reduced number of bias terms.

Acknowledgements

SJO acknowledges support from the US Planck Project, which is funded by the NASA Science Mission Directorate. Some of the results in this paper have been derived using the HEALPix [22] package. Bibliography

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2 A Flat Sky Expressions

In this appendix we rewrite the main equations used in this paper using the ﬂat-sky formalism.

A quadratic estimator,x ¯LM is given by

2~` d 1 x ~ ~ ~ ~ x¯(L) = W (`1, `2)T¯(`1)T¯(`2), (A-1) (2⇡)2 Z ~ ~ x ~ ~ x where `2 L `1. The weight functions W (`1, `2) are the ﬂat-sky analogues to the W ⌘ `1`2L weight functions. We have

W(~` , ~` ) = CTT(~` + ~` ) ~` + CTT(~` + ~` ) ~` (A-2) 1 2 `1 1 2 · 1 `2 1 2 · 2 and S 2 ~ ~ W (`1, `2) = 1. (A-3)

The connected trispectrum in the ﬂat-sky is denoted as

~ ~ ~ ~ 2 ~ ~ ~ ~ ~ ~ ~ ~ T(`1)T(`2)T(`3)T(`4) = (2⇡) (`1 + `2 + `3 + `4) T(`1, `2, `3, `4). (A-4) C D E As on the full-sky, it is useful to break the trispectrum into primary and secondary contractions as

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 2 ~ ~ ~ ~ `1`2 `1`3 `1`4 T(`1, `2, `3, `4) = d L (`1 + `2 + L)(`3 + `4 L) P~ ~ (L) + P~ ~ (L) + P~ ~ (L) (A-5) `3`4 `2`4 `3`2 Z ✓ ◆ The forms of the trispectra in Table 2.1 map directly onto the ﬂat-sky expressions, simply by replacing the full-sky weight functions with the corresponding ﬂat ones. If our data contains a trispectrum given by

cdP`1`2 ( ) = C Wc(~` , ~` )Wd(~` , ~` ), (A-6) `3`4 L L 1 2 3 4 then the bias to the (x, z) estimator for the (a, b) trispectrum (denoted by the quantity ˆxz CL,ab 84 BIBLIOGRAPHY

deﬁned in Eq. (2.25)) is given by

2 2 xz 1 d `1 d `2 x z ˆ = C + F ~ F ~ W (~` , ~` )W (~`0 , ~`0 ) L,ab L zb 2 2 `1 `2 1 2 1 2 C cd xa ⇡ ⇡ | | | | L L (2 ) (2 ) R R Z Z c ~ ~ d ~ ~ c ~ ~ d ~ ~ C ~` ~` W ( `1, `10 )W ( `2, `20 ) + C ~` ~` W ( `1, `20 )W ( `2, `10 ) . (A-7) ⇥ | 1 10 | | 1 20 | ⇢ 2 B. SUPPLEMENTARY PLOTS 85

2 B Supplementary Plots

This section contains several results not included in the preceding paper.

2 B 1 Estimator Sensitivity

The bias to the lensing estimator from extragalactic foregrounds can be reduced by limiting the range of scales that are used to reconstruct the lensing potential, at the cost of reduced sensitivity. To determine how the sensitivity of the estimator is reduced, we calculate the ˆ signiﬁcance of the lensing potential spectrum, C` , measured from a limited multipole ˆ range. In the approximation that the covariance of C` is diagonal, the signiﬁcance is:

` 2 1 max Cˆ 2 = (2` + 1) ` , (B-1) 2 Cˆ + A `=`min 0 ` ` 1 X B C B C ˆ @ A where A` is the noise bias. We replace C` with the theoretical spectrum, and calculate the noise bias from our Lmax = 1500 estimator. We show the result in Fig. 2.11, calculated using either the multipoles below a given ` (blue line) or the multipoles above (black line). This shows that the estimator is most sensitive in the multipole range 800 1700, ⇠ and so excluding data outside this range does not substantially reduce the sensitivity. The foreground bias, which is large at high `, and the ISW signal, which is large at low `, can therefore be reduced by omitting these multipoles from the estimator without a large reduction in sensitivity.

2 B 2 Frequency Dependence of the Foreground Bias

Using the extragalactic foreground simulations described in Sec. 2.5 we can calculate the expected bias to the lensing potential estimator. In Fig. 2.12 we show the lensing spectrum, ˆ C` , calculated from the simulated foreground components at 90, 148, 219, and 350 GHz using our Lmax = 1500 estimator. Before calculating the reconstruction the radio and infrared point source maps are masked by removing bright sources down to a ﬂux density

S max = 200 mJy (upper panel) or 6 mJy (lower panel). The thermal and kinetic SZ maps are masked by removing the 3,000 brightest clusters at 148 GHz (upper panel) or the 12,000 86 BIBLIOGRAPHY

35 ]

σ 30

5 Detection Significance [ 0 500 1000 1500 2000 Lmin or Lmax

Figure 2.11 Detection signiﬁcance as a function of the minimum (black) and maximum (blue) multipole used in the estimator. The black curve is calculated with `max = 3000 and the blue curve with `min = 2.

brightest clusters (lower panel). The spectra have been smoothed to reduce a large power asymmetry between the even and odd multipoles, caused by the way the simulated maps were constructed, as discussed in Sec. 2.5.

Based on these results we would expect the unresolved radio point source component to be the largest contribution to the bias in the 143 GHz Planck channel, and infrared sources in the 150 GHz SPT and ACT channels. However, as discussed in Secs. 2.6.1 and 2.6.2, the biases from radio point sources and thermal SZ are sensitive to the number of sources masked. As expected the 219 GHz channel contains a negligible thermal SZ contribution, and the kinetic SZ contribution is essentially the same at all frequencies. Since we only pass the separated foreground components through the estimator, any terms that correlate with the CMB or the lensing potential will not be present in Fig. 2.12. 2 B. SUPPLEMENTARY PLOTS 87

2 B 3 Bispectrum- and Trispectrum-Related Spectra

As described in Sec. 2.4, we can estimate the shot noise amplitudes from the data using the estimators:

3 (3) Sˆ = A3 (2` + 1) C = A3 T¯`mN⇤ T¯`mN⇤ ` `m `m D ` 0 `m 1 D E X X D E (B-2) B C 4 (4) B C Sˆ = A (2` + 1) C = A @ N N⇤ N N⇤ A , 4 ` 4 `m `m `m `m D ` 0 `m 1 D E X BX ⌦ ↵ C B C @¯ ¯ A where N`m = `imi dnˆ Y`⇤m(nˆ)Y`1m1 (nˆ)Y`2m2 (nˆ)T`1m1 T`2m2 . Although the bispectrum is not ˆ ˆT needed to calculateP R the bias to C` , it is needed to calculate the bias to C` , and provides additional information about the foregrounds. We test the estimators on the simulated 148 GHz map, applying the SZ and point source mask as described in Sec. 2 B 2. When we (3) (4) (3) use the Lmax = 1500 estimator we ﬁnd the C` and C` spectra shown in Fig. 2.13. The C` spectrum has no noise bias, since the CMB is Gaussian (in the approximation that lensing (4) can be neglected) and so has no bispectrum, but we have subtracted a noise bias from C` .

(4) There is an additional bias to C` caused by lensing of the CMB, which introduces a non-zero trispectrum into the data that overlaps with the point source trispectrum. As discussed in Sec. 2.4 the point source and lensing estimators can both be constructed to be unbiased. Here we will simply calculate the bias to the point source estimator, which is related to the response functions in Sec. 2.4, using the theoretical C` spectrum. The primary trispectrum contraction is:

˜TT 1 2 2 C` N(0) = C AS (2` + 1)(2` + 1) 1 ` ⇡2 ` ` 1 2 TT TT 16 0 ` ` C` C` BX1 2 1 2 ⇣ ⌘ B 2 B 2 @ `1 `2 ` [`(` + 1) + `1(`1 + 1) `2(`2 + 1)] , (B-3) 0 0001 1 B C C B C C B C C S 2 @ A AC where A` is the point source estimator normalization. The secondary contraction can be 88 BIBLIOGRAPHY

calculated in the ﬂat-sky approximation:

~ 2 2 d~` d`0 1 (1) = S 1 1 N` A` 2 2 TT TT TT TT (2⇡) (2⇡) C` C` C` C` ⇣ ⌘ Z Z 1 2 10 20 ~ ~ ~ ~ ~ ~ ~ ~ C f ( `1, `0 ) f ( `2, `0 ) + C f ( `1, `0 ) f ( `2, `0 ) , (B-4) ~` ~` 1 2 ~` ~` 2 1 1 10 1 20  | | | | where f (~`, ~`0) = ~L ~` C˜ + ~L ~`0 C˜ and ~L = ~` + `~ . We show the primary and secondary · ` · `0 0 contractions in Fig. 2.13 as the solid and dashed blue lines, respectively. The best-fit Sˆ 3 and Sˆ 4 amplitudes are found by subtracting the lensing bias and calculating the sumD overE ` inD Eq.E (B-2). For the Lmax = 1500 estimator and masking scheme 3 9 3 4 12 4 we find Sˆ = (3.89 0.0049) 10 µK and Sˆ = (1.27 0.058) 10 µK , obtaining ± ⇥ ± ⇥ the errorD estimatesE from the scatter in the spectrum.D E These results are comparable to the one-point amplitudes calculated using our foreground model in Sec. 2.6. For the Lmax = 3 11 3 4 3000 estimator we find Sˆ = (3.860 0.002) 10 µK and Sˆ = (1.43 0.048) ± ⇥ ± ⇥ 15 4 10 µK , where the errorD estimatesE assume a 6% observed sky fraction.D E If the foreground signal contained a large contribution from clustered sources then the best-fit line would be a poor fit to the data. In Fig. 2.13 the best-fit line gives a good fit, indicating that the contribution from clustered sources is small.

2 B 4 Results on Simulations

ˆ 4 Using the best-fit value of S we calculate the bias to C` , which is shown in Fig. 2.14. We compare this to the lensD reconstructionE calculated from the simulated 148 GHz foreground map (containing no CMB or noise), which includes several of the bias terms. The green line in Fig. 2.14 shows the lens reconstruction (the “true” bias, with the exception of the S 2 term which is not included), and the solid blue line shows the bias calculated from the point source estimator. The dashed blue line is the disconnected Gaussian noise in the reconstruction, which in practice is automatically subtracted if the C` noise bias is calculated from the measured map spectrum. The bias calculated from the point source estimator is a good fit to the true bias at ` & 100, but at ` . 100 the fit is poor. This is likely due to systematic e↵ects in the simulated maps, since the full-sky simulations are constructed 2 B. SUPPLEMENTARY PLOTS 89

by mirroring structure from an octant of the sky, introducing spurious large-scale power as discussed in Sec. 2.5. 90 BIBLIOGRAPHY

ˆ Figure 2.12 Bias to C` calculated using the Lmax = 1500 (upper panel) or Lmax = 3000 (lower panel) estimator and masking scheme from the simulated radio point source (red), infrared point source (blue), thermal SZ (green), and kinetic SZ (purple) maps. The gray bars show the error bars from a Fisher matrix calculation. 2 B. SUPPLEMENTARY PLOTS 91

4 Bispectrum ] 3 K µ

12 3 * [x10

LM 2 S 2 LM S M Σ

) 1 B (1/N 0

500 1000 1500 2000 2500 3000 Multipole

Trispectrum

] 0.8 4 K µ

15 0.6 * [x10 2 LM

S 0.4 2 LM S M Σ )

T 0.2 (1/N

0.0

1000 2000 3000 4000 Multipole

Figure 2.13 The bispectrum and trispectrum-related spectra calculated from the 148 GHz simulated map. The red line shows the best-ﬁt spectrum, which depends on a single parameter: S 3 for the bispectrum, and S 4 for the trispectrum. The noise bias has been subtracted from the trispectrum estimate, and the bias terms caused by CMB lensing are D E D E shown in blue. 92 BIBLIOGRAPHY

0 10

] −1

7 10 [x 10

π −2 10 /2 φ

φ l C 2 −3 10 +1)] l ( l [ −4 10

−5 10 10 100 500 1000 1500 2000 2500 Multipole

Figure 2.14 The green line shows the bias from the unresolved foreground components in the masked 148 GHz simulated map. The solid blue line is the measured bias, the dashed blue line is the disconnected term, and the red line is the sum of the blue lines. The gray error bars are from a Fisher matrix calculation for the Lmax = 1500 estimator. Chapter 3

Gravitational Lensing by Dusty Star-Forming Galaxy Halos

“All men have stars, but they are not the same things for di↵erent people. For some, who are travelers, the stars are guides. For others they are no more than little lights in the sky. For others, who are scholars, they are problems.”

Le Petit Prince Antoine de Saint-Exupery´

The multi-frequency capability of the Planck satellite provides information both on the integrated history of star formation (via the cosmic infrared background, or CIB) and on the distribution of dark matter (via the lensing e↵ect on the cosmic microwave background, or CMB). The conjunction of these two unique probes allows us to measure directly the connection between dark and luminous matter in the high redshift (1 z 3) Universe. We use a three-point   statistic optimized to detect the correlation between these two tracers. Follow- ing a thorough discussion of possible contaminants and a suite of consistency tests, using lens reconstructions at 100, 143 and 217 GHz and CIB measurements at 100–857 GHz, we report the ﬁrst detection of the correlation between the CIB and CMB lensing. The well matched redshift distribution of these two

93 94 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

signals leads to a detection signiﬁcance with a peak value of 42 at 545 GHz and a correlation as high as 80 % across these two tracers. Our full set of multi- frequency measurements (both CIB auto- and CIB-lensing cross-spectra) are consistent with a simple halo-based model, with a characteristic mass scale for

the halos hosting CIB sources of log10 (M/M ) = 10.5 0.6. Leveraging the ± frequency dependence of our signal, we isolate the high redshift contribution to the CIB, and constrain the star formation rate (SFR) density at z 1. We measure directly the SFR density with around 4 signiﬁcance for three redshift bins between z = 1 and 7, thus opening a new window into the study of the formation of stars at early times.

3.1 Introduction

This paper, one of a set associated with the 2013 release of data from the Planck1 mission [1], presents a first detection of a strong correlation between the infrared background anisotropies and a lensing-derived projected mass map. The broad frequency coverage of the Planck satellite provides two important probes of the high redshift Universe. In the central frequency bands of Planck (70, 100, 143, and 217 GHz), cosmic microwave background (CMB) fluctuations dominate over most of the sky. Gravitational lensing by large-scale structure produces small shear and magnification e↵ects in the observed fluctuations, which can be exploited to reconstruct an integrated measure of the gravitational potential along the line of sight [2]. This “CMB lensing potential” is sourced primarily by dark matter halos located at 1 . z . 3, halfway between ourselves and the last scattering surface (see Blandford and Jaroszynski 3, Blanchard and Schneider 4, or Lewis and Challi- nor 5 for a review). In the upper frequency bands (353, 545, and 857 GHz), the dominant extragalactic signal is not the CMB, but the cosmic infrared background (CIB), composed of redshifted thermal radiation from UV-heated dust, enshrouding young stars. The CIB contains much of the energy from processes involved in structure formation. According

1Planck (http://www.esa.int/Planck) is a project of the European Space Agency (ESA) with instruments provided by two scientific consortia funded by ESA member states (in particular the lead countries France and Italy), with contributions from NASA (USA) and telescope reflectors provided by a collaboration between ESA and a scientific consortium led and funded by Denmark. 3.1. INTRODUCTION 95

to current models, the dusty star-forming galaxies (DSFGs), which form the CIB have a redshift distribution peaked between z 1 and z 2, and tend to live in 1011–1013M dark ⇠ ⇠ matter halos [see, e.g., 6, and references therein]. As first pointed out by [7], the halo mass and redshift dependence of the CMB lensing potential and the CIB fluctuations are well matched, and as such a significant correlation between the two is expected. This point is illustrated quantitatively in Fig. 3.1, where we plot estimates for the redshift- and mass- kernels of the two tracers. In this paper we report on the first detection of this correlation. Measurements of both CMB lensing and CIB fluctuations are currently undergoing a period of rapid development. While the CIB mean was first detected using the FIRAS and DIRBE instruments aboard COBE [8–10], CIB fluctuations were later detected by the Spitzer Space Telescope [11] and by the BLAST balloon experiment [12] and the Herschel Space Observatory [13, 14], as well as the new generation of CMB experiments, including Planck, which have extended these measurements to longer wavelengths [15–18]. The Planck early results paper: [17] (henceforth referred to as PER) presented measurements of the angular power spectra of CIB anisotropies from arc-minute to degree scales at 217, 353, 545, and 857 GHz, establishing Planck as a potent probe of the clustering of the CIB, both in the linear and non-linear regimes. A substantial extension of PER is presented in a companion paper to this work [19, henceforth referred to as PIR]. The CMB lensing potential, on the other hand, which was first detected statistically through cross-correlation with galaxy surveys (Smith et al. 20, Hirata et al. 21, and more recently Bleem et al. 22, Sherwin et al. 23), has now been observed directly in CMB maps by the Atacama Cosmology Telescope and the South Pole Telescope [24, 25]. Planck’s frequency coverage, sensitivity and survey area, allow high signal-to-noise measurements of both the CIB and the CMB lensing potential. Accompanying the release of this paper, [26] reports the first measurement and characterisation of the CMB lensing potential with the Planck data, which has several times more statistical power than previous measurements, over a large fraction (approximately 70% of the sky). We will use this measurement of the lensing potential in cross-correlation with measurements of the CIB in the Planck HFI bands to make the first detection of the lensing-infrared background correlation. In addition to our measurement, we discuss the implications for models of the 96 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

CIB ﬂuctuations. The outline of this paper is as follows. In Sect. 3.2 we describe the data we will use, followed by a description of our pipeline for correlating the CIB and lensing signals in Sect. 3.3. Our main result is presented in Sect. 3.4, with a description of our error budget, consistency tests and an array of systematic tests in Sect. 3.5. We discuss the implications of the measured correlation for CIB modelling in Sect. 3.6.

3.2 Data Sets

3.2.1 Planck Maps

Planck [29, 30] is the third generation space mission to measure the anisotropy of the CMB. It observes the sky with high sensitivity in nine frequency bands covering 30–857 GHz at an angular resolution from 310 to 50. The Low Frequency Instrument (LFI; Mandolesi et al. 31, Bersanelli et al. 32, Mennella et al. 33) covers the 30, 44, and 70 GHz bands with radiometers that incorporate ampliﬁers cooled to 20 K. The High Frequency Instrument (HFI; Lamarre et al. 34, Planck HFI Core Team 35) covers the 100, 143, 217, 353, 545, and 857 GHz bands with bolometers cooled to 0.1 K. Polarization is measured in all but the highest two bands [36, 37]. A combination of radiative cooling and three mechanical cool- ers produces the temperatures needed for the detectors and optics [38]. Two data processing centres (DPCs) check and calibrate the data and make maps of the sky [39, 40]. Planck’s sensitivity, angular resolution, and frequency coverage make it a powerful instrument for Galactic and extragalactic astrophysics as well as cosmology. Early astrophysics results are given in Planck Collaboration VIII–XXVI 2011, based on data taken between 13 August 2009 and 7 June 2010. Intermediate astrophysics results are now being presented in a series of papers based on data taken between 13 August 2009 and 27 November 2010. This paper uses data corresponding to the second Planck data release, with data acquired in the period up to 27 November 2010 and undergoing improved processing. We use the Planck HFI temperature maps at all six frequencies, i.e., 100, 143, 217, 353, 545, and 857 GHz. The maps at each frequency were created using almost three full-sky surveys. Here we give an overview of the HFI map-making process with additional details given in Planck HFI Core Team [39], Planck Collaboration VI [41]. The data are organized 3.2. DATA SETS 97

Figure 3.1 Redshift- and mass- integrand for the CIB and CMB lensing potential power spectra at ` = 500, calculated using the CIB halo model of [17], evaluated at 217 GHz. The good match between the redshift and halo mass distributions leads to an expected correlation up to 80 %. The sharper features in the CIB kernel are artefacts from the [27] model. We note that the low mass, high z behavior of our calculation is limited by the 5 accuracy of the mass function we use [28]. All of our mass integrals use Mmin = 10 M . 98 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

as time-ordered information, hereafter TOI. The attitude of the satellite as a function of time is provided by two star trackers on the spacecraft. The pointing for each bolometer is computed by combining the attitude with the location of the bolometer in the focal plane, as determined by planet observations. The raw bolometer TOI for each channel is first processed to produce cleaned timelines and to set flags that mark bad data (for example data immediately following a cosmic ray strike on the detector). This TOI processing includes: (1) signal demodulation and filtering; (2) deglitching, which flags the strong part of any glitch and subtracts the tails; (3) conversion from instrumental units (volts) to physical units (watts of absorbed power, after a correction for the weak non-linearity of the response); (4) de-correlation of thermal stage fluctuations; (5) removal of the systematic e↵ects induced by 4 K cooler mechanical vibrations; and (6) deconvolution of the bolometer time response. Focal plane reconstruction and beam shape estimation is made using observations of Mars. The simplest description of the beams, an elliptical Gaussian, leads to full-width at half- maximum (FWHM) values of 9.65, 7.25, 4.99, 4.82, 4.68 and 4.33 0as given in Table 4 of Planck Collaboration VI [41]. As explained in this paper, the inter-calibration accuracy between channels is better than the absolute calibration. This leads us to adopt conservative absolute calibration uncertainties of 0.64, 0.53, 0.69, 2.53, 10., 10. % at 100, 143 217, 353, 1 545 and 857 GHz respectively. We convert between emissivities given in MJy sr (with the photometric convention ⌫I⌫ = constant) and temperatures in µK, using the measured bandpass filters (see PER and PIR for details). For the sake of consistency testing (presented in particular in Sect. 3.5), we will sometimes use temperature maps where only a fraction of the TOI is used to generate the sky map. In particular, throughout this paper we use the terminology “half-ring” (HR) maps to refer to maps made using the first and second half of the stable pointing period, “survey” for individual full-sky survey maps (note that the third survey is incomplete for the particular data release used in the intermediate papers), and “detset” for maps made using two independent sets of detectors per frequency [for details see 39]. We create three masks to exclude regions with bright Galactic and extragalactic foreground emission. The first mask accounts for di↵use Galactic emission as observed in the Planck data. To allow us to test for the e↵ects of residual Galactic emission on our results we create three di↵erent versions of this mask, each with a di↵erent masked area, such 3.2. DATA SETS 99

that 20, 40 or 60 % of the sky is unmasked. Each version of this mask is created directly from the Planck 353 GHz map, from which we remove the CMB using the 143 GHz channel as a CMB template before smoothing by a Gaussian with FWHM of 5. The map is then thresholded such that the mask has the required sky fraction. Although the Galactic emission is stronger at 857 GHz, we expect the 353 GHz mask to better trace dust emission at the lower frequencies we use. The mask therefore accounts for Galactic dust and Galactic CO emission as explained in Planck Collaboration XII [42]. We will not worry about synchrotron emission, which is important at low frequencies, since its contribution at 100 GHz and at high Galactic latitudes is small, and, as with the dust component, will be uncorrelated with the lensing potential. The second mask covers bright point sources. This mask is created using algorithms tailored to detect point sources in the Planck data and is optimized for each frequency, as detailed in [43] and [44]. The third mask is designed to remove extended high-latitude Galactic dust emission (“cirrus”), as traced by external Hi data, as we will describe in Sect. 3.2.2. While the ﬁrst two masks are described in [42], the latter is speciﬁc to our cross-correlation analysis, as it provides a method to reduce the large-scale noise in our measurement, and the 3-point nature of our estimate ensures that it will not introduce a bias (although we test for this in Sect. 3.5). Ultimately, when we combine the three masks we obtain an e↵ective sky fraction of 16, 30 and 43 % for the 20, 40 and 60 % Galactic masks, respectively.

3.2.2 External Data Sets

Hi Maps

We use measurements of 21-cm emission from Galactic neutral hydrogen (Hi) as a cirrus monitor. Outside of our Galactic and point source masks we use the Hi data to construct a template of the dust emission in regions where the Hi column density is low (less than 20 2 N 2 10 cm ), and we mask regions where it is high, since in these regions the Hi HI  ⇥ and dust emission are not well correlated [45, 46, PER]. The masking procedure that we use is described in detail in [47]. It consists of subtracting the Hi dust template from the Planck temperature map at 857 GHz and calculating the skewness of the residuals in 5 deg2 regions. If the skewness is larger than a given value then the region is masked. This is an 100 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

Figure 3.2 Combined Galactic, point-source and Hi mask with sky fractions 16, 30 and 43 %. improvement over the usual cut-o↵ in Hi column density. We use the latest release from the Leiden/Argentina/Bonn (LAB) survey [48], which consists of the Leiden/Dwingeloo

Survey (LDS) [49] north of 30 declination, combined with the Instituto Argentino de Radioastronomia Survey [50, 51] south of 25 declination. The angular resolution of the combined map is approximately 0.6 FWHM. The LAB Survey is the most sensitive Milky Way Hi survey to date, with the greatest coverage both spatially and kinematically. We make use of projections of the LAB maps onto Nside = 512 HEALPix maps performed by Land and Slosar [52] and made available through the LAMBDA website2. The local stan- 1 1 dard of rest velocity coverage spans the interval 450 km s to +400 km s , at a resolution 1 of 1.3 km s , with an rms brightness-temperature noise of 0.07–0.09 K, and with additional errors due to defects in the correction for stray radiation that are less than 20–40 mK for most of the data.

IRIS/IRAS Maps

As a consistency test we will use an additional tracer of the CIB that derives from re- processed IRAS maps at 60 and 100 µm. This new generation of IRAS maps, known as IRIS, beneﬁts from improved zodiacal light subtraction, a calibration and zero level

2http://lambda.gsfc.nasa.gov/product/foreground/ 3.3. CROSS-CORRELATION FORMALISM AND IMPLEMENTATION 101

compatible with DIRBE, and an improved de-striping procedure [53]. IRAS made two full-sky maps (HCON-1 and HCON-2), as well as a ﬁnal map that covers 75 % of the sky (HCON-3). The three maps had identical processing that included deglitching, checking of the zero-level stability, visual examination for glitches and artifacts, and zodiacal light removal. The three HCONs were then co-added, taking into account the inhomogeneous sky coverage maps, to generate the average map (HCON-0). Note that the Finkbeiner et al. [54] maps are also constructed from the IRAS 100 µm data, and as such we will not investigate their cross-correlation properties since the IRIS map contains the same information. For simplicity we will assume that the e↵ective IRIS beam is uniform across the sky and described by a Gaussian with FHWM of 4.30.

3.3 Cross-Correlation Formalism and Implementation

We now describe our statistical formalism and its implementation, with additional technical details given in the appendices. Our analysis consists of cross-correlating a full-sky reconstruction of the CMB lensing potential with a temperature map.

3.3.1 Reconstructing the CMB Lensing Potential

The CMB is lensed by the gravitational potential of all matter along the photon trajectory from the last scattering surface to us. The lensed CMB is a remapping of the unlensed CMB with the lensed temperature equal to ⇥˜ (nˆ) = ⇥(nˆ + ), where ⇥(nˆ) is the unlensed CMB r temperature and is the lensing potential. We use the methodology described in Planck

Collaboration XVII [26] to obtain estimates ˆ LM of the lensing potential in harmonic space, using the standard Okamoto and Hu [2] quadratic estimator. Complete details on the lens reconstruction procedure, which we use are given in Planck Collaboration XVII [26], although we review it brieﬂy in point form here. Our estimates of ˆ are obtained by the following set of steps:

1. Inverse variance ﬁlter the CMB map.

2. Use the ﬁltered CMB map as the input to a quadratic lensing estimator, which is 102 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

designed to extract the o↵-diagonal contributions to the CMB covariance matrix induced by lensing.

3. Subtract a “mean-ﬁeld bias”, which corrects for known non-lensing contributions to the covariance matrix, including instrumental noise inhomogeneity, beam asymmetry, and the Galaxy+point source mask.

The output from this pipeline is an estimate of the lensing potential in harmonic space ˆ LM and an associated noise power spectrum NL , which we use to weight our cross-correlation estimates. We also produce a set of simulated lens reconstruction, which we use to establish our statistical error bars. Our nominal lens reconstructions use the 143 GHz channel, however there is almost equivalent power to measure lensing using the 217 GHz channel. Combining both channels would reduce the noise power spectrum of our lens reconstruction by approximately 25 %, compared with using either individually (the improvement is significantly less than 50 % because a significant portion of the lens reconstruction noise is due to the finite number of CMB modes, which we are able to observe, and is correlated between the two channels). We choose to focus on 143 GHz here because it is significantly less susceptible to CIB contamination. We will use lens reconstructions based on the 100 and 217 GHz data for consistency tests.

3.3.2 Decreasing the Foreground Noise

An important source of noise (but, as we will explain below, not bias) in our cross-correlation measurement is Galactic foreground emission. Dust emission is the dominant Galactic component at HFI frequencies above 217 GHz (see Sect. 3.5.1 for a quantitative discussion). In order to reduce the Galactic dust emission we create a dust template and subtract it from the temperature maps described in Sect. 3.2.1. At 100 and 143 GHz the CMB signal is signiﬁcantly brighter than the dust emission outside the Galactic mask. We therefore do not create and subtract a dust template at these frequencies. Note that while we could use other frequency maps to trace the CMB and remove it, to quantify the non-negligible amount of CIB that would be removed this way is not easy given the uncertainties in the cross-frequency CIB correlation structure. 3.3. CROSS-CORRELATION FORMALISM AND IMPLEMENTATION 103

We rely on the well documented (but complex) correlation between Galactic Hi and dust [e.g., 45, 46, 55, PER] to reduce the contamination by subtracting the Hi-correlated dust component. As was performed in PER, we split the Hi map into two velocity components: a low-velocity local component (LC) typical of high-latitude Hi emission, and a component of intermediate-velocity clouds (IVC). We found that the inclusion of a high-velocity component makes a negligible di↵erence to the dust-cleaned map. Unlike the dedicated high-resolution Hi observations used in PER and PIR that only partially cover the sky, here we use the full-sky, low resolution LAB survey introduced in Sect. 3.2.2 as our Hi tracer. Although the resolution of this survey is lower than the Planck resolution, it allows us to perform dust cleaning on large scales, where our cross-correlation measurement has high signal-to-noise ratio. The emissivity of the dust varies across the sky, and so the correlation between the dust and Hi emission is expected to vary. To account for this we divide the sky into regions where we assume that the dust-Hi correlation is constant. For the sake of convenience, we use regions of approximate size 13 (52) deg2 deﬁned by the Healpix pixels at resolution Nside = 16 (8) that are outside the Galactic mask. We test that our conclusions do not depend on this resolution. The details of our procedure is as follows. We subtract the 143 GHz Planck temperature map from each of the 217–857 GHz temperature maps to remove the CMB signal (this CMB subtraction is only done for the purposes of creating the dust template). We upgrade each of the Nside = 512 LAB maps compiled in Land and Slosar [52] to the Planck map resolution of Nside = 2048. Within each region we then simultaneously ﬁt for the amplitude of each Hi velocity component in the CMB-subtracted maps, and use the two coecients per region to assemble a full-sky (minus the mask) dust template for each of the 217–

857 GHz channels. We smooth each template with a Gaussian of FWHM 100 to remove the discontinuity at the patch boundaries, and then subtract the template from the original (CMB-unsubtracted) Planck maps. We note that the accuracy of this procedure would be dicult to evaluate for all possible uses of the map, i.e., whether it might constitute a robust component separation method remains to be demonstrated. However, in the case of our cross-correlation analysis the dust-removal requirements are less severe, since the dust emission only contributes to our measurement as noise. We will describe later in Sect. 3.5.2 the e↵ect on the cross-spectrum 104 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

Figure 3.3 Angular cross-spectra between the reconstructed lensing map and the temperature map at the six HFI frequencies. The error bars correspond to the scatter within each band. The solid line is the expected result based on the PER model and is not a ﬁt to these data (see Fig. 3.16 for an adjusted model), although it is already a satisfying model. In each panel we also show the correlation between the lens reconstruction at 143 GHz and the 143 GHz temperature map in grey. This is a simple illustration of the frequency scaling of our measured signal and also the strength of our signal as compared to possible intra-frequency systematic errors.

of removing this emission, and will place limits on the residual Galactic contamination in Sect. 3.5.3. 3.3. CROSS-CORRELATION FORMALISM AND IMPLEMENTATION 105

3.3.3 Measuring Cross-Correlations

To estimate the cross-correlation between the lensing potential and a tracer t, we calculate

t 1 Cˆ = tˆ` ˆ⇤ . (1) ` 2` + 1 m `m Xm 1 2 As the CIB has an approximately ` dependence and the lensing potential has an ` dependence, we multiply the cross-spectra by `3, and then bin it in 15 linearly spaced bins between 100 < ` < 2000. As we will discuss in Sect. 3.5, modes with ` < 100 are not considered, due to possible lens mean-ﬁeld systematic e↵ects, and modes with ` > 2000 are removed due to possible extragalactic foreground contamination. We have tested that our results are robust to an increase or decrease in the number of `-bins.

We expect the error bars to be correlated across bins to some extent, due to pseudo-

C` mixing induced by the mask, and between frequencies, because the lens reconstruction noise is common. In addition, any foregrounds that are present in multiple channels will introduce correlated noise. The foreground mask will also induce a coupling between dif- ferent modes of the unmasked map. This extra coupling can be calculated explicitly using the mixing matrix formalism introduced in Hivon et al. [56]. Using this formalism and our best-ﬁt models we have evaluated the correlation between di↵erent bins of the cross- correlation signal for our nominal binning scheme. We ﬁnd that the mask-induced correlation is less than 2 % across all bins at all frequencies. We will thus neglect it in our analysis. For this reason, and based on the results we obtain from simulations, we do not attempt to “deconvolve” the mask from the cross-spectrum [see e.g., 56] and instead correct for the power lost through masking the maps by a single sky fraction, fsky, ignoring the mode coupling.

As will be discussed later in Sect. 3.6.1, when we ﬁt models to the cross-spectrum we will assume that the noise correlation between bins can be neglected and that the band- powers are ﬂat. 106 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

3.3.4 Simulating Cross-Correlations

In order to validate our measurement pipeline and to conﬁrm that our estimate of the cross- spectrum is unbiased we create simulated maps of the lensed CMB and CIB that have the expected statistical properties.

Using the Planck only favored ⇤CDM cosmology as described in Planck Collaboration XVI [57] we generate a theoretical prediction of the lensing potential spectrum using CAMB [58], from which we generate 300 maps of that are used to lens 300 CMB realizations using the approach described in [26]. We then use the PER best-ﬁt CIB model to generate CIB auto- and CIB- cross-spectra, from which we create CIB realizations that are correctly correlated with in each HFI band. The PER model that we use describes the CIB clustering at HFI frequencies using a halo approach, and simultaneously reproduces known number count and luminosity function measurements. At each frequency we add a lensed CMB realization to each of the CIB realizations and then smooth the maps using a symmetric beam with the same FWHM as the beam described in Sect. 3.2.1. Once this set of realizations has been generated we apply the reconstruction procedure described above to produce an estimate of the lensing potential map, and then calculate the cross-power spectrum using our measurement pipeline.

These simulations will miss some complexities inherent in the Planck mission. They do not take into account inhomogeneous and correlated noise, and we do not simulate asym- metric beam e↵ects. In addition, we do not simulate any foreground components, and we instead take a di↵erent approach to determine their contribution. While simplistic, we believe that our simulations are good enough for the purposes of this particular measurement. In Sect. 3.5 we will discuss possible limitations, as well as how we test for systematic e↵ects that are not included in the simulations.

We use the simulated maps to check that our pipeline correctly recovers the cross- spectrum that was used to generate the simulations. For the `-bins used in our analysis, we ﬁnd that the recovered spectrum is unbiased (to within the precision achievable with 300 simulations), and with a noise level consistent with expectations. The noise in the recovered spectrum is discussed in Sect. 3.5.1. 3.4. A STRONG SIGNAL USING PLANCK HFI DATA 107

Figure 3.4 Temperature maps of size 1 deg2 at 545 and 857 GHz stacked on the 20,000 brightest peaks (left column), troughs (centre column) and random map locations (right column). The stacked (averaged) temperature maps is in K. The arrows indicate the lensing deflection angle deduced from the gradient of the band-pass filtered lensing potential map stacked on the same peaks. The longest arrow corresponds to a deflection of 6.300, which is only a fraction of the total deflection angle because of our filtering. This stacking allows us to visualize in real space the lensing of the CMB by the galaxies that generate the CIB. A small and expected o↵set ( 10) was corrected by hand when displaying the deflection field. '

3.4 A Strong Signal Using Planck HFI Data

We now describe the result of applying our pipeline to our nominal data set, i.e., the lens reconstruction at 143 GHz and the foreground reduced Planck HFI temperature maps with a 40 % Galactic mask, which when combined with the point source mask and Hi mask leaves 30.4 % of the sky unmasked. The results are presented in Fig. 3.3. The error bars 108 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

correspond to the naive scatter measured within each bin. The thin black line corresponds to the expected CIB-lensing correlation predicted using the PER CIB model (the HOD parameters of the PER 217 GHz best-ﬁt model were used at 100 and 143 GHz since no CIB clustering measurement at these frequencies is available). As can be seen from these plots, the noise is strongly correlated across frequencies, especially at the lowest frequencies where the CMB dominates the error budget. A detailed analysis of the uncertainties and potential systematic errors attached to this result is presented in Sect. 3.5.

As clearly visible in Fig. 3.3, a strong signal is detected. To set a reference point and naively quantify its statistical significance when taken at face value, we define a detection significance as follows. We count the number of standard deviations as the quadrature sum of the significance in the di↵erent multipole bins:

2 15 CT s = i . (2) ⌫ v T tu i=1 0Ci 1 X B C B C For our nominal parameters this gives us 3.6 @,4.3 A,8.3 , 31 , 42 , and 32 , at, respectively, 100, 143, 217, 343, 545 and 857 GHz. Note that these numbers include an additional 20 % contribution to the statistical error to account for mode correlations (which we discuss in Sect. 3.5.1), but do not include systematic errors or our point source correction. As a comparison, in each panel we plot the correlation between the lens reconstruction at 143 GHz and the 143 GHz map in grey. This shows the frequency scaling of our measured signal and also the strength of the signal, as compared to possible intra-frequency systematic e↵ects. This will be studied in depth in Sect. 3.5.

This first pass on our raw data demonstrates a strong detection that is in good agreement with the expected CIB-lensing signal. To get a better intuition for this detection, we show in Fig. 3.4 the real-space correlation between the observed temperature and the lens deflection angles. This figure allows us to visualize the correlation between the CIB and the CMB lensing deflection angles for the first time. These images were generated using the following stacking technique. We first mask the 545 and 857 GHz temperature maps with our combined mask that includes the 20 % Galaxy mask, and identify 20,000 local maxima and minima in these maps. We also select 20,000 random locations outside the masked 3.5. STATISTICAL AND SYSTEMATIC ERROR BUDGET 109

region to use in a null test. We then band pass filter the lens map between ` = 400–600 to remove scales larger than our stacked map as well as small-scale noise. We stack a 1 deg2 region around each point in both the filtered temperature map and lensing potential map, to generate stacked CIB and stacked lensing potential images. We take the gradient of the stacked lensing potential to calculate the deflection angles, which we display in Fig. 3.4 as arrows. The result of the stacking over the maxima, minima and random points is displayed from left to right in Fig. 3.4. The strong correlation seen already in the cross-power spectrum is clearly visible in both the 545 and 857 GHz extrema, while the stacking on random locations leads to a lensing signal consistent with noise. From simulations, we expect a small o↵-set ( 100) in the deflection field. This o↵set was corrected for in this plot. We ' have verified in simulations that this is due to noise in the stacked lensing potential map that shifts the peak. As expected, we see that the temperature maxima of the CIB, which contain a larger than average number of galaxies, deflect light inward, i.e., they correspond to gravitational potential wells, while temperature minima trace regions with fewer galaxies and deflect light outward, i.e., they correspond to gravitational potential hills.

3.5 Statistical and Systematic Error Budget

The ﬁrst pass of our pipeline suggests a strong correlation of the CIB with the CMB lensing potential. We now turn to investigate the strength and the origin of this signal. We will ﬁrst discuss the di↵erent contributions to the statistical error budget in Sect. 3.5.1, and then possible systematic e↵ects in Sect. 3.5.2. Although the most straightforward interpretation of the signal is that it arises from dusty star-forming galaxies tracing the large-scale mass distribution, in Sect. 3.5.3 we consider other potential astrophysical origins for the observed correlation.

3.5.1 Statistical Error Budget

In this section we discuss any noise contribution that does not lead to a bias in our measurement. The prescription adopted throughout this paper is to obtain the error estimates from the naive Gaussian analytical error bars calculated using the measured auto-spectra 110 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

of the CIB and lensing potential. We ﬁnd that these errors are approximately equal to 1.2 times the naive scatter within an `-bin, and we will sometimes use this prescription where appropriate for convenience (as will be stated in the text). This is justiﬁed in Appendix 3 A where we consider six di↵erent methods of quantifying the statistical errors using both ˆT simulations and data. The Gaussian analytical errors, C` , are calculated using the naive prescription

ˆT 2 ˆTT ˆ T 2 fsky (2` + 1) ` C` = C` C` + C` , (3) ⇣ ⌘ ⇣ ⌘ where as before fsky is the fraction of the sky that is unmasked, ` = 126 for our 15 linear ˆTT ˆ bins between ` = 100 and ` = 2000, C` and C` are the spectra measured using the data, T and C` is the model cross spectrum. This last term provides a negligible contribution due ˆ to the large noise bias on C` , as we now describe.

The statistical error has two sources: instrumental and astrophysical. The measured ˆXX XX XX,N auto-spectra in Eq. 3 contain a signal and noise contribution: C` = C` + C` . It is informative to split the right hand side of Eq. 3 into four pieces:

CIB CIB 2 ,N CIB,N CIB,N ,N CIB rhs = C` C` + C` + C` C` + C` C` + C` C` . (4)  ⇣ ⌘ Here the ﬁrst term is a signal-only piece, the second is a noise-only piece, and the remaining two terms are mixed signal and noise pieces. To discuss the relative importance of these terms, we will use for the signal terms the model spectra, and for the noise terms we subtract the model spectra from the measured spectra: CˆXX,N = CˆXX CXX. With this def- ` ` ` inition, the noise will include the instrumental contribution, as well as other astrophysical signals including the CMB, which we do not remove from our data for reasons previously mentioned. We show the di↵erent terms in Fig. 3.5. Up to 353 GHz the measured tem- ˆTT perature spectrum, C` , is dominated by the CMB at low ` and the instrumental noise at high `. At higher frequencies Galactic emission dominates at low ` and the CIB at high `. For all frequencies up to 353 GHz the dominant contribution to the errors in our signal comes from the noise-only term (in blue), which is proportional to the temperature noise ,N CIB spectrum. At 353 GHz and above the mixed signal-noise term C` C` (orange) becomes 3.5. STATISTICAL AND SYSTEMATIC ERROR BUDGET 111

T Figure 3.5 Naive analytical estimates of the contribution to the C` variance as a function of multipole and frequency as given in Eq. 4. We assume the same bin sizes as in Fig. 3.3. The CIB di↵erent lines are the contribution to the analytical error from the signal only: C` C` + CIB 2 ˆ,N ˆCIB,N C` (green), noise only: C` C` (blue), and the mixed signal and noise terms: ˆCIB,N ˆ,N CIB C⇣ ` C` ⌘ (yellow) and C` C` (orange). The total contribution is the solid black line, CIB 2 and the theory spectrum, C` , is the dashed line. ⇣ ⌘ important and is the largest contribution at 545 and 857 GHz at high `.

3.5.2 Instrumental and Observational Systematic Effects

A number of systematic errors a↵ect the Planck HFI analysis and we brieﬂy discuss some of them here. A more complete discussion can be found in [39]. We will illustrate how the very nature of our measurement, a 3-point function, makes it particularly robust to many systematic e↵ects, and we will check for their signatures using null tests. For example, there is no noise bias in the 3-point measurement, and many e↵ects that can lead to biases in the auto-spectrum of do not a↵ect us. 112 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

Potential Sources of Systematic Error

We begin by describing our knowledge of known systematic e↵ects, before discussing others that could bias our result. To account for an error in the calibration of the temperature maps, we simply add in quadrature a calibration uncertainty to our error bars. In Sect. 3.5.2 we use null tests to check that these errors are consistent with the data. In addition we use the null tests to search for evidence that the calibration has changed between surveys, for example due to gain drifts. We account for beam errors in a conservative manner by using a constant error at each frequency equal to the maximum error in the beam multipoles, B`, at any ` (see PIR for details). The B` uncertainty is larger at high-` with, for example, values at ` = 1500 of 79.5 %, 8.2 %, 0.53 %, 0.95 %, 0.31 %, and 0.70 %, at 100–857 GHz, respectively. The calibration error is therefore larger than the beam error at all ` between 217 and 857 GHz but smaller at high ` in the 100 and 143 GHz channels. We add the beam error in quadrature in an `- and frequency-dependent manner. As discussed in Planck Col- laboration XVII [26], uncertainties in the beam transfer (as well as the ﬁducial CMB power T spectrum C` T) propagate directly to a normalization uncertainty in the lens reconstruction. Based on the beam eigenmodes of [59], it is estimated in [26] that beam uncertainty leads to an e↵ective normalization uncertainty of approximately 0.2% and 143 and 217 GHz, and 0.8% at 100 GHz. To be conservative, on top of the calibration and beam error we will add in quadrature a 2 % uncertainty on the overall lens normalization.

CMB lens reconstruction recovers modes of the lensing potential through their anisotropic distorting e↵ect on small-scale hot and cold spots in the CMB. The quadratic estimator, which we use to reconstruct the lensing potential is optimized to measure the lensing induced statistical anisotropy in CMB maps. However, other sources of statistical anisotropy, such as the sky mask, inhomogeneous noise, and beam asymmetries, produce signals, which can potentially overlap with lensing. These introduce a “mean-ﬁeld” bias, which we estimate using Monte Carlo simulations and subtract from our lensing estimates. In- naccuracies in the simulation procedure will lead to errors in this correction, particularly if the correction is large. The mean-ﬁeld introduced by the application of a Galaxy and point-source mask, for example, which can be several orders above magnitude larger than the lensing signal at ` < 100. This is discussed further in Appendix B of [26]. The mask 3.5. STATISTICAL AND SYSTEMATIC ERROR BUDGET 113

mean-field is a particular concern for us because it has the same phases as the harmonic transform of the mask. If our masked CIB maps have a non-zero monopole, for example, it will correlate strongly with any error in the mask mean-field correction. For this reason we do not use any data below ` = 100 in our analysis. To summarize, we do not expect these known systematic e↵ects to be present at a significant level. Nevertheless, we still perform a set of consistency tests that would be sensitive to them or other unknown e↵ects.

Null Tests

The Planck scanning strategy, its multiple frequency bands and its numerous detectors per frequency, o↵er many opportunities to test the consistency of our signal (see Sect. 3.2.1). We focus on such tests in this section. Our aim is to reveal any systematic e↵ects that could lead to a spurious correlation. For all of the tests presented, we will quote a 2 value as well as the number of degrees of freedom (Ndof) as a measure of the consistency with the expected null result. Throughout this section, black error bars in plots will correspond to the measured scatter within an ` bin multiplied by 1.2, as was justified in Sect. 3.5.1 and Appendix 3 A, and will also include a CIB calibration error and a beam error, while the coloured boxes correspond to the analytical errors of the corresponding signal (i.e., not the di↵erence corresponding to the null test). Plotting these two error bars illustrates how important any deviation could be to our signal. Throughout this section, we will illustrate our findings with the 545 GHz correlation, since it is our prime band for this measurement, but our conclusions hold at other frequencies. The first test we conduct is to take the temperature di↵erence between the two half- ring (HR) maps to cancel any signal contribution, and therefore investigate the consistency of our measurements with our statistical errors on all time scales. We null the temperature maps and correlate with our nominal lensing map. The results are shown in the left panel of Fig. 3.6. We see a significant deviation from null only when considering survey di↵erences. This particular failure can probably be explained by apparent gain drifts due to nonlinearity in the analog-digital conversion [41, 60], not yet corrected at this frequency. Note however that the predicted variation is about 1% while the deviation from null would call for a variation of 1.5-2%. But in any case, its amplitude is too small to significantly a↵ect our 114 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

measurement.

We see a signiﬁcant deviation from null only when considering survey di↵erences. This particular failure can probably be explained by apparent gain drifts due to nonlinearity in the analog-digital conversion [41, 60], not yet corrected at these frequency. But in any case, its amplitude is too small to signiﬁcantly a↵ect our measurment.

The second test uses multiple detectors at a given frequency that occupy di↵erent parts of the focal plane. These detector sets are used to construct the “detset” maps that were described in Sect. 3.2.1. The two “detset” maps are subtracted and then correlated with our nominal lens reconstruction. This test is particularly sensitive to long term noise properties or gain variations, as we do not expect these to be correlated from detector to detector. Since this detector division breaks the focal plane symmetry, it is also a good check for beam asymmetry e↵ects. Here again, we do not ﬁnd any signiﬁcant deviation, as illustrated in the middle panel of Fig. 3.6.

The third test we conduct makes use of the redundant sky coverage, using multiple surveys to cancel the signal. As above, we null the temperature signal and correlate with the nominal lens reconstruction. This test is particularly sensitive to any long term, i.e., month timescale drifts that could a↵ect our measurement. It is also a good test for any beam asymmetry e↵ects, as individual pixels are observed with a di↵erent set of orientations in each survey. Since only the first two surveys are complete for this particular data release, we only use the two full survey maps to avoid complications with the partially completed third survey. Here again, we do not find any significant deviation, as illustrated in the right panel of Fig. 3.6.

To conclude, this ﬁrst set of stringent consistency tests have shown that there is no obvious contamination of our measurements due to instrumental e↵ects. In addition, the rea- 2 sonable /Ndof obtained gives us conﬁdence in our statistical noise evaluation. Although we measure the noise directly from the data, this success was not guaranteed. 3.5. STATISTICAL AND SYSTEMATIC ERROR BUDGET 115

Figure 3.6 Null tests at 545 GHz. Left: di↵erence spectra obtained by nulling the signal in the HR temperature map before correlating it with our nominal reconstruction. Middle: temperature signal nulled using di↵erent detectors at 545 GHz. Right: temperature signal nulled using the ﬁrst and second survey maps. The black error bars correspond to the scatter measured within an `-bin, while the coloured bands correspond to the analytical estimate. Except for the survey null test (see text for details), these tests are passed satisfactorily 2 except, as illustrated by the quoted and Ndof, thus strengthening conﬁdence in our signal.

3.5.3 Astrophysical Contamination

We now turn to possible astrophysical biases to our measurement. We will discuss suc- cessively known astrophysical contaminants that can either come from Galactic or extragalactic origin. Once again, besides our knowledge of these signals, we will rely heavily on consistency tests made possible by having multiple full sky frequency maps.

Galactic Foregrounds

Galactic foregrounds have two possible e↵ects on our measurement. The first is the introduction of an extra source of noise. The second is that contamination of the lensing reconstruction by any Galactic signal, e.g., synchrotron, free-free or dust, which could then correlate with foreground emission present in the temperature maps, remains a source of bias that has to be investigated. We will show that this bias is small. To do so, we take three approaches. We first investigate various Galactic masks, then perform the lensing reconstruction at various frequencies, and finally investigate the e↵ect of a dust-cleaning 116 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

Figure 3.7 Left: di↵erence between the cross-spectra measured using the 20 % Galactic mask (20 % is the unmasked sky fraction) from that measured with our default 40 % Galac- tic mask. Middle: spectra obtained when di↵erencing the 60 % and 40 % Galaxy mask measurements. For both left and middle panels and all Galactic masks, the same point source and Hi masks are used, which removes an additional fraction of the sky. Right: di↵erence between the cross-spectra calculated with the Hi cleaned temperature maps from those with no Hi cleaning. This cross-spectrum is thus the correlation between the Hi template and the reconstruction. The error bars are calculated in the same way as in Fig. 3.6. Again, the null tests are passed with an acceptable 2. procedure. First, we consider two additional masks, either more aggressive or more conservative than our fiducial one. Both were introduced in Sect. 3.2.1. The first one leaves approximately 20 % of the sky unmasked, while the second one leaves approximately 60 % of the sky. Given the strong dependence of Galactic foregrounds on Galactic latitude, any Galac- tic contamination should vary strongly when we switch between masks. Comparing the measurements using these masks with our fiducial 40 % mask in the left and centre panels of Fig. 3.7, we do not see any substantial deviation from our fiducial measurements. This excludes strong Galactic contamination of our results. Second, we perform the lens reconstruction at 100 and 217 GHz, di↵erent from the fiducial frequency of 143 GHz, and compare their correlation with the temperature maps. 3 Given the strong dependence of the Galactic emission with frequency, T ⌫ for syn- / chrotron and T ⌫2 for dust in this frequency range, any contamination of our signal / 3.5. STATISTICAL AND SYSTEMATIC ERROR BUDGET 117

Figure 3.8 Left: di↵erence between cross-spectra calculated using the lens reconstruction at 100 GHz with the nominal 143 GHz reconstruction. We see an overall shift, which leads to a high reduced 2. This shift can be explained by the expected overall normalization uncertainties of the 100 GHz and 143 GHz reconstructions. While this uncertainty is not included in the 2 or the solid bars, it is included later in our analysis in Sect. 3.6.1. Middle: same as the left panel, but the 217 GHz reconstruction is used instead of the 100 GHz reconstruction. Right: di↵erence between cross-spectra when we consider the 143 GHz lens reconstruction calculated with a less restrictive Galaxy mask (that excludes only 20 % of the sky) and the nominal reconstruction mask that excludes 40 % of the sky. would have a strong frequency dependence. The comparison with the 100 GHz (217 GHz) reconstruction is presented in the left (centre) panel of Fig. 3.8. The right panel shows the di↵erence of the cross-spectra calculated using the 143 GHz reconstruction with a more aggressive Galaxy mask (20 % instead of 40%), to reduce possible Galactic contaminants in the reconstruction, and the nominal reconstruction. The three di↵erences are consistent 2 with null as demonstrated by the quoted and Ndof. Third, we investigate more speciﬁcally how cirrus, the dominant Galactic contaminant for our higher frequency channels, a↵ects our measurements. We rely on the dust cleaning procedure detailed in Sect. 3.3.2 that aims to remove the Hi-correlated dust component. This procedure leads to a decrease in the variance measured outside the mask of 22, 65, 73 and 73 % in the 217, 353, 545 and 857 GHz maps, respectively. This frequency dependence is expected given the dust scaling. However, in Fig. 3.7, where we show the di↵erences between the cleaned and non-cleaned cross-spectra, we observe that the large scale Hi 118 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

cleaning, even though it makes a substantial impact on the power within our map, only makes a small change at low-` in the cross-spectrum, as well as reducing the noise at all multipoles. If we quantify the e↵ect of our “local” Hi cleaning on the detection significance level computed using only statistical errors, we find that the significance is increased by 4, 4, 28, and 36 % at 217, 353, 545 and 857 GHz, respectively. Also, not surprisingly, we observe that for frequencies up to 353 GHz where the statistical errors are dominated by the CMB, the Hi cleaning has almost no e↵ect on the cross-spectra. From the three studies in this section we conclude that there are no obvious signs of Galactic foreground contamination in our cross-correlation.

Point Source Contamination

We now discuss another well-known potential source of contamination, namely the contribution of unresolved point sources visible either through their radio or dust emission. Our concern is that a correlation between a spurious lens reconstruction caused by unresolved point sources can correlate with sources in the temperature map. Although in Sect. 3.5.3 our null test using lens reconstructions at di↵erent frequencies suggests that unresolved point sources are not an obvious contaminant, we will now perform a more detailed test designed specifically to search for point source contamination. Following [61, 62], we will construct a point source estimator designed to be more sensitive than the lensing estimator to point source contamination. Our focus here will be on possible contamination from the point source shot-noise bispectrum. In Sect. 3.5.3 we will discuss contamination from a scale dependent bispectrum. Our (unnormalized) quadratic estimator, which is designed to detect point source contributions is given by 143 2 143 2 ⇥¯ (ˆn) Y⇤ (ˆn)(⇥¯ (ˆn)) , (5) LM ⌘ LM LM ⇣ ⌘ X where ⇥¯ is the inverse-variance filtered sky map. This estimator is simply the square of the inverse-variance filtered sky map, which is a more sensitive probe of point sources than the standard lensing estimator. 2 In Fig. 3.9 we plot the cross-spectrum of ⇥¯ 143(ˆn) measured at 143 GHz and ⇥¯ ⌫ LM LM for the full set of HFI channels. This cross-spectrum⇣ ⌘ is probing the same point source 3.5. STATISTICAL AND SYSTEMATIC ERROR BUDGET 119

T contributions that could bias our estimates of C` , however with a greater signal-to-noise ratio.

There is one complication here, which is that just as lens reconstruction may be biased by point source contributions, the point source estimator is correspondingly biased by lensing. The bias to the plotted cross-spectra is given by

⌫ Mm m C G 1 2 143 2 ⌫ L L`1`2 m1 m2 M m1 m2 m1 M m2 (⇥¯ (ˆn)) ⇥¯ ⇤ = ( 1) I C˜` + ( 1) I C˜` , LM LM C⌫⌫ CtotCtot `2L`1 1 `1L`2 2 L `1m1 `2m2 `1 `2 D E X X ⇣ ⌘ (6)

m1m2m3 tot where G = dnYˆ ` (ˆn)Y` (ˆn)Y` (ˆn), C˜` is the unlensed CMB spectrum, C is `1`2`3 1m1 2m2 3m3 ` the spectrum of ⇥143(ˆn), Im1m2m3 is as defined in Okamoto and Hu [2], and we have used R `1`2`3 ¯ tot the fact that for our inverse-variance filtering, ⇥`m ⇥`m/C` . We have calculated this T ⇡ contribution using our measured CL and subtracted it from the data points of Fig. 3.9. We can consider the e↵ect of shot noise on this cross-spectrum. With the shot-noise bispectrum defined by

⌫ m1m2m3 3 ⇥` m ⇥` m ⇥ = G S (7) 1 1 2 2 `3m3 S 3 `1`2`3 D E D E the bias to the plotted cross-spectrum is given as

M Mm1m2 Mm1m2 ( 1) G` ` G ` ` (⇥¯ 143(ˆn))2 ⇥¯ ⌫ = S 3 L 1 2 L 1 2 . (8) LM LM S 3 C⌫⌫ CtotCtot L ` m ` m `1 `2 D E D E X1 1 X2 2 This bias is plotted for best-fit values of S 3 as the black lines in Fig. 3.9. To minimize h i systematic e↵ects that might bias the S 3 estimator, we have estimated S 3 from the spectra of Fig. 3.9 between multipoles betweenD È = 500 and 2000. The fitted S 3 amplitudes are given in Table 3.1. D E These amplitudes match our expectations, for example see Planck Collaboration XIII [63]. We observe a decrease in the amplitude of the point source contribution going from 100 to 217 GHz, which corresponds to a dominant contribution from radio point sources. We do not see any evidence of a dusty galaxy contribution to the shot-noise bias. These conclusions have been verified using less restrictive point source masks that cover fewer 120 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

Figure 3.9 Results from the point source contamination estimator of Eq. (5). The best-ﬁt cross-spectra associated with shot noise are plotted in black. We do not show the best-ﬁt at 545 and 857 GHz since the signal-to-noise ratio is low. The grey line is a prediction for the bias from the CMB lensing - infrared correlation, and has been subtracted from the spectra (plotted as black points). We see that with the subtraction of the bias from CMB lensing, the measured bispectrum-related spectrum is generally consistent either with zero, or with the shape expected for shot noise.

sources.

3 T With estimates of S in hand, we estimate a corresponding bias to C` , given by

S 3 Wm1m2 M ⌫ M `1`2L m1m2 M ˆ LM⇥¯ ⇤ = ( 1) G (9) LM S 3 C⌫⌫ CtotCtot `1`2L D L E ` m ` m `1 `2 D E X1 1 X2 2 3.5. STATISTICAL AND SYSTEMATIC ERROR BUDGET 121

m1m2 M m m2 Mm1 m1 M m2 where W = ( 1) 2 (I C˜` + I C˜` )/2 with deﬁned in Planck Col- `1`2L `2L`1 1 `1L`2 2 RL RL laboration XVII [26]. We show this contribution later in Fig. 3.11 as the dotted line. While non-zero, we see that the point source shot noise contribution is always negligible in the ` range we consider, except at lower frequencies where the radio point sources are important (but still not strong enough to lead to any clear signal in the cross-spectra).

SZ Contamination

A fraction of CMB photons travelling from the surface of recombination are scattered by hot electrons in galaxy clusters. In the most massive clusters approximately 1 % of CMB photons passing through them get scattered. On average, their energy will be increased, which leads to a measurable spectral distortion. This is the so-called thermal Sunyaev- Zeldovich (SZ) e↵ect [65]. At the location of a galaxy cluster the CMB appears colder at frequencies below about 220 GHz and hotter at higher frequencies, with a temperature change proportional to the cluster optical depth to Compton scattering and to the electron temperature. Since hot electrons in clusters also trace the large scale matter potential that is traced by CMB lensing, we expect an SZ-induced contamination in our measurement at some level. We will show below that the level of contamination is negligible. In these calculations we ignore the small relativistic corrections to the thermal SZ spectrum [e.g., 66]. We also ignore the kinetic SZ signal coming from the bulk motion of hot electrons in clusters, since it is subdominant to the thermal signal [18, 67, 68].

Table 3.1 Point source estimator. The measured quantity S 3 , as deﬁned in Eq. 9, is given as a function of frequency. D E

Frequency [ GHz] S 3 [ 109µK3] (No. of ) ⇥ 100 ...... D 11E .7 5.8 (2.0) ± 143 ...... 4.3 1.8 (2.3) ± 217 ...... 3.7 1.6 (2.2) ± 353 ...... 6.1 3.9 (1.6) ± 545 ...... 79 39 ( 2.0) ± 857 ...... ( 1.9 2.1) 103 ( 0.9) ± ⇥ 122 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

Figure 3.10 Frequency spectrum of our cross-spectra averaged within `-bins (black points with associated error bars). The light shaded regions correspond to the HFI frequency bands. The solid black curve corresponds to the best-fit CIB assuming a [64] spectrum, while the dot-dashed line assumes a [9] spectrum. The dashed black line corresponds to the best-fit model when allowing for an SZ component in addition to the [64] CIB shape. The blue dots correspond to the associated absolute value of the best-fit SZ component. We conclude from this plot that the SZ e↵ect is not an important contaminant.

The frequency dependence of the SZ signal in our map depends on the detector band- passes and is d⌫ h(⌫) g(⌫) f (⌫) = , (10) R d⌫ h(⌫) where h(⌫) is the detector bandpass and g(⌫)R is the SZ frequency dependence, which in the non-relativistic limit is g(⌫) = x (ex + 1)/(ex 1) 4, with x = h⌫/k T . The e↵ect B CMB of the bandpass only makes a large di↵erence at 217 GHz near the null of the SZ signal. The thermal SZ a↵ects our measurement in two ways. First, since the SZ emission in our maps is not a Gaussian random ﬁeld [e.g., 69] it introduces a spurious signal into our lens reconstruction that will correlate with the SZ signal in our CIB map. As shown in Osborne 3.5. STATISTICAL AND SYSTEMATIC ERROR BUDGET 123

et al. [62], this is well approximated by a Poisson noise term and is thus already addressed by our treatment of point sources in Sect. 3.5.3. The spurious lensing signal will also correlate with other components in our map such as the CIB. However, we ignore these terms since they will be smaller than those that correlate directly with the SZ emission. Additionally, a contribution comes from SZ emission in our CIB map that correlates with the lensing potential itself. The latter is the dominant term and we discuss it in this section. To measure a contribution from the SZ-lensing correlation we attempt to separate the SZ and CIB emission based on their di↵ering spectral shapes. We consider all frequencies from 100 to 857 GHz, but we will illustrate this procedure by considering only two ` bands: ` = 300–450; and 1200–1450. The first is well inside the linear regime, while the second receives a more important non-linear contribution. However, we have checked that if we consider di↵erent `-bins we obtain similar conclusions. We model the signal within each ` band as s`(⌫) = a1,`c(⌫) + a2,` f (⌫), where c(⌫) and f (⌫) are, respectively, the CIB frequency dependence (as proposed in Fixsen et al. 9 or Gispert et al. 64) and the SZ frequency dependence obtained from Eq. 10. For each ` band, we will solve for a1,` and a2,` minimizing the associated 2 while forcing both amplitudes to be positive. As an approximation to the error in each multipole band we calculate the scatter of the signal within the band and multiply it by 1.2, as discussed in Sect. 3.5.1. In Fig. 3.10 we show the measured frequency spectrum within each ` band, along with the best-fit SZ-lensing and CIB-lensing spectra. For the CIB-only fit with the [64] frequency dependence we find a relatively poor fit in the lowest `-bin, 2 (dof) = 15.5 (5), but an improved fit in the higher `-bin, 2 (dof) = 4.15 (5). Including the SZ component gives 2 = 0.52 and 1.34 in the low and high ` bins for one extra degree of freedom. When we use the [9] frequency dependence we find an improved fit, with 2 (dof) = 2.25 (5) and 5.49 (5) in the low and high-`-bins, respectively. Overall, the improvement in the 2/dof when including the SZ component does not justify inclusion of the SZ component in the model, with the poor fit driven by the lowest frequency bands where the CIB scaling is rather unconstrained. In fact, our measurements might constitute the first constraints to date on this scaling. From these results we conclude that including the SZ-lensing correlation in our data does not improve the fit in the ` range of interest to us and thus we do not consider it necessary to correct for. 124 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

As an extra validation of this result, we now verify its consistency with current models of the CIB and SZ emission. For this purpose, we use the calculation of the correlation from Osborne et al. [62], based on Babich and Pierpaoli [70], which models the SZ emission as a statistically isotropic signal modulated by a biased density contrast, where the bias depends on the cluster mass and redshift. To obtain an estimate of the contribution to the cross-spectrum at 217–857 GHz we assume that the measured cross-spectrum at 143 GHz is entirely due to thermal SZ emission (note that we do this to find what we believe to be an upper limit on the SZ contribution at 217–857 GHz; for the reasons stated above we do not expect the 143 GHz correlation to be due to SZ). Since the SZ signal at 143 GHz gives a decrement in the CMB, and the CIB emission gives an enhanced signal, it is possible that this approach could still underestimate the SZ signal. We find that in order to fit the cross-spectrum at 143 GHz using only the SZ-lensing correlation requires an amplitude of (2.4 1.6) times our calculated SZ-lensing cross-spectrum. In Fig. 3.11 the dashed line ± shows the magnitude of this SZ signal scaled to each frequency using Eq. 10. The small contribution it makes at 217–857 GHz further suggests that we can neglect this component. At 217 GHz the signal is negative, while at higher frequencies it is positive.

ISW Contamination

The Integrated Sachs-Wolfe (ISW) e↵ect describes the redshifting (blueshifting) of photons travelling through gravitational potential wells (hills) that decay as the photons travel through them [71]. The induced modulation of the CMB mean by the gravitational potential generates CMB fluctuations that correlate with the lensing potential, which also traces out the gravitational potential perturbations [72–74]. Note that because the mean of the CIB is relatively much smaller than its fluctuation, the ISW induced CIB fluctuations make a negligible change to total CIB anisotropy. The CMB ISW induced signal has the same frequency dependence as the CMB and so is only a significant contaminant for us at low frequencies. We evaluate this signal using a theoretical calculation performed in CAMB [58]. The results are shown as the solid line in Fig. 3.11. It is a negligible contribution at all frequencies, except in the lowest `-bin of the lowest frequencies, where the measured cross-spectrum is consistent with zero. 3.5. STATISTICAL AND SYSTEMATIC ERROR BUDGET 125

Figure 3.11 Foreground components at each frequency. The data points and error bars show our results. The dashed line is an estimated upper limit on the magnitude of the SZ contamination derived in Sect. 3.5.3. We show the absolute value of this contribution, which is negative at frequencies less than 217 GHz. The dot-dashed line is the extragalactic point source contribution, with an amplitude measured from our data as derived in Sect. 3.5.3. Again we show the absolute value, with the signal being negative below ` 1200. The ⇠ oscillating solid line corresponds to the calculated ISW contamination.

CIB Bispectrum

Having calculated the bias from the point source shot noise in Sect. 3.5.3, we now discuss a more complicated form of the unresolved point source 3-point function that could be present in our data, namely the clustering contribution. While unknown (although the ﬁrst detection was recently reported in Crawford et al. [75]), the CIB bispectrum is potentially a direct contaminant to our measurement. Because of the non-linear clustering of DSFGs 126 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

Figure 3.12 Cross-spectrum of the 545 GHz lens reconstruction correlated with the 545 GHz temperature map with di↵erent Galactic masks. The legend gives the visible sky fractions. The solid line represents the analytic unclustered shot-noise contribution ﬁt to the fsky = 0.09 points above ` = 1300.

(PER), it has to exist. But because of the very large redshift kernel that characterizes the CIB, this non-Gaussian e↵ect will be washed out, reducing its importance. Nevertheless, we ought to study it carefully. If important, this e↵ect would show up as a departure of the data from the best-fit curve in Fig. 3.9, since the best-fit model that we used assumes only a Poissonian shot-noise contribution. We do not see any significant deviation in Fig. 3.9. Still, in order to isolate this e↵ect we create cross-spectra with increased sensitivity to the clustered point source signal. We do this by calculating the lens reconstruction at 100 GHz and 545 GHz, where, respectively, the radio and dusty point source contribution is stronger. The 545 GHz map has a much larger Galactic dust signal than our nominal 143 GHz map. However, unlike in our fiducial estimates, here we do not attempt to project out dust contamination from the map used to perform our lens reconstruction as this would also remove some of the CIB signal in the bispectrum. As was found in Sect. 3.5.3 the cross-correlation between the 100 GHz 3.5. STATISTICAL AND SYSTEMATIC ERROR BUDGET 127

reconstruction and the 100 GHz temperature map does not show any large di↵erence with the cross-spectrum obtained using the 143 GHz signal. We are thus not sensitive enough to detect a bias from the clustering of radio sources using this method. However, we do detect a strong cross-correlation between the raw 545 GHz lens reconstruction and the 545 GHz temperature map. This cross-spectrum is shown in Fig. 3.12 for three di↵erent Galaxy masks. The line shows the point source shot-noise template derived in Sect. 3.5.3, ﬁt to the cross-spectrum with the 10 % Galaxy mask at ` above 1300. If the signal were entirely due to extragalactic point sources, then the signal would be independent of masking, and we do see a convergence of the signal at high ` as the size of the Galactic mask is increased. At low `, however, there is a large Galactic contribution and the convergence with the reduced mask size is less clear. We thus conclude that a strong contribution from Galactic dust is present in this measurement at all `. We do not attempt to calculate accurately the shape of the clustering contribution to the CIB bispectrum here, since it is beyond the scope of this work, even though a simple prescription for it has recently been proposed in [76]. To separate the Galactic from non- Galactic contributions in our bispectrum measurement is dicult, even if a strong Galactic signal is clearly present, given the strong dependence of the signal on variations of the Galactic mask in Fig. 3.12. However, the combination of dust cleaning that we perform in our nominal pipeline, coupled with the fact that our nominal pipeline uses the 143 GHz map for lens reconstruction, means that we do not observe any dependence with masking in our measurement, as seen in Fig. 3.7. Because of this, the CIB bispectrum is unlikely to be a large contribution to our measurement. Furthermore, even if we were to assume that all of the signal seen in Fig. 3.12 was extragalactic in nature, using the [64] frequency scaling for the CIB (also appropriate for the Galactic dust in fact, Planck Collaboration XXIV 47), the roughly 1700 µK.sr observed at ` = 400 for the 40 % Galactic mask would only lead to a 0.02 µK.sr signal in Fig. 3.11, which is an order of magnitude smaller than our measured signal. To conclude, although our analysis does not lead to a clean measurement of the CIB bispectrum, we can safely assume that it is not a contaminant to our measurement. 128 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS 1.56 10.01 0.39 2.05 0.65 0.15 1.84 3.16 2.63 2.96 2.61 0.39 2.21 1.13 K.sr. The extragalactic foreground contribu- µ 1.72 0.59 1.16 0.04 0.10 0.41 0.960.07 1.83 0.17 3.09 0.68 4.83 7.10 1.59 3.02 5.10 7.97 11.72 0.69 0.14 0.33 1.31 3.050.05 5.79 0.12 9.78 15.29 0.48 22.48 1.13 2.14 3.61 5.65 8.31 0.09 0.15 0.64 0.28 0.60 0.18 0.10 0.08 0.13 0.37 0.25 0.09 0.09 0.05 0.46 0.16 0.06 0.03 0.04 0.08 0.03 . Both statistical and systematic errors are given (see Sect. 3.5 for details). 0.01 0.01 0.03 0.01 1.88 1.870.69 1.75 0.70 1.81 0.53 1.79 0.522.09 1.81 0.48 1.972.95 0.41 1.96 1.91 2.96 0.38 2.13 1.87 2.38 1.94 0.50 2.34 2.051.40 2.00 2.18 0.48 2.55 1.271.26 1.87 2.12 0.33 2.78 1.21 1.46 1.57 2.31 0.35 3.07 1.14 1.13 1.14 1.98 3.44 2.44 0.32 0.89 1.14 0.92 0.37 1.94 2.60 0.83 1.18 1.36 2.73 0.75 1.25 1.36 2.87 1.01 3.04 1.30 1.30 0.88 0.96 1.36 0.46 1.39 0.60 1.43 1.48 0.64 0.45 ` 21.47 21.79 16.56 16.08 14.83 12.76 11.7628.92 15.60 29.05 14.98 23.38 10.44 20.12 21.37 11.33 18.32 10.67 15.38 12.76 12.34 19.36 14.32 18.78 11.08 12.94 8.73 12.67 9.00 11.70 8.19 7.13 7.37 9.85 9.35 4.42 5.75 5.99 4.09 C 0.01 ······ ··· ··· ······ ··· ··· ······ ··· ··· ······ ··· ······ ······ ··· ··· ··· ······ ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· 163 290 417 543 670 797 923 1050 1177 1303 1430 1557 1683 1810 1937 10000 1.0710000 1.90 -0.00 -0.00 0.681000 0.791000 0.461000 0.4110000 0.24100 0.31100 0.14100 0.441 0.651 0.391 0.101 0.291 0.52 1000 1.321000 5.44 2.4810000 0.26 2.15 0.2810000 0.54 1.30 1.14 -0.00 0.05 1.331000 0.12 1.17 2.051000 0.09 6.28 1.27 2.4910000 1.11 0.07 2.16 0.42 1.1710000 1.37 1.30 1.30 0.13 -0.00 1.12 -0.00 0.23 0.09 1.321000 0.14 1.29 0.28 0.05 1.15 5.081000 0.37 0.02 1.51 8.99 1.23 2.49 3.19 0.10 0.01 2.17 1.07 1.97 3.75 1.31 0.12 0.62 0.94 2.66 2.15 1.34 0.04 0.50 0.98 3.78 1.91 1.16 0.24 0.34 0.93 1.15 5.69 1.24 0.09 0.65 1.07 8.93 0.99 1.47 1.63 0.43 0.93 0.99 0.66 0.19 0.97 1.05 2.11 0.21 0.90 1.22 3.15 0.20 1.48 0.95 2.04 0.25 0.90 0.89 0.94 1.02 ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ mean ` ` ` ` ` ` ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ⌘ ` C C C C C C fore fore fore fore fore fore ` ` ` ` ` ` sys sys sys sys sys sys ` ` ` ` ` ` stat stat stat stat stat stat ` ` ` ` ` ` 3 3 3 3 3 3 ` ` ` ` ` ` C C C C C C C C C C C C C C C C C C 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 ` ` ` ` ` ` has been removed from ` ` ` ` ` ` ` ` ` ` ` ` ⇣ ⇣ ⇣ ⇣ ⇣ ⇣ ⇣ ⇣ ⇣ ⇣ ⇣ ⇣ fore ` C 857 GHz 100 GHz 143 GHz 217 GHz 353 GHz 545 GHz Table 3.2 Cross-spectrum detection band-powers.tion, All values are in units of 3.6. INTERPRETATION AND DISCUSSION 129

3.5.4 Final Statistical and Systematic Error Budget

Throughout the suite of tests for instrumental and observational systematic errors presented in Sect. 3.5.2, as well as the suite of tests for possible astrophysical contaminants presented in Sect. 3.5.3, we have established the robustness of our measurement. The fact that our consistency tests do not lead to any significant deviation gives us confidence in our error budget. As described in Sect. 3.5.2 we add to them an overall calibration uncertainty, beam uncertainty, and lens normalization uncertainty, consistent with the Planck data processing paper [39]. We gather the measured band-powers in Table 3.2, along with our statistical and systematic errors. These band-powers have been corrected for the point source component measured in Sect. 3.5.3, whose amplitude is also given in Table 3.2. Once all systematic e↵ects are factored in, we claim a detection significance of 3.6 (3.5), 4.3 (4.2), 8.3 (7.9), 31 (24), 42 (19), and 32 (16) statistical (statistical and systematic) at 100, 143, 217, 353, 545 and 857 GHz, respectively.

3.6 Interpretation and Discussion

The correlation we have investigated leads to a very strong signal at most HFI frequencies. After a thorough examination of possible instrumental and astrophysical origins, we interpret it as originating from the spatial correlation between the sources of the CIB and the matter responsible for the gravitational deflection of CMB photons. In this section, we build on this result and interpret the measurement using both angular and frequency information. Before doing so, we highlight the spectral information contained in the signal. It was shown in PER that both the frequency spectrum of the CIB mean and fluctuation rms are well approximated by the two modified blackbody spectra proposed by [9] and [64], with a slight preference for the latter. We expect our measurements to follow the same spectral energy distribution (SED). Following the procedure outlined in Sect. 3.5.3, we plot in Fig. 3.10 the best-fit CIB component with either a [9] – dot-dashed black line – or [64] – solid black line – SED. We do so for two `-bins. We can see that, indeed, for a given `-bin, our measurements qualitatively follow the expected CIB spectrum. Unlike PER, we 130 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

do not ﬁnd a preference for the [64] shape in our low `-bin and only a slight one in the high ` bin. It is worth emphasizing that by carrying out a cross-correlation measurement we can obtain constraints at the lowest frequency, which is usually heavily contaminated by the CMB (see PIR for discussion). This is particularly interesting, because these measurements are simultaneously the most sensitive to high-z star formation processes and the most discrepant with either of the SEDs, i.e., they are both systematically low by about 0.5 . The models presented in this section will allow us to use both the spectral dependence and the relative amplitudes of the `-bins that was lost in Fig. 3.10. We now describe the general methodology we will use, before describing our models in detail.

3.6.1 Model Comparison Methodology

For the purpose of model ﬁtting, we will utilize both the CIB-lensing cross-spectra measured in this paper and the CIB auto-spectra obtained from PIR. We use the CIB-lensing cross-spectra for two purposes: to improve constraints on the model parameters; and to provide a consistency test of models ﬁt to the CIB auto- and frequency cross-spectra alone. As will be seen in PIR, the cross-spectra of the CIB at di↵erent frequencies provide powerful constraints on the CIB emissivity. We use the log-likelihood,

1 1 ln (p) = Cˆi C˜i (p) Nij Cˆ j C˜ j(p) † , (11) L ` ` ` ` ` 2 ` ij X X h i⇣ ⌘ h i where Cˆ and C˜ are the data and theory spectra with parameters p, the i and j indices denote the type of spectra (e.g., 100 GHz or 100 GHz 100 GHz), and N is the covariance ⇥ ⇥ matrix that includes both statistical and systematic errors. We make the approximation that the covariance matrix is diagonal, i.e., we treat the errors for di↵erent bins of each auto- and cross-spectrum as being uncorrelated. The small ( 2 %) mask-induced mode- ' coupling between neighbouring bins supports this approximation. However, calibration and beam errors (which are correlated between the auto- and cross-spectra at a given frequency), as well as the lens normalization error (which is also correlated across spectra) are not accounted for in this approximation. In addition, the lens reconstruction has some 3.6. INTERPRETATION AND DISCUSSION 131

sensitivity to all modes of the temperature maps, and so di↵erent modes are correlated to some degree. We also neglect the fact that the contribution to the error from the CIB signal itself (the orange line in Fig. 3.5) is also substantially correlated from frequency to frequency. However, our evaluation using simulations suggests that these e↵ects are too small to signiﬁcantly a↵ect our procedure. We thus resort to simply adding the beam, calibration and normalization uncertainties in quadrature to the statistical errors. The posterior probability distributions of model parameters are determined using now standard Markov Chain Monte Carlo techniques [e.g., 77, 78].

3.6.2 Two Modelling Approaches

The strength of the correlation signal should come as no surprise, given our current knowledge of CMB lensing and the CIB. The PER model predicts a high correlation between the CIB and the lensing potential. As clearly illustrated in Fig. 3.1, the broad overlap of the redshift distributions of the CIB with the lensing kernel peaking at z 2–3 leads to a cor- ⇡ relation of 60–80 %. This is comparable to our measurements at all of the HFI frequencies, as illustrated in Fig. 3.13, In models of the cross-correlation, the underlying properties we can probe come from a combination of the lensing potential and the characteristics of the DSFGs, in particular their emissivity and clustering properties. Mostly driven by linear physics, the former is well understood theoretically, as confirmed by recent observations [21, 24, 25, 79]. Assum- ing the currently favored ⇤CDM cosmology, we can consider it to be known to better than 10 % in the multipole range of interest to us, an uncertainty dominated by the uncertainty in the normalization of the primordial power spectrum. Given that this is much smaller than the theoretical uncertainties related to DSFGs, we will fix the cosmology to the currently favoured Planck alone flat ⇤CDM model in [57], and will focus our analysis on the modelling of the DSFGs. At a given redshift we model the fluctuations in the mean CIB emission, ¯j, as being proportional to the fluctuations in the number of galaxies, ng [80],

ng j ¯j . (12) / ng 132 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

Figure 3.13 Cross-correlation coecients calculated from the model spectrum and best- ﬁt halo model at each frequency. The CIB is a spectacular tracer of CMB lensing, and vice-versa. The data points represent the measured cross-correlation divided by the best-ﬁt auto power spectra models at 545 GHz.

With this hypothesis, the goal of the CIB modelling becomes twofold: ﬁrst, to better understand the statistical properties of the dusty galaxy number density, ng; and second, to reconstruct the mean emissivity of the CIB as a function of redshift. In this paper we will use two di↵erent models of the CIB emission. The ﬁrst model (described in Sect. 3.6.2 and inspired by Hall et al. 15) uses a single bias parameter at all frequencies with the mean CIB emissivity modelled as a two parameter Gaussian. This model is not designed to be physically realistic, and furthermore we will marginalize over an arbitrary amplitude in this case. Nevertheless, we present this simple model to show that our measurements are quite consistent with broad expectations of the CIB. The second model, described in Sect. 3.6.2, is a natural extension of the Halo Occupation Density (HOD) approach used in PER [see also 81, and references therein]. But unlike the results obtained in PER we now use a single HOD to describe the spectra at all frequencies. This is possible by allowing for deviations from the [82] model (hereafter B11) that was used to 3.6. INTERPRETATION AND DISCUSSION 133

Figure 3.14 Marginalized 2-D distributions of zc and z for the linear bias model, ﬁt to all frequencies simultaneously. The orange dots indicate the parameter values at the minimum 2.

ﬁx the emissivity.

Linear Bias Model

As a ﬁrst pass at interpreting our measurement, we will consider a redshift independent linear bias model with a simple parametric SED. This model was found to provide a reasonable ﬁt to the auto-spectra in the linear regime in PER. Throughout this paper we use the Limber approximation, and in this section, since we are using a linear model, we write 134 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

the relevant angular spectra as

XY ⇤ X Y C` = d W ()W () P(k = `/, ), (13) Z0 where X and Y are either the CIB at frequency ⌫ or the lensing potential , the integral is over , the comoving distance along the line of sight, is the comoving distance to the ⇤ X last scattering surface, P(k, ) is the matter power spectrum at distance , and the W functions are the redshift weights for each of the signals X:

a¯j () W⌫() = b ⌫ ; (14) 3 W () = ⌦ H2 ⇤ . `2 m 0 a ⇤ ! Here b is the DSFG bias that we assume to be redshift independent, a is the scale factor,

¯j⌫() is the mean CIB emissivity at frequency ⌫, as deﬁned in PER, ⌦m is the matter density today in critical density units and H0 is the Hubble parameter today. We use the Hall et al. [15] model for the CIB kernel, which treats the CIB emissivity as a Gaussian in redshift:

¯j () a 2 exp (z z )2/22 f (15) ⌫ / c z ⌫(1+z) h i where we use a modiﬁed blackbody frequency dependence

f ⌫ B (T ). (16) ⌫(1+z) / ⌫ d

We ﬁx the dust temperature to Td = 34 K, the spectral index to = 2 [83], and assume a constant bias b. We include a single normalization parameter for j, which we marginalize over. Since the normalization and bias parameters are degenerate in Eq. 13, if we were to only use the measured auto- and cross-spectra this approach would be equivalent to marginalizing over a frequency independent bias parameter. However, we will further constrain our model using the FIRAS data, which breaks this degeneracy. We constrain the zc and z parameters at each frequency, giving us a total of 13 free parameters.

For 217–857 GHz, we use the FIRAS measurements of the CIB mean intensity from 3.6. INTERPRETATION AND DISCUSSION 135

[84] as an additional constraint to our model. The mean intensity is simply

⇤ I⌫ = d a¯j⌫() . (17) Z0 Using this equation and the measured FIRAS mean and uncertainty we calculate a 2 value and add this to the 2 in Eq. 11. Since there are no FIRAS constraints at 100 and 143 GHz, as well as no CIB auto-spectra measurements, and noisier cross-spectra measurements at these frequencies, our constraints for the 100 and 143 GHz redshift parameters are weaker than for the other parameters. The linear bias model considers only linear clustering, and so when fitting the auto- spectra we restrict ourselves to ` < 500, where non-linear contributions are negligible. Because we do not consider the high-` modes, we also neglect the shot-noise contribution to the auto-spectra. The best-fit model is shown as the coloured dashed lines in Fig. 3.15, with 2 values of 13.4, 16.8, 25.2, 21.8, 9.1, and 9.4 if we break up the 2 contribution per 2 frequency from 100 to 857 GHz, leading to an overall of 95.7 for Ndof = 59. We see that the model captures some features of the data, but we also have evidence it is significantly missing some as well. This is perhaps not surprising given the simplicity of the model. The two-dimensional marginal distributions of the zc, z parameters are shown in Fig. 3.14. Although we allowed for these parameters to be frequency dependent we note that the point zc = 1 and z = 2.2 is in a region of high probability at all frequencies, and gives a redshift distribution for the emissivity density roughly consistent with our expectations, rising toward z = 1 due to the 2 term and then only slowly falling o↵ toward higher redshifts. Althought it is useful to see the extent to which such a simple model can explain our data, we now turn to make a stronger connection between the properties of the infrared light and the distribution of the underlying dark matter applicable into the non-linear regime.

An Extended Halo Model Based Analysis

In this section we use a description of the CIB motivated by the halo model, which has been used successfully to describe the transition between the linear and non-linear clustering regimes for optical galaxies. We use the halo model to attempt to reconstruct the CIB 136 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

Figure 3.15 Measured cross-spectra with the best-fit j reconstruction model fit to both the CIB auto- and CIB-lensing cross-spectra (solid coloured), and the best-fit linear bias model (dashed coloured). The 2 values quoted in each panel are the contribution to the global 2 from the data in the panel for the halo model, and loosely indicate the goodness of fit (see text for details). The one and two-halo contributions are shown as the dashed and solid black lines, respectively. A light dashed black horizontal line is indicating the zero level. emissivity as a function of redshift. This is an extension of the approach taken in PER, where the modelled CIB emissivity at high redshift was treated as a single bin with the amplitude constrained by the data. The goal of this approach is to isolate the high-redshift contribution to the CIB, which is dicult to probe using observations of individual galaxies, due to their low brightness. The power of such an approach is further demonstrated in PIR. We replace the linear bias used in Sect. 3.6.2 with a halo model and an HOD prescription that assigns galaxies to host dark matter halos (see PER for references and definitions). 3.6. INTERPRETATION AND DISCUSSION 137

Figure 3.16 PIR auto-spectra with the best-fit mean emissivity j reconstruction model fit for the CIB auto and CIB-lensing cross-spectra (solid coloured). The 2 values are defined as in Fig. 3.15. The one and two-halo contributions are shown as the dashed and solid black lines, respectively, while shot noise is the dot-dashed black line.

It allows a consistent description of the linear and non-linear part of the galaxy power spectrum and its redshift evolution. Because it is built on the clustering of dark matter halos, the halo model allows us to describe the clustering of DSFGs and the gravitational lensing caused by the halos in a consistent way. However, it is important to realise that the HOD prescription was developed to describe stellar mass within dark matter halos – an application for which it has been thoroughly tested – while here we are applying it to star formation within halos. The accuracy of this approach needs to be further quantiﬁed. However, it provides a good phenomenological description of our data as well as other CIB measurements, but also of other astrophysical probes of the relation between dark matter and light (e.g., Leauthaud et al. [85], Hikage et al. [86]). Unlike the model presented in PER we use a single HOD to describe our data at all frequencies. This is made possible by allowing for a deviation from the B11 emissivity 138 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

model. Note however that we will still consider the CIB emissivity to depend only on redshift and not on the galaxy host halo mass, a simpliﬁcation highlighted in Shang et al. [87] that will be relaxed in the PIR model. The emissivity of the CIB is inhomogeneous, due to spatial variations in the number density of galaxies:

j⌫ ng (nˆ, z) = (nˆ, z) g(nˆ, z). (18) ¯j⌫ n¯g ⌘

Here j(nˆ, z) is the CIB emissivity at redshift z with mean ¯j(z), ng(nˆ, z) is the number density of DSFGs with meann ¯g(z), and g(nˆ, z) is the DSFG overdensity, with power spectrum 3 g(k, z) g(k0, z)⇤ = (2⇡) (k k0) Pgg(k, z). We calculate this power spectrum, including Dthe constituent 1E and 2-halo terms, using the procedure described in Appendix C of PER, with the constraint ↵sat = 1, a theoretically favoured value [28]. We remove the relationship between Msat, a characteristic satellite mass scale, and Mmin, the halo mass at which a halo has a 50 % probability of containing a central galaxy that was enforced in PER (i.e.,

Msat = 3.3Mmin), and allow both Msat and Mmin to vary independently.

At redshift z < 1 we fix the emissivity to the B11 value, but at higher redshift we assume that the emissivity is constant within z-bins and solve for the amplitude of the bins. Two factors a↵ect the number of bins that we choose. The auto-spectra have a ¯j2 dependence, and so if the true value of ¯j has a strong z dependence within a bin then the best-fit emissivity in the bin will be dicult to interpret. The best-fit bin values could be significantly di↵erent from those that would be obtained by binning the true emissivity. However, as more bins are used and the number of parameters increases, it becomes more dicult to determine the best-fit parameters and the parameters will be highly correlated. After investigation using simulations, we found that three bins was a good compromise, given the expected slow redshift evolution. The bins are defined by: 1 < z 1.5; 1.5 < z   3; and 3 < z 7. As in Sect. 3.6.2 we use the FIRAS results at 217–857 GHz to add an  integral constraint on the emissivity. The CIB auto and lensing cross-spectra are [7]:

⌫⌫0 ⌫ ⌫0 C` = d W ()W () Pgg(k = `/, ); Z (19) ⌫ ⌫ C` = d W ()W () Pg(k = `/, ) . Z 3.6. INTERPRETATION AND DISCUSSION 139

Figure 3.17 Marginalized 2-D distribution of log10 (Mmin/M ) and log10 (Msat/M ) for our overall HOD model when the CIB-lensing cross-spectra are combined with the CIB auto- spectra and FIRAS measurements. The orange dot shows the best-ﬁt value. The contours correspond to 68%, 95% and 99.7% conﬁdence intervals.

Since we ﬁx ¯j at z < 1, the model spectra consist of a low redshift part that is independent ¯ ¯ of the emissivity parameters, and a contribution from z > 1 that is proportional to j⌫ j⌫0 for the auto-spectra and ¯j⌫ for the lensing cross-spectra.

Overall, the halo-based model contains two halo parameters that describe the galaxy clustering and are independent of frequency, and three j amplitudes at each frequency, giving a total of 20 parameters for the six frequencies of interest to us. The auto and cross- spectra have a total of 120 `-bins, with four additional FIRAS data points. Solving for the likelihood described in Sect. 3.6.1, gives the best-ﬁt models shown in Figs. 3.15 and 140 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

3.16 as solid lines. The 2 values in each panel are the contribution to the total 2 from 2 the data within the panel. The combined reduced is 102.1 for Ndof = 104, indicating a good fit. The constraints we find on Msat and Mmin are shown in Fig. 3.17. We force M M in the MCMC fitting procedure, with the dashed line in Fig. 3.17 showing sat min equality. The orange dot corresponds to the parameter values that give the minimum 2 in the fit, and are log10 (Mmin/M ) = 12.18 and log10 (Msat/M ) = 12.76, which gives Msat/Mmin = 3.80. The mean parameter values are log10 (Mmin/M ) = 10.53 0.62 and ± log10 (Msat/M ) = 10.80 0.74. The best-fit value of Mmin is consistent with those derived ± in PER at multiple frequencies, even though we now set ↵sat = 1 and reconstruct the mean emissivity as a function of redshift. The associated mean emissivity parameters are given in Table 3.3 and displayed in Fig. 3.18, where we also plot the B11 model for reference. As can be seen in Fig. 3.18, we remain consistent with the B11 model in most redshift bins.

3.6.3 Interpreting the Reconstructed Emissivities

We now illustrate one interesting consequence of this measurement and show how using the constrained emissivities, j⌫(z), we can estimate the star formation rate (SFR) density at di↵erent redshifts. Following Penin´ et al. [81], we begin by writing the emissivity as an integral over the galaxy ﬂux densities:

1 d d2N j (z) = a S dS . (20) ⌫ dz ⌫ dS dz ⌫ ! Z ⌫

2 Here S ⌫ is the ﬂux density, and d N/dS ⌫ dz is the number of galaxies per ﬂux element and redshift interval. The galaxies contributing to the CIB can be divided into various populations (labelled as p) based on the galaxy SED, e.g., according to galaxy type or dust temperature: 1 2 d d Np j⌫(z) = a S ⌫ dS ⌫ . (21) dz dS ⌫ dz ! Xp Z 3.6. INTERPRETATION AND DISCUSSION 141

If we deﬁne s⌫ as the ﬂux density of an LIR = L source with the SED of a given population, i.e., S ⌫ = s⌫LIR (with LIR in units of L ), then we can write Eq. 21 as [81]:

1 2 d dV d Np j⌫(z) = a s⌫ LIR dLIR. (22) dz dz dLIR dV ! Xp Z The contribution to the infrared luminosity density from a given population is

2 d Np ⇢ = L dL . (23) IR,p IR dL dV IR Z IR

We assume a simple conversion between LIR and the star formation rate density, ⇢SFR, using the Kennicutt constant K [88]. Since by definition ⇢SFR = K p ⇢IR,p, we can rewrite Eq. 22 as: P ⇢ p s⌫ ⇢IR,p j (z) = (1 + z) 2 SFR , (24) ⌫ K ⇢ P p IR,p ! where the final term in brackets is the e↵ective SEDP of infrared galaxies, which we write as s⌫,e↵. We derive these SEDs following the evolution model of Bethermin´ et al. [6] using Magdis et al. [89] templates. The construction of these e↵ective SEDs will be explained in detail in future work. Finally, we obtain the conversion factor between mean emissivity and SFR density, K ⇢ = . SFR(z) 2 j⌫(z) (25) (1 + z) (z) S ⌫,e↵(z) Using Eq. 25 we find the coecients for each of the redshift bins and frequencies used in Table 3.3.

3.6.4 Discussion and Outlook

In the previous section we described a model that simultaneously ﬁts the CIB auto-spectra and the CIB-lensing cross-spectra, at all frequencies and with a single HOD prescription. Given that we use an emissivity function that is close to the B11 emissivities (to within our uncertainties), we expect predictions of the galaxy number counts derived from our best-ﬁt emissivity to agree with current estimates [90]. The fact that our measurement is consistent with previous models of the CIB lends support to our current understanding of 142 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

Figure 3.18 Reconstructed mean emissivity, ¯j, for each frequency as a function of redshift. The solid line at low z and the dashed line at higher z correspond to the B11 model. The B11 emissivity model at z > 1 is not used, and is shown only for reference. The black error bars correspond to the 68% C.L. while the color shading display the full posterior distribution.

its origin. For example, the characteristic mass scale at which halos host galaxies, Mmin, is consistent with values derived in PER, and is consistent with, but slightly higher than, the value derived more recently in [14], log10 (Mmin/M ) = 9.9 0.5 (although a direct ± comparison could be misleading given the di↵erent model assumptions). In particular, it is clear that our model has limitations, some of which have been partially addressed in recent work [6, 14, 27, 87, 91, 92] and are points of focus in PIR, amongst them the mass independence of the emissivity. Another question worth further investigation is the 3.6. INTERPRETATION AND DISCUSSION 143

Figure 3.19 Marginalized 1-D distribution of the emissivity in the high redshift bin at 353 GHz with (black line) or without (blue line) including the CIB-lensing correlation. Its inclusion helps to constrain the emissivity at high redshift, transforming an upper limit into a detection.

Table 3.3 Reconstructed emissivity as a function of redshift and associated star formation rate. At each frequency and for each of the three redshift bins the ﬁrst quantity corresponds 1 1 to the mean emissivity in the corresponding redshift bin, ¯j(z), in Jy Mpc sr , while the 3 1 second corresponds to the SFR density, ⇢SFR, in M Mpc yr . 1 < z 1.51.5 < z 33< z 7    ¯j(z) ⇢SFR ¯j(z) ⇢SFR ¯j(z) ⇢SFR 100 GHz . . . 7.16 5.77 1.96 1.58 3.53 3.05 0.655 0.564 5.49 4.78 0.271 0.236 ± ± ± ± ± ± 143 GHz . . . 12.7 9.60 1.37 0.964 6.82 5.46 0.438 0.351 10.5 9.05 0.178 0.153 ± ± ± ± ± ± 217 GHz . . . 11.9 6.33 0.310 0.165 17.3 7.23 0.282 0.118 36.6 13.8 0.182 0.068 ± ± ± ± ± ± 353 GHz . . . 116 17.1 0.671 0.099 75.5 27.5 0.286 0.104 164 47.3 0.320 0.092 ± ± ± ± ± ± 545 GHz . . . 185 106 0.320 0.183 224 148 0.317 0.210 417 251 0.659 0.396 ± ± ± ± ± ± 857 GHz . . . 193 139 0.144 0.104 354 212 0.317 0.190 609 359 1.37 0.809 ± ± ± ± ± ± dependence of our results on the binning scheme chosen for the emissivity, which will be addressed in a future paper. 144 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

Figure 3.20 Correlation between the lensing potential and the IRIS map at 100 µm using our nominal lens reconstruction. We clearly see a correlation and estimate the signiﬁcance to be 9 , ignoring possible systematic e↵ects. The solid line represents a simple reasonnable prediction for this signal.

Given the consistency of our model with the PER results, the information added by our cross-spectrum measurement is worth quantifying. As an example, we show in Fig. 3.19 the highest redshift emissivity bin in the 353 GHz band. Adding the CIB-lensing cross- spectrum information tightens the constraint on the high-redshift part of the emissivity. This statement also holds for the other frequencies and stems from the fact that the CMB lensing kernel peaks at high redshift, making the cross-correlation more sensitive to the high-redshift CIB signal than the CIB auto-spectrum, as is illustrated in Fig. 3.1. Although this gain does not translate into a substantial improvement in Mmin, it leads to interesting constraints on the SFR density, as can be seen in Table 3.3.

The results at frequencies above 217 GHz each lead to around 2 evidence for a non- zero SFR density for 1.5 < z < 3 and for 3 < z < 7. The values inferred are consistent 3.6. INTERPRETATION AND DISCUSSION 145

with other probes of the SFR in these redshift ranges, as compiled for example in Fig. 1 of Hopkins and Beacom [93]. Assuming that each frequency is independent, we obtain SFR densities for the three redshift bins of 0.423 0.123, 0.292 0.138 and 0.226 0.100 ± ± ± 3 1 M Mpc yr , respectively where the errors are 68% C.L.. We note that the j distributions are rather non-Gaussian so that the 95% C.L. become 0.228, 0.246 and 0.191 respectively. This roughly 2 detection per bin compares very favourably with other published measurements. These constraints clearly illustrate how this particular correlation can be used to better isolate the high redshift component of the CIB and improve our constraints on the star formation rate at high redshift. Such constrains will improve with future measurements, in particular if we can increase the signal-to-noise ratio in our lower frequency channels, where the high redshift contribution is the greatest. This will likely require an accurate removal of the CMB, our dominant source of noise at low frequencies. A more thorough discussion of this possibility will be given in PIR. To fully utilize the richness of the correlation will require more studies. Future work could involve using more sophisticated halo models specifically designed to model star formation within halos, as well as relaxing some of the assumptions made here, such as the mass independent luminosity function. In addition the use of map-based methods that enable estimates of the galaxy host halo mass by stacking the lensing potential maps is worth pursuing, as is the extension to other data-sets. For illustration purposes, we show in Fig. 3.20 raw measurements of the correlation between our lensing potential map and the IRIS map at 100 µm that was introduced in Sect. 3.2.2. We use our nominal mask and lens reconstruction, with no Hi cleaning performed on the IRIS map. We clearly see a strong correlation, whose significance we estimate to be 9 , ignoring any possible systematic e↵ects. To guide the eye we have added a prediction (not a fit) based on the HOD model presented in [81]. The full analysis of this signal is beyond the scope of this paper, but it illustrates possible future uses of the lensing potential map. In this case the IRIS wavelength range will help us to isolate the low-redshift contribution to the CIB. To conclude, we have presented the first measurement of the correlation between lensing of the CMB and the CIB. Planck’s unprecedented full-sky, multi-frequency, deep survey enables us to make an internal measurement of this correlation. Measurements with high statistical significance are obtained, even after accounting for possible systematic errors. 146 CHAPTER 3. GRAVITATIONAL LENSING BY CIB GALAXY HALOS

The high degree of correlation that is measured (around 80 %) allows for unprecedented visualization of lensing of the CMB and holds great promise for new CIB and CMB focused science. CMB lensing appears promising as a probe of the origin of the CIB, while the CIB is now established as an ideal tracer of CMB lensing.

Acknowledgements

Based on observations obtained with Planck (http://www.esa.int/Planck), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada. The development of Planck has been supported by: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN and JA (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzer- land); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and PRACE (EU). A description of the Planck Collaboration and a list of its members, including the technical or scientiﬁc activities in which they have been involved, can be found at http://www. sciops.esa.int/index.php?project=planck&page=Planck_Collaboration.We acknowledge the use of the HEALPix package, and the LAMBDA archive (http://lambda. gsfc.nasa.gov). Bibliography

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Figure 3.21 Ratio of various error estimation procedures to the errors obtained with the data-based analytical estimate. At each frequency the numerator is given by: (i) the scatter within an `-bin in simulations (solid black line); (ii) the scatter within an `-bin in the data (solid dashed black line); (iii) the scatter of bins across simulated realizations (solid coloured line); (iv) the analytical errors calculated from the simulations (dashed coloured line); (v) the scatter across realizations for the cross-correlation between the simulated temperature map and the lensing potential reconstructed from the data (coloured dot-dashed line). The grey envelope is the precision of the simulated errors expected from 100 simulations (shown as a spread around unity).

3 A Statistical Errors

In this section we compare six di↵erent methods to estimate our statistical errors. This comparison is used to validate our claim (presented in the main text) that we can obtain our errors from the naive analytical errors calculated from the data, i.e., method 1 below. The 164 BIBLIOGRAPHY

six di↵erent methods we compare are:

1. The naive, Gaussian, analytical errors estimated from the data through the measured ˆTT ˆ total power of the T and ﬁelds, respectively C` and C` and the model cross- T spectrum C` .

2. As above but instead of the data maps we use one of our simulations of the CIB and CMB temperature maps described in Sect. 3.3.4. The lens reconstruction is obtained from the simulated maps using the same procedure that we use for the data.

3. The scatter directly within individual `-bins in the data-determined cross-spectrum.

4. As above but the scatter is measured in each `-bin for each simulation realization, and the errors are averaged over 100 realizations.

5. The scatter of the bins is calculated using the cross-spectra measured from our simulated maps. This is a direct measurement of the statistical error we require (to the extent that our simulations are realistic), and will di↵er from the scatter within the `-bins, for example due to noise correlations between di↵erent multipoles of the cross-spectrum.

6. The error in the cross-spectrum of the reconstructed lensing potential in simulated maps with the measured temperature maps. This will only give part of the contribution to the error since the temperature maps are ﬁxed, but it is still a useful cross-check.

In Fig. 3.21 we show a comparison of the errors found from our six measurement methods. The precision achievable with 100 simulations is indicated by the grey envelope. We show the errors in each ` bin from the di↵erent methods divided by the data-based analytical estimate. To discuss the implications of these results we shall focus on the 100 GHz panel ﬁrst. The scatter measured within an ` bin is fairly consistent in the simulations (method 4, black solid line) and in the data (method 3, black dashed line) giving us conﬁdence in our simple simulation procedure. This rules out important systematic contributions and shows that our signal is mostly Gaussian, as expected. Note that the consistency with the simulations is not surprising, since at low frequencies we are dominated by CMB and instrumental 3 A. STATISTICAL ERRORS 165

noise, which are well understood. In addition, the fact that the analytical errors calculated on the simulations (method 2, coloured dashed line) are mostly within the grey shaded region, and are therefore close to the analytical errors calculated from the data (method 1), gives us further conﬁdence in the simulations. To the extent that the simulations are accurate, the scatter of the `-bins in simulations (solid coloured line) is the error that we require. The fact that it is essentially all within the shaded envelope means that this method gives errors that are consistent with the analytical errors measured using the data, justifying our nominal choice for calculating the errors at low frequencies. However, comparing the black lines with the coloured lines clearly indicates that in the data and simulations obtaining the error bars by measuring the scatter within the `-bins leads to an underestimation of the errors by approximately 20 %. Given the fact that this di↵erence is observed in both the data and the simulations, we exclude any instrumental systematic e↵ect as its cause and explain it as being due to noise correlations within the `-bins. Such a correlation is expected, since most of the lens reconstruction signal in the `-range of interest to us comes from modes in the CMB map within a relatively narrow range at ` 1500, and so multipoles in the lens reconstruction are correlated. We have also ' checked that the mask induced `-bin coupling is negligible, given the bin width we have chosen, and is always smaller than 2 %. All of these conclusions remain valid up to 353 GHz. However, at 545 and 857 GHz, we see by looking at the errors measured using simulations (solid black for method 4, solid coloured for method 5, and dashed coloured for method 2) that the errors deduced from the analytical estimates measured from the data are substantially higher than those we measure in the simulations. This is easily explained through the fact that we are omitting any foreground emission in our simulations. The relative contribution of Galactic foreground emission is higher at low `, which is expected because the Galactic cirrus emission has a steep power spectrum. Overall, the amplitude of this contribution is also consistent with what is seen in Fig. 3.5. Since the scatter within the `-bins measured in simulations (black dashed line) is about 20 % lower than the data-based analytical estimates at all frequencies, we use the data- based analytical estimates as the basis for our statistical errors. We could alternatively scale the scatter-determined errors by 20 % and obtain consistent results. This approach accounts 166 BIBLIOGRAPHY

for the foreground emission seen at 545 GHz and 857 GHz, but will in practice neglect the contribution to the errors from the non-Gaussian part of the foregrounds. However, we show in Sect. 3.5.3 that this contribution is small enough that we can ignore it. The remaining method to discuss is obtained from the cross-spectrum of the reconstructed lensing potential in simulated maps with the data-measured temperature maps (method 6, coloured dashed-dotted line in Fig. 3.21). At 545 and 857 GHz the CIB signal is dominant over a large ` range, and so the error obtained from this method is equal to the “signal” terms in Eq. 4, which are the orange and green lines in Fig. 3.5. These two lines make up a signiﬁcant fraction of the total error and provide a reasonable approximation to the true error at high-`. However, at low-` where Galactic emission is important, and at 100–353 GHz where the CMB and instrumental noise are the largest components, the orange and green curves do not accurately describe the total error. We can see from Fig. 3.21 that the errors obtained using this method are close to the errors measured using the other techniques. However, this method will underestimate the true errors, since the scatter in the CMB and noise components is neglected. Note that the results presented in Fig. 3.21 are all computed using the 40 % Galaxy mask, but we have checked that they hold when using the 20 % and 60 % Galaxy masks

(which are discussed in detail in Sect. 3.5.2) and that the results show the appropriate fsky scaling. Chapter 4

Reconstructing The Redshift Distribution Of The CIB Emissivity

“To consult the statistician after an experiment is ﬁnished is often merely to ask him to conduct a post mortem examination. He can perhaps say what the experiment died of.”

Ronald Fisher

Fluctuations in the cosmic infrared background light (CIB) have now been observed by several experiments, and have been shown to be correlated with the structure that lenses the CMB. Measurements of the angular spectra of the ﬂuctuations contain information about the history of star formation in the universe, and can potentially be used to measure the star formation rate at high redshift. We show how measurements of the CIB spectra and CIB lensing potential cross spectra can be exploited to enable the redshift dependence of the CIB emissivity to be reconstructed, from which the star formation rate can be estimated under certain assumptions.

167 168 CHAPTER 4. RECONSTRUCTING THE CIB REDSHIFT DISTRIBUTION

4.1 Introduction

The cosmic infrared background light (CIB) is emitted as graybody light from dust that is heated by young UV-luminous stars residing in dusty star-forming galaxies, and redshifted to microwave and infrared wavelengths today. The emission predominantly comes from galaxies at z > 1 and contains much of the energy emitted during star formation processes [e.g. 1–3]. The statistical properties of fluctuations in the CIB emission contain information about how galaxies cluster and trace the dark matter, and can be probed by measurements of the angular power spectrum [4–6]. Above 400 GHz the CIB is the ⇠ dominant far-infrared extragalactic background, allowing the spectra to be cleanly measured over a wide frequency range by, for example, Herschel [7–9] and Planck [10–12]. Spatial fluctuations in the CIB emission arise from spatial fluctuations in the density of dust, which we will model as arising from fluctuations in the galaxy number density: ¯j (1 + j (~x)/¯j ) n¯ (1 + n (~x)/n¯ ), where j is the dust emissivity at frequency ⌫, ⌫ ⌫ ⌫ ⇠ gal gal gal ⌫ ~x is position,n ¯gal is the mean galaxy number density, is the over-density, and the mean CIB intensity is d a¯j(), where is the comoving distance and a is the scale factor. From this point forwardR we will drop the bar over j. We follow Haiman and Knox [4] and write j as a function of redshift, with no dependence on the overall mass density [see 13, for a discussion of this assumption] In this model the CIB auto power spectra depend on the spatial power spectrum of the galaxy density and a redshift dependent emissivity. We aim to show how well the latter can be recovered from CIB observations. The former we calculate by assuming that each galaxy resides in a discrete dark matter halo, with the dark matter halo correlation structure described by the halo model [for a review of the halo model see e.g. 14]. The halo model has been used to model both the CIB power spectra [10, 11, 15, 16], and the CIB lensing potential cross spectra [6, 12]. We estimate the number of galaxies occupying each halo using a halo occupation distribution model [HOD 17–22]. The CIB spectra measured at Planck frequencies in Ade et al. [12] are consistent with a population of galaxies described by a single HOD model. If this remains true for higher sensitivity measurements, then the redshift dependent emissivity can be determined using multi-frequency measurements of the CIB spectra, since the spectra at di↵erent frequencies are sensitive to the CIB emission at di↵erent redshifts. 4.2. RECONSTRUCTING THE CIB EMISSIVITY 169

At CMB frequencies the CIB signal is sensitive to star formation at high redshift. Due to the large telescope beam size common with experiments in this frequency range the CIB emission is largely unresolved, with many galaxies within the beam. The high-redshift contribution can be enhanced by correlating CIB maps with a tracer of the high redshift dark matter density. The CMB lensing potential is an ideal tracer [6, 12]. The CMB is lensed over a broad redshift range, with the largest deflection angles oc- 2 1 curring around z 2 3, caused by structure on scales of k few 10 Mpc , and an ⇠ ⇠ ⇥ rms deflection of 20.7 [23–26]. The auto spectrum of the CMB lensing potential has been ⇠ measured by ACT, SPT and Planck [12, 27, 28], with a detection significance of 30 ⇠ in Planck data. The cross spectrum of the CMB lensing potential with the CIB signal was recently measured in Ade et al. [12]. Our goal in this paper is to determine how well the CIB emissivity can be reconstructed as a function of redshift using the CIB spectra and the CIB lensing potential cross spectra. In particular we wish to determine:

The relative importance of the di↵erent CIB spectra. • The sensitivity of the results to the number of observation frequencies. • For simplicity, we will take a Fisher matrix approach.

4.2 Reconstructing the CIB Emissivity

Spatial fluctuations in the CIB brightness trace fluctuations in the galaxy number density. In this section we outline how the CIB angular spectra can be calculated from the statistical properties of the galaxy number density. Let gal(~x) = ngal(~x)/n¯gal. The two-point statistics of gal are described by: Pgg(kˆ) = gal(kˆ)gal(kˆ)⇤ and Pg(kˆ) = gal(kˆ)(kˆ)⇤ , where is the dark matter over-density. To calculateD these spectraE we use theD model in AdeE et al. [10], with the following parameters that determine the number of galaxies per dark matter halo: 1 1 log10 Mmin/h Msun = 10.75, log10 Msat/h Msun = 11.0, and ↵ = 1, with definitions found⇣ in Ade et al. [10].⌘ We use the same⇣ parameters⌘ at all CIB frequencies, which assumes that for the galaxy population sourcing the CIB emission the number of galaxies per dark matter halo of a given mass is independent of redshift. 170 CHAPTER 4. RECONSTRUCTING THE CIB REDSHIFT DISTRIBUTION

We calculate the CIB lensing potential cross spectra using the same halo model parameters, following Song et al. [6]. Lensing of the CMB can be described by a lensing potential , with the lensed CMB a remapping of the unlensed CMB: T(nˆ) = T˜(nˆ + (nˆ)), where r T is the lensed CMB, T˜ is the unlensed CMB, and the lens deﬂection angle is d~ = . r The lensing potential can be estimated from CMB data by measuring non-Gaussian correlations that lensing introduces [29, 30]. The lensing potential is related to the gravitational potential of the dark matter density projected along the line of sight, and at the scales we consider can be calculated using linear theory [for a review see e.g. 26].

With these assumptions the CIB cross spectrum between frequencies ⌫ and ⌫0 is:

⌫⌫0 ⌫ ⌫0 C` = d W ()W ()Pgg(k = `/, ), (1) Z and the CIB lensing potential cross spectrum is:

⌫ ⌫ C` = d W ()W ()Pg(k = `/, ), (2) Z ⌫ 2 where W () = aj⌫()/, W () = 3⌦0H ( )/( a), ⌦0 is the matter density, H0 is 0 ⇤ ⇤ the Hubble constant, and is the distance to recombination. For our ﬁducial model of j ⇤ we use Bethermin et al. [2], [see also 31]. We do not use the auto spectrum of to constrain j, since our lack of knowledge of j introduces a greater source of uncertainty in Eq. (2) than our lack of knowledge of cosmological parameters. We ﬁx the cosmological parameters to the WMAP 9-year values [32]. We wish to estimate j from measurements of the spectra in Eq. (1) and Eq. (2). The simplest parameterization of j is as a set of amplitudes within redshift bins:

j⌫(z) = j⌫(z) H(z z0) + H(z zi 1) H(z zi) j⌫,i, (3) i=1 X ⇣ ⌘ where H is the Heaviside function, zi are the redshifts deﬁning the Nz bins, and j⌫,i are free parameters. Since we are primarily interested in the high-redshift emissivity, we separate the low-redshift part which has been measured using galaxy counts [e.g. 9]. In Fig. 4.1 we show the derivatives of the spectra in Eqs. (1) and (2) with respect to the 4.2. RECONSTRUCTING THE CIB EMISSIVITY 171

Figure 4.1 Derivatives of the CIB auto spectrum (top and middle panels) and CIB lensing potential cross spectrum (bottom panel) with respect to the amplitudes, ji, of the binned emissivity.

j⌫,i parameters in four bins. The auto spectrum at 217 GHz (top panel) has greater sensitivity to the high-redshift bins of j than the auto spectrum at 545 GHz (middle panel), as we would expect since at 217 GHz the CIB has a greater contribution from high redshift. The CIB cross spectrum (lower panel; independent of frequency due to the linear j dependence in Eq. (2)) also has a signiﬁcant contribution from the highest redshift bin, again, as expected since the lens deﬂection angles are largest around z 2 3 with a broad redshift distribution. ⇠ As can be seen from the shape of the curves in Fig. 4.1, the dependence of the CIB spectra on j⌫,i is similar for neighboring redshift bins. Estimates of two di↵erent j⌫,i parameters obtained from the CIB spectra are therefore likely to be highly correlated. This makes 172 CHAPTER 4. RECONSTRUCTING THE CIB REDSHIFT DISTRIBUTION

it dicult to interpret the estimated ˆj⌫,i parameters, and we therefore choose a di↵erent parameterization of j. We describe j by its principal components, calculating a set of basis functions which depend on the signal-to-noise ratio of the measured spectra. We expect the component best constrained by the data to be a smooth non-oscillating function of redshift, with subdominant components that describe features with sharper redshift dependence.

4.3 Fisher Matrix Analysis

The principal components are calculated from the Fisher matrix [e.g. 33]:

X Y @C` 1 @C` F = , (4) ij @ X,` Y,` @ ` pi C p j X XXY where the X,Y indices denote the particular auto or cross spectrum, is the covariance C matrix of the spectra, and pi are the parameters describing the emissivity. For the purpose of calculating the Fisher matrix we take pi to be the amplitude of j at each observation frequency in 30 redshift bins between z = 1 7, and we denote this as p = ⌫, z . Each i { }i dimension of the Fisher matrix has size equal to the number of observed frequencies multiplied by the number of redshift bins. We discuss the case where the frequency dependence of j is parameterized in Sec. 4.4.

The derivatives in Eq. (4) can be calculated analytically:

@C⌫⌫0 z=zi+1 2 ` ⌫0 ⌫ a = ⌫⌫00 ji + ⌫0⌫00 ji d Pgg(k = `/, z) @ j⌫00 2 i Zz=zi ⌫ (5) ⇣ z=zi+1 ⌘ @C` a = ⌫⌫ d W (z)Pg(k = `/, z), @ j⌫00 00 i Zz=zi and are shown in Fig. 4.1. 4.3. FISHER MATRIX ANALYSIS 173

4.3.1 Noise Covariance Matrix

The covariance matrix in Eq. (4) contains three types of element: the variance of the CIB spectra, the variance of the CIB cross spectra, and the covariance between the two. De- noting the observation frequencies with the indices i, j, and p the three elements are:

ij kp i j k p ik jp ip jk ⇤ ⇤ C` C` = a`ma`m ; a`m⇤ a`m = C` C` + C` C` ij k i j k ik j i jk ⇤ ⇤ DC` C` E = Da`ma`m ; a`m⇤ `mE = C` C` + C` C` (6) i k⇤ i k ik i k DC` C` E = Da`m`⇤m ; a`m⇤ `mE = C` C` + C` C` . D E D E The covariance matrix is then calculated from, e.g.:

1 ij kp⇤ ij,kp = C` C` , (7) C (2` + 1) fsky D E where fsky is the observed sky fraction, which for simplicity we will take to be unity. Al- though in practice experiments cannot observe the CIB over the full sky due to the large dust signal from the Galaxy, this assumption is equivalent to observing a smaller sky area but with reduced noise. The CIB auto spectra contain a contribution from instrumental noise [34]: ⌫⌫,n 2 2 C` = w exp `(` + 1) /8 log 2 , (8) ⇣ ⌘ where is the beam FWHM and w is the white noise level. The lensing potential auto spectrum has a contribution from instrumental and CMB noise. We describe the noise ,n ,n spectrum of , C` , by a single parameter Lmax. C` is the noise spectrum that would be found if is reconstructed from noiseless CMB data using only data modes up to multipole

Lmax. Lmax is chosen based on the instrumental noise level of the experiment. For Planck data L 1500 and for ACT/SPT data L 3000. max ⇠ max ⇠ For simplicity we do not include systematic sources of error such as calibration errors, beam uncertainties, which can be important at high-`, or errors associated with CMB removal. At frequencies below 400 GHz the CMB is the dominant background to mea- ⇠ surements of the CIB auto and cross spectra, as well as a large source of noise in the CIB lensing potential cross spectra. To constrain the emissivity at low frequency, it is likely that the CMB must be subtracted from the CIB auto and cross spectra, however this increases 174 CHAPTER 4. RECONSTRUCTING THE CIB REDSHIFT DISTRIBUTION

Figure 4.2 CIB auto and CIB lensing potential cross spectra relative to our ﬁducial model at 217 GHz assuming 10 µK-arcmin noise, 5’ beam FWHM and Lmax = 3000 (dark gray bars), and 100 µK-arcmin noise, 5’ beam FWHM and Lmax = 1500 (light gray bars). The colored lines show the spectra if the emissivity in a single redshift bin is doubled while holding the mean CIB intensity constant. the complexity of the modeling [12], and so for simplicity we do not consider it here.

4.3.2 Experimental Model

We consider two experimental models:

An experiment observing at the Planck frequencies: 100, 143, 217, 353, 545, and • 857 GHz (3000-250 µm) with a noise level of 10 µK-arcmin and beam FWHM of 5’ at all frequencies.

An experiment observing at the Herschel frequencies: 545, 857, and 1200 GHz (550, • 350, and 250µm), with no instrumental noise, but only using the spectra below `max = 3300. 4.3. FISHER MATRIX ANALYSIS 175

Figure 4.3 The ﬁve (normalized) principal components of the CIB emissivity with the largest eigenvalues, obtained with experimental model 1. In the upper panel we assume no knowledge of the frequency dependence of j, while in the lower panel we assume that the frequency dependence is known exactly. Note the di↵erent y-scale in the two panels.

We will refer to these cases as model 1 and model 2. We restrict ourselves to `max = 3300 since the CIB shot noise becomes important at ` & 3300, depending on frequency, and the data at higher ` does not tighten the constraint on j. For both experiments we consider

Lmax = 1500 and 3000, and assume that the experiments observe the entire sky.

In Fig. 4.2 we show the CIB spectra at 217 GHz with error bars calculated from the diagonal elements of the covariance matrix. The colored lines show the theoretical spectra if the value of the emissivity in a single redshift bin is doubled, while ﬁxing the mean CIB intensity, I⌫ = d aj⌫(). R 176 CHAPTER 4. RECONSTRUCTING THE CIB REDSHIFT DISTRIBUTION

4.3.3 Principal Components

The principal components S µ are the eigenvectors of the Fisher matrix [e.g. 35]:

Nc 2 Fij = S µ( ⌫, z i)µ S µ( ⌫, z j), (9) µ= { } { } X1 where Nc is the number of components, equal to the dimension of the Fisher matrix, and

µ is the theoretical minimum error on the amplitude of component µ that can be achieved if the measured spectra have covariance . C In Fig. 4.3 we show the principal components at 857 GHz for experimental model 1, calculated assuming that all of the CIB spectra from 100-857 GHz are used. The upper panel shows the components if all of the j⌫,i amplitudes are free parameters. In this case the components at each frequency are di↵erent. At frequencies where the CIB spectra are measured with high signal-to-noise ratio the components describe sharper features in j, and have a greater contribution at high redshift. In the lower panel we assume that the frequency dependence of j is known exactly. This reduces the number of free parameters, since j at one frequency determines j at all other frequencies. The components have sharper features than in the upper panel, and so j will be better constrained by the data. In general, we ﬁnd that the dominant principal components have a smoother shape with fewer oscillations, and so sharper features in the emissivity are less constrained by the data than smoother features. We expand the emissivity in terms of the principal components: j (z ) = a S ( ⌫, z ), ⌫ i µ µ µ { }i 2 where aµ is the amplitude of component µ and can be measured with varianceP µ. Assum- ing a Gaussian error distribution forp ˆi (valid when j is measured with high signal-to-noise ratio), the joint probability distribution for the parameters is:

Nc 1 1 2 2 exp (pi p¯i)Fij(p j p¯ j) = exp (aµ a¯µ) /µ , (10) 2 2 µ= Xij ! X1 ! where bars denote the parameter values at the peak of the distribution, and measurements of the component amplitudes will be uncorrelated. We reconstruct the emissivity using all of the components whose amplitudes are measured with signal-to-noise ratio above a threshold of 3. If the signal-to-noise threshold is lowered and more components are used, 4.4. RESULTS 177

Figure 4.4 Reconstructed emissivity relative to the true emissivity for experimental model 1 (solid) and model 2 (dashed), assuming that the frequency dependence of j is precisely known. The colored lines give the reconstruction calculated using only the principal components measured with signal-to-noise ratio greater than 3, as determined from a Fisher matrix calculation, using only the CIB auto spectra, auto and cross frequency spectra, or CIB lensing potential cross spectra. the error on the reconstruction increases. However, if too few components are used then the accuracy of the reconstruction decreases.

4.4 Results

To estimate how accurately the CIB emissivity can be reconstructed we calculate j using only the principal components measured with signal-to-noise ratio greater than 3, and we ﬁx the amplitude of these components to the true value:

⌫, true ⌫,z i j⌫(zi)S µ( z i) aµ = { } { } . (11) ⌫,z S µ( ⌫, z i)S µ( ⌫, z i) {P}i { } { } P 178 CHAPTER 4. RECONSTRUCTING THE CIB REDSHIFT DISTRIBUTION

We find that if no assumptions are made about the frequency dependence of j then the reconstruction is poor with both experimental models, even if all of the CIB spectra are used. If the frequency dependence is precisely known, then the reconstruction is reasonable, with an accuracy of 25% at 857 GHz. The reconstruction at 857 GHz is shown in Fig. 4.4. ⇠ At the noise levels we consider the CIB cross spectra provide a stronger constraint than either the CIB auto or CIB lensing cross spectra, with the lensing cross spectra not substantially improving the reconstruction. Experimental model 1 provides a better reconstruction than model 2, since the larger number of frequency channels allow more CIB cross spectra to be measured. With the frequency dependence fixed the CIB cross spectra provide a powerful constraint, since each cross-spectrum is determined entirely by the parameters of j at a single frequency. Since the frequency dependence of j is not precisely known, Fig. 4.4 is an upper limit on the sensitivity to j that could be achieved with this experimental configuration. If additional variables are introduced to parameterize the frequency dependence, then the sensitivity will be reduced. The shape of the fiducial emissivity a↵ects the reconstruction accuracy. Sharp features in the emissivity require a greater number of components for an accurate reconstruction, and our results are sensitive to the choice of fiducial emissivity. The star formation rate can be calculated from the emissivity using the method described in Ade et al. [12], which scales j by a redshift dependent factor. This assumes that the dust temperature and rate of dust production are proportional to the global star formation rate. Since the star formation rate and emissivity at a given redshift are proportional under these assumptions, then the star formation rate can be determined with the same precision as the emissivity in Fig. 4.4. Bibliography

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Measuring The Galaxy Cluster Bulk Flow From WMAP Data

“The history of astronomy is a history of receding horizons. Knowledge has spread in successive waves, each wave representing the exploitation of some new clew to the interpretation of observational data.”

The Realm Of The Nebulae Edwin Hubble

We have looked for bulk motions of galaxy clusters in the WMAP 7 year data. We isolate the kinetic Sunyaev-Zeldovich (SZ) signal by filtering the WMAP Q, V and W band maps with multi-frequency matched filters, that utilize the spatial properties of the kinetic SZ signal to optimize detection. We try two filters: a filter that has no spectral dependence, and a filter that utilizes the spectral properties of the kinetic and thermal SZ signals to remove the thermal SZ bias. We measure the monopole and dipole spherical harmonic coecients of the kinetic SZ signal, as well as the ` = 2 5 modes, at the locations of 736 ROSAT observed galaxy clusters. We find no significant power in the kinetic SZ signal at these multipoles with either filter, consistent with the ⇤CDM prediction. Our limits are a factor of 3 more sensitive than the claimed bulk flow ⇠ 184 5.1. INTRODUCTION 185

detection of Kashlinsky et al. [1]. Using simulations we estimate that in maps filtered by our matched filter with no spectral dependence there is a thermal SZ dipole that would be mistakenly measured as a bulk motion of 2000 4000 ⇠ km/s. For the WMAP data the signal to noise ratio obtained with the unbiased filter is almost an order of magnitude lower.

5.1 Introduction

Verifying the depth of convergence of the 627 km/s motion of the Local Group (LG) with respect to the frame defined by the Cosmic Microwave Background (CMB) has been a long-standing goal of observational cosmology. Cosmographers seek to identify the region in which the local universe is at rest with respect to the CMB frame, and the LG motion has converged to the CMB dipole direction. This region would contain all of the mass sources responsible for the motion of the LG and would define a sample of the universe representative of the whole. The coherent motion of matter caused by gravitational potentials is termed a bulk flow [e.g., 2]; if the flow covers the entire universe it is sometimes referred to as a dark flow [e.g., 3]. Such a flow is equivalent to an intrinsic dipole of the universe. Models of inflation can be constructed to explain such a measurement [e.g, 4, 5]. To date there is no consensus on the depth of convergence. The CMB dipole magnitude and direction have been precisely measured by WMAP: (3.355 0.008) mK in the direction ± of galactic longitude l = 263.99 0.14 and latitude b = 48.26 0.03 [6]. The LG ± ± motion towards the Great Attractor was measured by Dressler et al. [7] in the direction of 1 1 l = 312, b = 6 and found to be coherent out to 30h Mpc and near zero by 40h Mpc. 1 1 Using a full-sky peculiar velocity survey, with a depth ranging from 80h Mpc to 110h

Mpc, Lauer and Postman [8] found a dipole toward (l = 220, b = 28) 27, inconsistent ± with the CMB dipole direction at 99.99% conﬁdence and generated by mass concentrations 1 beyond 100h Mpc. An analysis of IRAS indicated a dipole direction within 13 of the 1 CMB dipole that has not fallen to zero by 300h Mpc [9]. Kocevski et al. [10] used an all-sky X-ray selected cluster catalog and found that the direction of the LG’s peculiar 1 velocity is well aligned with the CMB as early as the Great Attractor region 40h Mpc away. However, most of the Kocevski et al. [10] dipole signal is attributed to the Shapley 186 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

1 supercluster 150h Mpc away, as well as a handful of massive clusters behind our galaxy. 1 Using 56 SMAC clusters within 120h Mpc, Hudson et al. [11] found a bulk ﬂow of

687 203 km/s toward l = 260 13, b = 0 11 that did not drop o↵ with depth. They ± ± ± found that it could not be caused by the Great Attractor, but could be caused by the Shap- ley Concentration (at marginal significance). They argue that multiple data sets exclude 1 1 convergence to the CMB frame by 60h Mpc, and that at depths of 60 120h Mpc the flow is limited to 600 km/s. Using a new method of optimally weighting peculiar velocities, Watkins et al. [12] compiled all major peculiar velocity surveys (including SMAC) 1 and found them to be highly consistent. Within a region of radius 100h Mpc they find ⇠ a flow of 407 81 km/s toward l = 287 9, b = 8 6 implying that 50% of the LG ± ± ± motion is generated beyond this depth. Extending this analysis Feldman et al. [13] find a consistent flow, 416 78 km/s toward l = 282 11, b = 6 6. These results are found ± ± ± to be in disagreement with the ⇤CDM model with WMAP 5 year cosmological parameters at a high confidence level. Using the 2MASS Redshift Survey Lavaux et al. [14] find that less than half of the am- 1 plitude of the CMB dipole is generated within 40h Mpc, and that most of the amplitude ⇠ 1 of the dipole is recovered by 120h Mpc, although the directions of the two flows do not agree. Colin et al. [15] use Type Ia supernovae (SNe 1a) to find that there is a bulk flow 1 of around 260 km/s at 180h Mpc, which disagrees with ⇤CDM at the 1 2 level. ⇠ 1 However, at 435h Mpc they find improved agreement between the SNe Ia data and the ⇠ isotropic ⇤CDM model. The kinetic Sunyaev-Zeldovich (SZ) e↵ect was first used to place a limit on the bulk 1 flow velocity by Benson et al. [16] using 10 clusters between 300 600h Mpc. They found no detection but limited the flow in the direction of the CMB dipole to 1410 km/s  at 95% confidence. Kashlinsky and Atrio-Barandela [17] found that by utilizing the kinetic SZ e↵ect in WMAP data, flows as small as 200 km/s (or 30 km/s for the Planck experiment [18]) could in principle be measured. Using the WMAP 5 year data Kashlinsky et al. 1 [3] [also 1, hereafter KAKE] found a coherent dipole out to at least a distance of 300h 1 Mpc. When all clusters in their sample beyond 300h Mpc are combined the dipole aligns well with the CMB dipole and is within 6 of the dipole found by Watkins et al. [12]. How- ever, the magnitude of the KAKE dipole is considerably larger, estimated to be between 5.1. INTRODUCTION 187

600 1000 km/s and is detected on much larger scales with most of the signal coming from 1 a region between 120 600h Mpc away. Using a larger cluster sample Kashlinsky et al. 1 [19] find that the flow has constant velocity out to approximately 575h Mpc. However, using the same method as KAKE, Keisler [20] do not find a significant detection of a bulk flow. They find that residual CMB in the filtered KAKE maps, that is correlated between the WMAP channels and not accounted for in the errors, decreases the significance of the cluster dipole. Atrio-Barandela et al. [21] present an analysis of the errors in the cluster dipole measurement. They do not reproduce the errors in Keisler [20] unless the monopole and dipole are removed from the full sky CMB map instead of only in the region outside of the galactic mask, suggesting this as a reason for the null result of Keisler [20]. We measure the galaxy cluster peculiar velocity distribution using the WMAP 7 year data. The main di↵erence between our method and that of KAKE and Keisler [20] is the filter we use to suppress the CMB signal in the WMAP maps. We use a multi-frequency matched filter that optimizes detection of the cluster signal. As well as providing a higher signal to noise measurement, our filter automatically accounts for CMB and noise correlations between the channels. Using simulations we find that if no attempt is made to reduce the thermal SZ signal in the maps, our results would be biased. We use a modified version of the matched filter that incorporates the thermal SZ spectrum, to reduce the thermal SZ signal by over an order of magnitude. In addition, we subtract the monopole and dipole from the region outside of the galactic mask, and use an identical pipeline for the data and error analysis. We also measure the temperature monopole at the locations of galaxy clusters. The cluster monopole contains a contribution from the thermal and kinetic SZ e↵ects. The kinetic SZ contribution is expected to be zero if there is no cluster monopole velocity, as we expect. In addition we measure the quadrupole, octupole, and ` = 4 and 5 modes. Our method can be applied to data from the upcoming Planck experiment to achieve a result with greater sensitivity (Mak et al. 2010). In Sec. 5.2 we present all necessary theoretical background information. We describe the SZ e↵ect and how it can be used to measure the bulk flow, and present the ⇤CDM predictions for the velocity of the flow. In Sec. 5.3 we describe the data that we use to make our measurement, and explain how we construct our cluster sample. In Sec. 5.4 we 188 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

explain our method, describing how we model the SZ emission, how we construct filters to optimize cluster detection, and how we determine the amplitude and direction of any bulk flow. In Sec. 5.5 we describe the level of contamination of thermal SZ, unresolved radio point sources, and galactic emission, which we determine using simulations. In Sec. 5.6 we present our results and in Sec. 5.7 give our conclusions. Throughout this paper we assume a spatially flat ⇤CDM cosmology with the WMAP 7 year cosmological parameters [6].

5.2 Theory

5.2.1 SZ Clusters as Tracers of the Velocity Field

The SZ e↵ect can be used to measure the deviations of galaxy clusters from the Hubble ﬂow [22]. Approximately 1% of CMB photons traveling through galaxy clusters are scattered by electrons trapped in the gravitational potential of the cluster. If the galaxy cluster is moving with respect to the CMB rest frame, the scattered photons are red or blue shifted. This is the kinetic Sunyaev-Zeldovich (kSZ) e↵ect [22] and results in a fractional temperature change in the radiation of

TKSZ vp = dl neT , (1) T · c Z where dl is the line of sight distance through the cluster, vp is the cluster peculiar velocity, ne is the electron density and T is the Thompson scattering cross section. The quantity n dl is the optical depth to Thompson scattering, ⌧, with typical value 0.01 for the e T ⇠ Rclusters we observe. By observing the temperature increment or decrement in the direction of a galaxy cluster, the line of sight cluster peculiar velocity can be estimated. For clusters 5 in our sample we expect T /T 10 . To date no detection of a cluster velocity has KSZ ⇠ been made using the kSZ e↵ect. A bulk motion of mass in the universe will cause a dipole pattern in temperature at the 5.2. THEORY 189

cluster positions. The kSZ signal from each cluster that is part of a bulk motion is given by,

T v KSZ = bulk ⌧ cos ✓ , (2) T c where ⌧ is the optical depth of the cluster and ✓ is the angle between the cluster position and the bulk ﬂow direction. Cluster motions with more structure than monopole and dipole terms can be described by decomposing the cluster velocity ﬁeld into spherical harmonics.

The electrons within massive clusters have temperatures that are typically a few keV. The thermal motion of these electrons causes an increase in the energy of the scattered photons, which is the thermal Sunyaev-Zeldovich (tSZ) e↵ect [23]. The fractional temperature change in the radiation in the non-relativistic limit is

T ex + 1 k T TSZ = x 4 dl B e n , (3) T ex 1 m c2 e T ! Z e where x = h⌫/kBTCMB, ⌫ is the radiation frequency, dl is the line of sight distance through the cluster and Te is the electron temperature. To determine the importance of relativistic corrections to Eq. 3, we have calculated the thermal SZ spectrum of all 736 clusters in our cluster sample (we describe the cluster modeling we use in Sec. 5.4.2), using Eq. 3 to calculate the tSZ emission and also including relativistic corrections using the model of [24]. We combine the spectra, weighting the spectrum from each cluster with the same weight that we use when we measure the bulk flow signal (we describe how we calculate the bulk flow signal in Sec. 5.4.4). We find that the di↵erence between the flux calculated using Eq. 3 and the flux from the relativistic formula is less than 5% at all of the frequencies we use, and so for our purposes we can safely ignore the relativistic corrections.

4 For typical clusters T /T 10 at 90 GHz, and the ratio of the kinetic signal to the TSZ ⇠ thermal signal is (assuming the clusters are isothermal),

1 T vpmec vp T KSZ 0.1 e . (4) TTSZ ⇠ kBTe ⇠ 300km/s 5keV ! ✓ ◆ The thermal signal is therefore a signiﬁcant contaminant to any measurement of the kinetic signal. However, since there are not expected to be intrinsic large scale moments in the tSZ 190 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

signal, the tSZ dipole amplitude is expected to be smaller than the monopole amplitude. Unlike the tSZ signal, the kSZ signal has an identical frequency spectrum to the CMB. The kSZ signal can therefore only be distinguished from CMB ﬂuctuations by the di↵erent spatial properties of the two signals. The CMB power peaks on degree scales, whereas the SZ emission from galaxy clusters typically varies on arcminute scales. The signal we detect is smeared out by the instrumental beams of the experiment, 300.6 in the WMAP Q band, 210 in the V band and 130.2 in the W band.

5.2.2 Expected Signal for a Generic Velocity Tracer

1 On scales larger than 10h Mpc, fluctuations in the matter density of the universe are ⇠ Gaussian distributed [25, and references therein]. The inhomogeneities in the matter density cause galaxy clusters to have peculiar velocities, with typical values of 300 km/s [e.g., ⇠ 26]. If a sample of galaxy clusters within a region is chosen, that sample will have a non- zero bulk motion due to the non-uniform matter density on even larger scales. Although the direction of this bulk motion is not determined by the ⇤CDM model, the rms amplitude can be calculated given a set of cosmological parameters. We now calculate this amplitude. We can estimate the expected cluster bulk flow velocity by expanding the line of sight peculiar velocity distribution in spherical harmonics. The rms bulk flow velocity will be the power in the first multipole. We follow the derivation for the density distribution given in Peebles [27]. We decompose the line of sight peculiar velocity into spherical harmonics,

Y`m(rˆ), with the amplitude of the `,m mode given by:

2 a`m = drr (r) d⌦rY⇤ (rˆ) v(r) rˆ , (5) `m · Z Z where r is the comoving radial distance, is the comoving number density of objects in the sample, v is the object peculiar velocity and ⌦r is the solid angle. We approximate (r) by an isotropic function, which we estimate by calculating the number density of clusters in our sample within radius r. We normalize such that:

R drr2(r) = 1 , (6) Z0 5.2. THEORY 191

100

Number of clusters 20

0 0.0 0.1 0.2 0.3 0.4 0.5 z

Figure 5.1 Redshift distribution for the cluster sample used in this paper.

where R is the comoving distance within which the a`m are calculated. Fig. 5.1 shows the redshift distribution of our cluster sample used to calculate .

The power in multipole ` is deﬁned as C = a 2 where the average is over all values ` | `m| of m within multipole `. If the line of sight peculiarD velocityE over the whole sky is a Gaus- sian random ﬁeld, as expected in the ⇤CDM model, then it is completely described by C`. The power in the dipole (` = 1) is (the details of the derivation are given in Appendix 5 A):

2 2 C = a 2 = f 2H2 dkP(k) drr2(r) ( j (kr) 2 j (kr)) , (7) 1 | 1m| 9⇡ 0 1 2 D E Z Z ! where P(k) is the matter power spectrum, f = (a/D) dD/da, a is the scale factor, D is the growth function, H0 is the Hubble constant and j1 and j2 are spherical Bessel functions. Since the clusters we observe are all at redshift less than one, we approximate f by its 0.6 value today, taking it to be equal to ⌦m . We obtain a prediction of the rms dipole velocity shown in Fig. 5.2. Since our cluster sample contains few clusters with redshifts greater than 0.3, the bulk velocity of our entire cluster sample is largely determined by clusters ⇠ 192 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

150

100

50 RMS velocity [km/s]

0 0.01 0.10 1.00 Upper redshift of shell

Figure 5.2 Expected bulk flow velocity in ⇤CDM cosmology with a selection function calculated from our cluster sample in redshift shells extending from z = 0 to the specified x-axis value. The dashed line shows the result with a uniform selection function. The dark and light shaded areas are the 68% and 95% confidence limits from cosmic variance. with redshifts less than this. The e↵ect of the selection function is therefore to increase the expected dipole velocity in shells extending to redshifts greater than 0.3. The shaded area in Fig. 5.2 is the uncertainty from cosmic variance. The additional uncertainty on C1 from sample variance is inversely proportional to the number of clusters that are used to measure the dipole. For redshift shells extending beyond z = 0.05 our cluster sample contains over one hundred clusters and so the sample variance in Fig. 5.2 will be more than ten times smaller than the cosmic variance. We therefore do not include it in the figure. A measurement of a kSZ bulk flow with an amplitude larger than that expected from cosmic variance would be an important result if confirmed, requiring modifications of inflation to explain it [e.g., 4, 5]. A measurement of a kSZ monopole with a velocity greater than that expected by sample variance would be a violation of the Copernican principle, which states that we do not live in a specially favored place in the universe. However, a measurement of the kSZ monopole is more susceptible to contamination. The thermal 5.3. DATA 193

SZ signal, as well as the radio point source signal from clusters, are both expected to be isotropic, with any dipole signal coming from sample variance. Any dipole signal from tSZ or radio point sources is therefore expected to be weaker than any corresponding monopole signal. The radio source monopole at the cluster locations could be removed by observing and subtracting the emission, or by utilizing the di↵erent spectra of the radio source and kSZ signals. For thermal SZ the monopole can be suppressed spectrally.

5.3 Data

We have searched for a cluster dipole in the WMAP 7 year maps1 [6]. We use the foreground reduced maps produced by WMAP. These were generated by removing a foreground model from each ‘unreduced’ map [28]. Both the K and Ka band maps were used to produce the galactic foreground model, and so there are no K or Ka band foreground reduced maps. The maps are pixelized in the Healpix format [29] with a pixel size of 70. The CMB dipole has been removed from each map. Removing the CMB dipole from the maps does not significantly change the cluster dipole that we search for. We have tested the e↵ect of removing the CMB dipole by simulating kSZ maps (our simulation procedure is described in Sec. 5.4.2), adding a CMB dipole signal, and then removing it using a least squares fit to all map pixels. We find that the amplitude of the cluster dipole in the resulting Q band maps is changed by less than 1%, and the direction of the dipole is changed by less than 250. In the V and W band channels the change is smaller due to a stronger cluster signal relative to the CMB dipole signal. We use the two Q (41 GHz), two V (61 GHz) and four W (94 GHz) band maps from each di↵erencing assembly. We exclude the K and Ka band maps from the analysis since no foreground reduced maps are available. The galactic synchrotron and extra-galactic ↵ radio emission has a spectral index ↵ > 0 (with source flux, S ⌫ ), and the emission is ⇠ stronger in the K and Ka band than in the Q, V or W bands. Although the instrument noise is lower in the K and Ka bands than in the Q, V or W bands, the cluster signal is weaker, due to dilution by the larger beam FWHM. Including the K and Ka band channels therefore

1The WMAP maps, beam functions and galactic masks are available at http://lambda.gsfc.nasa. gov/product/map/current/m_products.cfm 194 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

only increases the signal to noise of the cluster dipole measurement by 10% and so we ⇠ exclude them from our analysis. The noise per pixel is 65µK (Q band), 80µK (V band), and 137µK (W band). The ⇠ ⇠ ⇠ noise is lower near the ecliptic poles due to the WMAP scan strategy. The beam FWHM are 300.6 (Q band), 210 (V band), and 130.2 (W band). We generate noise simulations using the prescription described in Limon et al. [30]. The noise is assumed to be uncorrelated between pixels, which is a good approximation for the temperature maps that we use [30]. We use the WMAP beam transfer functions, which are the square roots of the window functions used for the power spectrum analysis, to construct our matched ﬁlters and smooth our simulated maps. The WMAP extended temperature analysis mask is used to exclude the galaxy and other known regions of high foreground emission. Any cluster that lies behind this mask is not included in our cluster sample.

5.3.1 The X-ray Selected Cluster Sample

Following KAKE we derive our cluster sample from the REFLEX [31], BCS [32], Ex- tended BCS [33] and CIZA [35, 34] cluster catalogs. These catalogs contain 447, 206, 107, and 130 clusters respectively. After removing 20 overlapping clusters we obtain a sample of 870 clusters. After removing clusters whose center lies behind the WMAP galactic mask we find 736 clusters (when using the WMAP Kp0 galactic mask that was used in the KAKE analysis we find 771 clusters in the sample). The redshift distribution of the cluster sample is shown in Fig. 5.1. We convert the luminosities in the BCS, BCSe and CIZA catalogs to the values for a ⇤CDM universe. For the REFLEX sample we use a catalog with the luminosities already converted to an h = 0.7, ⇤CDM cosmology so no correction is necessary for these clusters. To derive average electron density and temperature profiles for our cluster sample we use the Archive of Chandra Cluster Entropy Profile Tables [ACCEPT, 36, 37]. This is a sample of 239 clusters observed by the Chandra X-ray Observatory [38], with electron density and temperature profiles for each cluster derived from a deprojection analysis. 145 of the ACCEPT clusters overlap with our sample. 5.4. METHOD 195

5.3.2 External Simulations

To test for contamination from galactic emission, thermal SZ and unresolved radio sources, we use simulated maps produced using the Planck Sky Model2 (PSM). The PSM is a set of programs and data used for the simulation of full sky microwave maps that includes galactic synchrotron, dust and free-free emission, kinetic and thermal SZ, radio and infrared point sources and CMB. The PSM simulations do not include a kinetic SZ monopole or cluster bulk flow velocity. We use one realization of the sky generated at the WMAP Q, V and W band center frequencies. To test the e↵ect of radio point sources we use 100 simulated maps of the unresolved radio background produced by Colombo and Pierpaoli [39]. They combine sources from the NRAO-VLA Sky Survey (NVSS) catalog [40] with higher frequency surveys to extrapolate the radio point source emission at 1.4 GHz to the WMAP frequencies. The radio source flux distribution in the maps at each frequency therefore reflects the probability distribution for the flux of each NVSS source.

5.4 Method

5.4.1 Outline of the Method

The bulk motion of many galaxy clusters relative to the CMB rest frame creates a signal in the WMAP temperature maps that has a dipole pattern at the location of clusters. We now describe the method we use to recover the amplitude and direction of any dipole motion using the WMAP CMB maps. In addition to kSZ the dipole signal at the position of clusters includes contributions from CMB, instrumental noise, tSZ, galactic emission and infrared and radio point source emission. The first step is to filter the maps to enhance the kSZ signal relative to the other terms. We use two filters, one of which enhances the cluster signal relative to the CMB and instrument noise, the other filter also removes the thermal SZ term. The galactic term is suppressed by masking the WMAP maps with the WMAP extended temperature analysis mask, and only fitting for a cluster dipole outside of this region. In Sec. 5.5.2 we give upper

2http://www.apc.univ-paris7.fr/APC_CS/Recherche/Adamis/PSM/psky-en.php 196 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

limits on the radio point source contamination and in Sec. 5.5.3 we give upper limits on the galactic signal. Using the PSM simulations we find a negligible contribution from infrared point source emission. The next step is to calculate the monopole and dipole at the locations of the ROSAT observed clusters in the filtered maps. We use simulated kinetic SZ maps to convert the amplitude of the dipole at the cluster locations to a bulk flow velocity using a method we describe in Sec. 5.4.5. Our simulated kinetic and thermal SZ maps include signal only from the ROSAT observed clusters that we use, and will be described in Sec. 5.4.2. There are two ways we could calculate the cluster velocity dipole. We could calculate the optical depth through each cluster and use Eq. 1 to estimate the velocity of each cluster. We could then fit for a dipole in the cluster velocity distribution. Alternatively, we could calculate the dipole at the cluster positions in the WMAP maps and convert the dipole signal to a velocity using an estimate of the average optical depth of the clusters in our sample. The two methods di↵er in the weight given to each cluster in the dipole fit. We expect the signal to noise of the latter method to be higher since all clusters are given equal weight in the fit, whereas in the former method clusters with a larger optical depth (and hence a greater kSZ signal in the map) are given less weight. We follow KAKE and choose the latter method. We split the cluster sample into di↵erent redshift bins (some of which overlap) and calculate the cluster dipole in each. To calculate the errors on the monopole, dipole and higher order modes, we generate 100 realizations of the CMB, convolve the maps with the beams from each of the eight channels and add instrument noise using the procedure described in Limon et al. [30]. We pass these maps through our pipeline to find the distribution of monopole, dipole and higher order modes.

5.4.2 Cluster Modeling

Our measurement of the WMAP temperature dipole does not depend on any cluster simulations. However, in order to convert the cluster temperature dipole measurement into a velocity dipole we require simulated maps of the kSZ emission from our cluster sample. In addition, we require tSZ realizations to estimate the level of tSZ contamination in our 5.4. METHOD 197

results.

The clusters in our sample have typical angular sizes of 10, which is small compared ⇠ with the WMAP beam FWHM, 300.6 in Q band and 130.2 in W band. For this reason we model the clusters as point sources convolved by the WMAP beams.

Optical Depth Determination

In order to generate kSZ realizations with a simulated bulk flow velocity we require the optical depth to Thompson scattering for every cluster in our sample. We calculate this using average electron density and temperature profiles that we derive from the ACCEPT catalog (described in Sec. 5.3.1). To calculate the electron density and temperature profiles we select 145 ACCEPT clusters that overlap with our sample. We average the electron density and temperature profiles of the overlapping clusters within radial bins. We use bins with a smaller size nearer to the center of the cluster where the error on the electron density and temperature is lower (the error is provided in the catalog). The mean bin size is 10 kpc, and the largest radial bin extends to 1 Mpc from the cluster center. Each profile is normalized so that it has unit value at the center of the cluster. The densities and temperatures of all of the clusters are then averaged within each radial bin. We compute the kSZ and tSZ signals using these profiles, giving the profiles a di↵erent normalization for each cluster, as described in the following paragraph. Extrapolating the density profile to larger radii using a model [41], we find a negligible contribution to the optical depth from beyond this region. The profiles are not significantly changed when all of the clusters in the ACCEPT sample are used to calculate them. Since we use the same electron density profile for each cluster, we only need to calculate the normalization to that profile to estimate the optical depth. The optical depth is calculated by integrating the electron density along the line of sight,

fe(r)rdr ⌧(R) = T Ne , (8) pr2 R2 Z where R = dA✓ is the distance from the cluster center perpendicular to the line of sight, dA is the angular diameter distance, T is the Thompson scattering cross section, fe(r) is the 198 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

normalized electron density profile and Ne is the normalization to that profile. We calculate the normalization to the electron density profile by requiring that the cluster bolometric luminosity in the ROSAT catalogs is equal to the luminosity from Bremsstrahlung emission, which depends on Ne [42],

1 2 6 32⇡ 2⇡kB Q 2 2 2 1 L = Z gN N d r 4⇡r f (r) f (r)T(r) 2 erg/s , (9) 3 3m m c3h e i e i e ! e Z where kB is the Boltzmann constant, me is the electron mass, Q is the electron charge, c is the speed of light, h is Planck’s constant, Z is the number of protons per nucleus, g is the frequency averaged Gaunt factor (of order unity), T(r) is the electron temperature, fe(r) and fi(r) are the normalized electron and ion density proﬁles (which we take to be equal) and

Ne and Ni are the normalizations to those profiles. We normalize the temperature profile such that the X-ray emission weighted temperature is equal to TX, which we calculate from 0.33 0.01 the X-ray luminosity of each cluster using the relation T = (2.76 0.08)L ± [43]. X ± X Using this procedure we find that the mean central optical depth of our cluster sample 3 is (4.9 0.9) 10 . The mean central optical depth of the 145 clusters that overlap with ± ⇥ 3 the ACCEPT sample is ⌧ = (6.1 1.0) 10 which is in agreement with the value 0 ± ⇥ calculated using each individual electron density profile in the ACCEPT catalog, ⌧0 = 3 (6.7 0.9) 10 . This gives us confidence that we can calculate accurate values of the ± ⇥ optical depth for clusters that are not in the ACCEPT sample. The dominant source of uncertainty in our optical depth calculation is the cluster X-ray luminosity, from the X-ray catalogs. Propagating all errors through to the optical depth gives uncertainties of 15%. ⇠ These errors are propagated through to the simulated kSZ maps which allows us to calculate the precision of our method. We have not accounted for any anisotropy in the optical depth distribution of our clusters or accounted for the di↵erent methods used to calculate the X-ray luminosities in the di↵erent X-ray catalogs we use. Both of these steps would require an independent analysis of the X-ray data. However, since our dipole fit does not depend on the X-ray luminosities our null detection is una↵ected. Any error in the X-ray luminosities will only a↵ect the conversion of the dipole amplitude from µK to km/s. Since the X-ray luminosities are used to calculate the optical depth any asymmetry in the optical depths will be accounted for 5.4. METHOD 199

when converting from µK to km/s.

KSZ and TSZ Simulations

The kSZ signal is calculated from Eq. 1. We give each cluster in our simulated maps a bulk velocity using Eq. 2 and a random velocity drawn from a Gaussian distribution, with variance [e.g., 44] 2 2 f H 1 2 = 0 dk P(k) W(kR) 2 , (10) v 6⇡2 | | Z0 where W(kR) 2 is the Fourier transform of the window function, and again we use f ⌦0.6. | | ⇡ m 1 To select cluster scales we use a top hat window function with R = 10h Mpc, and ﬁnd

v = 272 km/s. We ignore correlations between the velocities of di↵erent clusters, which we expect to be small for our cluster sample [eg. 26, 44]. The expected tSZ signal is calculated by integrating Eq. 3 through the cluster using the same electron density and temperature profiles calculated in Sec. 5.4.2. Since WMAP is sensitive enough to detect the tSZ signal in stacked images we have checked that the average tSZ signal in our simulations agrees with WMAP observations. Following the method in Atrio-Barandela et al. [45] (also 46), we have stacked maps centered on each cluster to produce a coadded map in the Q, V and W bands from the WMAP maps and our simulated tSZ maps. We find that the residual after subtracting the two is consistent with CMB and instrument noise. In Table 5.1 we show the simulated thermal SZ monopole in maps filtered by the matched filter we use in our analysis. We compare this to the monopole found in the WMAP maps filtered by the same filter. Although the monopole in our simulated maps is biased high it is within 2 of the measured monopole in all except the two lowest redshift shells. This gives us confidence that our optical depth calculation is reasonable.

5.4.3 Filters

The distribution of our cluster sample on the sky is not isotropic and so measurements of the peculiar velocity multipoles will be contaminated by higher order CMB multipoles. The higher order CMB multipoles couple to the signal we are trying to measure through the 200 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

Table 5.1 Simulated thermal SZ monopole and measured monopole.

zmin zmax Ncl Simulated tSZ Monopole, a0 [µK] Measured Monopole, a0 [µK] 0.0 0.04 95 -78.7 -22 33 ± 0.0 0.05 139 -84.2 -41 27 ± 0.0 0.06 192 -97.0 -62 24 ± 0.0 0.08 294 -97.9 -63 18 ± 0.0 0.12 445 -99.5 -73 13 ± 0.0 0.16 546 -103.4 -81 12 ± 0.0 0.20 619 -109.1 -85 12 ± 0.0 1.0 736 -123.9 -91 11 ± 0.05 0.30 578 -128.6 -99 12 ± 0.12 0.30 271 -152.2 -112 17 ± mask we use to cover non-cluster regions of the WMAP maps. To reduce the contamination from the CMB we ﬁlter the maps.

Two filters are used, and described here. The first is an optimal matched filter that removes low-order CMB multipoles by utilizing information at locations other than the cluster position to distinguish a CMB multipole from a kSZ multipole. The second filter uses both spatial information and the spectral di↵erence between the thermal and kinematic SZ e↵ects to remove the contaminating tSZ signal.

An optimal filter is constructed for each channel such that when the filtered maps from each channel are combined, the kSZ signal to noise ratio is maximized. In order to construct CMB noise our filters we require a matrix of cross spectra, C` = C` + C` , that describes the statistical properties of the maps to be filtered. We assume that the noise in map ⌫, n⌫(✓), is a homogeneous and isotropic random field. The cross power spectrum of maps ⌫1 and ⌫2 is equal to n` ,⌫ n⇤ , where n` ,⌫ are the coecients in the spherical harmonic expansion m 1 `m,⌫2 m of n⌫(✓),D and the averageE is over m. We generate C` by averaging the cross power spectra of 100 CMB and noise realizations.

The maps in each channel are ﬁltered and combined to produce the spherical harmonic coecients of the ﬁltered map as follows,

nchannels i i f`m = `a`m , (11) = Xi 1 5.4. METHOD 201

i i where a`m are the spherical harmonic coecients of the WMAP channel i map and ` is the ﬁlter for channel i. We include the factor of p4⇡/2` + 1 from spherical convolution in

`. The first filter we use has a filter function of the form [47]:

1 m C` = B` , (12) `

T 1 where m indicates that the ﬁlter is a matched ﬁlter, = (1/npix) `(2` + 1)B` C` B`, npix is ⇡/ 2 2 the number of map pixels and (4 npix)B` is an estimate of theP kSZ power spectrum from a single cluster, in each channel (we assume the beam convolved sources are symmetric).

Since the WMAP beams are much larger than galaxy clusters we approximate B` by the WMAP beam function. We have calculated the filter using the WMAP beam profile convolved with an average cluster profile and also a cluster profile modeled as a model [41] and find no significant di↵erence in the results. A simple modification to this filter allows the thermal SZ bias to be removed [e.g., 48, 49, for the flat and curved sky cases respectively],

1 u C` = (↵B` F`) , (13) ` where u indicates that the filter is an unbiased filter, F` is an estimate of the thermal SZ sig- 2 T 1 nal in each channel, = ↵ is a normalization factor, ↵ = (1/npix) (2` + 1)F C F`, ` ` ` T 1 T 1 = / ` + = / ` + (1 npix) `(2 1)B` C` F` and again (1 npix) `(2 1)B` CP` B`. We are using x x maps with units of thermodynamic temperature and so F` = [x(e + 1)/(e 1) 4.0] B`. P P m u Figs. 5.3 and 5.4 show the functions ` and ` for each channel. These filters have very di↵erent behaviors and we now discuss why this arises. The matched filter (hereafter MF) suppresses the input map at multipoles less than 300 ⇠ (large scales) due to the strong CMB signal at these multipoles. The filter function also suppresses the map at multipoles greater than 1000 (small scales), where the expected kSZ ⇠ signal to noise ratio is low. The W band filter is non-zero up to the highest multipoles since the W band beams are smaller, and so the signal extends to higher multipoles. Between multipoles 300-1000 the signal to noise ratio is highest and so these scales are retained in the filtered map. The bumps near multipoles 500 and 700 in the MF correspond to the 202 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

3.5 Q1 3.0 Q2 V1 V2 2.5 W1 W2 W3 m l 2.0 W4 Φ 1.5

1.0

0.5 0.0 0 200 400 600 800 1000 1200 1400 l

Figure 5.3 Multi-frequency matched ﬁlters for the WMAP channels.

0 u l Φ −20 Q1 Q2 V1 −40 V2 W1 W2 −60 W3 W4 −80 0 200 400 600 800 1000 1200 1400 l

Figure 5.4 Unbiased multi-frequency matched ﬁlters. A discussion of the ﬁlter shape is given in the text. 5.4. METHOD 203

troughs in the CMB spectrum the filter amplifies scales where the contaminating CMB emission is weaker. The tSZ bias removing filter (hereafter UF) in Fig. 5.4 is less intuitive. At low multipoles the CMB signal in each channel is almost identical because each channel observes the same CMB sky, and at low multipoles the beam functions are all close to unity. At low multipoles the CMB can therefore be removed by subtracting the Q and W band maps. Since there are two Q band channels and four W band channels the CMB is removed by giving the Q band maps double the weight of the W band maps in the subtraction. This explains why the two Q band filters have an absolute value double that of the four W band filters at low multipoles. This is reflected in Eq. 13 by the o↵-diagonal elements being almost equal to the diagonal elements in the cross-power spectra matrix at low multipoles. If the CMB sky in each channel were di↵erent then the filter would look similar to the MF but with di↵erent amplitudes for each channel. In Fig. 5.5 we show what the tSZ bias removing filters would look like for an experiment with four frequency channels: 100 GHz,

143 GHz, 217 GHz and 353 GHz, with beam FWHM of 100,70,50 and 50 respectively, and white noise levels: 25, 15, 25 and 75 µK/K arcmin. The fact that there are no peaks in the WMAP filters is a reflection of the limited frequency coverage and large beams. Since the UF combines channels in a way that removes the tSZ signal, the filters do not increase the kSZ signal to noise ratio as much as the MF. The signal to noise of the cluster temperature dipole measurement is therefore lower. Fig. 5.6 shows the spectra of a simulated map containing only the kSZ signal at the ROSAT cluster locations with and without a bulk flow component of 1000 km/s, and convolved by the WMAP beams. The maps with and without a bulk flow both have spectra that are identical to the WMAP beam function at high multipoles. The map with the bulk flow component contains more power at all multipoles as well as additional power at the dipole and octupole modes. Although the di↵erence between the spectra with and without a bulk flow looks predominantly like an amplitude shift, the bulk flow signal is recoverable by information contained in the spherical harmonic m modes, which are averaged to create the spectra. Our filters are designed to optimize the detection of sources shaped like the WMAP beams. The similarity of the simulated kSZ spectrum with a bulk flow component to the beam profile indicates that our filter will have the desired e↵ect of increasing the 204 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

100 GHz 15 143 GHz 217 GHz 353 GHz

10 Φ

0 500 1000 1500 2000 2500 3000 l

Figure 5.5 Thermal SZ bias removing ﬁlters for an experiment with four frequencies: 100 GHz, 143 GHz, 217 GHz and 353 GHz, beam FWHM of 100,70,50 and 50, and white noise levels 25, 15, 25 and 75 µK/K arcmin. 5.4. METHOD 205

10−6

10−8 [µK] l

C Q1 dipole V1 dipole 10−10 W1 dipole Q1 V1 W1 10−12 1 10 100 1000 l

Figure 5.6 Spectra of a simulated kSZ map (with signal only at the locations of our cluster sample) convolved with the WMAP beams. The simulations were performed both with (top 3 spectra) and without (bottom 3 spectra) a bulk flow component of amplitude 1000 km/s. The dashed lines are the WMAP beam functions, scaled to the signal amplitude. signal to noise of the cluster dipole measurement. Figs. 5.7 and 5.8 show the WMAP maps after filtering with the MF and UF respectively. Both filters have suppressed the CMB signal and the only visible structure in the map is the low noise region around the ecliptic poles. The maps are noise dominated; the larger values of the map in Fig. 5.8 reflects the lower signal to noise ratio obtained with the UF. Since we are only calculating the dipole in a small region at the center of the clusters, we could filter the maps only in this region. This would save computation time but care would need to be taken to avoid ringing e↵ects around the edge of the mask. For this reason we filter the maps in all regions outside the WMAP galactic mask.

Wiener Filters

We now compare our filter to the Wiener filter used by KAKE. A Wiener filter (hereafter WF) minimizes the squared di↵erence between the filtered map and an estimate of the 206 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

Figure 5.7 Sum of the eight WMAP maps in the Q, V and W bands ﬁltered by the matched ﬁlters in Fig. 5.3. The map is noise dominated.

Figure 5.8 Sum of the eight WMAP maps in the Q, V and W bands filtered by the tSZ removing filters in Fig. 5.4. The values in this map are larger than in Fig. 5.7 since the map is noise dominated and the unbiased filters give a lower signal to noise measurement of the kSZ signal. 5.4. METHOD 207

signal. Although a multi-frequency WF can be constructed, we follow KAKE and create a ﬁlter for each map that is designed to suppress the CMB component. The KAKE ﬁlter is [1]: Csky/ f CCMBB2 w = ` sky ` ` , ` sky (14) C` / fsky

CMB sky where C` is the CMB power spectrum, B` is the WMAP beam function, C` is the spectrum of the WMAP map estimated outside of the galactic mask and fsky is the fraction of the sky outside of the mask. The full sky spectra could alternatively be calculated by unmasked 1 masked masked deconvolving the mask: C = ` M C C / fsky, where M`` is the multi- ` 0 ``0 ` ⇡ ` 0 pole mixing matrix which accounts forP the cut sky and is calculated from the galactic mask (see eg. Appendix A of Hivon et al. [50] for details). The KAKE filter we use is shown in Fig. 5.9. We have determined that this is the same Wiener filter that was used in the KAKE analysis and give details of our pipeline in Appendix 5 B. The KAKE filter is constructed from the spectrum of the map that is to be filtered, and maps filtered with it su↵er from low multipole fluctuations, caused by cosmic variance. We create our Wiener filter in such a way that it does not su↵er from this problem [51],

Cnoise/ f w = ` sky , ` CMB 2 noise (15) C` B` + C` / fsky

noise where C` is the power spectrum of the instrument noise plus all foreground components in the masked map. This filter is not normalized and so pixels in the filtered map are not equal to the estimated amplitudes of the kSZ signal in the unfiltered map pixels. Since we model our clusters as point sources convolved by the beam the normalization factor would be npix/ `(2` + 1)B``. The normalization is instead applied later when the dipole amplitude is convertedP from Kelvin to km/s. Our Wiener filter is shown in Fig. 5.10. Since the noise in the WMAP maps has a flat spectrum and the CMB signal decreases at high multipoles, the WF acts like a high pass filter for each of the WMAP channels. At low multipoles the shape of this filter is similar to our MF (although the relative normalizations of the channels is not), but at high multipoles the shapes di↵er since the WF does not take into account the fact that the cluster signal decays exponentially with increasing multipole, while the noise spectrum remains flat. We find that both our Wiener filter and 208 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

1.0

0.5 w l Φ Q1 Q2 V1 0.0 V2 W1 W2 W3 W4 −0.5 0 200 400 600 800 1000 l

Figure 5.9 Wiener ﬁlter used by KAKE.

1.0

0.8

0.6 w l Φ Q1 0.4 Q2 V1 V2 0.2 W1 W2 W3 0.0 W4

0 200 400 600 800 1000 l

Figure 5.10 Wiener ﬁlter for each of the WMAP channels. 5.4. METHOD 209

0.5 Our Wiener Filter 0.4 KAKE Filter

0.3 band) − (V l

Φ 0.2

l B

0.1

0.0 200 400 600 800 1000 l

Figure 5.11 Suppression of the kSZ signal by the beam and ﬁlter in the V band.

the KAKE filter suppress the cluster signal approximately equally as shown in Fig. 5.11. At the multipoles where the signal peaks ( 400 in the V band) the two filters suppress ⇠ power by a similar amount. We find that our Wiener filter reduces the noise in the a1m by a factor of 2 relative to the KAKE filter by reducing CMB noise at multipoles less than ⇠ 100 that is correlated between the WMAP channels (we give a more detailed comparison ⇠ of the filters in Appendix 5 B), and is therefore more sensitive to the cluster signal.

For the matched filter and unbiased filter we calculate the cluster dipole in the pixels that overlap with the centers of each cluster. For our Wiener filter we instead use a 150 aperture around each cluster, in order to increase the signal to noise of the measurement. We have tried fitting for the dipole using di↵erent aperture sizes out to 300 as well as di↵erent weighting schemes for the pixels within the aperture: uniform weighting, weighting by the inverse noise variance and weighting by the expected cluster signal to noise ratio. The results presented in Table 5.4 have a uniform weighting within the aperture, which was chosen to give a good signal to noise measurement while keeping our results free from any bias caused by incorrect signal or noise estimates. 210 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

The WF that we use is not a multi-frequency filter (although in principle a multi- frequency WF could be used) and so there is a filtered map for each of the WMAP channels. We filter each map and compute the cluster dipole in each. We then average the dipoles from the di↵erent channels with uniform weights:

8 1 (i) a1m = a1m . (16) 8 = Xi 1 The signal to noise could be increased by weighting the results from each channel by the expected signal to noise ratio of the kSZ signal. In this scheme the weights given to each channel are approximately equal the noise at higher frequencies is compensated for by the decreased beam FWHM, and therefore larger cluster signal.

5.4.4 Dipole Fitting Procedure

To measure the cluster dipole we perform a weighted least squares fit to a dipole function at the locations of the clusters in the filtered map. We also fit for the monopole and, separately, we perform the fit for modes up to ` = 2, and up to ` = 3, 4, 5. Our dipole fitting method is based on the Healpix IDL procedure remove dipole [29]. We fit for a vector of coecients, , of the real spherical harmonics using the least squares formula,

T 1 T = (X WX) X Wy , (17) where y is the data map, W is a diagonal matrix with diagonal values equal to a weights map, and X is a matrix giving the contribution of the fitting function to each pixel. If a monopole and three dipole coecients are fit to the data, then X is an n 4 matrix. We pix ⇥ choose the weights map to have non-zero values only in the central cluster pixels, and equal to the inverse noise variance of the filtered map. The noise variance is calculated from 100 filtered WMAP CMB and noise realizations. The vector are the coecients of the real 5.4. METHOD 211

spherical harmonics, Rlm, deﬁned by

Y`0 if m = 0

8 1 m Rlm = (Y` + ( 1) Y` ) if m > 0 (18) > p2 m m > > 1 m <> (Y` m ( 1) Y`m) if m < 0 . > i p2 > > The matrix XT WX is a mixing:> matrix that couples di↵erent spherical harmonic modes together. The e↵ect of the mask W is that a least squares fitting process can then be used on the masked map to determine the dipole from clusters alone. Caution is needed because information is lost when the map is masked. Consequently if too few modes are fitted, power will ‘leak’ from higher order modes into the fit parameters, corrupting the result.

Since the kSZ signal is proportional to the optical depth, in principle the signal to noise ratio of the measurement can be increased by weighting each pixel by the optical depth. We ﬁnd that the increase in signal to noise is approximately 10%. However, when ﬁltering with the MF this weighting scheme increases the contamination from thermal SZ, since the thermal SZ signal is also proportional to the optical depth. We therefore do not weight by the optical depth to reduce the risk of contaminating our result.

In principle the correlated errors on the three dipole m values can be calculated, with covariance matrix T 1 T T T 1 N = (X WX) X WCW X(X WX) , (19) where C is the pixel-pixel noise covariance matrix. Since it is not practical to compute a matrix as large as C, we calculate the covariance using simulations. The dominant source of error is CMB and instrument noise, which we estimate by passing 100 CMB and noise realizations through our pipeline, performing the least squares fit on each. There are additional sources of error from our uncertainty in the optical depth, and from the random component of the galaxy cluster peculiar velocities, which are calculated from Eq. 10. We estimate the errors from both of these terms by passing simulations of the kSZ signal through our analysis pipeline, performing the least squares fit on each realization. The scatter in the derived values of then provides an estimate of the noise correlations between the dipole directions, which is then used to calculate the 2 significance of the measured dipole in 212 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

each redshift shell. We find that the errors from the latter two terms are negligible compared to the CMB and noise error, and that the errors on the dipole directions are Gaussian with a narrower distribution in the direction perpendicular to the galactic plane. This is due to clusters lying within the galactic mask being removed from the fit. As a check on this process, we also estimate the errors from the WMAP maps by re- calculating the dipole fit after rotating our weights map away from the clusters. We rotate the weights map in increments of 100 in the galactic longitudinal direction. We remove points within 1 of the cluster center to reduce any residual SZ emission. This leaves us with 2147 di↵erent directions. This process allows us to preserve the angular distribution of the cluster sample in the fit. When either filter is used we find errors that are consistent with those found from the CMB noise realizations. The errors we find by rotating the weights map are less reliable than the those calculated from the CMB and noise realizations, since when we rotate our weights map some regions get rotated into the galactic mask, which we then exclude from the fit. The dipole is therefore calculated from fewer clusters in this scheme, and so the errors we quote are from the CMB and noise realizations. We find that 100 CMB and noise realizations is sucient to estimate our errors. Taking a typical error for one of the dipole coecients in the Wiener filtered maps to be 0.8µK we find that the standard deviation of the sample variance is 0.09µK for 100 noise realizations. Even if our errors are 10% too large we would still not detect a significant cluster dipole with our filters.

5.4.5 Conversion to Velocity Dipole

We create a conversion matrix to calculate the velocity dipole from the temperature dipole,

av = MaT , (20)

where av are the velocity monopole and dipole coecients, aT are the temperature coef- ﬁcients and M is a 4 4 conversion matrix. We calculate M by creating four simulated ⇥ maps of kSZ signal alone from our cluster sample. In one map all of the clusters are given a monopole velocity. Each of the other three maps has a bulk ﬂow velocity in one of three 5.4. METHOD 213

basis directions. These maps are passed through our analysis pipeline to use the same filtering and velocity fitting processes that are used for the real data. The four fit coecients 1 from each of the input maps are the elements of each row of M . The matrix M is close to diagonal in all redshift shells, and the velocity and temperature dipoles are close to align- ment, as they should be. To convert from µK to 1000 km/s we use the following values for the diagonal entries of M: 4.07 (our Wiener filter), 3.41 (KAKE Wiener filter), 0.075 (matched filter), and 0.12 (unbiased matched filter). For the dipole contribution:

100 12 10 c M = 0 0101 , (21) TCMB ⌧e↵ B C B 001C B C B C @B AC where ⌧e↵ is the e↵ective optical depth after filtering and we have used the same normalization as KAKE (which makes M a factor of p3/4⇡ smaller than for the conventional 5 definition of a ). For our Wiener filter we find ⌧ = 2.8 10 . This value is smaller than 1m e↵ ⇥ 3 the average optical depth of our cluster sample (⌧ = 4.9 10 ) due to the filtering, as well ⇥ as the fact that the signal is diluted by beam smoothing and averaging over the aperture used to calculate the cluster dipole. We find a lower e↵ective optical depth than the value 4 used in the KAKE analysis of ⌧ 10 , which could explain why we find velocity limits e↵ ⇠ 10 times larger. We are confident that we correctly recover the cluster velocities and ⇠ show in Fig. 5.13 that we correctly recover the velocities in simulations.

5.4.6 Tests of the Method

We have performed tests to check that the results from our filtering and dipole fitting procedures are not contaminated by systematic e↵ects. First we verify that we can correctly recover a known cluster bulk flow from simulated kSZ maps alone, then we consider maps containing kSZ, CMB and instrument noise. In Sec. 5.5 we describe the e↵ects of thermal SZ, unresolved sources, and galactic emission on this process. Using simulated maps with a kSZ component only, but that include a cluster bulk flow we have verified that we recover the input bulk flow velocity using our method. We assign all of the clusters in the simulated maps a kSZ signal from both the bulk flow and a random 214 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

velocity drawn from the expected ⇤CDM distribution in Eq. 10. For each choice of input bulk velocity we generate 25 realizations, filter the maps, and fit for a cluster dipole using the same pipeline that is used for the WMAP data. We repeat the process with three di↵erent input bulk flow directions: (galactic latitude, longitude) = (0,0), (0, 90) and (90,

0) and bulk flow velocities of 0 km/s, 500 km/s and 1000 km/s (7 cases with 25 realizations each). Fig. 5.12 shows the results of the fits. We do not plot the recovered velocity for the cases where the simulated bulk velocity is non-zero because we use our kSZ simulations to calibrate the recovered velocity, and so the mean recovered velocity is exact by design. The scatter in the recovered dipole direction is due to the random component of the cluster line of sight velocity (with variance given by Eq. 10) and uncertainty in the cluster optical depth. The recovered amplitude in the maps with no bulk flow velocity has significantly larger scatter when it is calculated using the UF. This reflects the lower signal to noise of this filter caused by the extra degree of freedom. We have repeated the test using simulations that additionally contain CMB and instrument noise. We create 100 realizations for each WMAP channel and add a bulk flow signal to the kSZ component, with a range of velocities logarithmically spaced between 100 and

100,000 km/s in the direction of galactic latitude 11 and longitude 103, which is the bulk flow direction found by KAKE in the redshift shell extending to z = 1. Fig. 5.13 shows the recovered bulk flow amplitude and Fig. 5.14 shows the error in the direction of the recovered dipole when all clusters are included in the fit. The lower signal to noise of the UF is apparent in both of these figures. Fig. 5.15 shows the recovered monopole velocity in a di↵erent set of 100 simulated WMAP realizations.

5.5 Systematic Effects

5.5.1 Thermal SZ

There are not expected to be intrinsic large scale moments in the tSZ signal, but due to the limited size of our cluster sample the tSZ dipole signal caused by random scatter is not negligible. We have examined the e↵ects of thermal SZ on our results using our own simulations of the tSZ emission from our cluster sample, which are described in Sec. 5.4.2, as 5.5. SYSTEMATIC EFFECTS 215

Matched Filter Unbiased Filter 1000 0 km/s 1000

100 Recovered 10 Amplitude [km/s] 100 6 500 km/s 5 4 3 2 1 0 6 1000 km/s 5 Recovered 4 Angle Error [deg] 3 2 1 0 0 0.05 0.10 0.15 0 0.05 0.10 0.15 0.20 Mean redshift Mean redshift

Figure 5.12 Left: Error in the recovered dipole direction from simulated kSZ maps with a bulk ﬂow velocity of 0 km/s (top), 500 km/s (middle) and 1000 km/s (bottom) in the directions: (latitude, longitude) = (0,0) black lines, (0, 90) blue lines and (90,0) red lines, using the MF. Right: Same for the UF. The arrows are the 95% conﬁdence limits. 216 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

100.0

10.0

1.0

Matched Filter Unbiased Filter Recovered velocity [1000 km/s] 0.1 0.1 1.0 10.0 100.0 Input velocity [1000 km/s]

Figure 5.13 Recovered bulk flow velocity in simulated maps containing CMB, noise and kSZ using the MF (black) and the UF (blue). All clusters are included in the fit. The x- axis is the bulk velocity input into the simulated maps, the y-axis is the recovered velocity. The red line indicates perfect recovery. When the input bulk flow is small, a dipole is not detected and the data points provide an estimate of the scatter in the recovered amplitude. 5.5. SYSTEMATIC EFFECTS 217

140 Matched Filter Unbiased Filter 120

100

40 Angle Error [degrees] 20

0.1 1.0 10.0 100.0 Input velocity [1000 km/s]

Figure 5.14 Error in the direction of the bulk ﬂow velocity in simulated maps containing CMB, noise and kSZ with the MF (black) and UF (blue), when all clusters are included in the ﬁt. 218 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

−5

Velocity error [1000 km/s] −10 Matched Filter Unbiased Filter −15 0.1 1.0 10.0 100.0 Input velocity [1000 km/s]

Figure 5.15 Error in the recovered monopole velocity in simulated maps containing CMB, noise and kSZ with the MF (black) and UF (blue). well as a simulated tSZ map produced using the Planck Sky Model, described in Sec. 5.3.2. We find that when we fit for the cluster dipole amplitude in our tSZ simulated maps filtered with the MF, using all clusters in our sample, there is a 3 bias in the y and z directions, which equates to a dipole amplitude of approximately 4000 400 km/s in the ± direction of l, b = (88,54) 40, which is 66 from the KAKE bulk flow direction in the ± same redshift shell. However, in the PSM tSZ map we find a reduced dipole of 2500 km/s, in the direction of l, b = (327,10), which is 137 from the KAKE bulk flow direction. These simulations suggest that contamination to the cluster dipole measurement is at the 1 level and so cannot be ignored. As expected, we find that the tSZ monopole in the simulated maps is large and equivalent to a kSZ signal from clusters with velocities of 12000 km/s in our simulations, and 10000 km/s in the PSM maps. The simulated tSZ maps filtered with the UF have a cluster monopole and dipole amplitude that are more than an order of magnitude lower. In our simulated maps containing only tSZ we find a recovered cluster dipole amplitude of approximately 200 km/s. In the PSM 5.5. SYSTEMATIC EFFECTS 219

simulated tSZ map we find a recovered cluster dipole amplitude of 70 km/s, significantly less than the measured cluster dipole and noise. The monopole amplitude is 200 km/s in our simulated tSZ map and 100 km/s in the PSM tSZ map. Although the UF removes the tSZ signal suciently for our purposes, it does not remove it to the ⇤CDM limit. The residual tSZ in the filtered maps, and the cause of the non-zero cluster monopole and dipole is source confusion. The emission from clusters with small angular separation overlaps more strongly in the Q band where the beam is large, than in the W band. When the channels are subtracted the tSZ signal does not subtract perfectly in the central cluster pixel and there is residual tSZ in the map. Since this only a↵ects a small number of clusters, it leaves a dipole signal in the map.

5.5.2 Radio Point Sources

At frequencies below 100 GHz extragalactic radio sources can be a significant contribution to CMB maps. We use simulations of the unresolved extra-galactic radio point source emission produced by Colombo and Pierpaoli [39] and summarized in Sec. 5.3.2, to verify that our bulk flow measurement is not significantly a↵ected by sources. Because the model is based on sources observed by NVSS, the simulated maps retain information about the distribution of sources on the sky and account for the increased radio emission at galaxy cluster positions, caused by clustering of radio galaxies. The uncertainty on the radio point source monopole and dipole in the maps recovered using the UF are larger than in the maps filtered with the MF. In the maps filtered with the MF we find a non-zero dipole introduced by radio sources with a signal of amplitude 0.2 in each of the dipole directions. In the redshift shell encompassing all clusters we find signals of 0.2,0.1 and 0.3 in the x, y and z directions previously defined. We find a monopole signal of 2, equivalent to a kSZ monopole velocity of approximately 2000 km/s. In the maps filtered with the UF we find a larger signal with an amplitude of 0.9, 1 and 1.1 in each of the dipole directions. The monopole signal is 6, as expected it is large because the radio sources all add signal to the maps whereas the dipole is caused by sample variance. Because the UF is designed to remove tSZ by a specific weighting of channels, the radio 220 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

source contribution is actually amplified by the filter. In the UF the Q band channel is given relatively more weight, so that emission with a tSZ spectrum is canceled when the channels are combined. The radio point source signal is strongest in the Q band and weakest in the W band. The UF therefore has a large contribution from the Q band radio signal that is not o↵set by the W band signal, where the radio point source signal is much weaker. The resulting map generated using the UF therefore has a greater residual point source signal than the map generated with the MF. The radio point source signal and the tSZ signal could in principle both be suppressed simultaneously by using a filter designed to remove both signals [51, e.g.,]. However, the increase in the number of degrees of freedom that this would introduce would further reduce the signal to noise of the measurement.

5.5.3 Galactic Emission

For our analysis we use the WMAP foreground reduced maps which have been processed by the WMAP team to suppress the galactic signal outside of the WMAP galactic mask [6]. We test the e↵ect that any residual galactic foreground would have on our results by repeating the analysis using maps that have not had the galactic components suppressed. The cluster monopole and dipole that we obtain from these maps is therefore an upper limit on any residual galactic contamination in our results. Figs. 5.16 and 5.17 show the sum of the filtered PSM simulated maps. Both maps show residual galactic emission that is not visible in the WMAP maps (Figs. 5.7 and 5.8) due to their foreground reduction procedure. The larger visible foreground signal in the PSM simulated map filtered with the UF is caused by galactic synchrotron. The synchrotron signal is smaller in the map filtered with the MF because the large scale galactic signal is removed at each frequency by the MF, but remains in the maps filtered by the UF. When the filtered maps at each frequency are combined, the signals from the galactic components are not canceled since the filters are designed to only cancel a signal that has a thermal SZ spectrum. Because the maps containing only the individual contribution of a particular foreground component are available for the PSM simulations, we have also run our analysis pipeline on each component separately to identify the source of any bias. The results when all clusters 5.5. SYSTEMATIC EFFECTS 221

Figure 5.16 Sum of the PSM simulated maps convolved with the WMAP beams and ﬁl- tered with the MF. This map shows galactic emission around the edges of the mask that is absent from the WMAP foreground reduced maps due to WMAP’s foreground reduction method [6]. The color scale is altered to better show the galactic emission by mimicking 1 the e↵ect of replacing the data by sinh (data) [29].

Figure 5.17 Sum of the PSM simulated maps convolved with the WMAP beams and ﬁltered 1 with the UF. The sinh color scale that was used in Fig. 5.16 is used here. 222 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

are included in the dipole fit are shown in Table 5.2. We find no significant bias in the monopole or dipole of the maps filtered by the MF. The combined map filtered by the UF has the largest galactic contribution from synchrotron emission, with a 5.7 signal in the monopole (labeled † in the table) and a 4 signal in the dipole direction that points towards the center of the galaxy (labeled †† in the table). The dust and free-free monopole signals largely cancel resulting in a 6.2 monopole signal. The significant dipole from synchrotron emission is partially canceled by the dust and free- free signals, resulting in a dipole signal that is not significant and would not be considered a detection. As a further check, we repeat the cluster dipole analysis with the cluster coordinates rotated by one degree away from the true cluster locations. We find that the emission from the galactic components is not significantly changed. Since the emission is not localized at galaxy cluster locations, we expect the galactic signals we find to be suppressed by the WMAP foreground reduction procedure, and so we expect the galactic signals in Table 5.2 to be an upper limit on residual galactic emission in the WMAP maps. 5.5. SYSTEMATIC EFFECTS 223 K] µ K] 95% Confidence Limit [ µ [ 1 C p K] µ z[ ). 1 shell. 90 b = 0 b , = z K] 0 µ 156 1.0 -120 -1.0 743 432 = y[ l a †† ), z ( 0 . 0 4 = b , K] µ 90 -717 x[ = l † a 7 . 5 ), y ( 0 = b K] µ , [ 0 0 a = l Dust tSZ Bias Removing 58 0.70 170 0.94 -159 -1.1 70 0.58 244 432 Dust Matched 0.36 0.034 0.034 0.0013 0.38 0.019 0.018 0.0010 0.38 62 Free-free MatchedFree-free tSZ Bias Removing 0.042 -101 0.0040 -1.2 -0.46 -0.017 188 0.38 1.0 0.019 -0.042 177 -0.0023 1.2 0.60 -29 -0.24 62 260 432 All galactic MatchedAll galactic tSZ Bias Removing 0.80 -513 0.077 -6.2 -0.37 -357 -0.014 0.92 -2.0 0.046 174 -0.11 1.2 -0.0061 -80 1.00 -0.66 405 62 432 Component Filter Monopole Dipole Synchrotron tSZ Bias Removing -469 Synchrotron Matched 0.40 0.038 0.054 0.0020 0.16 0.0079 -0.086 -0.0048 0.19 62 The uncertainty estimates are obtained fromThe CMB dipole and basis noise directions realizations. are x ( Table 5.2 Monopole and dipole in the PSM simulations for the a b 224 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

5.6 Results

5.6.1 Dipole

Table 5.3 and Fig. 5.18 show our KAKE filter pipeline results, Table 5.4 and Fig. 5.19 show the WF results, Table 5.5 and Fig. 5.20 show the MF results and Table 5.6 and Fig. 5.21 show the UF results. The points in Figs. 5.18–5.21 are from the WMAP 7 year data. The green line is the noise bias, the red and blue lines are the 95% and 99.7% confidence limits that there is no bulk flow, which are estimated from our realizations containing CMB and instrumental noise. We find smaller errors for our Wiener filter than for the KAKE filter since our filter suppresses more power at ` . 100 as shown in Fig. 5.24. The noise at these multipoles is dominated by the CMB and is correlated between channels [20]. We do not find a significant dipole in any of the redshift shells using any of the filters. In Fig. 5.21 some of the points are close to the 99.7% confidence limit. Since this filter is much less sensitive than the other filters which have results that are consistent with noise, this result cannot be due to a bulk flow. Instead it is likely that it is at least partly due to radio point source contamination which in Sec. 5.5.2 we estimated to cause a 1 bias ⇠ in each of the a1m components in maps filtered by the UF. We have repeated our analysis for the WMAP 5 year data and find no significant cluster dipole. We find that for the MF the CMB and instrument noise contribute approximately equally. For the UF the noise is dominated by the instrument noise, with approximately 90% contribution. The UF is an order of magnitude less sensitive than the MF. This is due to the channels being combined in a way that is not optimal for maximizing the kSZ signal, but results in cancellation of the tSZ signal. The reduction in sensitivity of the UF is largely due to the limited frequency coverage of the maps we use, and so an experiment with greater frequency coverage, such as Planck, would perform better with this filter. Although the dipole amplitude we find is consistent with zero, our limits to the bulk flow velocity are tighter in some directions than in others. Fig. 5.22 shows the 95% confidence upper limit to the bulk flow over the whole sky using the results from the MF with all clusters included in the fit. Table 5.7 gives the upper limit to the flow in the direction of other well-known dipoles. The upper limit in all directions is above the limit expected by cosmic variance in the ⇤CDM model, and is above the measured low redshift flow. 5.6. RESULTS 225

7 0.00 0.05 0.10 0.15 0.20720 20

6 6

15 15 5 5

4 4

[µK] 10 1

3 C 3 |v| [1000 km/s]

2 2 5 5

1 1

0 0 0 0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15 0.20 Mean redshift Mean redshift

Figure 5.18 Left: Cluster dipole amplitude in the maps filtered by the KAKE filter. Points with a plus sign have redshift shells with minimum redshift of 0, those with a cross have shells with minimum redshift of 0.05 and 0.12. The green line is the noise bias, the red line is the 95% confidence limit that there is no bulk flow and the blue line is the 99.7% confidence limit. Right: Cluster dipole amplitude in km/s.

7 0.00 0.05 0.10 0.15 0.20720 20

6 6

15 15 5 5

4 4

[µK] 10 1

3 C 3 |v| [1000 km/s]

2 2 5 5

1 1

0 0 0 0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15 0.20 Mean redshift Mean redshift

Figure 5.19 Cluster dipole amplitude in the maps ﬁltered by the WF. 226 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

0.00 0.05 0.10 0.15 0.2012 12 150 150

10 10

8 8 100 100

[µK] 6 1 C

4 |v| [1000 km/s] 50 50 4

2 2

0 0 0 0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15 0.20 Mean redshift Mean redshift

Figure 5.20 Cluster dipole amplitude in the maps ﬁltered by the MF.

1200 0.00 0.05 0.10 0.15 0.201200

100 100

1000 1000

80 80

800 800

60 60

600

[µK] 600 1 C

40 40

400 400 |v| [1000 km/s]

20 200 200 20

0 0 0 0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15 0.20 Mean redshift Mean redshift

Figure 5.21 Cluster dipole amplitude in the maps ﬁltered by the UF. 5.6. RESULTS 227

Figure 5.22 95% confidence upper limit to the bulk flow in the redshift 0 1 shell. We find a bias in simulated thermal SZ maps equivalent to a bulk velocity of 2500 km/s. ⇠

5.6.2 Monopole and Higher Moments

The monopole result from the MF is expected to be strongly contaminated with thermal SZ emission, and we find a result consistent with tSZ contamination: 6868 838 km/s. In ± the maps filtered with the UF we find a monopole consistent with zero: 4692 8947 km/s. ± We find no moments between multipoles 2 and 5 that are significantly di↵erent from zero.

Table 5.7 95% Conﬁdence Limits on the Dipole.

Dipole Name l [deg] b [deg] Velocity [km/s] Reference CMB 263.99 48.26 3485 Jarosik et al. [6] Lauer & Postman 220 28 4625 Lauer and Postman [8] Watkins 287 8 4509 Watkins et al. [12] KAKE 267 34 3886 Kashlinsky et al. [1] 228 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

5.6.3 Comparison with SuZIE Measurements

The SuZIE II experiment has placed limits on the bulk ﬂow using pointed observations of galaxy clusters [16, 52]. SuZIE II made simultaneous measurements of the SZ e↵ect in three frequency bands, centered at 145 GHz, 221 GHz, and 355 GHz, from the Caltech

Submillimeter Observatory, with a 10.5 beam FWHM at all frequencies. A significant detection of the thermal SZ signal was made in 15 clusters; no significant detection of the kinetic SZ signal was made in any of the clusters observed. Benson et al. [16] fit a thermal and kinetic SZ spectrum to the measured cluster temperature at each frequency. No significant cluster dipole was found in the SuZIE cluster sample. We find that the SuZIE bulk flow limits are more sensitive than our results with the MF. We have combined the cluster dipole from the SuZIE data with the dipole limit we find in the WMAP data. When combining the data we weight the dipole amplitude in each of the x, y, and z directions by the inverse noise variance. We find that the 95% confidence limit in the direction of the CMB dipole is increased from the SuZIE result of 1500 km/s to 1800 km/s with the combined data, because our best fit dipole points in a di↵erent direction to the SuZIE dipole. With the UF our result is given less than 1% weight, and the combined result is unchanged from the SUZIE dipole measurement. The cluster monopole velocity in the SuZIE data using all of the clusters is 570 320. The result with the UF is not ± sensitive enough to tighten this constraint.

5.7 Conclusions

We have used the kinetic Sunyaev-Zeldovich e↵ect to look for large scale moments in the galaxy cluster line of sight velocity distribution in the WMAP 7 year data. We use a multi- frequency matched filter that maximizes the cluster signal to noise ratio. We use a sample of 736 clusters derived from the ROSAT X-ray catalogs, and calculate the dipole at the locations of the clusters in the filtered maps. We find no evidence of a cluster dipole in the WMAP 7 year data in any of the redshift shells we use, consistent with predictions from the ⇤CDM theory. Using kSZ simulations we create a matrix to convert the temperature dipole amplitude to a velocity amplitude. We 5.7. CONCLUSIONS 229

find a 95% confidence upper limit to the flow of 3485 km/s in the direction of the CMB dipole, and 3886 km/s in the direction of the KAKE claimed flow (galactic longitude 267, latitude 34). We have performed our analysis on the WMAP 5 year data used in the KAKE analysis and find no evidence of a cluster bulk flow. We find that results obtained using our Wiener filter have greater sensitivity than the results produced using the KAKE filter, and our matched filter gives results approximately 3 times more sensitive than those from the ⇠ KAKE filter. The reason for the di↵erence between our velocity limits and those of KAKE is due to the increased sensitivity of our pipeline and the di↵erent factor we use to convert from µK to km/s, which depends on the kSZ simulations used.

Our analysis di↵ers from that of KAKE in several ways. We use a di↵erent filter to suppress the CMB component. We have tried three filters: a Wiener filter, a matched filter and a matched filter that suppresses the tSZ emission. We use a di↵erent cluster sample. Our sample contains 736 clusters outside of the galactic mask. The sample described in KAKE has 674 clusters and the sample in Kashlinsky et al. [19] contains 985 clusters. For our Wiener filter we do not detect a cluster signal with any aperture we use. The numbers we quote in Table 5.4 are from a 150 aperture which we found gives a higher signal to noise measurement than larger apertures.

Using simulations of the tSZ signal we find that the results with our matched filter are contaminated by thermal SZ, although the contamination is below the WMAP sensitivity. The ⇤CDM model does not predict an intrinsic dipole in the tSZ emission, but due to the relatively small size of our cluster sample, we find a non-zero signal. We estimate the signal to have an amplitude equivalent to a kSZ signal with a bulk flow velocity of 2000 4000 ⇠ km/s. The KAKE analysis is performed with large 300 apertures around each cluster and ⇠ so the tSZ signal is diluted, as seen by the small monopole values KAKE observe.

When the maps are filtered with the MF we find a monopole which is consistent with thermal SZ simulations, with a magnitude of ( 91.2 11.1) µK, equivalent to a kSZ signal ± of 6868 838 km/s. We use a modified multi-frequency matched filter that utilizes the ± di↵erent spectral shapes of the kSZ and tSZ signals to remove the thermal SZ bias. How- ever, the signal to noise of the cluster dipole measurement is reduced by almost an order of magnitude, a consequence of using only three frequency bands in our analysis. 230 CHAPTER 5. MEASURING THE GALAXY CLUSTER BULK FLOW

The tSZ bias removing filter also has increased contamination from extra-galactic unresolved radio emission. A filter could be constructed to suppress this signal, however, since this would further reduce the signal to noise of the measurement, we do not further modify our filters to remove it. The limits we place on the cluster bulk velocity can be decreased by using higher signal to noise measurements of the kinetic SZ signal, as well as by using a larger cluster catalog. We expect that our method can be applied to data from the upcoming Planck experiment.

Planck is currently surveying the sky at a resolution of 50 at 217 GHz. In addition Planck will itself produce a cluster catalog, which can be used for the analysis. Planck’s wide frequency coverage from 30-857 GHz, including a channel at the thermal SZ null of 217 GHz, will allow the tSZ signal to be removed with a smaller impact on the kSZ signal to noise ratio. The e↵ect of radio point sources will likely be negligible because the strongest cluster signal is at frequencies where the radio point source signal is small. However, extragalactic infrared sources with a rising spectrum will probably be an important contaminant. Our ﬁlter could be modiﬁed using spectral information about the IR source population to suppress emission with this frequency dependence. A paper describing the application of our method to the Planck experiment is in preparation [53].

Acknowledgements

We acknowledge useful discussions with Eiichiro Komatsu. We would like to thank Kris Gorski´ and the JPL data analysis group for fostering initial conversations among the authors. We thank Loris Colombo for giving us simulations of the unresolved radio point source emission. SJO would like to acknowledge useful discussions with Neelima Seh- gal. EP thanks Stefano Borgani for useful conversations at the beginning of this work and thanks the Aspen Center for Physics for hospitality. SJO and SEC acknowledge support from the US Planck Project, which is funded by the NASA Science Mission Directorate. DSY Mak acknowledges support from the USC Provost’s Ph.D Fellowship Program. EP is an ADVANCE fellow (NSF grant AST-0649899). She also acknowledges support from NASA grant NNX07AH59G and JPL-Planck subcontract 1290790. Some of the results in this paper have been derived using the HEALPix [29] package. The authors acknowledge 5.7. CONCLUSIONS 231

the use of the Planck Sky Model, developed by the Component Separation Working Group (WG2) of the Planck Collaboration. Bibliography

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BIBLIOGRAPHY 239 b b K] µ K] [ µ [ 1 1 C C p p 1.5 1.6 5.01.0 5.0 1.2 3.0 1.5 3.72.81.90.78 5.4 0.68 1.4 0.63 2.4 0.610.68 2.0 1.7 1.3 1.4 1.6 K] Dipole Amplitude, 1.7 1.9 1.3 3.6 2.8 5.5 1.91.71.6 3.0 2.6 2.2 ± 2.0 3.4 µ ± ± ± 8.75.54.43.13.02.5 18. 2.2 15. 5.2 8.1 6.6 4.5 3.0 K] Dipole Amplitude, ± ± ± ± ± ± ± ± [ ± ± ± µ ± ± ± ± ± ± ± ± ± ± ± z [ 1 z a 1 a 5.63.9 -0.97 -1.4 5.02.5 -3.2 2.01.7 -1.6 1.5 0.096 1.1 2.1 -0.59 -0.29 0.910.82 -0.59 0.91 -0.47 1.4 -0.74 -0.99 K] 8.77.6 -6.6 -5.6 6.34.3 -2.4 3.8 -5.6 3.5 -1.5 3.0 0.017 2.3 -1.9 2.1 -1.1 2.0 -0.86 1.8 -0.31 1.9 -0.26 2.4 -0.098 0.56 K] 0.96 -0.41 µ µ ± ± ± ± ± ± ± ± ± ± ± ± [ ± ± ± ± ± ± ± ± ± ± ± ± ± [ ± y y 1 1 a a 1.9 -2.6 2.52.2 -2.7 -2.7 1.7 -2.1 10.8.7 -12. 6.3 -11. 4.9 -4.3 4.9 -4.9 3.8 -6.0 3.0 -4.9 -3.9 K] 3.9 0.15 1.72.2 -1.4 -1.8 2.32.1 -1.7 -1.3 1.6 -1.2 5.34.7 0.73 -2.1 1.61.41.1 -1.8 -1.9 0.85 -1.1 0.80 -0.66 -0.95 K] ± µ ± ± 0.91 -1.1 0.96 -0.93 ± ± ± ± ± ± ± ± [ ± ± ± µ ± ± ± ± ± ± ± ± ± ± [ x ± ± 1 x 1 a a a a K] µ K] [ µ [ 0 3.83.53.12.2 -12. 2.1 -8.6 1.9 -1.8 1.7 -3.2 1.5 -2.1 1.5 -2.4 1.4 -1.2 1.2 -0.82 1.3 -0.77 1.5 0.024 0.55 1.2 2.9 a 0 2.5 -4.8 2.21.91.31.0 -3.8 0.066 -0.58 -0.92 a ± ± ± ± ± ± ± ± ± ± ± ± ± 0.910.790.620.55 -2.3 0.51 -2.3 0.45 -1.6 0.50 -1.3 -0.91 0.66 -0.87 -1.1 -0.19 ± ± ± ± ± ± ± ± ± ± ± ± ± Monopole, Monopole, of the cluster centers. In Tables 5.5 and 5.6 the monopole and dipole are calculated in the central cl 0 ⇤ cl N ⇤ N z z median z median z i z i h z h max z max z min 0.00.0 0.020.0 0.0250.0 0.014 0.030.0 0.016 0.040.0 0.015 0.019 0.016 0.050.0 0.027 0.0048 0.060.0 0.019 0.0057 0.033 0.080.0 0.030 28 0.039 0.0075 0.120.0 36 0.035 0.050 0.0100 0.160.0 0.041 50 0.067 0.20 0.012 0.052 95 0.080 -1.4 0.015 1.0 0.066 -6.2 0.092 139 0.019 0.076 192 -5.8 0.029 0.12 0.083 294 -1.6 0.039 445 -2.3 0.097 0.049 546 -3.9 619 0.079 -4.6 -4.4 736 -3.9 -3.6 -3.6 z 0.050.12 0.30 0.30 0.13 0.19 0.12 0.18 0.064 0.048 578 271 -3.9 -2.1 min 0.00.0 0.020.0 0.0250.0 0.014 0.030.0 0.016 0.040.0 0.015 0.019 0.016 0.050.0 0.027 0.0048 0.060.0 0.019 0.0057 0.033 0.080.0 0.030 28 0.039 0.0075 0.120.0 36 0.035 0.050 0.0100 0.160.0 0.041 50 0.067 0.20 0.012 0.052 95 0.080 0.40 0.015 1.0 0.066 -1.9 0.092 139 0.019 0.076 192 -3.0 0.029 0.12 0.083 294 -2.5 0.039 445 -2.2 0.097 0.049 -3.4 546 -3.8 619 0.079 -3.5 736 -3.7 -3.7 -3.7 z 0.050.12 0.30 0.30 0.13 0.19 0.12 0.18 0.064 0.048 578 271 -4.1 -4.0 Table 5.4 Results from our Wiener filter. The monopole and dipole are calculatedpixel within of 15 the clusters. Table 5.3 Results from the KAKE filter. ⇤ 240 BIBLIOGRAPHY b b K] K] µ µ [ [ 1 1 C C p p 20 23 537 1373 133135128119134184 653 636 603 522 492 588 602418 137 1717 55 19 18 11 K] Dipole Amplitude, 314241 992 710 7040 152 110 1927 19 32 746420224 1073 1259 726 K] Dipole Amplitude, 3629 65 43 166 699 101 69 ± µ µ ± ± ± ± ± ± ± [ ± ± ± ± ± ± ± ± ± ± ± ± ± ± [ ± ± ± ± z z 1 1 a a 37 -30 866862 -29 -449 70 -16 28 -0.059 21 -8.7 384298 159 263 170 228 -74 180159 15 -131 154 -182 -214 K] 101 -1.8 554332 13 33 -7.1 2523 -1.4 24 -9.3 -16 K] 230 -418 610 -93 140 -178 140126 -56 -68 150 -320 ± µ ± ± µ ± ± ± ± ± ± ± ± ± ± [ ± ± ± ± ± ± ± [ ± ± ± ± ± ± y y 1 1 a a 2426 -1.3 9.5 11097 -79 60 -110 5849 -107 37 -44 28 -23 28 -38 26 -6.1 1.8 5.7 K] 598 -29 37 -0.13 869 -1045 117 -10 K] 756 -1268 410394 -281 344 -310 280 -200 225 -230 199 -200 183 -147 165 -123 185272 -68 66 343 µ ± ± µ ± ± ± ± ± ± ± ± ± [ ± ± ± ± [ ± ± ± ± ± ± ± ± ± ± ± x x 1 1 a a a K] K] µ µ [ [ 0 0 17 11 44332724 -82 18 -20 13 -45 12 -16 12 -39 11 -22 12 -18 -14 -5.7 -2.2 a 6557 -39 -111 a 326193169 269 938 616 9589 591 549 425 -242 7792129 486 367 230 140122102 694 660 607 267 1256 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Monopole, Monopole, cl cl N N z z median median z z i i z z h h max max z z min min 0.00.0 0.020.0 0.0250.0 0.014 0.030.0 0.016 0.040.0 0.015 0.019 0.016 0.050.0 0.027 0.0048 0.060.0 0.019 0.0057 0.033 0.080.0 0.030 28 0.039 0.0075 0.120.0 36 0.035 0.050 0.0100 0.160.0 0.041 50 0.067 0.20 0.012 0.052 95 0.080 0.015 1.0 0.066 59 0.092 139 55 0.019 0.076 192 0.029 -15 0.12 0.083 294 0.039 -22 445 0.097 0.049 -41 546 -62 619 0.079 -63 -73 736 -81 -85 -91 0.00.0 0.020.0 0.0250.0 0.014 0.030.0 0.016 0.040.0 0.015 0.019 0.016 0.050.0 0.027 0.0048 0.060.0 0.019 0.0057 0.033 0.080.0 0.030 28 0.039 0.0075 0.120.0 36 0.035 0.050 0.0100 0.160.0 0.041 50 0.067 0.20 0.012 0.052 95 0.080 118 0.015 1.0 -156 0.066 0.092 139 0.019 0.076 192 73 0.029 0.12 0.083 -223 294 0.039 -164 445 0.097 0.049 546 -22 619 -60 0.079 -38 736 -56 -14 40 z z 0.050.12 0.30 0.30 0.13 0.19 0.12 0.18 0.064 0.048 578 271 -99 -112 0.050.12 0.30 0.30 0.13 0.19 0.12 0.18 0.064 0.048 578 271 98 189 The monopole is contaminated by foregroundsThe with errors levels have consistent a with one simulations. sided distribution and so we use a 95% confidence limit and do not state error bars. Table 5.6 Results from the tSZ bias removing filter. Table 5.5 Results from the matched filter. a b 5 A. EXPECTED CLUSTER VELOCITY DIPOLE 241

5 A Expected Cluster Velocity Dipole

We follow the derivation for the density distribution in Peebles [27]. The line of sight velocity ﬁeld can be expanded as

2 a`m = drr d⌦r(r)Y⇤ (rˆ)v(r) rˆ , (A-1) `m · Z where r is the comoving radial distance, (r) is the comoving number density of objects in the sample, which we assume to be isotropic, v is the object peculiar velocity and Y`m(rˆ) are the spherical harmonics. The peculiar velocity at a given wavenumber is related to the over-density by kˆ v(k) = ifH0(k) , (A-2) k where f = (a/D) dD/da ⌦0.55, ⌦ is the matter density parameter, a is the scale factor, D ⇡ m m is the growth function and H0 is the Hubble constant. The expansion in terms of spherical harmonics is given by

3 2 2 2 2 d k P(k) 3 ik r a`m = f H d r(r)Y`m(rˆ) kˆ rˆ e · | | 0 (2⇡)3 k2 · Z Z 2 D E 3 2 2 2 d k P (k) 3 `0 = 16⇡ f H d r(r)Y`m(rˆ) kˆ rˆ ( i) j` (kr)Y` m (kˆ)Y⇤ (rˆ) , 0 (2⇡)3 k2 · 0 0 0 `0m0 ` m Z Z X0 0 (A-3) where P(k) is the matter power spectrum and j`(kr) are the spherical Bessel functions. Then

2 2 C = a 2 = f 2H2 dkP(k) drr2(r)g (kr) , (A-4) ` | `m| ⇡ 0 ` D E Z Z ! where 1 g`(kr) = ` j`(kr) (` + 1) j`+ (kr) . (A-5) 2` + 1 1 The selection function, (r), is estimated⇥ by calculating the number⇤ density of clusters in our sample in radial bins, which we then interpolate to give a smooth function and normalize. 242 BIBLIOGRAPHY

5 B Comparison with KAKE Results

5 B 1 Filter Pipeline Comparison

We have verified that we can reproduce the KAKE pipeline by comparing our filtered maps with the publicly available maps used for the Kashlinsky et al. [54] analysis. We find that the mean di↵erence between the W band maps passed through our pipeline and the filtered 10 maps from the KAKE pipeline is 1.5 10 mK indicating that our pipeline can reproduce ⇥ their results. As a further check we calculate the spectrum of the filtered W1 channel map, and the absolute value of the di↵erence between our filtered map spectra and the spectra from the KAKE maps in Fig. 5.23. In Fig. 5.24 we show the spectra of the WMAP maps passed through the two pipelines. The signal to noise is low at ` < 100, which is why both filters suppress these multipoles. The KAKE filter is larger at ` < 100 because the filter does not remove the fluctuations in the map that are due to cosmic variance. This leaves noise in the maps that is correlated between channels [20]. By further suppressing the map at ` < 100 our filter reduces the noise in our dipole measurement by a factor of 2 while ⇠ not significantly a↵ecting the signal, as can be seen in Fig. 5.11.

Since we model our clusters as point sources we can calculate the profile of the cluster signal in the filtered maps. In Fig. 5.25 we show the integrated signal out to a given radius in the Q1, V1, and W1 channels after filtering with the Wiener filter. The signal is normalized so that if no filtering were applied the result would approach unity at large radius. The fact that the signal peaks at 10 150 justifies our choice of a 150 aperture within which to calculate the monopole and dipole. 5 B. COMPARISON WITH KAKE RESULTS 243

W1 Map Spectrum 100 W1 difference W2 difference W3 difference 10−2 W4 difference ] 2

10−4 /2 [mK l 10−6 +1)C l ( l 10−8

10−10 200 400 600 800 1000 l

Figure 5.23 Spectrum of the W1 channel map filtered by the KAKE filter (black) and the di↵erence between the filtered W band map spectra from our KAKE filter pipeline and the spectra of the publicly available maps used in the Kashlinsky et al. [54] analysis.

10−2

10−3 ] 2 10−4

−5 /2 [mK

l 10

+1)C −6 l

( 10 l

−7 10 KAKE Filter Our Wiener Filter 10−8 10 100 1000 l

Figure 5.24 Filtered W1 channel map spectrum from the KAKE pipeline and our pipeline. 244 BIBLIOGRAPHY

0.5 Q1 0.4 V1 W1 0.3 0 τ 0.2 (R)/ eff τ 0.1

0.0

−0.1

0 20 40 60 80 100 R [arcmin]

Figure 5.25 Integrated signal from a beam smoothed and ﬁltered cluster in our simulated maps.

5 B 2 Cluster Dipole Comparison

In this section we remove clusters with bolometric luminosity L < 2 1044erg/s from our X ⇥ sample and use the WMAP Kp0 mask instead of the 7 year extended temperature analysis mask to better compare with the KAKE results. As before we remove the monopole and dipole of the map outside of the mask before fitting for the cluster monopole and dipole. We find that the sample of clusters satisfying the luminosity cut and the redshift cut of z 0.16, 0.2, 0.25 is almost identical to the sample used in the KAKE analysis. There are  some di↵erences in the REFLEX, BCS and BCSe catalogs for clusters with luminosities close to L = 2 1044erg/s, and some di↵erences for clusters in the CIZA catalog that are X ⇥ near to the WMAP galactic mask boundary. We show the di↵erences we find in Table 5.8. 5 B. COMPARISON WITH KAKE RESULTS 245

Table 5.8 Number of clusters satisfying the luminosity and redshift cut in our sample and the KAKE sample.

zmax Number of clusters Number of clusters KAKE Number of clusters we with L > 2 1044erg/s include but we do not include but KAKE do not X ⇥ 0.16 131 13 11 0.20 205 14 18 0.25 269 14 20

To check that these di↵erences do not change the results by a large amount we have calculated the cluster dipole using the KAKE ﬁlter with both the KAKE cluster sample and our own sample. We ﬁnd the results shown in Table 5.9 when averaging over the Q, V and W bands.

Table 5.9 Comparison of the cluster monopole and dipole in our cluster sample with the KAKE sample.

Temperature [µK]

Cluster sample zmin zmax m m x x y y z z Ours 0.0 0.16 -0.74 2.05 1.44 2.87 -7.04 2.98 0.04 2.68 Ours 0.0 0.20 0.02 1.48 3.04 2.21 -6.23 2.52 1.93 1.97 Ours 0.0 0.25 0.04 1.21 1.56 1.92 -4.17 2.14 2.08 1.72 KAKE 0.0 0.16 -1.07 2.07 1.51 2.96 -7.00 2.86 0.48 2.72 KAKE 0.0 0.20 -0.22 1.46 3.06 2.22 -6.20 2.45 2.08 2.02 KAKE 0.0 0.25 -0.20 1.22 1.68 1.99 -4.25 2.12 2.19 1.76

All of the results have less than 3 significance with some of the a1y components having 2.5 significance. The results averaged over the W band channels only (but with the ⇠ same filter and cluster samples used to produce Table 5.9) are shown in Table 5.10. 246 BIBLIOGRAPHY

Table 5.10 Same as table 5.9 for the W band channels only.

Temperature [µK]

Cluster sample zmin zmax m m x x y y z z Ours 0.0 0.16 -1.64 2.11 1.11 3.12 -8.63 3.24 0.01 2.84 Ours 0.0 0.20 -0.41 1.59 2.71 2.42 -7.73 2.67 2.08 2.04 Ours 0.0 0.25 -0.49 1.30 1.03 2.06 -5.53 2.28 2.10 1.81 KAKE 0.0 0.16 -1.64 2.13 0.96 3.20 -8.15 3.11 0.54 2.88 KAKE 0.0 0.20 -0.48 1.55 2.60 2.43 -7.45 2.60 2.28 2.12 KAKE 0.0 0.25 -0.60 1.30 1.06 2.13 -5.43 2.26 2.25 1.87

These results have greater significance than the combined channel results. We still find nothing with more than 3 significance, although the z = 0.0 0.20 shell a value has 1y 2.9 significance. The result from Kashlinsky et al. [54] is shown in Table 5.11, which is similar to the results in Table 5.10.

Table 5.11 Result from Kashlinsky et al. [54].

Temperature [µK]

zmin zmax mx y z 0.00 0.16 -1.47 1.20 -8.26 0.38 0.00 0.16 -0.32 2.83 -7.58 2.13 0.00 0.16 -0.44 1.30 -5.57 2.10

These results are more signiﬁcant than those obtained using the full cluster sample. This cannot be fully explained by the higher luminosity clusters having larger optical depths. In Table 5.12 we show the average optical depth and number of clusters in the sample both with and without the luminosity cut. 5 B. COMPARISON WITH KAKE RESULTS 247

Table 5.12 Mean optical depth of our cluster sample.

With LX cut No LX cut

z Number of Clusters ⌧ 103 Number of Clusters ⌧ 103 max avg ⇥ avg ⇥ 0.16 131 6.3 573 3.7 0.20 205 6.6 649 4.1 0.25 269 7.0 713 4.5 All z 327 7.0 771 4.9

Although the average optical depth is higher with the luminosity cut, the noise (which is roughly proportional to 1/ pNclusters) is also higher and so we expect a similar sensitivity both with and without the luminosity cut. However, when we calculate the cluster dipole in maps filtered with our Wiener filter with our cluster sample we find much lower significance as shown in Table 5.13.

Table 5.13 Result from our ﬁlter pipeline with the luminosity cut.

Temperature [µK]

Cluster sample zmin zmax m m x x y y z z Ours 0.00 0.16 -1.58 0.45 -0.21 0.90 -1.75 1.09 -0.35 0.58 Ours 0.00 0.20 -1.01 0.40 -0.08 0.68 -1.05 0.76 -0.51 0.48 Ours 0.00 0.25 -1.15 0.34 -0.84 0.57 -0.98 0.68 -0.62 0.49

We conclude from these results that we can accurately reproduce the KAKE pipeline and do not detect a bulk flow with greater than 3 significance in the WMAP maps. The significance of the results is lower when we use our Wiener filter and the full cluster sample than when using the KAKE filter and the cluster sample that has low luminosity clusters removed. By comparing with Table 5.3 we see that the results with the smaller cluster sample are more significant than when those with the full cluster sample. However, since we would expect any bulk flow to be of similar significance with either of the cluster samples we conclude that there is no significant detection of a bulk flow. Chapter 6

Optimal Analysis of Azimuthal Features in the CMB

“Oh, those are fabulous!” cried Mr. Wonka. “They ﬁll you with bubbles, and the bubbles are full of a special kind of gas, and this gas is so terriﬁcally lifting that it lifts you right o↵ the ground just like a balloon, and up you go until your head hits the ceiling – and there you stay. . . . But don’t drink it outdoors! There’s no knowing how high up you’ll be carried if you do that. I gave some to an old Oompa–Loompa once out in the back yard and he went up and up and disappeared out of sight! It was very sad. I never saw him again. . . . He must be on the moon by now.”

Charlie and the Chocolate Factory Roald Dahl

We present algorithms for searching for azimuthally symmetric features in CMB data. Our algorithms are fully optimal for masked all-sky data with inhomogeneous noise, computationally fast, simple to implement, and make no approximations. We show how to implement the optimal analysis in both Bayesian and frequentist cases. In the Bayesian case, our algorithm for evaluating the posterior likelihood is so fast that we can do a brute-force search

248 6.1. INTRODUCTION 249

over parameter space, rather than using a Monte Carlo Markov chain. Our motivating example is searching for bubble collisions, a pre-inﬂationary signal which can be generated if multiple tunneling events occur in an eternally in- ﬂating spacetime, but our algorithms are general and should be useful in other contexts.

6.1 Introduction

What happened at the beginning of the universe? How did the universe start? Are there other universes? What do they look like? Obtaining answers to these exciting questions is theoretically and experimentally challenging, but there are known signals that may be present in the Cosmic Microwave Background (CMB) data that can help us to answer them. One way to solve the horizon, flatness, and relic abundance problems is if the universe started with a phase of slow roll inflation. There are theorems [1] that show that inflation cannot be past eternal: there must be something before it. One possibility is that our current patch of universe was born as a nucleation bubble from a phase of false vacuum eternal inflation [2, 3]. In this phase the universe is thought to be trapped in an unstable high-energy vacuum. The vacuum energy makes the universe expand exponentially, but since the vacuum is quantum mechanically unstable to tunneling, new bubbles are continuously produced and start expanding at close to the speed of light. If the decay rate per unit spacetime volume is less than H4, with H the Hubble rate, then the expansion of the universe draws the bubbles far apart and they do not fill the universe. Inside the bubble the universe looks like an open FRW cosmology, with a big bang apparent singularity at the spacetime location of the nucleation. Inflation can occur within the bubble and produce a universe that looks locally like our own. In this scenario, the singularity in the past before inflation is an illusion, and we came from an eternally inflating space-time. Though bubbles do not percolate and fill the whole of space, there is a chance that bubbles collided before our present time, leaving a specific disk-shaped imprint in the CMB. Discovery of such a signature would have consequences that can hardly be overstated. 250 CHAPTER 6. OPTIMAL ANALYSIS OF AZIMUTHAL FEATURES

First, we would better understand what happened before the period of inflation in our recent past. We would learn that we are a bubble in an eternally inflating universe, and that eternal inflation consists of a new phase of our universe. Furthermore, detection of a bubble would provide indirect confirmation of the anthropic explanation of the cosmological constant [4], which states that the observed value of the cosmological constant is approximately the value required for structures to form in our universe. For structure formation the cosmological constant cannot be larger than a certain upper bound, and since it is most probable for the cosmological constant to be peaked at the highest possible value, then it follows that we should observe a cosmological constant close to its upper bound. And we did in 1998, confirming this prediction. This anthropic explanation relies on the fact that the fundamental theory of the universe has a landscape of vacua with di↵erent fundamental parameters, so that in one of them the right anthropic value of the cosmological constant can be found. String theory naturally provides such a landscape of vacua, and eternal inflation o↵ers a way of populating them all in the universe. While detection of a bubble collision in the CMB would strictly speaking teach us that the universe is described by a field theory with at least two vacua, one unstable and eternally inflating and the other the stable terminal one, it would give evidence towards the anthropic explanation of the cosmological constant, landscape of string theory, and ultimately string theory itself.

The bubble signal, if it exists, will likely be present at a low signal-to-noise ratio and so we will require sophisticated algorithms to search for it. The amplitude of the signal can be measured using statistically optimal estimators, however, the large amount of data required, coupled with potentially complex noise properties makes them computationally expensive to implement. Approximate estimators can be used but this reduces the sensitivity to the signal, e↵ectively throwing away information. We will discuss generally how optimal estimators can be implemented in a computationally ecient way and use as an example the search for the bubble collision signal in full-sky CMB data. Our approach is broadly applicable to many image processing problems, and we will present the methodology in a general manner wherever possible.

There are several challenges in implementing the optimal analysis. CMB maps have 6.1. INTRODUCTION 251

two components with very di↵erent statistical properties: the instrument detector noise, often most easily described in real space, and the CMB signal itself, which is more simply described in harmonic space. Furthermore, the noise can be inhomogeneous and anisotropic. The inhomogeneity means that the noise has a larger variance in some regions than others, and can be caused by, for example, observing some areas of the sky for longer than other areas. If the image has masked areas then the noise will be anisotropic, with pixels that are completely masked described as having infinite noise variance. It is dicult to avoid masking data in CMB analyses since some foreground emission, such as from the galactic plane, is many times brighter than the CMB at all frequencies. The mask has both large and small scale features, and so algorithms must account for the full range of scales when calculating the noise covariance. Estimators constructed using matched filters, for example [5], typically assume that the noise is isotropic, which is not the case in practice. The resolution of the images also presents computational challenges. To evaluate optimal estimators we must calculate the operation of the inverse signal+noise covariance matrix on a vector. While it is simple to design algorithms that improve upon the (N3 ) compute-time O pix complexity of the simplest algorithms, the compute times are still non-negligible even for more sophisticated algorithms at the resolutions that we consider: (106)( (108)) pixels O O for the WMAP (Planck) experiment. The image resolution is determined by the size of features in the signal, as well as the image noise properties, and so can be decreased for large-scale signals. An additional complication is the large number of parameter values that can be required to describe the signal being searched for. For example, the signal could be at any location on the sky, and have many possible angular profiles that could be a complicated function of the model parameters. These challenges appear to make calculating the exact likelihood for all possible parameter values very expensive.

In this paper, we present a complete solution to these computational problems. We use a simple methodology, parametrizing the searched-for signal and calculating the likelihood of the parameters, and ﬁnding algorithmic tricks to make the computations fast. We show how to implement the optimal analysis in both the Bayesian and frequentist statistical frameworks. The advantages of our method are simplicity, exactness, optimality, and minimal computational cost required. For example, in the Bayesian case, we can compute the exact all-sky posterior likelihood using a very straightforward procedure which is 252 CHAPTER 6. OPTIMAL ANALYSIS OF AZIMUTHAL FEATURES

so computationally fast that we can explore the parameter space by brute force, without MCMC-based sampling algorithms. Previous searches have looked for the bubble signal in the WMAP data. Ref. [6] finds no evidence for bubble collisions, quoting an upper limit N < 4.0 on the expected number of collisions. This analysis is based on exploration of the Bayesian likelihood and is similar in spirit to ours, but there are a few minor di↵erences. First, our fast algorithms allow the likelihood analysis to be simplified while remaining computationally a↵ordable, thus removing several steps in the analysis. Second, we evaluate the exact likelihood function which makes the analysis fully optimal (for example, our filter is optimally weighted in the presence of sky cuts and inhomogeneous noise, and we can use WMAP V-band data in addition to W-band). Finally, we prefer to reparametrize and quote the final result as an upper limit on the maxmimum amplitude A of a bubble which intersects our Hubble volume, rather than an upper limit on the expected number of bubble collisions N. The data analysis mainly constrains A, while the limit on N is somewhat dependent on the prior on A which is chosen. Notwithstanding these minor di↵erences, we agree with the recent conclusion of [6]: there is no statistically significant evidence for bubble collisions in WMAP. In ref. [7], the signal expected from a large number of bubble collisions has been studied, and it is found that the CMB data disfavor a bubble signal due to the low CMB quadrupole power. We will also study this case, finding similar results. In this paper we will focus on the technical details of our analysis, presenting the main results in a companion paper [8]. We restrict ourselves to the WMAP temperature data, and use the WMAP7+BAO+H cosmological parameters throughout [9]. In 6.3 we describe 0 § the calculation of the bubble signal, in 6.4 we describe our method, including the calcu- § lation of the likelihood, and describe the Bayesian and Frequentist approaches, in 6.5 we § extend the analysis to include a large number of bubbles, and we conclude in 6.6. §

6.2 Mini-Summary of the Theory of Bubble Collisions

In this section we brieﬂy review the theory of bubble collisions, focusing on aspects which will be needed for the data analysis. For a detailed review, see [10]. The spacetime diagram for a single bubble collision is shown in Fig. 6.1. When a bubble 6.2. MINI-SUMMARY OF THE THEORY OF BUBBLE COLLISIONS 253

Outside Within 8 Collided Collided Region Region

6 Present Earth Reionization Recombination θ Earth bubble

Reheating4 Ddc x y

2 Our bubble Colliding bubble t z z=0 z=r

0 x False vacuum 0 2 4 6 8 10 Figure 6.2 The Earth’s last scattering sur- Figure 6.1 A spacetime diagram show- face at the time of decoupling, based ing the causal structure of a single bubble on Fig. 4 from [15]. The color of the collision, based on Fig. 3 from [15]. Co- shaded region indicates the magnitude of ordinates are chosen so that light propa- the curvature perturbation, assumed to gating in the plane of the diagram moves be of the “ramp” form given below in along 45 lines. Eq. (4). nucleates, the region of spacetime contained in the future lightcone of the nucleation is an inﬂationary spacetime with small negative curvature. Our observable Hubble volume is a smaller region given by taking the intersection with the past lightcone of a present-day observer. If there is a second bubble nucleation, and its future lightcone intersects our observable Hubble volume, then the metric will be perturbed, and this will generate a disc- shaped CMB temperature perturbation [10–14]. The expected number N of bubble collisions is given by [16]

H2 ⌦ f . N k 2 (1) h i' Hi p

(Ni N?) Here p⌦k represents the curvature of the current universe. It scales as e , with Ni the number of e-foldings during inﬂation and N 63 the number of e-foldings since ? ⇠ reheating. Unless there is a mechanism forcing the universe to have the lowest possible number of inﬂationary e-foldings, then p⌦k is naturally an exponentially small number. 4 S is related to the decay rate per unit four-volume, , as /H e E , where H is the ⇠ f ⇠ f 254 CHAPTER 6. OPTIMAL ANALYSIS OF AZIMUTHAL FEATURES

Hubble rate in the false vacuum region, and S E is the Euclidan action of the instanton that mediates the decay of the false vacuum. is expected to be an exponentially small number.

Finally, Hi is the inflationary Hubble rate during the period of slow-roll inflation within our bubble. The ratio H f /Hi can be very large, if the vacuum energy of the false vacuum is much larger than the one that drives slow roll inflation in our bubble. Of course Hi could be close to H f , in which case there is no large factor. Summarizing, we have the product of two exponentially small numbers and one possibly large number. The zeroth order expectation is that N is exponentially small. A first order, more hopeful, expectation uses h i the possible enhancement of H /H to conclude that N is either very small or very large. f i h i It therefore appears unlikely to us that N should be of order 1. However, the theoretical h i understanding of eternal inflation and of the string landscape is still in a very preliminary stage. In particular it is possible that in the landspace the probability is dominated by decays with a small Euclidean action and by local inflationary patches with small number of e- foldings. It is therefore hard to make strong statements about the theoretical expectations on N . It is fair to say that the signal we are looking for is well defined, and since a h i detection would have such important theoretical implications, we proceed anyway.

Similar considerations apply for the expected typical size of the temperature perturbation induced by the bubble collision. The bubble collision can be thought of as inducing a discontinuity ()in in the initial conditions of the inflaton in our patch. This initial discontinuity evolves during inflation into what we later call the ramp profile, with typical

(N N?) temperature perturbation ⇣ () (H /H ) a x e i , where a and H are the current ⇠ in 0 i 0 c 0 0 values of the scale factor and Hubble rate, respectively, and xc is the comoving distance from the bubble wall to the interior [14]. As was the case for the expected number of bub- 5 ble collisions, the size of the perturbation can be very di↵erent from 10 , which is the range we will probe with a dedicated analysis. Conditioning on the fact that inﬂation needs to have happened can motivate excluding very large density perturbations, but it does not exclude very small ones. Still, as for the case of the number of collisions, the theoretical understanding of the probability for the amplitude of the signal is very preliminary, and the discovery of an event would have such importance that we implement a dedicated analysis. Note too that the analysis we perform involves the development of techniques that have a wide range of alternative applications. 6.3. BUBBLE SIGNAL 255

Throughout this paper, we will parametrize the bubble size either by the comoving distance r to the bubble wall, or by the angular radius ✓bubble, which we deﬁne by:

r = Ddc cos ✓bubble (2)

where Ddc is the comoving distance to last scattering. A bubble will be parametrized by its size (either r or ✓bubble), its angular location nˆ on the sky (a unit two-vector), and one or more amplitude parameters (to be defined in 6.3.1). § Although the number density of bubbles and their amplitudes depend on microphysics of the inflationary model, the random distribution of bubble locations and sizes is determined by symmetry alone [16]. The angular location nˆ of the bubble is uniformly distributed over the sky, and in the spatially flat limit ⌦ 1 (which we will assume through- k ⌧ out this paper), the size parameter r is a uniformly distributed random variable. Equiva- lently, the size parameter ✓bubble has the distribution:

dP dr sin(✓ )d✓ (3) / / bubble bubble

This distribution implies that most of the bubbles are expected to have a large angular size.

6.3 Bubble Signal

6.3.1 Ramp and Step Models

The bubble collision generates a contribution to the initial adiabatic curvature perturbation

⇣, which evolves to generate a contribution to the CMB temperature anisotropy a`m. The spacetime symmetries of the collision imply that ⇣ is invariant under a residual SO(2,1) symmetry. For a suitable choice of coordinates, and in the limit of zero spatial curvature, the symmetry generators can be taken to be rotation around the z-axis and translations in the x and y directions. Thus ⇣(x, y, z) must be a function of z alone. 256 CHAPTER 6. OPTIMAL ANALYSIS OF AZIMUTHAL FEATURES

In the simplest scenario, the bubble contribution to ⇣ is of the “ramp” form:

aramp(z r) if z r ⇣(x, y, z) = (4) 8 0 if z < r <> > where r is the comoving distance to the:> bubble wall and aramp is a free parameter with units 1 Mpc . The distance r is related to the angular size ✓bubble of the bubble by cos ✓bubble = r/Ddc. We will refer to model (4) as the “ramp model”. In addition to the ramp perturbation, there is a contribution from the signal at the boundary of the collision. Following [17, 18], we will allow for a simple step function:

aramp(z r) + astep if z r ⇣(x, y, z) = (5) 8 0 if z < r <> > where astep is dimensionless. We will:> refer to model (5) as the “ramp+step model”. The size of the boundary and the signal expected in the boundary region are not well understood theoretically, and there is some disagreement in the literature as to which of (4) or (5) is better motivated. In this paper, our focus is data analysis and we will not weigh in on this theoretical issue; we will simply show how to perform the optimal analysis for both the ramp and ramp+step models.

6.3.2 CMB Temperature Proﬁles

To calculate the angular CMB temperature proﬁle produced by the bubble, we must account for the acoustic and gravitational physics which generates the CMB temperature anisotropy from the initial curvature ⇣. We can deﬁne a transfer function `(k) which represents the contribution of a Fourier mode of ⇣ with wavenumber k to the CMB temperature anisotropy at angular wavenumber ` [19–21]. The CMB temperature, a`m, is related to the 3D curvature perturbation ⇣ by 3 ` d k a`m = 4⇡i ⇣˜(k) `(k) Y⇤ (kˆ) (6) (2⇡)3 `m Z 3 ik x where ⇣˜(k) = d x ⇣(x)e · is the 3D Fourier transform of ⇣. In the special case where ⇣ ˜ ˜ 2 ˜ ikzz is a function ofR z alone, we have ⇣(k) = ⇣(kz)(2⇡) (kx)(ky), where ⇣(kz) = dz ⇣(z)e R 6.3. BUBBLE SIGNAL 257

is the 1D Fourier transform of ⇣(z). Plugging into Eq. (6), we get:

2` + 1 a`m = b` m0 (7) r 4⇡ where we have deﬁned

1 ` ` b` = 2 dkz `(kz) i ⇣˜(kz) + ( i) ⇣˜(kz)⇤ (8) 0 Z h i If the bubble collision is in direction nˆ (rather than in the z-direction), then Eq. (7) for the temperature proﬁle generalizes to:

a`m = b` Y`⇤m(nˆ) (9)

This can be shown by starting with Eq. (7) for a bubble in the z-direction, and applying a rotation R(nˆ) which carries directionz ˆ to direction nˆ:

R a = Dl a [ (nˆ) ]lm mm0 (nˆ) lm0 (10) Xm0 where Dl is the Wigner D-matrix (for a deﬁnition see, for example, [5]). Using the mm0 l identity Dm0(nˆ) = p4⇡/(2` + 1)Y`⇤m(nˆ), we obtain Eq. (9) above.

As an aside, we note that the general form (9) is valid for any azimuthally symmetric proﬁle, even one which does not come from an initial curvature perturbation (e.g. SZ emission from a galaxy cluster). In general, the harmonic-space proﬁle b` and its real-space counterpart b(✓) are related by:

2` + 1 b(✓) = b P (cos ✓) ` ⇡ ` ` 4 X ! b` = 2⇡ d(cos ✓)P`(cos ✓)b(✓) (11) Z For the ramp and step bubble models, we can specialize the general expression (8) to obtain ik r 2 explicit formulas for b . The relevant 1D Fourier transforms are ⇣˜ (k ) = e z /k and ` ramp z z 258 CHAPTER 6. OPTIMAL ANALYSIS OF AZIMUTHAL FEATURES

ik r ⇣˜ (k ) = ie z /k . We get: step z z

`/2+1 1 2 ` ramp 4( 1) 0 dk k `(k) cos(kr) for even b` (r) = (12) (`+1)/2 1 2 ` 8 4( 1) R 0 dk k `(k) sin(kr) for odd > < `/2+1 1 > 4( 1) R 1 dk k `(k) sin(kr) for even ` bstep(r) = :> 0 (13) ` (` 1)/2 1 4( 1) 1 dk k (k) cos(kr) for odd ` 8 R 0 ` <> > R We compute these proﬁles:> numerically, using CAMB [22] to compute the transfer function

`(k). Note that our normalization convention is to parametrize the bubble amplitude by parameters aramp, astep which appear in Eqs. (4), (5) for the curvature perturbation ⇣, and have 1 units [Mpc ] and [dimensionless] respectively (no µK). In particular this means that a bubble with positive amplitude corresponds to a cold spot on the sky (provided the radius is

> 1), since positive ⇣ corresponds to negative T on large scales. We elaborate on the ⇠ relation between ⇣ and T in the next section.

6.3.3 Can CMB Transfer Functions be Neglected?

ramp step The expressions from the previous section for the profiles b` , b` are nontrivial to evaluate, and one may wonder whether it is a good approximation to simply assume that the temperature perturbation T is proportional to the value of ⇣ on the last scattering surface (the Sachs-Wolfe approximation). To study this question quantitatively, we define “cosine” and “disc” profiles in real space by:

cos ✓ cos ✓bubble if ✓ ✓bubble bcosine(✓) =  0 if ✓ ✓ 8 bubble <> > 1 if ✓ ✓bubble bdisc(✓) = :  (14) 0 if ✓ ✓ 8 bubble <> > where ✓bubble is the angular size of the: bubble. We note that the integral (11) can be evaluated 6.3. BUBBLE SIGNAL 259

analytically for these proﬁles, giving the following harmonic-space proﬁles:

P`+2(z) P`(z) P` 2(z) bcosine = 2⇡ 2 + ` (2` + 1)(2` + 3) (2` 1)(2` + 3) (2` 1)(2` + 1) ! P` 1(z) P`+1(z) bdisc = 2⇡ (15) ` 2` + 1 ! where z = cos ✓bubble. In Fig. 6.3, we show the ramp, step, cosine, and disc proﬁles for a bubble at a distance r = 13886.6 Mpc, corresponding to angular size ✓bubble = 11.39, with arbitrary normalizations. In the Sachs-Wolfe approximation, the CMB temperature in direction nˆ is given by

T(nˆ) = (Ddcnˆ)/3, where is the Newtonian potential. On large scales, is related to ⇣ by = (3 + 3w)/(5 + 3w)⇣, where w 0.11 is the equation of state parameter at ⇡ last scattering. Putting this together, the Sachs-Wolfe approximation applied to the bubble proﬁle reads:

1 3 + 3w bramp D bcosine (16) ` ⇡3 5 + 3w dc ` ! 1 3 + 3w bstep bdisc (17) ` ⇡3 5 + 3w ` ! Considering the case of the ramp profile first, we find that the Sache-Wolfe approximation (16) is excellent for most bubble sizes, but breaks down for bubbles which are very small or very large. For very small bubbles (i.e. ✓bubble < 1 or equivalently r Ddc), ⇠ ⇡ the bubble profile is widened by 1 due to acoustic physics encoded in the CMB transfer ⇡ function. For very large bubbles (i.e. ✓ 90 or r D ), the Sachs-Wolfe approxima- bubble ⇡ ⌧ dc tion breaks down due to the contribution from the ISW e↵ect, which turns out to partially cancel the Sachs-Wolfe contribution and reduces the bubble amplitude by 30%. ⇡ For the step profile, we find that the Sachs-Wolfe approximation (17) is not very accu- step disc rate. More quantitatively, the correlation coecient r(b` , b` ) between the step and disc profiles is never close to 1; we find 0.2 r 0.7 depending on the bubble radius. We   give the precise definition of r(b, b0) in Eq. (20) below, but for now we simply treat it as a metric that can take values between -1 and 1, with correlation 1 meaning that the profiles ± are identical up to rescaling. 260 CHAPTER 6. OPTIMAL ANALYSIS OF AZIMUTHAL FEATURES

When we search for the ramp or step profiles in CMB maps, the ramp profile gets most of its statistical weight from low `, whereas the step profile gets its statistical weight from the full range of ` values which are measured with appreciable signal-to-noise. Intuitively, when we search for the profiles against the CMB sky, most of the signal for the ramp profile comes from the “bulk” of the profile, whereas most of the signal for the step profile comes from the sharp drop near the edge. Summarizing, CMB transfer functions are an order-one e↵ect for the step profile and must be included, but are less important for the ramp profile. This makes intuitive sense because the ramp profile is mainly a low-` signal, the step profile is mainly a high-` signal, and transfer functions are unimportant at low `. However, even for the ramp profile, transfer functions can a↵ect the shape or amplitude of bubbles with either ✓ 0 or ✓ 90, bubble ⇡ bubble ⇡ and should be included in a precise analysis.a

6.3.4 Discretizing the Bubble Radius

The data analysis algorithms in the next section will require the bubble radius, or equivalently the distance r to the bubble wall, to be discretized to a finite set of values r , , r . 1 ··· N If we use too few r values then we will decrease our statistical power, since we will end up searching for an incomplete set of profiles in the data. If we use too many r values then the computational cost increases (the cost will turn out to be roughly proportional to N). We therefore require a procedure to determine the minimum number of r values which are needed. To motivate our choice for this procedure, we first consider the (unnormalized) minimum variance estimator for the amplitude of a bubble at known location nˆ 0:

b` = a`mY`m(nˆ 0) (18) + E ` C` N` Xm

aWe also note that for an experiment which measures the damping tail ` > 2000, the Sachs-Wolfe approximation for the ramp proﬁle cannot be used without modiﬁcation, since it would⇠ give b`’s which are not exponentially suppressed at high `, which leads to a spurious detectable high-` signal since the CMB power spectrum C` is exponentially suppressed. 6.3. BUBBLE SIGNAL 261

1.0 Cosine profile Disc profile 0.8 Ramp profile Step profile

0.6 ) θ b( 0.4

0.2

0.0 0 5 10 15 θ [deg]

Figure 6.3 Angular bubble profiles b(✓) defined in 6.3.2, for bubbles at comoving distance § r = 13886.6 Mpc (corresponding to angular size ✓bubble = 11.39) and arbitrary normalization. The ramp and step profiles are obtained by evolving an initial feature in the adiabatic curvature perturbation (Eqs. (4), (5)) forward to obtain a CMB temperature profile, using the CMB transfer function. The cosine and disc profiles are approximations to the ramp and step profiles in which the CMB transfer function is omitted (Eq. (14)). Qualitatively, the e↵ect of including the CMB transfer function is to smooth sharp features in the profile, with smoothing length 1 given by the CMB acoustic scale. ⇡

where we have assumed all-sky homogeneous noise with power spectrum N`. The covariance Cov( , 0) of two such estimators with proﬁles b and b0 is given by: E E ` `

b`b`0 b b0 = (2` + 1) (19) + · ` C` N` X where we have introduced a dot product notation for compactness.

We determine a minimal set of r values by requiring that adjacent r values be “close”, in the sense that the associated proﬁles have a correlation coecient greater than 0.97, where 262 CHAPTER 6. OPTIMAL ANALYSIS OF AZIMUTHAL FEATURES

the correlation is deﬁned using the dot product in Eq. (19):

b b0 r(b, b0) = · > 0.97 (20) p(b b)(b b ) · 0 · 0 This definition of closeness corresponds to observational indistinguishability: two profiles are close if they cannot be distinguished statistically given a noisy observation of the CMB. Note that we are implicitly approximating the real WMAP noise model by all-sky homogeneous noise with the same noise power spectrum, but this is a reasonable approximation if we just want to decide whether two profiles are highly correlated.

For the ramp profile we find we need 129 r values to cover the range rmin = 0 to rmax = 14165.64 Mpc. For the step profile, we find that 2002 r values are needed, due to smaller-scale features in the profile.

6.4 Data Analysis

In this section, we consider the question: what is the optimal statistic for detecting a bubble collision, and how can we evaluate it in a computationally feasible way? We will answer this question in both the Bayesian ( 6.4.3) and frequentist ( 6.4.4) statistical frameworks. § § In both cases, we will ﬁnd that there is a natural choice of optimal statistic, but computing it appears to be prohibitively expensive. However, we will ﬁnd computational tricks which will allow us to evaluate the optimal statistic with reasonable computational cost, and without making any approximations.

6.4.1 Deﬁnitions and Notation

We denote the noisy CMB data by a vector dµp, where the index µ runs over observing channels (in our WMAP pipeline, µ indexes one of the six di↵erential assemblies V1,V2,W1,W2,W3,W4) and p runs over sky pixels. We keep the observing channels separate, rather than combining them to a single map dp, to avoid making the analysis sub- optimal by degrading all channels to the resolution of the worst channel. Throughout this paper, we refer to a two-index object such as dµp as a “per-channel pixel-space map”, and 6.4. DATA ANALYSIS 263

a single-index object xp as a “pixel-space map”.

Let c`m denote the CMB realization, and let Aµp,`m be the operator which acts on the harmonic-space map c`m to produce a per-channel pixel-space map (Ac)µp. In detail, (Ac) is deﬁned by multiplying c by the beam transfer function for each channel (and the Healpix window function), then applying spherical harmonic transforms to obtain a per-channel pixel-space map.

We define a covariance matrix C = S + N for the data vector dµp, consisting of signal and noise contributions. The covariance matrix is defined in the per-channel pixel-space domain; thus it has indices Cµp,µ0 p0 . In WMAP, the noise in di↵erent pixels is uncorrelated to an excellent approximation, and so we treat the noise covariance as diagonal: Nµp,µ0 p0 = 2 2 µpµµ0 pp0 , where µp is a noise variance which can depend on channel µ and sky pixel p. The signal covariance is given by S = A⌃AT , where the matrix ⌃ is diagonal in harmonic space: ⌃`m,`0m0 = C```0 mm0 . Note that with this covariance matrix S , the signal in di↵erent channels µ, µ0 is 100% correlated (but convolved with di↵erent beams). 1 We will shortly encounter expressions involving the inverse covariance matrix C . The matrix size is much too large to invert (or even store) C in dense form. However, given a per-channel pixel-space map x, there are iterative algorithms which can eciently compute 1 the matrix-vector product C x. We will use the multigrid conjugate gradient algorithm 1 from [23], which can perform one C multiplication in approximately 15 core-minutes 1 for WMAP. Some technical details of the C filter are presented in Appendix 6 A. For now, we just remark on one important feature: we can include a pixel mask by assigning formally infinite noise variance to masked pixels, and similarly include monopole+dipole marginalization by suitably modifying the noise covariance so that the relevant modes are given infinite variance. In order to present our algorithms in maximum generality, we introduce a general notation for azimuthally symmetric profiles as follows. We assume that the profile is parametrized by a direction nˆ, a vector of “linear” parameters a1, , aM, and a discrete ··· index I = 1, 2, , N representing one or more “nonlinear” parameters which have been ··· discretized. For the bubble collision problem, the linear parameters ai represent profile amplitudes, i.e. either aramp or astep. The number of linear parameters M is equal to 1 for the ramp model and equal to 2 for the ramp+step model. The index I represents the bubble 264 CHAPTER 6. OPTIMAL ANALYSIS OF AZIMUTHAL FEATURES

radius, discretized as described in 6.3.4. We will construct optimal statistics to search for § proﬁles of the parametrized form:

M Ii a`m = aib` Y`⇤m(nˆ) (21) = Xi 1 This parametrization should apply generally to any azimuthally symmetric family of pro- ﬁles, for example textures [24] or SZ clusters. In the subsequent sections, we will indicate which parts of our analysis framework apply in this generality, and which parts are speciﬁc to the case of the bubble collision.

As a ﬁnal piece of notation, we deﬁne the harmonic-space map Iinˆ by:

Ii (Iinˆ )`m = b` Y`⇤m(nˆ) (22)

so that the harmonic-space proﬁle in Eq. (21) is equal to i aiIinˆ . Less formally, Iinˆ is the proﬁle of a bubble with location nˆ, size I, amplitude parameterP ai = 1, and amplitude parameters a j = 0 for j , i.

6.4.2 An Algorithm for Fast Calculation of 2

2 2 T 1 Given a data realization dµp, we define its by (d) = d C d. Note that the likelihood function for the signal+noise realization d is a multivariate Gaussian with covariance matrix C: 1/2 1 T 1 (d) = det(2⇡C) exp d C d (23) L 2 ! so the 2 and the likelihood are related by (d) exp( 2(d)/2). L / 2 Given data realization dµp and profile parameters ( ai , I, nˆ) we define (d, ai , I, nˆ) { } { } to be the change in 2 when the profile is subtracted from the data realization d:

M 2 2 2 (d, ai , I, nˆ) = d aiAIi (d) (24) { } nˆ 0 i=1 1 B X C B C The Bayesian and frequentist statistics in the@ next two sectionsA will require very fast eval- 2 uation of (d, ai , I, nˆ), so in this section we will give an algorithm for computing it. { } 6.4. DATA ANALYSIS 265

The algorithm is organized as a set of precomputations which subsequently allow 2 to be evaluated at a single point (d, ai , I, nˆ) via an (1) table lookup. { } O We begin by rearranging (24) to write 2 in the following form:

2 T T 1 T T 1 (d, ai , I, nˆ) = 2 ai A C d + aia j A C AIj (25) { } Iinˆ Iinˆ nˆ Xi Xij Let us consider the two terms separately. The ﬁrst term in Eq. (25), the “data-bubble” term, can be computed eciently as follows. Deﬁne pixel-space maps DIi(nˆ) by:

Ii T 1 DIi(nˆ) = b` (A C d)`mY`⇤m(nˆ) (26) ` Xm

We precompute the maps DIi(nˆ) and save them to disk, by ﬁrst computing the harmonic- T 1 1 space map A C d using the fast C multiplication algorithm, and then calculating the pixel-space map DIi(nˆ) directly from Eq. (26) using a spherical harmonic transform for each pair (I, i). The data-bubble term in Eq. (26) is equal to 2 aiDIi(nˆ), so after the i DIi(nˆ) maps have been precomputed, the data-bubble term can subsequentlyP be evaluated with (1) computational cost. O Note that in this algorithm, we discretize the bubble location parameter nˆ using a ﬁnite pixelization of the sphere. In principle this pixelization need not have the same resolution as the pixelization used to represent the CMB maps (although in our WMAP pipeline, we use an Nside = 512 Healpix pixelization throughout). The second term in Eq. (25), the bubble-bubble term, cannot be calculated using the 1 same trick since this would require computing C AIinˆ for every pair (I, i) and pixel nˆ. However, the bubble-bubble term can be calculated statistically. Suppose that we simulate a per-channel map xµp with the same statistical properties as the data, so that the covariance T matrix xx is equal to C. Let XIi(nˆ) be the pixel-space map which is obtained from the h i simulation x in the same way that the map DIi(nˆ) above is obtained from the data:

Ii T 1 XIi(nˆ) = b` (A C x)`mY`⇤m(nˆ) (27) ` Xm

Then the Monte Carlo average XIi(nˆ)XIj(nˆ) , taken over random realizations of x, is given h i 266 CHAPTER 6. OPTIMAL ANALYSIS OF AZIMUTHAL FEATURES

by: T T 1 XIi(nˆ)XIj(nˆ) = A C AIj (28) h i Iinˆ nˆ The right-hand side is precisely what is needed to compute the bubble-bubble term in Eq. (25). Therefore, we use the following algorithm to compute the bubble-bubble term. We do an outer loop over Monte Carlo simulations x, and for each simulation we com-

pute the maps XIi(nˆ) using the same algorithm that was used to compute the maps DIi(nˆ) above, and accumulate the simulation’s contribution to the Monte Carlo average maps

XIi(nˆ)XIj(nˆ) . After all Monte Carlos have been run, we write the maps XIi(nˆ)XIj(nˆ) h i h i to disk. After precomputing these maps, the bubble-bubble term can be evaluated at a point

(d, nˆ, ai , I) with (1) computational cost. { } O

This concludes our fast algorithm for calculating 2. The intuitive idea behind the

algorithm is that we avoid doing a brute force scan over model parameters (nˆ, ai , I) when- { } ever possible. An obvious optimization is to eliminate the scan over the linear parameters a by noting that (2) is a quadratic polynomial in these parameters, so it suces to { i} compute the coecients of the polynomial (i.e. the maps DIi(nˆ) and XIi(nˆ)XIj(nˆ) ). What h i is less obvious is that the scan over the location parameter nˆ can be eliminated. In our algorithm we e↵ectively evaluate all nˆ simultaneously, using some algebraic tricks and a Monte Carlo approach to the bubble-bubble term. We do need to scan the bubble radius parameter (or more generally, any nonlinear parameters represented by the discrete index I) but we can organize the computation so that we do not pay the computational cost of a 1 C multiplication for each value of I, only the cost of a single spherical transform.

A ﬁnal comment is that running our fast (2) algorithm on a Monte Carlo ensemble of data realizations, rather than a single realization d, is computationally feasible. This is because the bubble-bubble term is the same in every Monte Carlo iteration, so we only need to recompute the data-bubble term. Since the data-bubble term is much faster to compute than the bubble-bubble term (by a factor equal to the number of Monte Carlos), evaluating 2 on an ensemble of Monte Carlos has roughly the same computational cost as the precomputations which are needed to evaluate it once. 6.4. DATA ANALYSIS 267

6.4.3 Bayesian Analysis

In a Bayesian analysis framework, we start with a prior p( ai , I, nˆ) on the parameters of the { } model, and wish to compute the posterior likelihood ( ai , I, nˆ d) given data realization d. L { } | From the posterior, we can compute various derived quantities such as conﬁdence regions and evidence integrals. Using Bayes’ theorem and the form of the Gaussian likelihood in Eq. (23), the posterior likelihood is:

( ai , I, nˆ d) (d ai , I, n) p( ai , I, nˆ) L { } | / L |{ } { } 1 2 exp (d, ai , I, nˆ) p( ai , I, nˆ) (29) / 2 { } { } ! Therefore, our fast 2 algorithm from the preceding section lets us evaluate the exact posterior as a table lookup with very minimal computational cost (tens of CPU cycles). This makes the Bayesian analysis essentially trivial; for example conﬁdence regions can be determined by gridding the likelihood rather than using an MCMC. The rest of this section is devoted to some practical details of the analysis for the special case of a bubble collision (rather than an arbitrary azimuthally symmetric set of proﬁles).

In this case, the form of the prior p( ai , I, nˆ) is constrained by symmetry: all directions nˆ { } are equally likely, and the comoving distance r to the bubble wall is uniformly distributed (see Eq. (3) above). Since the index I represents distance to the bubble wall, discretized to some set of values r , r , , r , we take the prior on the discrete index I to be proportional 1 2 ··· N to (r r )/2 for I = 1 2 1 wI = 8 (rI+1 rI 1)/2 for 1 < I < N (30) > > > (rN rN 1)/2 for I = N < > which corresponds to a uniform> distribution in r, discretized with trapezoid rule weighting. :> The prior p( ai , I, nˆ) then factorizes: { }

wI p( ai , I, nˆ) = p( ai ) (31) { } N (r r ) { } pix N 1 where p( a ) is a prior on the amplitude parameters, which depends on detailed physics of { i} 268 CHAPTER 6. OPTIMAL ANALYSIS OF AZIMUTHAL FEATURES

the inﬂationary model and cannot be deduced from symmetry alone. If a model-independent analysis is desired, there seems to be no particularly well-motivated choice of prior p( a ), { i} and so we take a uniform prior for simplicity.

Given the form (31) for the prior, we can marginalize over the bubble location and radius to obtain the posterior likelihood for the amplitude parameters:

wI 1 2 ( ai d) p( ai ) exp (d, ai , I, nˆ) (32) L { }| / { } Npix (rN r1) 2 { } XInˆ ! In practical data analysis, this likelihood function needs a small modiﬁcation for the following reason. We must mask regions of high foreground emission such as the Galactic plane, and the above likelihood sums over all bubble radii and locations, including bubbles which are completely masked and therefore unconstrained by the data. One symptom of this disease is that as the amplitude parameters a are taken to inﬁnity, the likelihood is { i} not exponentially suppressed, but approaches a nonzero constant, since we can “hide” a bubble of arbitrary large bubble amplitude in the masked part of the sky.

For this reason, we omit pairs (I, nˆ) in the sum (32) for which the corresponding bubble is unconstrained. If we simply excluded all pixels nˆ which are in the Galactic mask, then we would be throwing away information. If a bubble is centered on a masked pixel but a signiﬁcant fraction of the bubble spills outside the mask, then there are still many unmasked pixels that can be used to calculate a 2 value. To account for this, we exclude pairs (I, nˆ) such that: T 1 T 1 C < 0.1 C (33) Iinˆ Iinˆ Iinˆ 0 Iinˆ 0 nˆ 0 D E where the mean is taken over high latitude pixels nˆ 0 away from the galactic plane. This h·inˆ 0 radius-dependent criterion for masking bubble centers means that we only mask a bubble location (I, nˆ) if 90% of the temperature proﬁle is covered by the Galactic mask. ⇡ Omitting pairs (I, nˆ) which correspond to masked bubbles simply means that our Bayesian likelihood is the posterior likelihood for a bubble which is constrained to lie in the observable part of our sky, in the same sense that we constrain the bubble wall to intersect our observable Hubble volume. Parameter constraints derived from this likelihood have a rig- orous Bayesian interpretation as constraints on bubbles which lie in our Hubble volume and 6.4. DATA ANALYSIS 269

are not obscured by the Galaxy, with bubbles outside this observable volume unconstrained by the analysis. We assign confidence regions using the posterior likelihood ( a d) defined in Eq. (32), L { i}| and statistically test for bubbles by asking whether the point a = 0 is contained in the { i} 95% (for example) confidence region. We define confidence regions corresponding to probability p = 0.95 as follows. If there is only one amplitude parameter a, we define the confidence region [amin, amax] by the requirement that the total probability in each one-sided tail be (1 p)/2. Formally, a min 1 1 p 1 (a d) = (a d) = (a d) (34) 0 L | a L | 2 L | Z Z max Z1 In two or more variables a , we threshold the likelihood so that the total likelihood above { i} threshold is p. More formally, we define the confidence region by solving for the value L0 such that the set of points a satisfying (a d) satisfies: i L | L0

dai (a d) = p dai (a d) (35) (a d) L | L | ZL | L0 Z and taking the confidence region to be the set of a-values such that (a d) . L | L0 Note that if we do not mask bubble centers, then the integrals in Eqs. (34), (35) di- verge and we cannot define confidence regions. Thus some prescription for bubble center masking seems to be necessary, although our prescription (33) is not the only possibility. An alternate statistical test for bubbles is to compute the Bayesian evidence for the bubble model, and compare it to the evidence for a no-bubble model with the usual Gaussian likelihood. The relevant Bayes factor is:

1 2 dai p( ai ) Inˆ wI exp( (d, ai , I, nˆ)) K = { } { } 2 { } (36) R dai p( ai ) I wI P{ } { } nˆ R P which is only defined if the prior p( a ) is normalizable (i.e. da p( a ) < ). For a { i} { i} { i} 1 given inflationary model, there should be a physically defined prior p( a ) which is nor- R { i} malizable, but there is no clear choice of normalizable prior which is model-independent. For this reason, our perspective is that the Bayesian evidence is not a good statistical test for 270 CHAPTER 6. OPTIMAL ANALYSIS OF AZIMUTHAL FEATURES

bubbles in a generic, model-independent analysis, and we use conﬁdence regions instead. The posterior likelihood ( a d) in Eq. (32) has the following, perhaps counterintu- L { i}| itive, property which deserves explicit comment. For a randomly generated realization, there is an order-one probability that the maximum likelihood is very close to the point a = 0, much closer than the width of the likelihood. We discuss this further in Ap- { i} pendix 6 B and develop an analytic model of the likelihood. We show that this behavior is expected, and such a realization should simply be interpreted as one which is consistent with a Gaussian ﬁeld, with no statistical evidence for bubbles.

6.4.4 Frequentist Analysis

In this section we will construct optimal frequentist statistics for the bubble collision problem, and find an algorithmic trick which makes the analysis computationally feasible. We should say from the outset that we will formulate the frequentist analysis as a procedure for defining confidence regions by hypothesis testing, using a likelihood ratio test which can be shown to be optimal; we will not use the term “estimator”. In cosmology, frequentist analyses are often formulated in a di↵erent way, by defining a global estimator for the model parameters (in this case the bubble amplitudes). This type of estimator-based analysis is optimal in many cases (the formal criterion for optimality is that the estimator should be a sucient statistic for the likelihood ratio test), but fails to capture key structure of the bubble collision problem. To see this intuitively, consider a realization with multiple statistically significant bubble-like features. The likelihood for the bubble amplitude now has multiple peaks and the correct confidence regions consist of multiple disconnected “is- lands”. This structure will not be captured if we try to compress the likelihood into a single number by defining an estimator for the bubble amplitude. Therefore, our focus will not be on estimators, but on identifying the optimal Monte Carlo procedure for testing whether the data are consistent with a given set of bubble amplitude parameters a . { i} First consider the question: how do we test whether the no-bubble model (i.e. a = 0) { i} is consistent with the data, at say 95% confidence level? In a frequentist analysis, confidence regions are defined by Monte Carlo based hypothesis testing. We construct a test 6.4. DATA ANALYSIS 271

statistic ⇢0 which statistically separates no-bubble realizations from realizations with bubbles, and evaluate ⇢0 on an ensemble of Monte Carlo simulations. If the value of ⇢0 on the data is larger than 95% of the simulations, then a = 0 is excluded at 95% CL. { i} Constructing optimal test statistics is sometimes a challenge in the frequentist approach, but for the bubble collision problem there is a natural choice, as we now explain. We will use the prior on the bubble location from the previous section: all directions nˆ are equally likely, and the distance to the bubble wall is distributed with PDF proportional to the quantity w deﬁned in Eq. (30). We will not need a prior p( a ) on the amplitude I { i} parameters. For a ﬁxed set of amplitude parameters a , the conditional likelihood (d a ) is given { i} L |{ i} by: wI 1 2 (d ai ) exp (d, ai , I, nˆ) (37) L |{ } / Npix (rN r1) 2 { } XInˆ ! Our frequentist test statistic ⇢0 will be a likelihood ratio statistic:

(d ai ) ⇢0(d) = max L |{ } (38) ai (d 0) { } L |

If ⇢0(d) is larger than the value found in a large fraction (say 95%) of the simulations, then we will reject the null hypothesis that there is no bubble in the data. The Neyman-Pearson lemma states that this test has the lowest probability of rejecting the null hypothesis when the null hypothesis is false, i.e. the test has the greatest sensitivity of detecting a bubble if one is present in the data. We evaluate ⇢ on the data realization d using the fast 2 algorithm from 6.4.2. As 0 § remarked there, it is also computationally feasible to evaluate ⇢0(x) for a Monte Carlo ensemble of no-bubble simulations x . To test whether a = 0 is excluded at a given 0 { i} conﬁdence level, we simply test whether the “data” value ⇢0(d) exceeds the appropriate fraction of the simulation values ⇢0(x0).

The test statistic ⇢0 suces for testing whether the data is consistent with the no-bubble model a = 0. Ideally, we would like to do more: we want to determine the full conﬁdence { i} regions in the a parameter space, rather than just being able to test whether the point { i} a = 0 is contained in a given conﬁdence region. We next present an algorithm for { i} 272 CHAPTER 6. OPTIMAL ANALYSIS OF AZIMUTHAL FEATURES

computing full conﬁdence regions.

Conceptually, we test whether a point ai , 0 is contained in a given frequentist con- { } fidence region using a Monte Carlo procedure similar to the one described above for the a = 0 case, but with two di↵erences. First, instead of the test statistic ⇢ , we use the { i} 0 statistic: (d ai 0) ⇢ ai (d) = max L |{ } (39) { } ai 0 (d a ) { } L |{ i} which is the appropriate likelihood ratio statistic for separating the model with given amplitude parameters ai from models with amplitude parameters ai 0 , ai . Second, instead { } { } { } of evaluating ⇢ a on a Monte Carlo ensemble of no-bubble simulations, we must use an en- { i} semble of simulations which contain a randomly located bubble with amplitude parameters a . This presents a computational problem: suppose we want to plot frequentist confi- { i} dence regions by looping over a grid of a values, and testing whether each grid point { i} is contained in a given confidence region. Naively, this requires running a new ensemble of Monte Carlos for each grid point a , since the simulations are a -dependent (they { i} { i} contain simulated bubbles with amplitude a ). Of course, this procedure is computation- { i} ally impractical; we cannot a↵ord to do an independent set of Monte Carlos for each grid point. However, there is a computational trick which produces a mathematically equivalent result but is computationally a↵ordable, allowing frequentist confidence regions to be determined.

The idea behind the trick is the following. The reason we need a di↵erent Monte Carlo ensemble for each grid point a is that the simulations are a -dependent: they contain a { i} { i} single bubble with amplitude parameters a . Using notation from 6.4.1, the single-bubble { i} § simulation x a can be written: { i}

x a = x0 + aiAIinˆ (40) { i} Xi where x0 is a no-bubble simulation. We can imagine generating the simulation in two steps:

ﬁrst we randomly generate x0, I, and nˆ, and then we take the linear combination in the above equation to get x a . Thus, for ﬁxed x0, I, nˆ, the simulations with di↵erent values of ai are { i} { } not fully independent; they are all linear combinations of (M +1) independent maps, where 6.4. DATA ANALYSIS 273

M is the number of linear parameters. A little thought shows that if we evaluate the “data- bubble” term from 6.4.2 for each of these (M + 1) maps, this suces to compute 2 for § an arbitrary linear combination. This in turn suces to evaluate the frequentist test statistic

⇢ a (x a ) for all values of ai , which is what we need to compute conﬁdence regions. In { i} { i} { } other words, we can evaluate ⇢ a (x a ) at all grid points ai simultaneously (for a ﬁxed { i} { i} { } choice of x0, I, nˆ) with computational cost proportional to (M + 1), not proportional to the number of grid points.

More formally, our algorithm for computing frequentist conﬁdence regions is as follows.

1. At the beginning of each Monte Carlo iteration, we simulate a CMB+noise realiza-

tion x0 and randomly choose a bubble radius I (with PDF wI) and center nˆ.

2. Compute the quantity T T 1 Q = A C x (41) I0i0nˆ 0 I0i0nˆ 0 0 using the fast algorithm for the “data-bubble” term from 6.4.2. Repeating the same § algorithm for each i = 1, , M, compute the quantity: ···

T T 1 R = A C A (42) iI0i0nˆ 0 I0i0nˆ 0 Iinˆ

Note that QI0i0nˆ 0 is the data-bubble term for the CMB+noise realization x0, and ai0 RiI0i0nˆ 0

is the data-bubble term for a realization containing a bubble with parameters ai, I, nˆ , { P } and no CMB or noise component.

3. For each grid point a , we can compute { i}

(x a ai 0) { i} ⇢ ai (x ai ) = max L |{ } (43) { } { } ai 0 (x a ai ) { } L { i}|{ } 274 CHAPTER 6. OPTIMAL ANALYSIS OF AZIMUTHAL FEATURES

by using the following expression to compute conditional likelihoods with (1) com- O putational cost:

wI0 (x ai ai 0) L { }|{ } / Npix (rN r1) XI0nˆ 0 1 T T 1 exp a0 QI i nˆ + aia0 RiI i nˆ a0 a0 A C AI j nˆ (44) i0 0 0 0 i0 0 0 0 2 i0 j0 I0i0nˆ 0 0 0 0 0 i0 ii0 i0 j0 1 BX X X C B C Note that the last@ term in the exponential is the bubble-bubble term from A 6.4.2, § which is the same in every Monte Carlo iteration and has been precomputed.

4. Save ⇢ a (x a ) to disk (evaluated on a grid of ai values) and proceed to the next { i} { i} { } Monte Carlo.

5. After all Monte Carlos have been run, for each grid point a we rank the “data” value { i} ⇢ a (d) relative to the ensemble of simulated values ⇢ a (x a ) to assign a p-value { i} { i} { i} p( a ). Confidence regions for a can be assigned by thresholding these p values, { i} { i} e.g. the 95% confidence region consists of all values of a satisfying p( a ) 0.95. { i} { i}  The result of this algorithm is mathematically equivalent to running an independent set of Monte Carlos for each grid point a . { i} In step 1, we constrain the random choice of bubble size I and center nˆ so that the bubble is not obscured by the galaxy, using the same criterion (33) that we used in the Bayesian analysis. This constraint is necessary to make confidence regions well-defined; otherwise we would not be able to rule out any model since we can “hide” an arbitrarily large bubble behind the galaxy. Frequentist confidence regions obtained using this constraint can be rigorously interpreted as constraints on bubbles which overlap our Hubble volume and observable sky. This is completely analagous to the Bayesian case discussed above.

6.4.5 Computational Cost

We brieﬂy summarize the computational and storage costs of the analysis for the WMAP data. We use 5000 Monte Carlos to calculate the bubble-bubble term and an additional 5000 Monte Carlos to compute conﬁdence regions in the frequentist analysis. For each Monte 6.5. LARGE NUMBERS OF BUBBLES 275

1010

108 ] 2

K 6 µ 10 [ l C 4 l

104

102

10 100 1000 Multipole

Figure 6.4 The WMAP 7-year best fit ⇤CDM CMB spectrum (black line), the spectrum of bubbles with the ramp profile normalized to the CMB spectrum at ` = 2 (blue), and the spectrum of bubbles with the step profile normalized in the same way (red).

1 Carlo we must perform the C operation which takes approximately 15 core-minutes per 1 Monte Carlo, giving a total compute time of 2500 core-hours. We save all of the C - filtered simulations to disk and only perform the T AT operation when needed. This operation only takes a few core-seconds to compute, but must be done for each Monte Carlo and each profile that we use. We have over 2000 profiles and so the operation takes a few 104 ⇥ core-hours in total. At WMAP resolution each map requires approximately 6 MB of disk space (we save the maps in harmonic space which results in a factor of 4 reduction in disk 1 space). We require 60 GB to save all of the C -filtered maps, and an additional 13 GB to save the maps representing bubble-bubble terms.

6.5 Large Numbers of Bubbles

In this section we consider the signal arising from a large number of overlapping bubbles. As the number of bubble collisions increase, the signal on the sky from each additional 276 CHAPTER 6. OPTIMAL ANALYSIS OF AZIMUTHAL FEATURES

collision overlaps with previous collisions. We assume that the direction of each bubble is independent of the other bubble directions. Then in the limit N 1 the signal becomes a Gaussian random ﬁeld by the central limit theorem. We calculate the expected signal by summing the contributions from individual bubbles:

T(nˆ) = Ti(nˆ) = Xi 1 (45) N

= aib`(ri)Y`⇤m(nˆ i) Y`m(nˆ) `m 0 i=1 1 X X B C @B AC Here, ai, ri and nˆ i denote the amplitude, comoving displacement, and angular position of the i-th bubble. The proﬁle b`(r) could be either the ramp or the step proﬁle (for simplicity we have not considered the case of a ramp+step model with two amplitude parameters, but this is a straightforward generalization). We calculate the two-point correlation function of Eq. (45) as follows:

2 (T) (T)⇤ = a b (r )b (r )Y⇤ ( )Y ( ) + a a b (r )b (r )Y⇤ ( )Y ( ) `m `0m0 i ` i `0 i `m nˆ i `0m0 nˆ i i j ` i `0 j `m nˆ i `0m0 nˆ j * i i, j + ⌦ ↵ X X (46) where the average is over realizations of the bubble random ﬁeld. The second term h·i contains the angular average Y⇤ (nˆ i)Y` m (nˆ j) . Since the locations of bubbles i , j are as- h `m 0 0 i 2 sumed independent, this is equal to Y⇤ (nˆ i) Y` m (nˆ j) = `0` 0/(4⇡) and only contributes h `m ih 0 0 i 0 a monopole. We therefore ignore the second term in Eq. (46) because a bubble contribution to the monopole is not measurable (since TCMB is a free parameter anyway).

We can evaluate the ﬁrst term in Eq. (46) as follows. The parameters ai, ri and nˆ i are assumed indepdendent, so we can average them separately. The average over ri can be performed using the uniform prior (Eq. (3)):

1 rmax b (r )b (r ) = dr b (r)b (r) (47) h ` i `0 i i r ` `0 max Z0 6.5. LARGE NUMBERS OF BUBBLES 277

The average over angular position nˆ i is straightforward to compute:

1 Y⇤ (nˆ i)Y` m (nˆ i) = `` mm (48) `m 0 0 4⇡ 0 0 ⌦ ↵ Plugging into Eq. (46), we can now read o↵ the power spectrum of the bubble contribution to the CMB temperature as follows:

2 N a rmax bub 1 2 C = dr b`(r) (49) ` 4⇡ r D E max Z0 where a2 = a2 and the expected number of bubbles N in the observable range r h i h i i max  r r , are model-dependent quantities. We see from this expression that the combination  max N a2 is constrained by the power spectrum. h i

The many-bubble power spectrum has also been calculated recently by Aguirre & Kozaczuk [7]. Our result agrees if we use the ramp profile and ignore transfer functions, i.e. make the Sachs-Wolfe approximation in Eq. (16). For the ramp profile, we find that the Sachs-Wolfe approximation is fairly accurate but overpredicts Cbub by 14%, due to ` ⇡ omitting ISW contributions to the bubble profile as described in 6.3.3. We note that in [7], § higher-order corrections to the linear CMB + bubble model are calculated, but shown to be negligible.

bub In Fig. 6.4, we show the power spectrum C` for both the ramp and step models. We use the denser sampling of r that we used for the step profile in order to calculate the integral more precisely. The power spectrum falls o↵ roughly as 1/`5 for the ramp profile, and 1/`3 for the step profile. Given the roughly 1/`2 dependence of the CMB spectrum, this implies that the step model is constrained by a wide range of ` values, while the ramp model is constrained mainly by the lowest values of ` (in fact we find that 92% of the statistical weight comes from the quadrupole). For the ramp model, the Sachs-Wolfe approximation is fairly accurate and Planck will not significantly improve WMAP constraints; the opposite statements are true for the step model. This parallels the discussion in 6.3.3 above. § 278 CHAPTER 6. OPTIMAL ANALYSIS OF AZIMUTHAL FEATURES

6.6 Conclusions

Searching for anomalous signals in CMB data has become a big industry, with a large number of di↵erent methodologies being employed [25]. Signals that may be in the CMB data o↵er the opportunity to learn about the inflationary, and possibly the pre-inflationary, epoch. In this paper, we have focused on an example signal that has recently been discussed in the theory literature. This theory states that we live in a bubble of low vacuum energy density surrounded by an infinite, eternally inflating spacetime with a higher vacuum energy density. If there are other bubble regions that were created near to our own then they could have collided with our bubble in the past leaving a distinct pattern in the CMB. We have developed a toolkit of algorithms which allow us to perform the exact, all-sky, optimal data analysis for the bubble signal. The main features of our analysis are:

We precompute the bubble proﬁles including CMB transfer functions. •

We precompute a large number of maps (the maps denoted DIi(nˆ) and XIi(nˆ)XIj(nˆ) • h i in 6.4.2) which permit very fast evaluation of 2, essentially as a table lookup § operation.

After these precomputations, the Bayesian posterior likelihood can be evaluated so • quickly that a Bayesian data analysis may be performed trivially by gridding the likelihood (i.e. without MCMC). We compute the exact all-sky posterior without making any approximations, such as neglecting small-scale anisotropy in the noise.

We identify the optimal frequentist statistic for the bubble collision problem, and • show how a full frequentist analysis, including calculation of conﬁdence regions, can be made computationally a↵ordable with some additional computational tricks.

Although we have focused on the bubble collision problem, our algorithms should apply to any parametrized family of azimuthally symmetric proﬁles, and we have presented them in a form which emphasizes this generality. Our algorithms should be useful for other problems, e.g. searches for other defects such as textures, or optimal detection of Sunyaev- Zeldovich clusters. We analyze WMAP data in a separate paper [8]. 6.6. CONCLUSIONS 279

We have also calculated the signal that would be expected from a large number of independent bubbles. The signals from di↵erent bubbles overlap, and the total signal tends to a Gaussian random field as N 1, which implies that the power spectrum is the optimal statistic in the many-bubble limit. We calculate the many-bubble power spectrum and find results consistent with [7]. Our methodology could be extended to the case of a small number of collisions. This is slightly di↵erent to the single-bubble analysis because the bubbles can overlap. The extra parameters for the additional bubbles can be included in the likelihood, which can be calculated in a similar way to the single-bubble case, albeit with increased computational requirements. We defer the details to future work. Future observations can improve WMAP constraints on bubble collisions in a few ways. Considering CMB observations first, the increased angular resolution of upcoming experiments such as Planck will improve constraints on the step model, but not on the ramp model where the signal is weighted toward large scales that are already sample variance limited in WMAP. Future measurements of CMB polarization may eventually improve constraints on both models [13]. Looking beyond the CMB, large-scale structure can potentially provide interesting constraints on bubble collisions, e.g. coherent galaxy flows [26, 27], Lyman- alpha forest measurements [28], or 21-cm radiation surveys [29, 30].

Acknowledgments

We thank Adam Brown, Matt Kleban, Ben Freivogel, Steve Shenker, and Lenny Susskind for helpful discussions. SJO acknowledges support from the US Planck Project, which is funded by the NASA Science Mission Directorate. LS is supported by DOE Early Career Award DE-FG02-12ER41854 and the National Science Foundation under PHY-1068380. KMS was supported by a Lyman Spitzer fellowship in the Department of Astrophysical Sciences at Princeton University. Research at Perimeter Institute is supported by the Gov- ernment of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation. Some of the results in this paper have been derived using the HEALPix [31] package. Computations were performed at the TIGRESS high performance computer center at Princeton University which is jointly supported by the 280 CHAPTER 6. OPTIMAL ANALYSIS OF AZIMUTHAL FEATURES

Princeton Institute for Computational Science and Engineering and the Princeton Univer- sity Oce of Information Technology. We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA), part of the High Energy Astro- physics Science Archive Center (HEASARC). HEASARC/LAMBDA is a service of the Astrophysics Science Division at the NASA Goddard Space Flight Center. Bibliography

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1 6 A Details of the C Filter

In this section we overview the calculation of the inverse-variance ﬁltering operation x ! 1 C x, where, following notation introduced in 6.4.1, x is a per-channel pixel-space map § µp and C = N + AS AT . Additional details can be found in [23] where the method that we describe was ﬁrst developed.

T 1 For purposes of this paper, it suces to implement the ﬁltering operation x A C x ! where the quantity on the RHS is a harmonic-space map. (Throughout the body of the 1 T 1 paper, C only appears as part of the combination A C .) Now consider the identity:

T 1 T T 1 A C = A (N + AS A ) 1/2 1/2 T 1 1/2 1 1/2 T 1 = S (1 + S A N AS ) S A N (A-1)

This identity is not obvious, but can be proved by mutiplying both sides on the right by (N+AS AT ). It will be convenient to use the RHS of Eq. (A-1), so that the ﬁltering operation 1 can be performed using the purely harmonic-space operation a X a, where we have ! deﬁned the operator: 1/2 T 1 1/2 X = 1 + S A N AS (A-2)

1 A further advantage of using the RHS of Eq. (A-1) is that the inverse noise covariance N appears instead of the noise covariance N. This allows us to incorporate a foreground mask, 1 by setting matrix entries of N to zero in masked pixels. Analagously, we marginalize the CMB monopole and dipole (independently in each channel µ) by modifying the operator 1 1 N so that N x = 0 for the four independent modes of the monopole or dipole. (This 1 implies that C x is also zero for these modes.)

1 We implement the ﬁltering operation a X a using the preconditioned conjugate gra- ! dient algorithm [32]. The preconditioner must be chosen with care since it has a signiﬁcant impact on the compute time. For example, a diagonal preconditioner gives extremely slow convergence of the algorithm because on scales ` . 500 the data is signal dominated [9, 33] and so the approximation that the covariance is diagonal is not valid. Instead, a block diag-

onal preconditioner P can be used that contains the full covariance on large scales ` < `split 1 6 A. DETAILS OF THE C FILTER 285

and is diagonal on smaller scales:

1 X 0 P = (0) (A-3) 1 0 0 X 1 B C B C @B AC 1 where the subscript (0) denotes that the full resolution is used and X is the diagonal approximation. The block diagonal preconditioner is found to have faster convergence than the diagonal preconditioner [23], but is still too slow for our purposes. A multigrid preconditioner is found to decrease the convergence time by an order of magnitude over the block diagonal preconditioner for the WMAP data [23]. The aim of the multigrid preconditioner is to perform the conjugate gradient descent at resolution Nside with a preconditioner that is itself calculated using the conjugate gradient descent algorithm but at a lower resolution 1 of Nside/2. The X(0) operation is therefore evaluated using conjugate gradient descent with 1 (1) (0) a preconditioner X(1) that is calculated at a lower resolution such that lmax < lmax. The lower resolution preconditioner can itself be calculated using an even lower resolution pre- 1 conditioner X(2) in a recursive manner. The preconditioner for the first layer of recursion is: 1 X 0 P = (1) (A-4) 1 0 0 X 1 B C B C where the subscript (1) denotes that the@B preconditionerAC is calculated at resolution Nside/2. The time required to calculate harmonic transforms depends on the resolution as N3 , O side and so operations at the coarser resolution are significantly faster. The inversion therefore⇣ ⌘ proceeds recursively with the coarsest resolution of Nside = 128 preconditioned using the block diagonal preconditioner [34, 35]. Using the multigrid preconditioner we find that 1 C x can be calculated in approximately 15 core-minutes, which together with the other computational tricks that we describe makes our likelihood analysis computationally feasible. 286 BIBLIOGRAPHY

1.2 20

) 1.0 d | a

15 ( L 0.8

Counts 10 0.6

Likelihood 0.4 5 0.2

0 0.0 0.4 0.2 0.0 0.2 0.4 0.4 0.2 0.0 0.2 0.4 1 Maximum likelihood amplitude aML (Mpc ) Amplitude a

Figure 6.5 Left panel: Histogrammed maximum likelihood bubble amplitude aML, for many Monte Carlo simulations of the data d. Counterintuitively, aML is within 0.001 of zero 11% of the time. Right panel: Posterior likelihoods (d a) for three randomly simulated L | data realizations d. It is seen that the derivative (@ /@a) is always zero at a = 0, but this L point can be either the global maximum likelihood (blue, solid curve), a local maximum which is not the global maximum (magneta, dotted curve), or a local minimum (red, dashed curve). Likelihoods in this ﬁgure were generated using a toy model with 1000 µK-arcmin isotropic noise, 5 Gaussian beam, no sky cut, and assuming ﬁxed bubble size ✓bubble = 30.

6 B Likelihood Model

The posterior likelihood ( a d) deﬁned in Eq. (32) is a complicated function of the data L { i}| that cannot be easily evaluated analytically. However, by making some approximations we can create a model of the likelihood that can be used to understand its main properties. In particular we wish to understand the following phenomenon. Suppose that we make Monte

Carlo simulations of the data d, and compute the value aML of the amplitude parameter which maximizes the likelihood (a d) in each simulation. Naively we might guess that L | aML, histogrammed over many Monte Carlo simulations, would be Gaussian distributed with mean zero and width roughly equal to the minimum detectable bubble amplitude.

Instead, aML is distributed as shown in the left panel of Fig. 6.5. There is an order-one probability for aML to be very close to zero. While this phenomenon may seem counterintuitive, we will show that it has a natural explanation in terms of analytic properties of the likelihood. A clue can be obtained by visually inspecting the likelihood (a d) for a few randomly L | 6 B. LIKELIHOOD MODEL 287

generated data realizations d (Fig. 6.5, right panel). It is seen that the derivative @ /@a is L always zero at a = 0, but this local extremum can be either a local maximum or a local minimum. These properties of the likelihood can be understood analytically as follows. For notational simplicity, consider the simplest case of a single linear parameter i, and non-linear parameters I which can only take a single value, i.e. a randomly located profile with arbitrary amplitude but fixed radius and shape. Suppressing superfluous indices, the posterior likelihood can be written:

2 1 T T 1 a T T 1 (a d) = exp a nˆ A C d nˆ A C Anˆ (B-1) L | Npix 2 Xnˆ ! The ﬁrst derivative at a = 0 is:

@ (a d) 1 T T 1 L | = nˆ A C d (B-2) @a a=0 Npix ! Xnˆ In the limit N of a ﬁnely pixelized map, the RHS approaches zero. To see this, we pix !1 1 ﬁrst take the N limit of the harmonic-space map N : pix !1 pix nˆ nˆ P 1 1 2 (nˆ )`m d nˆ (nˆ )`m Npix ! 4⇡ Xnˆ Z 1 2 = d nˆ b`Y⇤ (nˆ) 4⇡ `m Z b` = `0 (B-3) p4⇡

1 1 i.e. the map Npix nˆ nˆ is a pure monopole. When we apply C A to this map, we get zero since there is noP instrumental sensitivity to the monopole. (More formally, as explained in Appendix 6 A, the monopole has been marginalized by assigning it inﬁnite noise variance 1 in the pixel domain, which implies that we get zero when we multiply by C .) This implies that the RHS of Eq. (B-2) is zero.b

bStrictly speaking, this is only true in the high resolution limit N . In a ﬁnite pixelization, the RHS pix !1 will be nonzero due to pixelization artifacts, but small (of order 1/Npix). Our procedure for removing terms in the likelihood which correspond to bubbles which are hidden behind the galactic mask, given in Eq. (33) above, also gives a small nonzero contribution to the derivative on the RHS of Eq. (B-2). These e↵ects mean that the local extremum of the likelihood does not appear precisely at a = 0, but is perturbed to a value of a 288 BIBLIOGRAPHY

The second derivative of Eq. (B-1) at a = 0 is:

2 @ (a d) 1 T T 1 2 T T 1 = L 2| ( nˆ A C d) nˆ A C A nˆ (B-4) @a a=0 Npix ! Xnˆ Here, the RHS is nonzero on a per-realization basis. However, using the expectation value ddT = C, one sees that the expectation value of Eq. (B-4) is zero. This implies that the h i second derivative will be positive (or negative) some order-one fraction of the time, in order to get a zero expectation value. Summarizing, we have now shown that the likelihood (a d) always has a local ex- L | tremum at a = 0, which is a local maximum an order-one fraction of the time. If a = 0 is a local maximum, then there is a further order-one probability that it turns out to be the global maximum, which can be seen intuitively because the likelihood has a finite number of local maxima which are candidates for the global maximum. This explains the phenomenon seen in the histogram in the left panel of Fig. 6.5 above, where the maximum likelihood is at a = 0 an order-one fraction of the time. How does this phenomenon a↵ect our statistical analysis? We answer this question separately for the Bayesian ( 6.4.3) and frequentist ( 6.4.4) cases. In the Bayesian case, § § we infer confidence regions on the amplitude parameters ai from the posterior likelihood (a d), using the prescription in Eqs. (34), (35). In a data realization where (a d) has its L i| L i| global maximum at ai = 0, we will find that the point a = 0 is contained in every confidence region, i.e. if instead of computing the 95% CL region, we use a di↵erent p-value, we will

find that the confidence region contains ai = 0, irrespective of the p-value. The frequentist case is similar: if we have a realization d where (a d) has its global L i| maximum near ai = 0, we will find that the frequentist test statistic ⇢0(d) is nearly equal to 1. It follows that a = 0 is contained in every frequentist confidence region, regardless of the threshold (i.e. whether 68%, 95%, etc.) This is because we determine whether a = 0 is outside the confidence region by testing whether ⇢0(d) is anomalously high compared to an ensemble of simulated values ⇢ (d0), and the simulated values satisfy ⇢ (d0) 1 by 0 0 definition. Therefore, in either the Bayesian or frequentist framework, there is an order- one probability that a random data realization d will have a maximum likelihood amplitude which is nonzero but very small. 6 B. LIKELIHOOD MODEL 289

precisely at ai = 0, which implies that a = 0 is contained in every confidence region. This behavior is expected analytically and simply means that the realization is consistent with being a Gaussian field, and there is no statistical evidence for a profile with nonzero amplitude. Chapter 7

Collisions with Other Universes: the Optimal Analysis of the WMAP Data

“It may be – I hope it is – redemption to guess and perhaps perceive that the universe, the hell which we see for all its beauty, vastness, majesty, is only part of a whole which is quite unimaginable.”

A Moving Target William Golding

An appealing theory is that our current patch of universe was born as a nucleation bubble from a phase of false vacuum eternal inflation. We search for evidence for this theory by looking for the signal imprinted on the CMB that is generated when another bubble “universe” collides with our own. We create an ecient and optimal estimator for the signal in the WMAP 7-year data. We find no detectable signal, and constrain the amplitude, a, of the initial cur- 8 vature perturbation that would be generated by a collision: 4.66 10 < ⇥ 4/3 8 1 a (sin ✓ ) < 4.73 10 [Mpc ] at 95% confidence where ✓ is the bubble ⇥ bubble angular radius of the bubble signal.

290 7.1. INTRODUCTION 291

7.1 Introduction

Few things are more exciting than discovering what happened at the beginning of the universe, or understanding the structure of spacetime outside our observable universe. Quite remarkably, there exist cosmological signatures that would allow us to address exactly these questions. In this paper we concentrate on one of these, which is the signature imprinted on the Cosmic Microwave Background radiation (CMB) by primordial bubble collisions. An appealing theory for the origin of the universe is that it was created by quantum mechanical tunneling from a much larger eternally inflating spacetime. This larger spacetime expands exponentially, driven by the energy of the false vacuum. Occasionally one region of space will tunnel to a lower energy vacuum. Even though the resulting bubble-like region expands at the speed of light, and bubbles are continuously produced in many places, the false vacuum region expands quickly enough that the bubbles never percolate. This is the so-called False Vacuum Eternal Inflation [1]. In this scenario, we live in the interior of one of the bubbles, where our standard slow roll inflationary phase takes place. Al- though bubbles do not percolate and fill the whole of space, there is a chance that in the past another bubble collided with our own [2], leaving a specific disk-shaped signal in the CMB [3]. This is a localized and well defined signature. Discovery of such a collision would have tremendous implications for the whole field of high-energy physics. We would learn of a new cosmological epoch, eternal inflation, that happened before the epoch of standard inflation. Furthermore, we would learn that the field theory that describes the universe has at least two vacua, one false and one true. This would already be an important discovery. In addition, the detection of a bubble collision would hint that there are many other universes like ours, a whole landscape of vacua. The presence of a landscape of vacua would provide indirect evidence of string theory, which predicts such a landscape, and also of Weinberg’s anthropic explanation of the cosmological constant [4], which relies on a landscape of vacua each with a di↵erent vacuum energy. This motivates us to search for bubble collisions in the WMAP data. Implementing an exact, fully optimal likelihood analysis is still not a completely solved problem, although considerable progress has been made recently in [5–7]. In the companion paper [8], we introduce new algorithmic tricks 292 CHAPTER 7. COLLISIONS WITH OTHER UNIVERSES

which solve this problem, making the optimal analysis computationally a↵ordable and furthermore simplifying the methodology. Our tools are particularly powerful for bubbles with a sharp edge feature (the “step” profile defined below), where the typical bubble is large but the CMB maps must be kept at high resolution, since most of the signal-to-noise comes from small angular scales. For bubbles without a sharp edge feature (the “ramp” profile below), the statistical weight comes entirely from angular scales of order 1 degree or larger, where WMAP is sample variance limited and additional data (e.g. from Planck) will not improve the measurement. The statistically optimal constraints reported in this paper therefore represent the ultimate constraints which can be obtained for this profile using CMB temperature. Finally, we extend existing analysis techniques by complementing the Bayesian analysis by a Monte Carlo based approach which does not depend on an external prior for the bubble amplitude. In this paper we focus on the results and describe the technical details in [8], where we highlight how the analysis techniques that we develop in this context can be applied to all localised features in the CMB. We use the WMAP 7-year cosmological parameters throughout [9].

7.2 Bubble Signal

The theory of bubble collisions has recently been reviewed in [10]. While the amplitude of the signal depends on the details of the dynamics describing the collision, the shape of the signal does not, and it appears as a disk on the sky. We model the bubble collision as a perturbation to the initial adiabatic curvature ⇣(x), and consider two possible forms: either a “ramp” profile ⇣ (z) = aramp(z r) for z r and zero for z r (where r is the comoving r  distance to the bubble wall), or a “step” profile ⇣ (z) = astep for z r and zero for z r. r  We propagate this curvature perturbation to a CMB temperature perturbation using the full numerically computed CMB transfer function. As described in detail in [8], the transfer function corrects the ramp profile by 10% and the step profile at order unity, so including ⇡ it is necessary for a precise analysis. The theoretical distribution of bubble sizes is determined by the symmetry and geome- try of the bubble collision, and we incorporate this information into our analysis. For small curvature ⌦k, the comoving distance r to the bubble wall is uniformly distributed [10]. 7.3. METHOD 293

Equivalently, the size distribution is given by dP(✓ ) d cos ✓ = sin ✓ d✓ . bubble / bubble bubble bubble The “typical” bubble is large; its angular size is of order one radian. For the ramp proﬁle, most of the signal for detecting such a bubble comes from angular scales comparable to the bubble radius (roughly ` . 20), where WMAP is cosmic variance limited. For the step proﬁle, the signal comes mainly from high-` modes associated with the sharp edge feature, and Planck can potentially improve the optimal WMAP constraints presented here.

7.3 Method

We search for the bubble signature in the WMAP 7-year V-band and W-band data. To minimize foreground contamination, we use foreground reduced maps, and apply the WMAP KQ75 extended temperature analysis mask to exclude the Galaxy and bright point sources. We use the algorithmic machinery from [8] to perform all-sky exact evaluation of the Bayesian likelihood and optimal frequentist statistic; we summarize the key steps as follows. Since the CMB and WMAP noise are Gaussian to a very good approximation, the likelihood for obtaining data realization d has the form (d) exp( 2(d)/2), where we have L / 2 T 1 2 deﬁned (d) = d C d. Here, C is the data covariance matrix with dimension (nNpix) , where n and Npix are the number of di↵erencing assemblies (DA’s) and pixels. We use the exact CMB + noise covariance matrix throughout, which leads to optimal statistics by optimally weighting the maps in the presence of masking, noise inhomogeneity, and per-DA beams. We consider a single-bubble model, which has four parameters: the amplitudes aramp, astep of the ramp and step perturbations, the distance r to the collision wall, and the direction nˆ of the bubble center. All information is contained in the change of 2 when the bubble is subtracted from the data, deﬁned by

2(d, a, r, nˆ) = 2 d arampramp astepstep 2(d) (1) r,nˆ r,nˆ ⇣ ⌘ ramp step where a = (a , a ), and r,nˆ denotes the bubble proﬁle, including CMB transfer functions and beam convolution. Calculating 2 would be computationally prohibitive were 294 CHAPTER 7. COLLISIONS WITH OTHER UNIVERSES

it not for several computational eciencies that we make use of. The ﬁrst is the precondi- 1 tioned conjugate gradient descent algorithm of [11] that allows us to calculate the C op- T 1 eration e ciently. The second is a method that allows us to e ciently calculate r,nˆ C r,nˆ 1 (which naively would require performing the C operation for many values of r and for every pixel nˆ in the WMAP map).

7.3.1 Bayesian Analysis

Our starting point for Bayesian analysis of the bubble collision signal is the posterior likelihood for model parameters (aramp, astep, r), marginalized over the bubble location nˆ:

p(r) 1 (aramp, astep, r d) exp 2(d, a, r, nˆ) (2) L | / Npix 2 Xnˆ ! where p(r) denotes the uniform prior on the distance parameter r. Starting from Eq. (2) we can marginalize over di↵erent parameters or take slices through the likelihood to calculate the posterior probability distributions for each parameter. We begin by setting astep = 0, to obtain the 2D likelihood (aramp, r) appropriate for the ramp L model. If we now marginalize over the distance parameter r, we obtain the 1D likelihood (aramp) shown in the top panel of Fig. 7.1. Assuming a uniform prior on aramp, the 95% L 6 ramp 6 1 confidence interval is 1.76 10 < a < 4.27 10 [Mpc ]. This constraint is much ⇥ ⇥ weaker than would naively be expected from Fig. 7.1. The weak constraint arises because bubbles with small angular size are poorly constrained, even for fairly large values of aramp. When we marginalize over the angular size to obtain the likelihood (aramp), this leads to L tails which are slow to decay. The broadening of the likelihood caused by the small bubbles is mainly due to our use ramp 1 of the amplitude parameter a , the slope of the initial curvature perturbation in Mpc (as opposed to the peak temperature in the CMB maps, for example). For a fixed value of aramp, a small bubble corresponds to a much smaller CMB fluctuation on our sky than a large bubble, and the signal-to-noise is further diluted by having many small patches on ramp the sky. This leads to a poorly constrained region in the two-parameter space (a , ✓bubble) ramp where ✓bubble is small and a can be large. 7.3. METHOD 295

1.00

0.10 Likelihood

0.01

−1.0 −0.5 0.0 0.5 1.0 ramp 6 −1 a x 10 [ Mpc ] 1.000

0.100 Likelihood

0.010

0.001 −0.10 −0.05 0.00 0.05 0.10 aramp (sin θ)4/3 x 106 [ Mpc−1 ]

Figure 7.1 Bayesian analysis of the bubble parameter space, assuming the “ramp model” for the bubble profile. Top panel: Posterior likelihood (aramp d) for the amplitude parameter aramp, defined to L | 1 be the slope of the initial curvature perturbation in Mpc , given the WMAP data d (solid black), after marginalizing the bubble radius. As explained in the text, the tails of the likelihood are slow to decay, due to a poorly constrained region of parameter space with small bubble radius. We illustrate this by showing the likelihood calculated using bubbles with a subset of angular sizes: ✓bubble < 20 (blue short-dashed), and ✓bubble > 20 (red long- dashed). Bottom panel: Posterior likelihood (↵ramp d), obtained from the top panel by changing ramp ramp ramp L | 4/3 variables from a to ↵ = a (sin ✓bubble) . After this change of variables, the likelihood is narrower and less sensitive to marginalization over the bubble radius. Vertical lines are 95% confidence limits on the amplitude parameter ↵ramp. The likelihood is consistent with no bubbles (↵ramp = 0). 296 CHAPTER 7. COLLISIONS WITH OTHER UNIVERSES

This interpretation of the broadening e↵ect suggests reparametrizing by replacing the amplitude parameter aramp by a variable which is more closely matched to the statistical signiﬁcance of the CMB signal. We deﬁne

ramp ramp 4/3 1 ↵ = a (sin ✓bubble) , ✓bubble = cos (r/Ddc)

where Ddc is the comoving distance to last scattering, and the 4/3 exponent is empiri- cally chosen so that two bubbles with the same value of ↵ and di↵erent angular sizes have roughly equal statistical significance. We can now obtain a 1D likelihood (↵ramp) L by marginalizing over the bubble size parameter ✓bubble. This is analogous to our pre- ramp vious marginalization; note that the Bayesian prior in the new variables (↵ , ✓bubble) is 1/3 dP = da dr (sin ✓ ) d↵ d✓ . / bubble bubble The likelihood (↵ramp) is shown in the bottom panel of Fig. 7.1. The ↵ramp parameter L has a narrower distribution than aramp, and we can achieve a tighter confidence interval. We obtain the 95% confidence limits:

8 ramp 4/3 8 1 4.66 10 < a (sin ✓ ) < 4.73 10 Mpc (3) ⇥ bubble ⇥ which we take to be our “bottom line” constraints on the ramp model. A similar Bayesian analysis can be performed for the step model. Considering first the posterior likelihood (astep d) with the bubble radius and location marginalized, we L | find that the likelihood has slowly decaying tails leading to a weak constraint. Changing step step 1/3 variables from a to a (sin ✓bubble) , we find a well-behaved likelihood (Fig. 7.2), and the “bottom line” 95% confidence limits:

5 step 1/3 5 3.72 10 < a (sin ✓ ) < 4.09 10 (4) ⇥ bubble ⇥

For both the ramp model and step model, the maximum likelihood amplitude is very close to zero, much closer than the width of the likelihood. While this behaviour appears counterintuitive, we explore this phenomenon in detail in [8] and show that it has a natural explanation. In simulations we ﬁnd that the maximum likelihood amplitude is nearly zero in an order-one fraction of the realizations. The scatter between maximum likelihood 7.3. METHOD 297

1.000

0.100 Likelihood

0.010

0.001 −60 −40 −20 0 20 40 60 astep (sin θ)1/3 x 106

Figure 7.2 Bayesian analysis of the “step” bubble model. We show the posterior likelihood (↵step d) after changing variables from astep to ↵step = astep(sin ✓ )1/3 to remove degen- L | bubble eracies. The dashed coloured lines have the same meaning as in Fig. 7.1. The likelihood is consistent with no detection of the signal. estimates of the amplitude, taken over many simulations, is consistent with Fisher matrix forecasts and roughly equal to the width of the likelihood, as expected intuitively. For the ramp model, simulations with nearly zero maximum likelihood amplitude often have low quadrupoles, i.e. the preference for zero bubble amplitude in the WMAP data is statistically related to the low quadrupole. This can be understood intuitively: in realizations with large quadrupoles, a bubble can cancel large-scale quadrupole power, and so a nonzero bubble amplitude is preferred by the likelihood.

7.3.2 Frequentist Analysis

We can also use optimal frequentist statistics to determine whether a single-bubble model with a , 0 gives a significantly better fit to the data than a no-bubble model with a = 0. Frequentist confidence regions are defined by Monte Carlo hypothesis testing and do not use a prior on bubble amplitude parameters, although we do make use of the theoretical 298 CHAPTER 7. COLLISIONS WITH OTHER UNIVERSES

1.000

0.100

0.010 Number (normalized)

0.001 10−10 10−8 10−6 10−4 10−2 100

log ρ0

Figure 7.3 Distribution of the likelihood ratio from simulations. An x-axis value of zero means that the likelihood peaks at a = 0. The red up-shaded distribution is consistent with the likelihood peaking at a = 0 within numerical precision. The WMAP value is consistent with a peak at a = 0, which occurs in 51.4% of the simulations.

prior on the bubble radius to improve statistical power. For testing whether a = 0 is consistent with the data, the Neyman-Pearson lemma implies that the optimal frequentist test statistic is the likelihood ratio:

(d a) ⇢0(d) = max L | (5) a (d 0) L |

We evaluate ⇢0 on the WMAP data, and compare it to a histogram of ⇢0 values obtained from 5000 Monte Carlo simulations. The results for the ramp model are shown in Fig. 7.3; we ﬁnd that 48.6% of the simulations have ⇢0 values larger than WMAP, so WMAP is consistent with a = 0. 7.4. MANY BUBBLES 299

7.4 Many Bubbles

We now consider the possibility that the data contain a large number of low amplitude bubbles, none of which could be detected individually. This case was ﬁrst considered in [12] assuming the Sachs-Wolfe approximation. Theoretically it is expected that either bubble collisions are unlikely to be present in the data, or that a large number of collision walls have intersected the last scattering surface [10]. A large number N 1 of independent random bubbles will add a Gaussian signal to the data with power spectrum:

N a 2 Cbub = h i dr b2 (6) ` 4⇡r ` max Z where a is the bubble amplitude, which is now a random variable, N is the expected number of collisions with our Hubble volume, and b` is the harmonic-space profile (defined bub precisely in [8]) of a bubble at comoving distance r. For the ramp profile, C` falls o↵ roughly as 1/`5, and a constraint on the amplitude of the spectrum largely comes from a measurement of the CMB quadrupole. Since the WMAP quadrupole is measured to be lower than the best-fit CMB spectrum, we immediately find that there is no evidence for the multi-bubble spectrum in the WMAP data. Using the WMAP likelihood code [9], we find the upper limit:

2 1/2 8 1 Na 6.95 10 Mpc (95% CL) (7) h rampi  ⇥

bub 3 For the step proﬁle, C` falls o↵ roughly as 1/` . The power spectrum constraint comes from a wide range of ` and we obtain:

2 1/2 4 Na 3.52 10 (95% CL) (8) h stepi  ⇥

7.5 Conclusions

We have searched for evidence that our universe collided with a bubble universe born out of a nucleation bubble from a phase of false vacuum eternal inﬂation. We use an ecient, optimal estimator for detecting the bubble signal in CMB maps and discuss the technical 300 CHAPTER 7. COLLISIONS WITH OTHER UNIVERSES

details in a companion paper [8]. We ﬁnd no evidence for the bubble signal when applying our estimator to the WMAP 7-year data, and we place limits on the amplitude of the signal. The bubble signal comes mainly from low ` in the ramp model, and from intermediate ` in the step model. Therefore, Planck data is unlikely to improve constraints on the ramp model parameters (although polarization may help a little [3]), but will improve constraints on the step model. Large-scale structure surveys and 21-cm line surveys can potentially improve limits on both models. We thank A. Brown and M. Kleban. SJO is supported by the US Planck Project, which is funded by the NASA Science Mission Directorate. LS is supported by DOE Early Career Award DE-FG02-12ER41854 and the National Science Foundation under PHY-1068380. KMS is supported at Princeton University by a Lyman Spitzer fellowship, and at Perimeter Institute by the Government of Canada and the Province of Ontario. Some of the results have been derived using HEALPix [13]. We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Bibliography

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301 Chapter 8

Summary

“Maybe I’ll go where I can see stars, he said to himself as the car gained velocity and altitude; it headed away from San Francisco, toward the uninhabited desolation to the north. To the place where no living thing would go. Not unless it felt that the end had come.”

Do Androids Dream of Electric Sheep? Philip K. Dick

8.1 Lensing and the Cosmic Infrared Background

The cosmic microwave background contains information about the structure between us and the surface of last scattering, encoded in the CMB covariance. We use this information to reconstruct the deflection angles of CMB photons, caused by gravitational lensing from the intervening structure, in the Planck CMB data. We find that the deflection angles are strongly correlated with the infrared emission measured in the higher frequency channels of Planck, with a correlation coecient of 0.8 between 217–857 GHz, indicating that ⇠ the two signals originate in the same redshift range, with both tracing the large-scale dark matter distribution. The cross-power spectrum of the CMB lensing potential and infrared background is sensitive to the redshift distribution of the CIB emission, as well as the relationship between

302 8.1. LENSING AND THE COSMIC INFRARED BACKGROUND 303

the dark matter that lenses the CMB and the luminous matter in galaxies. In the Planck HFI bands the cross spectrum contains a greater contribution from high redshift than the CIB auto spectrum, allowing us to probe galaxies at higher redshift. Using a halo model fit to the cross spectrum we find the mass scale at which a dark matter halo is likely to host a galaxy: log10 (M/M ) = 10.5 0.6, and we reconstruct the redshift distribution of the ± CIB brightness, finding results similar to the predictions of Bethermin´ et al. [1], giving us confidence in our understanding of the origin of the CIB and its redshift distribution. The cross spectrum has been measured with the lens reconstructions from SPT and the CIB measured by the Herschel SPIRE instrument [2, and references therein], showing good agreement with the Planck measurements, as seen in Fig. 8.1. The addition of the lensing information to models of the CIB auto spectra is found to tighten the emissivity constraint in the highest redshift bin, as we would expect given that the lens deflection angles are large between redshift 2 3, with a broad redshift range. ⇠

There are several ways that this analysis could be extended. The lower HFI frequencies are more sensitive to emission at high redshift, however the CMB is a signiﬁcant source of noise at these frequencies, and will bias the CIB auto spectrum if not removed. The CMB emission can be removed by subtracting a CMB template, which can be obtained from the lower frequency channels. Since the low frequency channels also contain CIB emission, it is dicult to remove the CMB without also reducing the CIB power, although in principle this could be accounted for in the analysis. At higher frequencies the cross spectrum signal to noise ratio is limited by the lens reconstruction noise, which can be reduced if the estimator is applied to a higher resolution, lower noise CMB map, or by using polarization information. Additional information is available from the cross-frequency spectra, for example the 100 143 GHz or 100 217 GHz spectra. Since the CIB emission in neighbor- ⇥ ⇥ ing Planck frequency bands is highly correlated, with a correlation coecient of 0.8, it is ⇠ not clear if including additional frequency bands between the Planck frequencies will substantially improve the estimate of the CIB redshift distribution, however a wider frequency coverage may. Another possibility is to use observations of line emission from molecular clouds in galaxies, such as CO rotational transitions, which can be used as a tracer of star formation. 304 CHAPTER 8. SUMMARY

Figure 8.1 Comparison of the Planck and SPT-SPIRE CIB-CMB lensing potential cross spectrum. The SPT-SPIRE data is from Holder et al. [2].

8.2 Kinetic SZ Velocity Constraints

The motion of galaxy clusters relative to the CMB rest frame causes a Doppler shift in CMB photons scattered by electrons within the cluster, allowing us to measure the velocities of clusters by observing the CMB temperature in their direction. The bulk peculiar velocity of all clusters located within a sphere out to redshift 1 is expected to be small, with velocities 10 km/s (depending on the redshift distribution of the cluster sample), caused by over- ⇠ densities in the gravitational field. The bulk velocity can be larger if other e↵ects contribute to the motion, for example pre-inflationary inhomogeneities could leave a “tilt” across our horizon [e.g. 3, 4] giving galaxy clusters a bulk motion relative to the CMB rest frame. Although WMAP is not sensitive enough to measure individual cluster velocities, by averaging the signal from many clusters the signal to noise ratio can be increased so that interesting velocity limits are reached. We use a sample of 736 clusters with mean redshift 0.12 and find no evidence of a bulk flow relative to the CMB rest frame, limiting any flow to v . 5600 km/s at 95% confidence, with the constraint depending on the direction of the flow. We find only a weak constraint on the radial velocity of the sample: 4692 8947 km/s, ± limited by our ability to remove the thermal SZ signal from the three WMAP channels we 8.3. COSMIC BUBBLE COLLISIONS 305

use. Recent measurements in Ade et al. [5], using Planck data and a larger cluster catalog with mean redshift 0.18, do not find a bulk flow. The wider frequency coverage of Planck, smaller beam FWHM, and lower detector noise in the HFI channels results in a greater sensitivity to the cluster kinetic SZ signal, with a 95% confidence upper limit of 254 km/s. The thermal SZ signal can be removed from Planck data with a smaller loss in sensitivity to the bulk flow, which leads to a more precise radial velocity measurement of 72 60 km/s [5]. ±

8.3 Cosmic Bubble Collisions

The horizon, flatness, and defect abundance problems can be solved if the universe began in a phase of slow roll inflation. Theorems show that inflation cannot be past eternal [e.g. 6], suggesting that our universe was created from a larger eternally inflating spacetime. In this theory we live in a “bubble universe,” with bubbles continually produced and expanding at the speed of light, but never filling space due to the fast expansion of the false vacuum. It is possible that a bubble universe collided with our own in the past, imprinting a specific disk- shaped signal in the CMB, and o↵ering us the opportunity to learn about eternal inflation. We perform an optimal search for the bubble signal, calculating the exact likelihood for a bubble of amplitude a in the data, marginalizing over the location of the bubble on the sky and the angular size of the bubble signal. We find that the likelihood is consistent with a = 0, i.e. no evidence for a bubble signal in the data, and we place limits on the size of curvature perturbation generated by a collision intersecting our last scattering surface. If there are a large number of bubbles, then the combined bubble signal is a Gaussian random field [7]. The amplitude of the multi-bubble power spectrum is proportional to N a2 , where N is the number of bubbles and a2 is the variance of the curvature perturbationD am-E plitudes from the bubble collisions. FromD E geometrical arguments we expect larger bubbles to be more likely, finding that the bubble power spectrum has a 1/`5 dependence, and so the 95% confidence upper limit that we place on N a2 largely comes from the measured quadrupole power. D E The CMB temperature power spectrum measured by WMAP is cosmic variance limited on large scales, and so observations by Planck will not substantially improve the sensitivity to the disk-shaped bubble signal. However, if the collision wall generates an additional 306 CHAPTER 8. SUMMARY

signal, then we would expect it to have more small-scale power, and so higher resolution experiments such as Planck may have increased sensitivity. The bubble collision is expected to generate a polarized signal in the CMB [8], which could be used to further increase the detection sensitivity. Bibliography

[1] M. Bethermin,´ H. Dole, G. Lagache, D. Le Borgne, and A. Penin.´ Modeling the evolution of infrared galaxies: a parametric backward evolution model. A&A, 529:A4, May 2011. doi: 10.1051/0004-6361/201015841.

[2] G.P. Holder et al. A CMB lensing mass map and its correlation with the cosmic infrared background. 2013.

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low-⌦0 inﬂationary models and the scale of homogeneity in the Universe. Physical Review Letters, 73:1582–1585, September 1994. doi: 10.1103/PhysRevLett.73.1582.

[5] P.A.R. Ade et al. Planck intermediate results. XIII. Constraints on peculiar velocities. Astron.Astrophys., 2013.

[6] Steven Weinberg. Anthropic Bound on the Cosmological Constant. Phys.Rev.Lett., 59: 2607, 1987. doi: 10.1103/PhysRevLett.59.2607.

[7] Jonathan Kozaczuk and Anthony Aguirre. Multiple Cosmic Collisions and the Mi- crowave Background Power Spectrum. Phys.Rev., D87:023506, 2013. doi: 10.1103/ PhysRevD.87.023506.

[8] Bartlomiej Czech, Matthew Kleban, Klaus Larjo, Thomas S. Levi, and Kris Sigurdson. Polarizing Bubble Collisions. JCAP, 1012:023, 2010. doi: 10.1088/1475-7516/2010/ 12/023.

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