Observational Hurdles in Cosmology: the Impact of Galaxy Physics on Redshift-Space Distortions

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Observational Hurdles in Cosmology: The Impact of Galaxy Physics on Redshift-Space Distortions

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Daniel Taylor Martens, B.A., B.S., M.S.

Graduate Program in Physics

The Ohio State University

2018

Dissertation Committee:

Christopher M. Hirata, Advisor David Weinberg Klaus Honscheid Ezekiel Johnston-Halperin c Copyright by

Daniel Taylor Martens

2018 Abstract

The modern study of cosmology is centered around the mysteries of Dark Matter and Dark Energy. The observable Universe, although highly non-linear on small scales, can be described statistically on large scales. Careful study of the large- scale structure of the Universe can provide powerful insights into the structure and evolution of matter, allowing constraints to be placed on the nature of Dark Energy and Dark Matter. This thesis contains the work that I have done to contribute to the understanding of these mysteries, in several specific situations. In Chapter 2, I discuss the effect of [N ii] and Hα line blending on measured redshift-space distortion and baryon acoustic oscillation parameters for the upcoming WFIRST telescope, which will help to measure the evolution of Dark Energy. Chapter 3 presents a radial measurement of the tidal alignment magnitude, providing a complementary method to gauge how strongly galaxies couple with the local gravitational field during formation.

Finally, in Chapter 4 I lay the framework for a fully general calculation of the eﬀects of resonant line scattering on the polarization anisotropies of the Cosmic Microwave

Background.

ii to my companions on the journey of life: Morgan, Andrew, Darian and David

iii Acknowledgments

When I was a kid, I was fascinated by outer space. I would frequently look up space-related facts at home so I could impress my teachers at school to the point where they had begun to ask me what my ‘daily fact’ was. (Sometimes, I would repeat the fact from the day before, hoping that they would not remember). My interest in learning as much as I could led me to a Bachelor’s Degree in physics, and

I decided to continue to push my limits by earning a PhD.

Beginning my graduate school career, I had no idea what speciﬁc area of physics

I wanted to go into – I just knew that I loved to learn. As fate would have it, I ended up working in the ﬁeld of Cosmology, studying under Chris Hirata, bringing me back to the fascination of space from my childhood. My ﬁve years at The Ohio

State University have been an incredible experience, in no small part due to the atmosphere of the CCAPP department and Chris Hirata’s working group. I have nothing but the highest praise for Chris as both an advisor and an individual. I am grateful for his guidance and patience; his deep understanding of cosmology and physics is only surpassed by his humility and kindness.

My experience at Ohio State has been a diﬃcult but rewarding journey, and I could not have completed my doctorate without the help of many, many people. My incredible wife Morgan has been by my side throughout the entire journey, and has supported me through the good times and the bad. My mother Marilee ﬁrst gave me

iv the dream of getting my PhD when I saw her earn a PhD in psychology. My father

Craig taught me to be responsible, to work hard, and to be loving. My brothers

Andrew, Darian, and David; Andrew, who I will always look up to, thank you for being there for me whenever I needed it. Darian, I can’t put into words how much you have meant to me. David, your loyalty and friendship have helped me more than you know.

Still others have helped shape my career within physics. My AP physics teacher,

Doug Forrest, helped instill within me a love for physics which has lead me to where

I am now. Xiao Fang and Paulo Montero-Camacho helped to create the greatest graduate student oﬃce in history; Paulo kept me sane with frequent conversations about video games, and Xiao’s eternal optimism was a beacon through the shadows of confusing equations and diﬃcult projects. You are both destined for great things!

To the members of my committee, David Weinberg, Klaus Honscheid, and Ezekiel

Johnston-Halperin, thank you for your guidance and support.

I am lucky to have a large extended network of friends and family; the reason why I work is to spend time with these people! To my gaming buddies, Darian Richardson,

James Jin, Skyler Newcom, Scott Cecil, Adrian Richardson, Brandon Richardson,

Justin Newcom, and Alex Long - we have spent many hours together, and hopefully we will have many more in the future! To Aref Jadallah, Melvin Barnes, Justin

Khol, Antonio Atria, and the rest of the club basketball team - playing with you was a pleasure. We have made many memories that I will never forget. To my close family of Jim and Lisa Williams, Madeline Glodowski, Jen and Kris Richardson, my grandmother Elaine, my extended Martens’ family, and the vast network of Newcoms: you have given me a place to call home. Thank you.

v Finally, the incredible atmosphere of CCAPP is due to the great people we have that have helped me to learn and grow. Eric Huﬀ, Michael Troxel, Ashley Ross,

Ami Choi, and Niall MacCrann were the most amazing post-docs I could ever ask to work with. I am forever grateful for the countless questions you have ﬁelded for me over the last few years. John Beacom, you have helped to both begin and foster the great environment I worked in - thank you for keeping my spirits up! Other graduate students I have loved working with, including Eric Speckhard, Ben Buckman, Shirley

Li, Jahmour Givans, Joe McEwen, Matthew Digman, and Bianca Davis, I am so grateful for your companionship on the journey we chose to undertake. You all have great things waiting for you.

To everyone above, thank you. In all of my achievements, past and present, you are both the ‘why’ and the ‘how.’

vi Vita

2013 ...... B.A., B.S. University of Cincinnati

2015 ...... M.S. The Ohio State University

Publications

Research Publications

“A Radial Measurement of the Galaxy Tidal Alignment Magnitude with BOSS Data” Martens D., Hirata C. M., Ross A. J., Fang X. Submitted to Monthly Notices of the Royal Astronomical Society

“Eﬀects of [NII] and Hα Line Blending on the WFIRST Galaxy Redshift Survey” Martens D., Fang X., Troxel M. A., DeRose J., Ross A. J., Hirata C. M., Wechsler R. H., Wang Y. Submitted to Monthly Notices of the Royal Astronomical Society

Fields of Study

Major Field: Physics

vii Table of Contents

Page

Abstract...... ii

Dedication...... iii

Acknowledgments ...... iv

Vita ...... vii

ListofTables...... xi

List of Figures ...... xiii

1. Introduction...... 1

1.1 The Science of Cosmology ...... 1 1.1.1 StateoftheField...... 1 1.1.2 Thesis Overview ...... 3 1.2 Introduction to Cosmology ...... 6 1.3 Correlation Functions and Power Spectra ...... 8 1.4 Redshift Space Distortions ...... 12 1.5 Gravitational Lensing ...... 17 1.6 Cosmic Microwave Background ...... 20

2. Eﬀects of [N ii] and Hα Line Blending on the WFIRST Galaxy Redshift Survey...... 25

2.1 Introduction ...... 25 2.2 Objectives...... 30 2.3 Calculation Outline ...... 34 2.4 Simulation ...... 37 2.4.1 Catalog generation ...... 37

viii 2.4.2 Assignment of the [N ii]/Hα ratio ...... 38 2.4.3 Data binning and redshift distributions ...... 42 2.4.4 Correlation functions ...... 45 2.5 Parameter fitting ...... 49 2.5.1 Covariance matrices ...... 51 2.5.2 RSD parameter fitting ...... 54 2.5.3 BAO Parameters ...... 56 2.5.4 FittingResults ...... 58 2.6 Measuring and Correcting Error Terms ...... 61 2.6.1 Effect of correlations between [N ii]/Hα and large-scale environment...... 65 2.6.2 Effects of the one-point redshift error PDF ...... 67 2.7 Discussion ...... 69

3. A Radial Measurement of the Galaxy Tidal Alignment Magnitude with BOSSData ...... 74

3.1 Introduction ...... 74 3.2 Theory ...... 79 3.2.1 Redshift space distortions ...... 79 3.2.2 Intrinsic alignment effects on RSD measurements ...... 80 3.2.3 Sub-samples using the Fundamental Plane offset ...... 85 3.2.4 Redshift space distortions of sub-samples ...... 88 3.2.5 Theoretical expectations ...... 90 3.3 Surveydata...... 92 3.3.1 Sample definition and characteristics ...... 92 3.3.2 Data preprocessing ...... 95 3.3.3 Sample splitting by orientation ...... 98 3.3.4 Systematics tests, random splits, and blinding procedures . 103 3.4 Clustering statistics ...... 112 3.4.1 Correlation functions ...... 112 3.4.2 Covariance matrices ...... 115 3.5 Analysis...... 117 3.5.1 Parameter fitting methods ...... 117 3.5.2 Finger of God Effects ...... 120 3.5.3 Full sample results ...... 123 3.5.4 Phase I: Statistical uncertainties ...... 125 3.5.5 Systematics-biased subsamples ...... 126 3.5.6 PhaseII...... 134 3.5.7 Phase III: Final results ...... 137 3.6 Conclusion ...... 142

ix 4. The Eﬀects of Resonant Line Scattering During Recombination on the CMB Power Spectrum ...... 144

4.1 Introduction ...... 144 4.2 Formalism...... 147 4.3 Atomic State Equations ...... 152 4.3.1 Ground State Change Due to Absorption ...... 153 4.3.2 Excited State Change Due to Absorption ...... 163 4.3.3 Excited State Change Due to Emission ...... 165 4.3.4 Ground State Change Due to Emission ...... 168 4.3.5 Total Atomic Equation ...... 169 4.4 Photon Phase Space Density ...... 170 4.5 Results ...... 183 4.6 FutureWork ...... 185

Appendices 188

A. Radial Tidal Alignment ...... 188

A.1 Eﬀect of a diﬀerence in bias on the random-split error bars . . . . . 188 A.2 Comparison of Intrinsic Alignment Values ...... 190

B. Resonant Line Scattering ...... 192

B.1 Irreducible Moments ...... 192 B.2 Spherical Basis ...... 194 B.3 Spin-Weighted Spherical Harmonics ...... 194 B.4 3-jSymbols...... 196 B.5 6-jSymbols...... 197 B.6 9-jSymbols...... 199 B.7 Stokes Parameters ...... 200 B.8 Delta Deﬁnitions ...... 200 B.9 Rotations ...... 203

Bibliography ...... 205

x List of Tables

Table Page

2.1 The number of galaxies, in millions, within each bin. This is the sum total of galaxies within each geometric region for the data counts. The galaxy ﬂux F is measured in units of 10−17 erg cm−2 s−1. In parenthesis is the label we use to reference each bin throughout the text...... 42

2.2 The median redshift of each bin...... 42

2.3 Parameters used in generating P (k, µ) for both the covariance matrix generation and the BAO models used for ﬁtting. These are calculated

using formulae from Seo and Eisenstein (2007). For Σs, the ﬁrst value indicates the value used for the True catalogs, while the second value was used for the Observed and Shuﬄed catalogs; it was increased due to the extra variance added to simulate photon noise...... 53

2.4 Here we display the best ﬁt parameters for the growth of structure pa-

rameter fv and the galaxy bias bg, for each ZH bin and redshift catalog. We calculate % error (Eq. 2.34) to indicate the agreement between the shuﬄed and observed parameters. The uncertainties provided by the jack-knife sampling have a fractional error of p2/5, due to the sample size of 6 regions...... 60

2.5 The best-ﬁt parameters for the BAO shift parameters αk and α⊥, and their χ2 values. We calculate % error (Eq. 2.34) to indicate the agreement between the shuﬄed and observed parameters. The uncertainties provided by the jack-knife sampling have a fractional error of p2/5, due to the sample size of 6 regions...... 62

3.1 Redshift cut statistics for the BOSS CMASS/LOWZ subsamples. . . 98

xi 3.2 CMASS and LOWZ medians and standard deviations for both the

W33–divided and total samples. Note that the median of W33 for the full samples is zero by construction...... 98

3.3 Values of mθ, for each combination of survey and systematic...... 107

3.4 The parameter diﬀerences between identical samples which we counted using both weighted and non-weighted methods. See Sec. 3.4 for details.114

3.5 Fitting statistics for the 1000 random splits of each survey...... 126

3.6 The average of parameter diﬀerences for our systematic-injected subsamples. Consistency with zero implies that the systematic in question will not disguise itself as galaxy orientation for our sample splitting.

Here, the σ∆f are the uncertainties in the systematic error from each respective systematic, not the uncertainty in our measurement of ∆fv. 132

2 3.7 Errors on fv due to diﬀerences in σFOG for the true divided subsamples.137

4.1 A reference for the labelling of states throughout this chapter. Al- though we combine all of these effects with a unified nomenclature in Sec. 4.3.5, each specific calculation requires names for different numbers of states in the excited and ground configurations...... 155

4.2 C1(χ, α, β)...... 182

xii List of Figures

Figure Page

1.1 Here, we demonstrate how galaxies are counted in the expression DD(r, µ) from Eq. 1.12. We choose a galaxy, shown here in red. We then count all the galaxies surrounding our chosen galaxy which fall within the µ bin (red dotted lines), where µ = cos(θ), and r bin (red solid lines). In the ﬁgure, the galaxies which meet both criteria are shown in green, while all other galaxies are shown in black. This process is repeated, with all galaxies at one time serving as the central red galaxy. The numbers of galaxies in each r and µ bin are then summed to calculate the correlation function...... 10

1.2 Here we demonstrate the ‘Finger of God’ effect. In the left panel, we have some random selection of galaxies with varying velocities, denoted by black arrows, which depend on specifics that we are not interested in. Note that the blue galaxies have velocities which are generally toward the observer, the red galaxies have velocities away from the observer, and the green galaxies have velocities that are transverse to the observer. Because the observer cannot distinguish the cosmic velocity from each specific galaxy’s peculiar velocity, it assumes that the red galaxies are farther away, the blue galaxies are closer, and the green galaxies are in the correct position. In the right panel, we show an exaggerated version of the result, where groups of galaxies are smeared in a direction which appears to be pointing toward the observer. . . 15

xiii 1.3 Here we demonstrate the Kaiser eﬀect. In the left panel, we show some galaxies (colored points) with velocities denoted by the black arrows, surrounding a large mass (central black point). Although the galaxy’s velocities will vary, in general they will be more likely to point toward the central mass because of gravitational attraction. Note that the blue galaxies have velocities which are generally toward the observer, the red galaxies have velocities away from the observer, and the green galaxies have velocities that are transverse to the observer. Because the observer cannot distinguish the cosmic velocity from each speciﬁc galaxy’s peculiar velocity, it assumes that the red galaxies are farther away, the blue galaxies are closer, and the green galaxies are in the correct position. In the right panel, we show an exaggerated version of the result, the group of galaxies appears to be ‘squashed’ transverse to thelineofsight...... 16

1.4 The intensity distribution of the CMB, as measured by the COBE satellite (Mather et al., 1994). It is important to note that the line is not a best-ﬁt line to the data; it is a prediction of the data given the temperature of the CMB, 2.7 Kelvin. Data for this plot is available online at https://lambda.gsfc.nasa.gov/product/cobe/firas_ monopole_get.cfm...... 22

1.5 The temperature anisotropies in the Cosmic Microwave Background as observed by Planck (Planck Collaboration et al., 2016). The color scale has been magniﬁed; variations from the mean temperature are of order 10−5. Photo: ESA and the Planck Collaboration...... 23

1.6 The Cosmic Microwave Background temperature power spectrum (Planck Collaboration et al., 2016). We can see on what scales the perturbations in temperature are most prevelant. Photo: ESA and the Planck Collaboration...... 23

xiv 2.1 A visual representation of the [N ii] lines and Hα line for the line blending scenario. The difference in strength between the two nitrogen lines is constant for each galaxy with a ratio of approximately 0.32, while the difference between the larger [N ii] line and the Hα line varies depending on the metallicity of the galaxy in question. Here, we show the median difference of our sample, with a ratio of [N ii]6585/Hα of 0.427. The Gaussian spreading centered at each line has a standard

deviation of σgrism, calculated generally in Eq. 2.1. In this example, we have shown the spreading due to a galaxy with radius 4 kpc, at redshift 1.5. The black dotted line is the resulting observed line, given that the constituent lines cannot be resolved...... 32

2.2 A flow chart describing the simulation pipeline; see Sec. 2.3 for details. The red hexagons indicate steps where data is input into the process, and yellow circles indicate steps where we output statistics, correlation functions, or fit parameters. Also included in the flowchart is a method for possible analysis of the ability of a ground-based spectroscopic observation to reproduce the observed catalog; although it is not implemented in this work, we discuss its potential in Sec. 2.6. . . 34

2.3 We display the 6 separate regions on the sky, within which we have independently calculated the correlation function, as discussed in Sec. 2.4.3. Note that although we must display a 2D projection, the sectors are congruent on the sphere. (The simulation does not cover the actual WFIRST footprint, which is likely to be placed in the Southern Hemisphere, but this does not matter in a statistically isotropic universe.) 36

2.4 Comparing the luminosity function of the Buzzard-v1.1 mock catalog to that predicted by Pozzetti et al. (2016). These are semi-analytic models made by fitting to observed luminosity functions from Hα surveys. Specifically, Models 1 and 2 feature a Schechter parameterization, while Model 3 was designed specifically for high-redshift slitless surveys such as WFIRST and Euclid, as it was fit directly to luminosity function data. This graphic is from one 54 deg2 section in the sky; the redshift range is from z = 0.7 to z = 1.5 (similar to our Z2 bins, but matching the exact range used for the Pozzetti et al. (2016) model). 39

2.5 Empirical relationship between [N ii]/Hα ratio and log stellar mass. Data being ﬁtted was obtained from Wuyts et al. (2016). These relationships were used to construct the relative [N ii] line strength within the mock galaxy sample...... 41

xv 2.6 The difference between the observed redshift and the true redshift for a sample of a little over 10 million galaxies. This does not include the Gaussian photon noise smearing, only the difference due to the line blending effect. This histogram is generated from one sky section of the parent Buzzard-v1.1 sample, cut from z = 0.7 to z = 1.3 (the approximate redshift range of the Z2 bins), and covers approximately 54 square degrees of sky...... 46

2.7 We display the physical positions of the galaxies given the redshifts of each of our true, observed, and shuﬄed catalogs. These images were generated using Regions 2 and 5 of the Z2H3 catalogs. We have selected all galaxies with a Cartesian x-coordinate between -50 and 50 Mpc, a z-coordinate between 1000 and 1100 Mpc, and a y-coordinate between 2300 and 2400 Mpc. The galaxies in the x-direction have been projected into the y-z plane, and the observer is located toward the bottom-left corner of the image (which can be seen by the ‘streaks’ due to the changes in redshift for the combination of the catalogs). . 47

2.8 For each ZH bin, we compare the monopole and quadrupole correlation functions for each of the true, observed, and shuﬄed redshift catalogs. The top panel of each plot displays the correlation function monopoles (solid lines) and quadrupoles (dashed lines). Both quadrupoles and monopoles are multipilied by r2, and the quadrupoles are multiplied by −1. The bottom panel of each plot shows the fractional comparison between the observed and shuﬄed correlation multipoles, as a function of separation r...... 50

2.9 For each ZH bin and Observed, True, and Shuﬄed catalog, we plot the

fitted values for fv and bg. Circles indicate the True catalog, V’s indicate the Observed catalog, and triangles indicate the Shuffled catalog. 2 −1 2 For this fitting calculation, σFOG was set to 20 [h Mpc] for the True catalog...... 63

2.10 For each ZH bin, we show the percent difference of α⊥ and αk relative to the systematic error budget (red box) of WFIRST for each parameter. The contours show the spread of fits for all 6 jack-knife combinations for that specific ZH bin, as referenced in Eq. 2.33, while the central values are calculated from the fits of the average of all regions. . . . . 70

xvi 2.11 For each ZH bin, we show the percent difference of fv and bg relative to the systematic error budget of WFIRST for fv, represented by the solid 2 −1 2 red lines. For this fitting calculation, σFOG was set to 20 [h Mpc] for the True catalog. The contours show the spread of fits for all 6 jack- knife combinations for that specific ZH bin, as referenced in Eq. 2.33, while the central values are calculated from the fits of the average of allregions...... 71

3.1 Here we show an exaggerated situation demonstrating a selection bias having an anisotropic effect on the resulting k-modes. If we assume that our observations are less likely to observe galaxies which are aligned perpendicularly to our line of sight, then some of the galaxies visualized above will not appear to the observer (the crossed-out galaxies). Since galaxies are more likely to be aligned with the higher density modes of the gravitational field, this changes the measured k-modes of the observed galaxies. Fourier modes perpendicular to the line of sight will have deeper troughs, and thus a higher inferred amplitude, while k- modes parallel to the line of sight will have shallower peaks, and thus a lower inferred amplitude. In this work, we intentionally group galaxies by their orientation, shown here by the color of green or blue. If we were to view all of these galaxies, but were less likely to see galaxies of the ‘’‘blue” type, then these distortions would dampen the linear Kaiser effect of clustering in our observation. (Note: this figure is similar to Figure 1 of Hirata (2009), but adapted to the situation with two sub-samples.) ...... 81

3.2 For the CMASS samples, we chose to ﬁt Eq. 3.16 separately to galaxies in ranges of 0.43 < z < 0.5, 0.5 < z < 0.6, 0.6 < z < 0.65, and 0.65 < z < 0.7. This helped mitigate an observed correlation between

galaxy redshift and median W33 for the CMASS sample. In this plot we compare to the same equation ﬁt over the full redshift range of 0.43 < z < 0.7 (‘single redshift bin’). Here we show the resulting

change in median W33 value by binned redshift for the CMASS SGC sample...... 89

3.3 For each sample, we show the fraction of positive W33 as a function of binned redshift. Note the good agreement of each distribution across the entire redshift range. For the LOWZ samples, this was achieved by using the model within Equation 3.16. For the CMASS samples, it was necessary to ﬁt this model separately to galaxies in ranges of 0.43 < z < 0.5, 0.5 < z < 0.6, 0.6 < z < 0.65, and 0.65 < z < 0.7. . . 100

xvii 3.4 Here, we plot the histograms for each survey’s distribution of W33.. 101

3.5 For each sample, we use relations of the Fundamental Plane to ﬁt an assumed galaxy radius, based on galaxy luminosity and redshift. Here we plot this assumed radius to the ’true’ galaxy radius which has been measured photometrically in the i-band. Galaxies are assigned into bins of positive and negative W33, based on whether their ﬁt radius is greater or less than their measured radius...... 102

3.6 Here we show the average W33 value, binned by the PSF FWHM. There is a clear correlation between the PSF FWHM value and a galaxy’s

calculated W33 value. This will be analyzed in more detail in Section 3.3.4. Further evidence of this can be seen in Table 3.3...... 104

3.7 For each sample, we show the fraction of galaxies with positive W33 as binned by the sky ﬂux at the time of observation. Note that there is no obvious correlation between the sky ﬂux on the night of observation,

and the calculated W33 value. Toward the edges of the range we can larger deviations from the center, which is due to smaller numbers of galaxies observed with those values of sky ﬂux...... 104

3.8 Here, we have created 500 random sample separations for each survey.

We have compared the parameter p(∆bg, ∆fv) to fv (Eq. 3.28). The Pearson correlation coeﬃcient is listed above each plot. We can see that although the coeﬃcient is non-zero, we believe that it is small enough to serve our purpose here as a blinding parameter which is independent of the change due to galaxy orientation...... 110

3.9 Top panel: We plot the correlation function monopoles for the CMASS and LOWZ. We compare our calculation, with normal lines, to the calculations of Chuang et al. (2017), in dotted lines. Bottom panel: The percent diﬀerence between the sample diﬀerences...... 124

3.10 Top panel: We plot the correlation function quadrupoles for the CMASS NGC and LOWZ SGC. We compare our calculation, with normal lines, to the calculations of Chuang et al. (2017), in dotted lines. Bottom panel: The percent diﬀerence between the sample diﬀerences...... 125

xviii 3.11 We show the results of ﬁtting clustering parameters of bg and fv to each sample of 500 random splits of each survey catalog, resulting in 1000points...... 127

3.12 The diﬀerences in parameter values for 500 random splits of the data. 128

3.13 Correlation functions from the best-fit parameters, found by the fitting methods in section 3.5.The shaded areas show the correlation functions from the data with error bars. The dark lines are produced by the best- fit parameters. This is from a random separation of the CMASS NGC catalog. This fit has a χ2 per dof of 1.034. The best-fit parameters are

fv = 0.3712 and bg = 1.9486...... 129

3.14 Here, we plot the final parameters of bg and fv for a variety of samples. Triangles indicate results from CMASS, and circles indicate results from LOWZ. The green points are the results from our full samples for CMASS and LOWZ. The red points are the results from the correlation functions from Chuang et al. (2017) being run through our pipeline. (Note that for LOWZ, the green and red points lie directly on top of each other). The black points are the full results from Chuang et al. (2017). Finally, the blue points are the results from Gil-Mar´ın et al. (2016), where we have shown several of their final parameters, which depend on details of the mocks used for fitting...... 130

3.15 For each survey, we show the diﬀerence in clustering parameters for

each pair of systematics-biased subsamples (i.e. with W33 scrambled but then each systematic template added in with the best-ﬁt coeﬃcient

mθ). A non-biased result should be consistent with zero, for both parameters. Plotted are the diﬀerences in measured bias and measured

fv for each survey, where the shaded regions represent 1 and 2 standard deviations. We see here that the systematics due to airmass, extinction, sky ﬂux, and PSF FWHM are consistent with zero...... 133

3.16 Here, we plot p(∆b, ∆f) by the χ2 per degree of freedom of the fits. The solid colored lines indicate the limits of acceptable χ2 for our fits; it can be seen that all of our fits meet this criteria. The ’X’ markers indicate the average χ2 for each survey, while the ’O’ markers indicate the the χ2 for each subsample. All fits result in p(∆b, ∆f) < 0.2, the limit discussed in Sec. 3.3.4. However, they all show a consistent offset of approximately 0.11 for LOWZ and 0.15 for CMASS...... 138

xix 3.17 Here, we display our ﬁnal results for the value of B along with theoretical predictions. The data point from Singh et al. (2015) was measured from the LOWZ sample using galaxy-galaxy lensing, and is labeled ‘Lensing.’ For each measurement we also show the theoretical prediction of a radial intrinsic alignment measurement at the calculated median luminosities of each respective sample, using the machinery of Hirata (2009). These predictions are shown as red ranges on the plot, overlapping the true measurement with the same sample luminosity. 140

3.18 For each subsample split, we show the values of ∆fv and ∆bg..... 141

4.1 Charts indicating particle interactions before and after recombination. Top panel: Protons and electrons interact via Coulomb scattering, while photons and electrons interact via Compton scattering. Neu- trinos and Dark Matter (essentially) do not interact with any of the other forms of matter. However, all forms of matter interact to some extent gravitationally, referenced by their relationship with the metric. Bottom panel: Protons and electrons recombine to form neutral hydrogen, while still Thomson scattering with the photons. Neutrinos and Dark matter are still essentially ignoring all other forms of matter. All forms of matter interact gravitationally, referenced by the relationship with the metric. The red arrow indicates the resonant line scattering interaction between neutral hydrogen and photons, which is the focus ofthisChapter...... 148

4.2 The ways in which light and matter interact for resonant line scattering. In absorption, a photon (red squiggly line) is absorbed by an atom. The atom increases in energy, jumping to a higher state. This process is discussed in Sections 4.3.1 and 4.3.2. For spontaneous and stimulated emission, the atom loses energy by photon emission and drops to a lower energy state. These processes are relevant for Sections 4.3.3 and 4.3.4...... 153

xx Chapter 1: Introduction

1.1 The Science of Cosmology

1.1.1 State of the Field

Cosmology is the study of the origin and evolution of the Universe. Given that

all of humanity is (essentially) conﬁned to a single planet, in a single star system,

within a single galaxy (of at least hundreds of billions of observable galaxies), it is

a resounding salute to the ingenuity of our species to realize that not only do we

have a ﬁeld of study devoted to the evolution of the Universe, but that we have made

incredible progress within that ﬁeld.

We know that our galaxy, full of hundreds of billions of stars, is not the only

galaxy; it is one of many others we have observed, which range in distance from the

Large Magellanic Cloud at ≈ 150,000 light-years to recent detections of galaxies which are tens of billions of light-years from us. We know that the Universe is expanding outward in all directions (Hubble, 1929; Lemaˆıtre, 1931) from a beginning coined the “Big Bang”. We both predicted (Alpher et al., 1948) and observed (Penzias and

Wilson, 1965; Dicke et al., 1965) the relic radiation from the Big Bang, emanating toward us from all directions. We have calculated the amount of hydrogen and helium in the Universe using theoretical models, and found agreement with observations. We

1 have engineered theories to explain the large-scale homogeneity and ﬂatness of our

Universe (Guth, 1981). We have even detected gravity waves from merging black holes and neutron stars billions of light-years away (Abbott et al., 2016).

These feats of ingenuity and perseverance have also led to many surprising discoveries, some of which have yet to be explained. We have seen that there appears to be much more matter in galaxies, and in the Universe as a whole, than simply luminous matter. Cluster masses inferred by galaxy velocities are much lower than that evidenced by luminous matter (Zwicky, 1937), and rotation curves of observed galaxies are diﬀerent than what is expected from their visible portions (Rubin and Ford,

1970). We have termed this material ‘Dark Matter’, and it is still unclear what type of particles make up Dark Matter, or even if a single particle is responsible for all of

Dark Matter’s mysterious eﬀects. Furthermore, although we know that the Universe is expanding, it wasn’t until relatively recently that we discovered that the Universe is accelerating in its expansion (Perlmutter et al., 1999; Riess et al., 1998), which requires either an extra component to our Universe (‘Dark Energy’) or a modiﬁcation to gravity in order to be explained.

To solve these mysteries, we must be productive both theoretically and experimentally. We need to devise models which can explain our data, and can incorporate our discoveries into a coherent theory of the Universe. Experimentally, we must both increase the amount of data we gather and the range of redshifts it is gathered from.

There are many types of data which can help answer these questions. Mapping out the observed positions of galaxies in redshift-space can give insight into how galaxies cluster in our Universe (described in Sec. 1.4), which can constrain models of Dark

Energy and Dark Matter. Using gravitational lensing (1.5), we can directly probe the

2 matter distribution of the Universe, which provides information about the clustering of both luminous matter and Dark Matter. Analyzing the Cosmic Microwave Back- ground (CMB), the relic radiation from the Big Bang, gives a detailed picture of the

Universe at its very early stages (1.6), which can be compared with snapshots of the

Universe at later redshifts to examine how large-scale structure has evolved, placing further constraints on Dark Energy and Dark Matter models. These are only a few of the methods cosmologists employ to study the Universe, but we introduce them here to provide context to the work presented in this Thesis.

1.1.2 Thesis Overview

In this Thesis, I present three separate contributions to the ﬁeld of Cosmology, which encompass unique projects and focuses. Here I will brieﬂy describe the motivation for each of these projects: what they seek to measure, and why that measurement is important. I will also indicate where useful background information can be found.

First, we present the basic cosmological model in Sec. 1.2. We brieﬂy discuss the cosmological assumptions, and how they combine with general relativity to present the foundational equations of cosmology.

Sec. 1.4 describes the eﬀects of Redshift-Space Distortions – how determining a galaxy’s position can be distorted by deviations in its redshift from the Hubble ﬂow

(the redshift of the galaxy given only the expansion of the Universe). We can take advantage of RSDs on large spatial scales to make measurements of both how matter clusters and the rate of growth of structure within the Universe. This is important because alternate theories of gravity, or the inclusion of Dark Energy and Dark Matter in cosmological models, aﬀect the predicted matter clustering amplitude. There are

3 many experiments, past, present, and future, which measure RSDs by taking surveys of the locations of galaxies.

One of these experiments is the Wide Field Infrared Survey Telescope (WFIRST).

WFIRST will measure the locations of millions of galaxies, and use these locations to measure RSDs (among other things). However, the equipment planned for use on

WFIRST has a line resolution which could potentially confuse the location of the Hα line, the intended indicator of the galaxy redshift, with the nearby [N ii] lines. This confusion results in an apparent blending between the [N ii] and Hα lines, which can change the inferred redshift of the galaxy. Over a catalog of galaxies, biases of this kind can result in changes to the matter clustering parameters measured from the sample. In Chapter 2, we describe our use of a simulation of galaxies to measure this line blending eﬀect, and determine if any mitigation is necessary in order to meet the science requirements for WFIRST, and adequately oﬀset any cosmological parameter measurements.

Sec. 1.5 gives an introduction to gravitational lensing, and how it can be used to measure the matter distribution throughout the Universe. While gravitational

‘strong’ lensing results in multiple images of a single object due to light rays passing around intervening matter, cosmologists can make use of gravitational ‘weak’ lensing.

For weak lensing, although changes to individual galaxies are small and impossible to measure on a galaxy-by-galaxy basis, these small changes can be aggregated across a galaxy sample to measure the statistical changes in the distribution of galaxy at- tributes, such as their elliptical shapes. In order to use the statistics of lensed galaxy shapes to measure the intervening matter distribution, we must have some idea of the initial statistical distribution of galaxy shapes.

4 The intrinsic alignment magnitude B measures the strength of the link between elliptical galaxy formation and the gravitational fields surrounding the galaxy as it forms. Since these gravitational fields can affect the orientations of galaxies, we must understand the strength of B in order to get an accurate estimate of weak lensing. In Chapter 3, we present a radial measurement of B (measured with redshift- space distortions along the line-of-sight), which can be compared to existing non- radial measurements (measured from correlations between the matter field and galaxy shapes), and thus provide more information about galaxy intrinsic alignments.

Finally, in Sec. 1.6 we explain the relevance of another window into the evolution of the Universe: the Cosmic Microwave Background. It is particularly useful as an early-Universe probe, and can be compared to results of ‘late-Universe’ measurements, such as the observation of galaxies. Importantly, the CMB is partially polarized due to anisotropies in the photon temperature during the early Universe. Resonant line scattering between photons and neutral hydrogen produces frequency-dependent distortions to these anisotropies, and by measuring them we can get a better understanding of when the neutral hydrogen formed in the Universe, and how much hydrogen there was. In Chapter 4, we lay the mathematical groundwork for a general extension of the research by (Hernandez-Monteagudo et al., 2007), capable of incorporating all resonant scattering transitions of neutral hydrogen. This chapter contains extended technical material which is designed as a resource for future students to continue this project, intended to fast-track the reader to the coding/implementation stage without needing to ﬁrst re-derive many of the equations.

In the following sections of the introduction, we provide the background to understand the context of the work presented in this Thesis. It goes without saying that

5 many full treatments of the following topics exist: for a discussion of Gaussian ran-

dom ﬁelds and their relationships to correlation functions, see Bardeen et al. (1986);

for a review on redshift space distortions, see Hamilton (1998); for gravitational lens-

ing, refer to Troxel and Ishak (2015); for in-depth treatment of the CMB, see Scott

and Smoot (2010). Here, we write just enough so that the reader may get a sense of

what this research pertains to, and is able to understand the relevance of the topics

presented.

1.2 Introduction to Cosmology

The study of cosmology initially began with two assumptions: homogeneity and

isotropy. We assumed that there is no special point throughout space-time (homo-

geneity), and that statistically, on large enough scales, the Universe should look the

same in all directions (isotropy). Although also aesthetically pleasing, these assump-

tions have since been conﬁrmed to the extent of the Universe that we can observe;

the CMB is isotropic to one part in 105, and galaxy surveys have observed homogene-

ity on scales larger than 50-100 Mpc. These starting assumptions, when combined

with general relativity, provide us with a surprising amount of information. Einstein’s

equation can be written as:

1 R − g R − g Λ = 8πT . (1.1) ab 2 ab ab ab where Rab is the Ricci tensor, R is the Ricci scalar, gab is the metric, Tab is the stress-energy tensor, and Λ is the cosmological constant, which is directly related to the concept of Dark Energy. In fact, the statements in 1.1.1 about Dark Energy’s implications are immediately apparent in this equation: the fact that the Universe is

6 undergoing accelerated expansion means either that Λ is a non-zero value (spacetime

is curved even without the presence of matter), there is a new contribution to the

stress-energy tensor (other forms of Dark Energy), or this equation is wrong altogether

(general relativity is only an approximation to the true theory, and breaks down on

large scales).

If you’ve never seen general relativity before, and these equations are completely

foreign to you - that’s ok! Their physical interpretation here is all we require, to

the extent of understanding the basis of cosmology. Eq. 1.1 relates the geometry of

spacetime, on the left side of the equation, to the energy of the spacetime on the right

hand side. This equation is a tensor equation; since there are 4 spacetime dimensions,

a and b are indices that have four values. Therefore, Eq. 1.1 is actually 16 separate

equations, although they are not all completely independent.

An exact solution to Eq. 1.1 for an expanding or contracting Universe is called

the Friedmann-Robertson-Walker metric:

dr2 ds2 = −dt2 + a2(t) + r2dΩ2 (1.2) 1 − kr2 where dΩ2 = dθ2 + sin2(θ)dφ2, k is related to the curvature of the Universe (k = 0 in a ﬂat Universe), and a(t) is the scale factor, which describes the amount the Universe has expanded at a given time t. The scale factor is normalized to 1 today. If we assume homogeneity and isotropy, then given Equations 1.1 and 1.2, we can write down the Friedmann equations:

2 H (a) −3 −4 −2 2 = Ωma + Ωra + Ωka + ΩΛ (1.3) H0 and a¨ 4πG = − (ρ − p). (1.4) a 3 7 Here, we have used the Hubble expansion rate, H =a/a ˙ . H0 is the Hubble expansion rate today, and the energy densities Ωi are deﬁned as:

ρi Ωi ≡ (1.5) ρcrit where: 3H2 ρ = 0 (1.6) crit 8πG

These equations form the basis of cosmology. It can be seen from Eq. 1.3 that by

finding the energy densities of different components of our Universe, we can solve for how fast the Universe expands. When we speak of solving models of cosmology, we are specifically speaking of determining the time-dependence of different energy densities within the Universe, and solving for other parameters related to the clustering of objects which trace the matter density (such as galaxies) or the matter density itself.

1.3 Correlation Functions and Power Spectra

The study of large-scale structure in the Universe is confined to statistical measures of galaxy and matter distributions. Why? Firstly, quantum fluctuations in the early Universe caused the matter over-densities which led to all the structure we see today. The probabilistic nature of these fluctuations prevents us from predicting exactly where a galaxy will form at a specific time. Secondly, the apparent Gaus- sian distribution of the CMB anisotropies, as well as the homogeneity of observed large-scale structure, justifies the use of statistical measures to quantify Universal structure.

In much of the ﬁeld of Cosmology (and Chapters 2 and 3 in this Thesis), the two-point autocorrelation function fulﬁlls the role of this statistical measurement.

8 Speciﬁcally, the two-point autocorrelation function measures the excess probability

of ﬁnding a galaxy within a certain radial distance, compared to the probability of

ﬁnding a galaxy in a random distribution of the same geometry. First we deﬁne the

matter ﬁeld over-density: ρ(~x,t) δ(~x,t) ≡ − 1 (1.7) ρ¯(t)

where ρ(~x,t) is the matter density at a speciﬁc position ~x and time t, andρ ¯(t) is the

average matter density at time t. We can then write the correlation function as:

ξ(r) = hδ(~x)δ(~x + ~r)i (1.8)

Note that if the statistical ﬁeld is isotropic, then ξ is only a function of the magnitude

of the separation r, not its direction. We will see that this is not the case for the

correlation functions of galaxies. In fact, all of the “pair-counting” variables used to

calculate ξ depend on both r and µ, where µ is the cosine between a given direction and the line of sight. The dependence on r is simple: to calculate the correlation function for a speciﬁc scale range, we need to count all pairs of galaxies within that scale range, and that scale range only. These concepts are illustrated in Fig. 1.1. In the algorithms used for Chapters 2 and 3, this is done by setting up bins of speciﬁc sizes, where the correlation function is calculated independently for each bin.

The dependence on µ is a little more complicated, but essential to what the correlation function is used for in cosmology. Redshift space distortions (described in Sec.

1.4) modify the observed radial position of galaxies; however, RSDs do not aﬀect the observed angular position. Thinking three-dimensionally, we can see that the correlation function will be diﬀerent for radial bins (where µ = 1) than for non-radial bins

9 Observer

Figure 1.1: Here, we demonstrate how galaxies are counted in the expression DD(r, µ) from Eq. 1.12. We choose a galaxy, shown here in red. We then count all the galaxies surrounding our chosen galaxy which fall within the µ bin (red dotted lines), where µ = cos(θ), and r bin (red solid lines). In the ﬁgure, the galaxies which meet both criteria are shown in green, while all other galaxies are shown in black. This process is repeated, with all galaxies at one time serving as the central red galaxy. The numbers of galaxies in each r and µ bin are then summed to calculate the correlation function.

(µ < 1), in a non-trivial way. Therefore, Eq. 1.12 has a dependence on µ so that we

can extract information from it pertaining to RSDs.

For an exact Gaussian distribution, the correlation function contains all possible

information. In practice, since true galaxy distributions are never perfectly Gaus-

sian, and must be approximated by the speciﬁc galaxies we observe in the survey

(shot noise), we do care about higher-order eﬀects. One way to deal with this is to

decompose the correlation function into moments:

i 2l + 1 Xmax 1 ξ (r) = ξ(r, µ )L (µ ). (1.9) l 2 i i l i i=1 max

Here, l indicates the moment of the distribution, and Ll(µi) are the Legendre

Polynomials. In practice, the moments of l = 0 and l = 2 are commonly used to

10 perform ﬁts to survey observations. In tandem with the correlation function is its

Fourier Transform, the power spectrum:

Z P (~k) = d3rξ(~r)ei~k·~r (1.10) where we can convert back to the correlation function with:

Z d3k ξ(~r) = P (~k)e−i~k·~r (1.11) (2π)3

The power spectrum measures the power at each frequency scale. The relationship between fourier variables, k = 2π/x, indicates that structure on small spatial scales

corresponds to power in high values of k, while structure on large spatial scales corresponds to power in small values of k. By looking at the power spectrum, it is easy to see which three-dimensional modes have the most power in generating the over and under-densities of large scale structure. In general, since P(k) and ξ(r) are fourier transform related, by definition they contain the exact same information. However, this is only true if we have complete knowledge of either the power spectrum or correlation function – i.e., accurate down to any scale size or resolution. In reality, we make estimates of these values based on real surveys that we conduct. Because of this, different situations can find it advantageous to use one or the other to interpret large scale structure. We primarily make use of correlation functions for the research contained in this Thesis, calculating ξ using the Landy-Szalay (Landy and Szalay,

1993) estimator of the correlation function:

DD(r, µ) − 2DR(r, µ) + RR(r, µ) ξˆ(r, µ) = . (1.12) RR(r, µ)

Here, ξˆ indicates that this is an estimator; a way to approximate the true correlation

function given the data. DD refers to the number of pairs of galaxies in a given

11 dataset, where RR refers to the number of pairs of galaxies in the same geometry as the dataset, but where the galaxies are randomly placed. This allows the correlation function to be independent of geometry; it isolates the excess clustering of galaxies that is due to gravitational attraction from that due to an attribute of the survey area. DR counts pairs of one data galaxy and one random galaxy.

1.4 Redshift Space Distortions

In order to map out the structure of our Universe, it is necessary to estimate the distance to extragalactic objects. For stars and other objects within the Milky

Way, it is often possible to use parallax, among other methods, to estimate distances.

However, these techniques cannot extend much farther than our closest extragalactic neighbors, so cosmologists and astronomers make use of Hubble’s law (Hubble, 1929):

v ≈ H0r (1.13)

where v is the magnitude of the galaxy’s radial velocity, r is its distance, and H0 is the

Hubble constant, an indicator of the current rate of expansion of the Universe. The beauty of Eq. 1.13 is immediately apparent; if we can simply measure the recession velocity of a galaxy, we have an idea of its distance from us. I deliberately use an approximate sign in Eq. 1.13, since all objects have their own independent motions due to local gravitational ﬁelds, which result in ‘peculiar velocities’ unrelated to the cosmic expansion velocity. However, as we observe objects farther away, the cosmic expansion velocity becomes a larger fraction of a galaxy’s total recession velocity, and so Eq. 1.13 becomes more accurate.

In order to estimate an object’s recession velocity, we use observations of its spectra. Light is redshifted depending on the recession velocity of the galaxy, and so in

12 measuring the wavelength diﬀerences between spectra of galaxies and ideal spectra

at rest, we can estimate a galaxy’s recession velocity, and hence its distance from

us. This concept is one of the most foundational principles in cosmology, and it has

served the ﬁeld incredibly since the inception of Hubble’s law.

What we call ‘redshift-space distortions’ (RSDs) are due to the components of an

observed object’s velocity that are unrelated to the cosmic expansion of the Universe.

Since we cannot, on an object-by-object basis, tell what part of its redshift is due to

its peculiar velocity or its cosmological velocity, we end up with a biased estimate of

that object’s radial distance. The redshift we measure, in reality, is:

v z = z + pec , (1.14) measured cosmo aH where a is the scale factor measuring the amount the Universe has expanded (a = 1 today) and vpec is the radial peculiar velocity of the galaxy in question. The peculiar velocity corrupts our measurement of the cosmological redshift of the galaxy. On a galaxy-by-galaxy basis, this is simply an incorrect cosmological redshift. But for a catalog of galaxies, these eﬀects can become correlated in ways that change the structure that we attempt to observe.

In general, the results can be separated into two categories. First, as mentioned earlier, each galaxy has a peculiar velocity, which on small scales is dependent on local details of that speciﬁc galactic system. These can be treated as a random addition to the cosmic expansion velocity, since a galaxy is as likely to be moving toward us as it is away from us, on its local orbital trajectory. For a galaxy catalog, this results in a smearing of the observed positions of galaxies on small scales, demonstrated in Fig.

1.2, called the ‘Finger of God’ eﬀect, because in redshift space, galaxy positions are

13 smeared along a line pointed toward the observer. Usually this eﬀect is treated as a nuisance parameter, and cannot tell us much about large scale structure.

The other eﬀect of RSDs occurs on larger scales. It is the result of the infall of galaxies to local over-densities, and is shown in Fig 1.3. This eﬀect, recognized by

Kaiser (1987), is related to the growth of structure parameter fv; more matter results in a larger pull on local galaxies, which increases fv while also causing a larger redshift distortion. To linear order, in Fourier space, we can write:

~ 2 ~ δs(k) = (1 + fvµ )δr(k). (1.15)

Here, δs are the observed redshift-space overdensities, δr are the true position-space overdensities, µ is the angle between a given wavevector ~k and the line of sight, and fv is the rate of growth of structure, f ≡ dln G/dln a, where G is the growth function, describing how overdensities grow in the Universe. It is apparent that for wavevectors perpendicular to the line of sight, µ = 0, and the observed densities are equivalent to the position space densities, since redshift space distortions can only affect radial velocities; RSDs are maximized for µ = 1. We can also add to Eq. 1.15 the impact of galaxy bias bg. Since we can only observe luminous objects in the Universe, such as galaxies, but not the true matter density field which is generally not luminous, we must quantify how well a given object traces the matter density field. This is done with the bias: the galaxy bias bg describes how well galaxy clustering is related to the clustering of all matter; a value of bias equal to 1 indicates that it traces the matter

ﬁeld exactly. When we include the galaxy bias to Eq. 1.15, we have:

~ 2 ~ 2 ~ δg(k) = (bg + fvµ )δm(k) = bg(1 + βµ )δm(k) (1.16)

14 True Position Observed Position

Observer Observer

Figure 1.2: Here we demonstrate the ‘Finger of God’ effect. In the left panel, we have some random selection of galaxies with varying velocities, denoted by black arrows, which depend on specifics that we are not interested in. Note that the blue galaxies have velocities which are generally toward the observer, the red galaxies have velocities away from the observer, and the green galaxies have velocities that are transverse to the observer. Because the observer cannot distinguish the cosmic velocity from each specific galaxy’s peculiar velocity, it assumes that the red galaxies are farther away, the blue galaxies are closer, and the green galaxies are in the correct position. In the right panel, we show an exaggerated version of the result, where groups of galaxies are smeared in a direction which appears to be pointing toward the observer.

15 True Position Observed Position

Observer Observer

Figure 1.3: Here we demonstrate the Kaiser eﬀect. In the left panel, we show some galaxies (colored points) with velocities denoted by the black arrows, surrounding a large mass (central black point). Although the galaxy’s velocities will vary, in general they will be more likely to point toward the central mass because of gravitational attraction. Note that the blue galaxies have velocities which are generally toward the observer, the red galaxies have velocities away from the observer, and the green galaxies have velocities that are transverse to the observer. Because the observer cannot distinguish the cosmic velocity from each speciﬁc galaxy’s peculiar velocity, it assumes that the red galaxies are farther away, the blue galaxies are closer, and the green galaxies are in the correct position. In the right panel, we show an exaggerated version of the result, the group of galaxies appears to be ‘squashed’ transverse to the line of sight.

16 where δg is the galaxy overdensitiy, δm is the matter overdensity, and β ≡ fv/bg. This means that the galaxy power spectrum can be written as:

2 2 2 Pg(k, µ) = bg(1 + βµ ) Pm(k, µ) (1.17)

Finally, if we assume General Relativity is accurate, then fv can be related to the

matter density in the Universe today, Ωm,0:

0.6 fv ≈ Ωm,0 (1.18)

By measuring β, and hence fv, we can make statements about the Universe’s matter density. Using the following formula:

2 ξ2(r) (4/3)β + (4/7)β = 2 , (1.19) ξ0(r) 1 + (2/3)β + (1/5)β it can be seen that measuring moments of the correlation function can lead to measurements of the matter density and clustering in the Universe. In Chapter 2, we explore how galaxy line blending could corrupt estimates of fv, among other parameters, through the ﬁtting of RSDs. In Chapter 3, we use measurements of RSDs to calculate the intrinsic alignment magnitude B, introduced in Sec. 1.5.

1.5 Gravitational Lensing

Observing galaxies to infer the structure distribution in the Universe suﬀers from a drawback: it only samples the luminous matter of the Universe, which is related, but not identical to, the total matter distribution. Inferring the matter distribution from the locations of luminous objects such as galaxies involves assumptions about how well these objects truly trace the non-luminous matter.

Gravitational lensing avoids this diﬃculty entirely by directly measuring the curvature of spacetime, which (from General Relativity) is a direct result of mass, not

17 just luminous mass. A consequence of general relativity is the behavior of light rays to

have their paths altered by sources of gravity, in the same way that light is refracted

when viewed through a lens. In principle, this changes the observed angular position

of galaxies from their true positions, but a method has not yet been discovered to

take advantage of this eﬀect, since we don’t know the initial angular position of each

galaxy. However, lensing also changes an observed galaxy’s shape; this fact can be

used to great eﬀect, as I will brieﬂy describe.

The ellipticity of a galaxy has two components, which are complex in the conven-

tion we will use here. We can deﬁne the intrinsic ellipticity of an object eI as: a − b e = e2iφ (1.20) I a + b where a and b are the major and minor axes of the galaxy, respectively, and φ is its orientation relative to our vantage point. This intrinsic ellipticity is sheared by the intervening matter after being lensed, resulting in an observed ellipticity of:

eI + γ eobs = ∗ ∗ ≈ eI + γ (1.21) 1 + γ eI

where a superscript of ‘∗’ denotes a complex conjugate, and γ = γ+ + iγ×. The

approximation in Eq. 1.21 is true in the linear limit, where both the intrinsic ellipticity

and the shear are small. The observed ellipticity is the result of gravitational lensing

acting on the intrinsic ellipticity of the object in question. The γ terms are deﬁned by the distortion of light rays by intervening matter distributions; the solution in general relativity, by integrating over the line of sight χ, results in:

Z ∞ −2 2 2 γi = (γi,+, γi,×) = ∂ dχW (χ, χi)(∂x − ∂y , 2∂x∂y)δm(χnˆi) (1.22) 0

This relates the gravitational shear to the linear density perturbation ﬁeld, δm. Here,

W is a weighting function which is dependent on details of the lens, andn ˆi is the

18 direction of the galaxy in question. γ+ indicates image stretching in the y or x directions, while γ× refers to image stretching along the diagonals.

It is impossible to tell if a single galaxy has been lensed, and by how much, since we have no knowledge of what its intrinsic ellipticity was. However, if we understand the distribution of intrinsic alignments, then a measurement of a large number of galaxies can reveal a statistical eﬀect which encompasses the entire ensemble of galaxy ellipticities, and make statements about the matter distribution. We can make correlation functions (similar in concept to those in Sec. 1.3) between the angular positions and the density weighted mean intrinsic shear of galaxies:

I ξg+(~r) = δg(~x)¯γ+(~x + ~r) , (1.23) where the density weighted mean intrinsic shear is deﬁned asγ ¯I = (1 + δg)γI . Mea- surements of this quantity have been made for a variety of galaxy samples, and we can ﬁt them with theoretical models, which are normalized by the parameter B.

Specifically, B signifies the coupling between the ellipticity of a galaxy and the tidal gravitational fields around it; a value of B = 0.05 indicates that an order unity tidal

ﬁeld leads to a 5 percent change in the quadrupole moment of the shape of the galaxy.

While for shear-density correlation functions B is measured based on the observed shapes of galaxies, in Chapter 3, we use RSDs to generate a radial measurement of

B, which is dependent upon matter clustering along the line of sight. We compare our measurement to the values obtained from density correlation function methods.

We have only scratched the barest surface of gravitational lensing here; there is much more information that has not been included, in the interest of creating a simple introduction to the concept. Please see Troxel and Ishak (2015), and references therein, for a more detailed treatment of gravitational lensing.

19 1.6 Cosmic Microwave Background

We complete the introduction section by brieﬂy explaining the concept and significance of the Cosmic Microwave Background (CMB). Although not directly relevant for Chapters 2 and 3, this will lay the foundation for the motivation of the work presented in Chapter 4.

The hot, homogenous state of the Universe directly following the Big Bang contained a ‘particle soup’ scattering endlessly with itself; three minutes after the Big

Bang, this soup was composed of nuclei, electrons, and photons, keeping their respective distributions matched. However, as the Universe expanded, this plasma cooled adiabatically. Eventually (approximately 400,000 years after the Big Bang) the temperature decreased to the point where protons and electrons could recombine to form neutral hydrogen, during what is referred to as ‘recombination.’ This event had a pro- found eﬀect on the photons in the early Universe. Before recombination, the photon distribution was tightly coupled with the electron and proton distribution through the process of Thomson scattering, which occurs primarily between a photon and a free electron. However, after recombination has suﬃciently converted the populations of free charged particles into neutral hydrogen, there is no more Thomson scattering to keep the photons interacting with the matter. Hence, the photons essentially free-stream in all directions, ignoring the neutral hydrogen.

Today, billions of years after the Big Bang and recombination, we can still see these CMB photons free-streaming to us, from all directions. Although they have traveled many lightyears, and been (to an extent) lensed and spectrally distorted, they are a remarkably pristine picture of the early Universe. From 1989 to 1993, the

COBE satellite (Mather et al., 1994) surveyed the CMB, and this data resulted in two

20 major discoveries. First, the absolute spectrum of the CMB is very nearly a perfect

blackbody, pictured in Fig. 1.6. This emphasizes how homogenous the early Universe

was - everywhere we look, we see a blackbody temperature. Secondly, however, there

are directionally-dependent temperature diﬀerences, although they are very slight, on

the order of 10−5. These diﬀerences indicate the density perturbations in the early

Universe, which were the ﬁrst seeds of the galaxies we see around us today. In Fig.

1.6 we show a plot of these temperature anisotropies from observations by the Planck satellite (Planck Collaboration et al., 2016).

Observations of the CMB are incredibly valuable in constraining cosmological models. While galaxy surveys show how the Universe has evolved over billions of years, the CMB is a snapshot into the relative infancy of the Universe. Furthermore, there has not been time for gravitational instabilities to become non-linear, so perturbation theory is fully justiﬁed in this regime. In the standard scenario, Dark Energy is dominated by both matter and radiation in the early Universe, and so we do not need to worry about the unknown Dark Energy behavior when analyzing the CMB, unlike galaxy surveys. These factors make the CMB an incredibly useful tool for cosmological analysis.

The CMB power spectrum measures the correlations between the anisotropies in the CMB, and is displayed in Fig. 1.6. The power spectrum can be compared to theoretical predictions based on cosmological parameters such as the baryon density Ωb and the radiation density Ωr. Furthermore, the CMB is partially polarized, due to the combination of Thomson scattering and temperature anisotropies during recombination.

21 Figure 1.4: The intensity distribution of the CMB, as measured by the COBE satellite (Mather et al., 1994). It is important to note that the line is not a best-ﬁt line to the data; it is a prediction of the data given the temperature of the CMB, 2.7 Kelvin. Data for this plot is available online at https://lambda.gsfc.nasa.gov/product/cobe/firas_ monopole_get.cfm.

22 Figure 1.5: The temperature anisotropies in the Cosmic Microwave Background as observed by Planck (Planck Collaboration et al., 2016). The color scale has been magniﬁed; variations from the mean temperature are of order 10−5. Photo: ESA and the Planck Collaboration.

Figure 1.6: The Cosmic Microwave Background temperature power spectrum (Planck Collaboration et al., 2016). We can see on what scales the perturbations in temperature are most prevelant. Photo: ESA and the Planck Collaboration.

23 The largest, most important reaction during recombination is Thomson scattering, where an electron and a photon collide, which is a frequency-independent process.

However, resonant line scattering between neutral hydrogen and photons leads to frequency-dependent distortions in both the blackbody frequency of the CMB and the

CMB temperature and polarization anisotropy power spectra. Although small, these distortions can provide valuable insight into the exact redshift of recombination, and hopefully serve as a method to directly measure both the baryon density and helium abundance in the early Universe.

In Chapter 4, we lay the foundation for a general method to calculate changes in the polarization and temperature power spectra due to resonant line scattering.

Our method allows all transitions for hydrogen to be calculated, up to any maximum quantum state n desired. As mentioned earlier, this chapter is highly technical and in- depth, focused on providing the precise mathematical tools necessary for this project to be ﬁnished in the future.

24 Chapter 2: Eﬀects of [N ii] and Hα Line Blending on the WFIRST Galaxy Redshift Survey

2.1 Introduction

Observations of high-redshift supernovae in the 1990s provided the ﬁrst direct evidence for an acceleration in the expansion rate of the universe (Riess et al., 1998;

Perlmutter et al., 1999)1. Whatever ﬁeld or particle is responsible for this surprising acceleration has been dubbed “dark energy,” and one of the major observational programs in modern cosmology is to measure its properties. It is of particular interest to determine whether the dark energy is consistent with a cosmological constant; whether it requires new dynamical degrees of freedom; or whether cosmic acceleration arises instead from a modiﬁcation to the laws of gravity on large scales.

In order to measure the properties of dark energy, cosmologists employ a variety of techniques. Observations of supernovae provide information on the expansion rate of the Universe for diﬀerent redshifts, probing the eﬀects of dark energy throughout cosmic history. Weak lensing surveys probe the matter distribution, allowing measurements of clustering at various redshifts. Galaxy clusters are the most massive

1This chapter has been submitted for publishing to the Monthly Notices of the Royal Astronomical Society. The authors are Daniel Martens, Xiao Fang, M.A. Troxel, Joe DeRose, Christopher M. Hirata, Risa H. Wechsler, and Yun Wang.

25 collapsed objects produced by cosmological structure formation, and can be traced using a wide range of observables (the visible galaxy content, the hot gas via X-ray emission or the Sunyaev-Zel’dovich eﬀect, and weak lensing). Galaxy redshift surveys

– the subject of this paper – can trace cosmic structures in three dimensions, although their cosmological interpretation requires accurate modeling of the relation between the visible galaxies and the mostly unseen matter.

The size of redshift surveys have been steadily increasing in tandem with tech- nological improvements. A sample of over 200,000 galaxies was investigated in the

2dF Galaxy Redshift Survey, constraining cosmological parameters within speciﬁc cosmological models (Cole et al., 2005). This was followed by the 6dF Galaxy Survey

(Jones et al., 2009). A spectroscopic analysis of over 54,000 luminous red galaxies using the Sloan Digital Sky Survey (SDSS) found evidence for baryon acoustic oscillations, and provided additional constraints on the cosmological parameters (Eisenstein et al., 2005). Further analysis has been done combining SDSS with 2dFGRS, as well as analyzing data provided by SDSS DR7 (Percival et al., 2007, 2010). More recently, the WiggleZ Dark Energy Survey has been used to measure the BAO peak at diﬀerent redshifts (Blake et al., 2011), while others have analyzed the distance to these redshifts (Xu et al., 2012). The SDSS-III/BOSS project, which included an upgraded spectrograph with enhanced red sensitivity, collected spectra of over 2.4 million galaxies (Alam et al., 2015). The redshift range of SDSS-III is extended by

SDSS-IV (eBOSS) which is currently observing (Dawson et al., 2016; Blanton et al.,

2017).

This progress is expected to continue: the Taipan survey will look at low-redshift galaxies, over half the sky (da Cunha et al., 2017), and the Dark Energy Spectroscopic

26 Instrument (DESI) will conduct a comprehensive spectroscopic survey of galaxies and

quasars over the Northern sky (DESI Collaboration et al., 2016a,b). A substantially

deeper survey is planned by the Prime Focus Spectrograph (PFS) (Tamura et al.,

2016), which will extend ground-based spectroscopic coverage out to 1.26 µm, and

the 4m Multi Object Spectroscopic Telescope (4MOST Depagne, 2015) project will

conduct optical spectroscopy in the south. The Euclid mission will conduct a space-

based near-infrared grism survey over both hemispheres (Laureijs et al., 2011).

The Wide Field InfraRed Survey Telescope (WFIRST) will be a 2.4 m space telescope that carries out a wide range of investigations in cosmology, exoplanets, and other areas of astrophysics (Dressler et al., 2012; Green et al., 2012; Spergel et al., 2015) following its launch in the mid-2020’s. WFIRST will carry a Wide Field

Instrument (WFI; capable of imaging, grism spectroscopy, and with an integral ﬁeld

channel) and a coronagraph. The galaxy redshift survey program on WFIRST will

use the grism to observe emission lines in the 1.00–1.93 µm bandpass2 and obtain

redshifts for 1.8 × 106 galaxies per month of observations. The principal tracer of

large scale structure will be the Hα emission line (at 6565 A),˚ which is visible in

WFIRST out to a maximum redshift of z = 1.94; at higher redshift, other emission

lines will be used, most notably the [O iii] doublet (4960 and 5008 A),˚ which is visible

out to z = 2.85.

The grism spectroscopy technique has the advantage of simplicity (the grism occu-

pies one slot on the WFIRST ﬁlter wheel, with no additional moving parts); it provides

enormous multiplexing; and it does not require that targets be selected in advance

2This wavelength range was chosen for the System Requirements Review/Mission Deﬁnition Review, and is somewhat diﬀerent from that considered during previous iterations of the WFIRST design.

27 (thus providing operational ﬂexibility, and avoiding selection biases that are “baked

in” to traditional redshift surveys at the time of target selection). However, it does

have drawbacks. One is that without a slit, each pixel is exposed to the full dispersed

sky scene rather than only the targeted galaxy, which leads to higher backgrounds

and confusion from other sources. Grism spectroscopy also has some constraints:

WFIRST requires a wide-ﬁeld grism in a converging beam; it was a signiﬁcant de-

sign challenge for all ﬁeld positions and all wavelengths to focus simultaneously, and

solutions are only available at moderate spectral resolution (Gong et al., 2016). At

this resolution (R ∼ 690 per 2-pixel element at λ = 1.5 µm), the Hα line is partially

blended with the neighboring [N ii] lines. This is a similar situation to the WFC3

Infrared Spectroscopic Parallel (WISP) Survey, which used an even lower-resolution

grism on the Hubble Space Telescope where Hα+[N ii] was completely blended (Atek et al., 2010). The results of the WISP survey have been used to make predictions on the eﬀectiveness of future surveys (Colbert et al., 2013), particularly through the discussion of the changes in the luminosity function of galaxies when line blending issues are present. In the past, a correction factor of 0.71 has been applied to the Hα luminosities of galaxies to account for [N ii] contamination (Mehta et al., 2015). Further analysis of the luminosity function changes, as well as an empirical parameterization of the [N ii]/Hα ﬂux ratio, were completed by Faisst et al. (2017).

A separate concern arising from the line blending eﬀect is the change in the observed redshift. Since the [N ii] lines are asymmetric in emission strength, they will shift the “observed” redshift zobs (assuming that the line centroid is at the Hα wavelength in the rest frame), to be redder than the true redshift, ztrue (Faisst et al.,

28 2017). Since the line ratio is determined by H ii region physics and depends on metal-

licity, this eﬀect is not the same for all galaxies: it is not removable by subtracting a

mean bias. Furthermore, since metallicity is correlated with galaxy mass and hence

with large-scale structure, Hα+[N ii] line blending leads to a redshift bias that is correlated with large-scale structure and could have highly non-trivial eﬀects on the inferred cosmological parameters.

The purpose of this paper is to quantify the extent of this observational problem, understand how it will aﬀect the measured galaxy correlation function if left unmitigated, and discuss potential mitigation strategies, to the extent that they will be necessary. We will analyze a sample of > 108 galaxies using mock data from the

Buzzard-v1.1 simulation. We compute the correlation function and perform BAO and RSD fits with both the true redshift catalog and the observed redshift catalog and assess the differences. We also create a “shuffled” redshift catalog, where the redshift errors ∆z/(1 + z) are scrambled, which allows us to test which changes in the clustering properties are due to the correlations of redshift error with large-scale structure, and which are due to the distribution of redshift errors alone.

The outline of the paper is as follows. In Section 2.2, we discuss the line blending problem in more detail, and introduce the main questions which this paper will attempt to answer. In Section 2.3, we describe the general road map to answering these questions, discussing the analysis strategies we will use to dissect the simulation results. We discuss details of our simulation in Section 2.4, as well as the description of the catalogs. In Section 2.5, we discuss our ﬁtting strategies and methods, and display our ﬁts for redshift space distortion and BAO parameters to the data. In Section

29 2.6 we discuss our results and how they compare to the requirements for WFIRST,

with a brief discussion of possible mitigation strategies. We conclude in section 3.6.

For this analysis, we use a ﬂat ΛCDM cosmology with parameters of Ωm = 0.286,

ΩΛ = 0.714, σ8 = 0.82, h = 0.7, Ωb = 0.047, and ns = 0.96. This is consistent with

the parameters used in the generation of the Buzzard-v1.1 simulations. Finally, all

line wavelengths are referenced to their vacuum values.

2.2 Objectives

All spectroscopic galaxy surveys contain some redshift errors, in the sense that the

observed redshift zobs deviates from the true redshift ztrue. Among existing samples

used for large-scale structure analyses, this is most evident for quasars (e.g. Dawson

et al., 2016), since the most accessible lines are broad and often asymmetric, and

redshift errors of several hundred km s−1 are common.

In the case of WFIRST, photon noise in the centroid of the emission lines will be the dominant source of scatter in the zobs vs. ztrue relation. The inclusion of statistical noise results in both a decrease in the constraining power of the survey, and the suppression of power at large kk. This noise does not require any additional treatment to remove, since the central values are unchanged. However, in the WFIRST survey, blending of the Hα emission line with the neighboring [N ii] doublet (one doublet member is on each side of Hα) results in an oﬀset zobs > ztrue for objects of higher

[N ii]/Hα ratio (which are likely higher metallicity and hence probably found in denser environments). Other possible errors could involve the emission line strength and the angular size of a galaxy affecting the width of the zobs − ztrue distribution, and this may in turn affect the redshift offset, if it interacts with point spread function (PSF)

30 asymmetry. Both of these properties may be correlated with the galaxy environment.

They are, however, beyond the scope of this paper; we plan to revisit PSF asymmetry

and other instrument-related issues when the WFIRST grism simulation pipeline is

at a more advanced stage.

Figure 2.1 demonstrates the scale of the line blending problem. The WFIRST

grism has a spectral resolving power of

λ λ = 461 obs , (2.1) ∆λ 1 µm where λ is the observed wavelength (this is measured for a 2-pixel element; an extended galaxy will be bigger and hence have lower spectral resolution).3 It is seen that

at this resolution, the Hα line and surrounding [N ii] lines are blended, and a ﬁt to a

single line will ﬁnd something close to the centroid of the blended features. If there

was a known error probability distribution function (PDF) P (zobs − ztrue|ztrue) that was both uncorrelated with galaxy environment and valid for every type of galaxy in the sample, then we could incorporate this in the theoretical correlation function, and the only eﬀect of the redshift errors due to line blending would be a reduction in the statistical constraining power in the survey. If, however, the redshift error

PDF is either not known or is correlated with galaxy environment, further steps may be needed to maximize accurate redshift reconstruction. Such correlations may be problematic for a survey even if the systematic redshift error is small compared to the statistical errors for one single galaxy.

3See Gehrels et al. (2015); updated information can be found at the WFIRST project website: https://wfirst.gsfc.nasa.gov/science/WFIRST_Reference_ Information.html

31 1.0 Hα True [N II] Hα Obs. 0.8

0.6

0.4 Relative Line Strength 0.2

0.0 6500 6520 6540 6560 6580 6600 6620 6640 Wavelength (Angstroms)

Figure 2.1: A visual representation of the [N ii] lines and Hα line for the line blending scenario. The difference in strength between the two nitrogen lines is constant for each galaxy with a ratio of approximately 0.32, while the difference between the larger [N ii] line and the Hα line varies depending on the metallicity of the galaxy in question. Here, we show the median difference of our sample, with a ratio of [N ii]6585/Hα of 0.427. The Gaussian spreading centered at each line has a standard deviation of σgrism, calculated generally in Eq. 2.1. In this example, we have shown the spreading due to a galaxy with radius 4 kpc, at redshift 1.5. The black dotted line is the resulting observed line, given that the constituent lines cannot be resolved.

32 In this paper, we use the Buzzard-v1.1 simulation (described in Sec. 2.4.1) to

address several key questions regarding the impact of the line blending phenomenon

on the WFIRST redshift survey:

1. If we ignore the eﬀects of line blending, what biases are induced in the baryon

acoustic oscillation (BAO) and redshift space distortion (RSD) parameters?

How does this compare to the corresponding statistical errors on these param-

eters, or the errors required by the WFIRST Science Requirements Document

(SRD)?

2. Do we need to mitigate biases caused by correlations between galaxy environ-

ment and redshift oﬀsets due to line blending?

3. If we determine that the problems are signiﬁcant enough to require some level

of mitigation, what type of mitigations are necessary?

The WFIRST SRD deﬁnes required performance based on a Reference Survey of

0.70 years4, with 1σ statistical errors of 0.70% on the transverse BAO distance scale

(α⊥); 1.28% on the radial BAO distance scale (αk); and 1.28% on the rate of growth

of structure measured from RSD (fv). Observational systematic errors are allocated an error of 0.58 times the Reference Survey statistical errors, i.e. 0.41% (α⊥), 0.74%

5 (αk), and 0.74% (fv). Note that these are requirements – it is always desirable to have

smaller systematic errors, but if they exceed their allocation they must be mitigated.

Finally, note that the Hα+[N ii] blending is only one type of observational systematic

4The actual allocations will be determined in the future by the Implementation SWG.

5 √This means that the combination of statistical errors and observational systematic errors would be 1 + 0.582 = 1.16 times larger than the statistical errors alone. Note furthermore that the SRD allows for other sources of error as well.

33 Figure 2.2: A flow chart describing the simulation pipeline; see Sec. 2.3 for details. The red hexagons indicate steps where data is input into the process, and yellow circles indicate steps where we output statistics, correlation functions, or fit parameters. Also included in the flowchart is a method for possible analysis of the ability of a ground- based spectroscopic observation to reproduce the observed catalog; although it is not implemented in this work, we discuss its potential in Sec. 2.6.

error, and as such should consume only a fraction of the systematic error budget (the

exact percentage has not been set; part of the purpose of this paper is to inform this

discussion).

2.3 Calculation Outline

In this section, we explain the speciﬁc steps taken throughout the analysis in order

to fully understand the detailed eﬀects of line blending. Figure 2.2 shows a ﬂowchart of

the process, beginning with the Buzzard-v1.1 mock galaxy catalog, moving through the calculations we perform, and ending with our parameter estimates. Although more technical details for each process will be expanded later on in Section 2.4, we take a moment here to give a high-level overview of the entire pipeline.

34 We begin with the mock catalog from the Buzzard-v1.1 simulation. We use

a “true” redshift for each galaxy in the catalog that incorporates peculiar velocity

eﬀects, but does not yet contain any line blending or statistical errors. The catalog

and its generation are described in detail in Section 2.4.1. The catalog covers one

quadrant of the sky (π steradians).

Our ﬁrst step is to assign to each galaxy a [N ii]/Hα ratio based upon that galaxy’s

redshift and stellar mass. Once each galaxy has a [N ii]/Hα ratio, we can then calculate the observed redshift zobs for each galaxy, which incorporates the eﬀect of line blending. We also include a statistical oﬀset of the redshift due to photon noise in the center of the line. These redshifts form the “observed redshift” catalog.

We next create a separate redshift for each galaxy called the “shuﬄed” redshift, or zshf . The purpose of this is to provide a redshift catalog where the distribution of the observed redshift catalog is accounted for, but removes any correlation between the

[N ii]/Hα ratio and the galaxy environment. By later comparing clustering parameters from the “shuﬄed” redshifts to those from the observed redshifts, we can determine how much of the eﬀect we see is captured in the 1-point distribution of redshift errors, and how much depends on environmental correlations.

Next, the galaxies are binned, depending on the galaxy redshift (Z2, Z3) and line

ﬂux (H1, H2, H3). The group cuts are described in Table 2.1. Note that a given galaxy may be in one bin in the true redshift catalog, but in another bin in the observed redshift catalog, if the oﬀsets push the observed redshift into another bin.

Furthermore, to ease the computational burden, each bin is split into six congruent kite-shaped geometric regions (S1, S2, ... S6) on the sky, each of solid angle π/6

35 Dec 90°

45° (S3) (S6)

(S1) (S4) (S2) (S5)

0° 180° 135° 90° 45° 0° RA

Figure 2.3: We display the 6 separate regions on the sky, within which we have independently calculated the correlation function, as discussed in Sec. 2.4.3. Note that although we must display a 2D projection, the sectors are congruent on the sphere. (The simulation does not cover the actual WFIRST footprint, which is likely to be placed in the Southern Hemisphere, but this does not matter in a statistically isotropic universe.)

steradians, as shown in Figure 2.3. These sections are counted separately, and then recombined in the analysis of the correlation functions.

For each redshift/luminosity bin (which will be referred to as “ZH” bins) and geometric region, we ﬁt redshift-space distortion (RSD) parameters and baryon acoustic oscillation (BAO) parameters, and compare the resulting parameter shifts with the

WFIRST error budget to assess their signiﬁcance and the possible need for mitigation.

36 2.4 Simulation

2.4.1 Catalog generation

We make use of the Buzzard-v1.1 mock galaxy catalog that we describe brieﬂy here and refer the interested reader to more detailed descriptions in DeRose et al.

(2018) and Wechsler et al. (2018). Buzzard-v1.1 is a simulated galaxy catalog constructed from a set of three nested dark matter-only lightcone simulations which are progressively lower resolution at higher redshifts. The lightcones have volumes of

−1 3 −1 3 −1 3 10 −1 (1050 h Mpc) , (2600 h Mpc) , (4000 h Mpc) , particle masses of 2.7×10 h M ,

11 −1 11 −1 −1 −1 1.3 × 10 h M , 4.8 × 10 h M and force softenings of 20 h kpc, 35 h kpc,

53 h−1kpc respectively. The highest resolution (1050 h−1Mpc)3 simulation is used for z < 0.34, the (2600 h−1Mpc)3 for 0.34 <= z < 0.9 and the (4000 h−1Mpc)3 simulation for 0.9 < z < 2.35. These simulations are run using L-GADGET2, a version of GADGET2 (Springel, 2005) modiﬁed for memory eﬃciency with 2nd-order

Lagrangian perturbation theory (2LPT) initial conditions created using 2LPTIC

(Crocce et al., 2006). Lightcones are generated on the ﬂy as the simulations run.

Galaxies are added to the simulation using the ADDGALS algorithm (Wechsler et al., 2018). Assuming an input luminosity function, this algorithm uses a model for density given absolute magnitude, p(δ|Mr, z) measured from a subhalo abundance matching (SHAM) model run on a smaller, higher resolution simulation. This model is then applied to the lower resolution lightcone simulations by drawing magnitudes from the assumed luminosity function, drawing densities from p(δ|Mr, z), and assigning the galaxy to a particle in the lightcone with the correct density. After all rest frame r- band magnitudes are assigned to all galaxies, SEDs are then assigned from SDSS using the δg −Mr−SED relation from SDSS (Cooper et al., 2006), where δg is the distance to

37 the ﬁfth-nearest neighbor galaxy. The SEDs are represented by kcorrect templates

(Blanton et al., 2003) from which line ﬂuxes and stellar masses can be determined. In

Fig. 2.4 we compare the Buzzard-v1.1 luminosity function to several semi-analytic

models.

2.4.2 Assignment of the [N ii]/Hα ratio

Here we outline our method to generate a [N ii]/Hα ratio for each galaxy. It can

be seen from the original Baldwin-Phillips-Terlevich (BPT) diagram (Baldwin et al.,

1981) that there is signiﬁcant variation in the value of each galaxy’s [N ii]/Hα ratio.

The primary source of variation is metallicity, with higher-metallicity star-forming

regions showing increased [N ii]/Hα and decreased [O iii]/Hβ. Creating an accurate

representation of this distribution within our simulation – including the eﬀects of

environment – is critical to the goals of this paper. Furthermore, the [N ii]/Hα ratio is expected to vary as a function of redshift (Kewley et al., 2013; Masters et al.,

2014; Steidel et al., 2014; Shapley et al., 2015; Strom et al., 2017; Kashino et al.,

2017) similarly to the general mass-metallicity (MZ) relation (Savaglio et al., 2005;

Maiolino et al., 2008; Lilly et al., 2013).

For each galaxy in the Buzzard-v1.1 mock catalog, [N ii]/Hα line strength ratios were assigned based on the stellar mass and redshift of that galaxy; we assume no other environmental trend in the mass-metallicity relationship itself. The dependence of the galaxy metallicity on star-formation rate is accounted for through the correlation between star-formation rate and redshift. This is justified observationally by Wuyts et al. (2016), who used the KMOS near-infrared multi-integral field unit survey to find the [N ii]/Hα ratio for 419 star-forming galaxies, in the redshift range 0.6 < z < 2.7.

38 Figure 2.4: Comparing the luminosity function of the Buzzard-v1.1 mock catalog to that predicted by Pozzetti et al. (2016). These are semi-analytic models made by fitting to observed luminosity functions from Hα surveys. Specifically, Models 1 and 2 feature a Schechter parameterization, while Model 3 was designed specifically for high-redshift slitless surveys such as WFIRST and Euclid, as it was fit directly to luminosity function data. This graphic is from one 54 deg2 section in the sky; the redshift range is from z = 0.7 to z = 1.5 (similar to our Z2 bins, but matching the exact range used for the Pozzetti et al. (2016) model).

39 They find that there is no significant dependence of the ratio on star-formation rate, given fixed redshift and stellar mass. Furthermore, this assumption has support from hydrodynamical simulations; Hirschmann et al. (2017) found good agreement with observations – that the primary evolutionary trends were based on redshift and stellar mass, in a manner consistent with our model. Other effects such as specifics of their

AGN model and ionized-gas hydrogen/electron density were found to have a much smaller impact on the cosmic evolution than the galaxy mass.

We base our fits on the results from Wuyts et al. (2016), who grouped their data into galaxy mass sub-ranges spanning log10(M/M ) = 9.88 − 11.13, for redshift bins at z ≈ 0.9, z ≈ 1.5, and z ≈ 2.3. We used these empirical relationships to create three linear fits, one for each redshift bin, giving the [N ii]/Hα ratio as a function of log stellar mass. We then used these fits to provide our catalog with nitrogen line strength ratios, depending on the mass and redshift of each galaxy in our sample.

The ﬁts can be seen in Fig. 2.5.

Approximately 0.1% of objects in the mock catalog have very small stellar masses, which when combined with the linear ﬁt from Fig. 2.5 produce a negative [N ii]/Hα ratio. These ratios are set to zero. This happens for galaxies with log10(M/M ) <

(9.2, 8.9, 9.5) for z ≈ (0.9, 1.5, 2.3), respectively. We do not expect this simplification to have any effect on our full sample results, due to the relatively small number of galaxies affected, and because fitting these lower-mass galaxies with a more complex model would likely still result in [N ii]/Hα near zero.

40 Figure 2.5: Empirical relationship between [N ii]/Hα ratio and log stellar mass. Data being ﬁtted was obtained from Wuyts et al. (2016). These relationships were used to construct the relative [N ii] line strength within the mock galaxy sample.

41 Table 2.1: The number of galaxies, in millions, within each bin. This is the sum total of galaxies within each geometric region for the data counts. The galaxy ﬂux F is measured in units of 10−17 erg cm−2 s−1. In parenthesis is the label we use to reference each bin throughout the text. F > 8 F > 13 F > 25 0.705 ≤ z < 1.345 - 65.1 (Z2H2) 17.8 (Z2H3) 1.355 ≤ z < 1.994 59.1 (Z3H1) 24.9 (Z3H2) 5.2 (Z3H3)

Bin Median Redshift Z2H2 0.9116 Z2H3 0.8698 Z3H1 1.6136 Z3H2 1.6134 Z3H3 1.6169

Table 2.2: The median redshift of each bin.

2.4.3 Data binning and redshift distributions

In order to generalize our analysis to many possible future surveys with a range of

redshifts and luminosity thresholds, we binned our simulation catalog by both redshift

and luminosity, and computed the correlations for all galaxies within each bin. The

redshift bins are diﬀerential, while the luminosity bins are integral; their ranges are

listed in Table 2.1. In order to decrease computation time, each bin was further

divided into six congruent geometric regions on the sky (S1 to S6, displayed in Fig.

2.3). Correlation functions were independently generated using the counts from each

sector.

42 To ensure that the correlation functions for diﬀerent groups of measurements, i.e.

zobs and ztrue, can be compared within the exact same redshift limits, it is necessary to use slightly different subsets of galaxies for different calculations – for example, one specific galaxy whose true redshift falls in redshift bin 1, may have its observed redshift place it within bin 2. Because of this, the exact galaxy samples vary slightly between true and observed samples.

Each galaxy was given a set of redshifts: a true redshift, an observed redshift, and a shuﬄed redshift. The true redshift, ztrue, is simply the original redshift value from the catalog – that is, the redshift that would be observed if the [N ii] and Hα lines could be separately resolved. This redshift still incorporates the peculiar velocities of each galaxy in the redshift value.

We use the observed redshift value, zobs, to include two separate eﬀects. First, we insert the statistical error in the wavelength centroid. Each galaxy’s observed redshift is modiﬁed by adding to it a number generated from a Gaussian with mean zero, and standard deviation:

−3 σz = 10 (1 + z). (2.2)

This error is statistical, and thus not dependent on each individual galaxy’s metallic-

ity. This is the error speciﬁed by the WFIRST SRD. The real errors will also depend

on line ﬂux and galaxy size (σz is smaller for galaxies that have brighter lines or

smaller angular sizes), however assessment of this is outside the scope of this paper.6

The second eﬀect incorporated into zobs is the [N ii]+Hα line blending eﬀect. As long as the nitrogen line strength is non-zero, this will pull each galaxy’s redshift

6 2 Studying how the statistical variance σz depends on galaxy properties will involve both the grism image simulations and (ultimately) WFIRST deep ﬁeld data, which will empirically constrain the precision of the redshift measurement by repeating it many times for the same sample of galaxies.

43 toward a “redder” value, due to the anisotropy of the nitrogen line pair. This eﬀect

is calculated by ﬁnding the oﬀset from the Hα line center, in the rest frame of the

galaxy: ∆ F − ∆ F δλ = 1 6585 2 6550 . (2.3) FHα + F6585 + F6550

Here δλ is the oﬀset of the observed line from the true Hα line, F6550 and F6585 refer to the strengths of the respective [N ii] lines, FHα is the strength of the Hα line, ∆1 is the diﬀerence in wavelength between Hα and [N ii] 6585, and ∆2 is the

diﬀerence in wavelength between [N ii] 6550 and Hα. (Both ∆1 and ∆2 are deﬁned to

˚ ˚ 1 e 3 e be positive.) We use the vacuum values of 6549.86 A and 6585.27 A for the ( D2− P1)

1 e 3 e and ( D2− P2) transitions of N ii, respectively, and we use a value of Hα at 6564.61

A.˚ We take the ratio in strengths between the two nitrogen lines, [N ii] 6550/[N ii]

6585 to be 0.32567 (Storey and Zeippen, 2000). Note that Eq. (2.3) is valid in the extreme case of a completely unresolved line; the marginally resolved case (relevant for WFIRST) can lead to smaller shifts (Faisst et al., 2017), and hence our analysis is conservative.

Once the offset for a specific galaxy is calculated, we can find the observed redshift using δλ zobs = ztrue + (1 + ztrue) + δst, (2.4) λHα where δst is a realization of the statistical error (see Eq. 2.2). Most galaxies show differences between the true redshift and the observed redshift at or below δz = 10−3, due to the natural [N ii]/Hα ratio empirically found in galaxies. There is a small subset of galaxies, of order 0.1%, with no difference between observed and true redshift, due to the linear metallicity fit. However, the majority of this subset is eliminated during

44 the binning process, as we remove galaxies below a certain luminosity threshold, and

low luminosity is correlated with the low stellar mass used in metallicity ﬁtting.

After each galaxy has values for ztrue and zobs, we can then generate the shuﬄed redshift, zshf . This is done by creating a list of δz values:

z − z δz ≡ obs true . (2.5) 1 + ztrue

We then shuﬄe the list, matching each δz with a true galaxy to create a “shuﬄed” redshift:

zshf ≡ (1 + ztrue)δz + ztrue. (2.6)

This creates a galaxy catalog where the redshift error distribution is identical to that in the observed catalog, but where all correlations between δz and galaxy environment are destroyed. In this way, by comparing results from the observed distribution to the shuﬄed distribution, we can see whether parameter oﬀsets are due to the distribution of redshift errors, or correlations between the redshift error distribution and galaxy environment. This will have an important impact in how we approach mitigation. In

Fig. 2.7 we display the diﬀerences in inferred position for a subset of galaxies within the catalog.

2.4.4 Correlation functions

Here, we detail our pipeline for calculating the correlation functions within each

ZH bin. The pipeline uses the same code as that described in Martens et al. (2018).

Once pair counts were obtained for the true, observed, and shuﬄed samples within each luminosity and redshift bin, the redshift distributions of each were used to generate random catalogs. Randomly placed galaxies were created and given redshifts pulled from the distribution of the matching data bin. The random galaxy count is

45 106

105

104

103

102

101 Number of Galaxies

100 10-4 10-3 z z obs − true 1 + ztrue

Figure 2.6: The difference between the observed redshift and the true redshift for a sample of a little over 10 million galaxies. This does not include the Gaussian photon noise smearing, only the difference due to the line blending effect. This histogram is generated from one sky section of the parent Buzzard-v1.1 sample, cut from z = 0.7 to z = 1.3 (the approximate redshift range of the Z2 bins), and covers approximately 54 square degrees of sky.

46 1100 True Observed

1080

1060

1040 z [Mpc]

1020

1000

1100 Shuffled All

1080

1060

1040 z [Mpc]

1020

1000 2300 2320 2340 2360 2380 2400 2300 2320 2340 2360 2380 2400 y [Mpc] y [Mpc]

Figure 2.7: We display the physical positions of the galaxies given the redshifts of each of our true, observed, and shuﬄed catalogs. These images were generated using Regions 2 and 5 of the Z2H3 catalogs. We have selected all galaxies with a Cartesian x-coordinate between -50 and 50 Mpc, a z-coordinate between 1000 and 1100 Mpc, and a y-coordinate between 2300 and 2400 Mpc. The galaxies in the x-direction have been projected into the y-z plane, and the observer is located toward the bottom-left corner of the image (which can be seen by the ‘streaks’ due to the changes in redshift for the combination of the catalogs).

47 equal to 3 times the simulated real galaxy count, which was decided on with the intent

to minimize error associated with random galaxy shot noise, but also work within the

limits of computation time.7 Pair counts were done on the random catalogs in order

to construct the correlation function using the Landy-Szalay method (LS Landy and

Szalay, 1993; Peebles and Hauser, 1974). The correlations are calculated as a function

of redshift-space separation s and µ = cos θ, where θ is the angle with respect to the

line of sight. The LS estimator is

DD(s, µ) − 2DR(s, µ) + RR(s, µ) ξ(s, µ) = , (2.7) RR(s, µ) where DD refers to the number of pairs of galaxies within a speciﬁc distance shell, s + ∆s/2 and s − ∆s/2, and within a speciﬁc angular range µ + ∆µ/2 and µ − ∆µ/2, for the data sample. RR refers to the same, but for the random sample. DR refers to a combined catalog of data and randoms, where we are counting pairs of opposite types only. We use 120 logarithmically spaced radial bins, from s = 1 Mpc to s = 200

Mpc. Both DD and RR counts are normalized by the total number of galaxies in that bin, speciﬁcally:

DD RR DD → and RR → , (2.8) nD(nD − 1) nR(nR − 1)

while the DR counts are normalized by:

DR DR → . (2.9) nDnR

Our correlation function code calculates pairs in 20 µ-bins, from −1 to +1, with a separation of ∆µ = 0.1. Although the simulations were generated in µ-binned“wedge”

7For the ﬁnal WFIRST analysis, more resources will be available to devote to random pair counting. The random catalog shot noise is a fraction nD/nR of the shot noise in the data, so to make this negligible, we will need a nR/nD much greater than that used in this paper.

48 space, we convert them to multipole space for parameter ﬁtting. The formula for

conversion is the same as that used in SDSS BOSS analysis (Ross et al., 2017):

i 2l + 1 Xmax 1 ξ (r) = ξ(r, µ )L (µ ), (2.10) l 2 i i l i i=1 max

8 where µi = (i−1/2)/imax, Ll is the Legendre polynomial of order l, ξ is the µ-binned

correlation function, and imax = 20/2 = 10 is the number of µ-bins from 0 to +1. In

Fig. 2.8, we plot the resulting monopole and quadrupole correlation functions for the

Z2H2, Z2H3, Z3H1, Z3H2, and Z3H3 bins, respectively. In each plot, we show the

comparison of the quadrupole and monopole correlation functions between the true,

observed, and shuﬄed redshift catalogs. We also show the fractional comparison from

the true catalog to the observed and shuﬄed catalogs.

2.5 Parameter ﬁtting

In this section, we detail the process of ﬁtting our correlation functions from

the simulation with smaller scale RSD parameters and large scale BAO parameters.

We perform the RSD and BAO ﬁts completely independently, and in separate scale

ranges; although in projects whose primary goal is parameter measurement would ﬁt

both parameter sets together, here we separate them to provide greater sensitivity to

the systematic errors we are studying, resulting in the most conservative choice for

systematic error budgeting.

First, we use CLPT and GSRSD (Wang et al., 2014) to ﬁt the parameters bg and

2 fv, on scales of 42 to 200 Mpc (for varying values of σFOG). This will be discussed in more detail in Section 2.5.2, and closely follows the method detailed in Martens

8To avoid confusion with power spectra, we use L instead of the traditional P for Legendre polynomials.

49 Figure 2.8: For each ZH bin, we compare the monopole and quadrupole correlation functions for each of the true, observed, and shuﬄed redshift catalogs. The top panel of each plot displays the correlation function monopoles (solid lines) and quadrupoles (dashed lines). Both quadrupoles and monopoles are multipilied by r2, and the quadrupoles are multiplied by −1. The bottom panel of each plot shows the fractional comparison between the observed and shuﬄed correlation multipoles, as a function of separation r.

50 et al. (2018). This ﬁt focuses on small scale, clustering parameters, and is explained

in detail in Sec. 2.5.2. Second, we ﬁt BAO parameters on scales of 60 to 200 Mpc,

described in Sec. 2.5.3. In both cases, we use the correlation function monopole and

quadrupole to drive the ﬁtting function.

2.5.1 Covariance matrices

To provide a best-ﬁt to the observed correlation functions, we assume a Gaussian-

distributed likelihood for our vector of measured correlation functions:

L(p) ∝ e−χ2(p)/2, (2.11) where χ2 is given by:

2 X X i ˆi −1 j ˆj χ (p) = (ξ`(p) − ξ`)[C ]``0ij(ξ`0 (p) − ξ`0 ). (2.12) `,`0 i,j

Here p is a vector of parameters; ` and `0 are the moments of the correlation function

(here equal to 0 or 2); i, j refer to the separation bins; ξˆ is the measured correlation function; ξ is the model correlation function; and C is the covariance matrix (S´anchez et al., 2008; Cohn, 2006), which we calculate using the method from Grieb et al.

(2016). Grieb et al. (2016) generate a theoretical model for the linear covariance of anisotropic galaxy clustering observations, making use of synthetic catalogs. As input, the calculation of the covariances are based on an input linear galaxy power spectrum dependent on both the wavevector and the angle with the line of sight, P (k, µ). In order to calculate this, we ﬁrst calculate the linear matter power spectrum using

CLASS (Blas et al., 2011) and then compute the no-wiggle power spectrum from the formulae listed in Eisenstein and Hu (1998). This is done at the median redshift of each sample. We next account for redshift space distortion eﬀects to the power

51 spectrum using the procedure outlined in Ross et al. (2017):

h 2 2 i 2 2 −k σv P (k, µ) = b C (k, µ, Σs) (Pnonlin − Pnw)e + Pnw (2.13)

where the no-wiggle power spectrum is also generated with the nonlinear power spec-

trum from the HaloFit model (taking no-wiggle from Eisenstein and Hu (1998) as

input)(Smith et al., 2003). We have used

2 2 2 2 2 σv = (1 − µ )Σ⊥/2 + µ Σk/2 (2.14)

and 1 + µ2β C(k, µ, Σs) = 2 2 2 (2.15) 1 + k µ Σs/2 We deﬁne the spreading due to photon noise:

1 + z Σ = (300 km s−1) × , (2.16) s,phot H(z)

2 2 2 where we set Σs = Σs,phot + (2.26 Mpc) . For the True catalog, Σs,phot is set to 0. This eﬀectively incorporates the spreading we have added in the shuﬄed and observed

catalogs due to uncertainty in photon noise, and is displayed in Table 2.3.

In these equations, our values for β,Σs,Σk and Σ⊥ depend on median redshift of the ZH bin we are ﬁtting to, and are listed in Table 2.3. In order for the matter power spectrum to be used in our covariance matrix calculation, it must be converted to a galaxy power spectrum using a bias appropriate for the tracers which are outlining the dark matter.

To estimate the galaxy bias for each of our samples, we used two separate methods.

For the ﬁrst ﬁtting run only, we made use of results from the HiZels survey (Cochrane et al., 2017), who perform measurements of the Hα emitting galaxies at bins of z = 0.8,

1.7, and 2.23, binning further by the mean luminosity of galaxies in separate bins. We

52 Bin β Σs(Mpc) Σ⊥(Mpc) Σk(Mpc) Z2H2 0.5762 4.97 (5.46) 7.422 13.709 Z2H3 0.5800 4.99 (5.48) 7.520 13.847 Z3H1 0.4382 4.64 (5.16) 5.679 10.956 Z3H2 0.4424 4.64 (5.16) 5.681 10.959 Z3H3 0.4362 4.64 (5.16) 5.679 10.957

Table 2.3: Parameters used in generating P (k, µ) for both the covariance matrix generation and the BAO models used for fitting. These are calculated using formulae from Seo and Eisenstein (2007). For Σs, the first value indicates the value used for the True catalogs, while the second value was used for the Observed and Shuffled catalogs; it was increased due to the extra variance added to simulate photon noise.

use their estimates of the bias in our covariance matrix (via the power spectrum) to perform the first set of fits. We then record the best-fit values of the biases from these

ﬁts; these values were used for the covariance matrix generation for our second ﬁtting run. This process gave us galaxy biases of approximately [1.47, 1.45, 2.12, 2.10, 2.13]

for ZH bins of Z2H2, Z2H3, Z3H1, Z3H2, and Z3H3, respectively. These power spectra

are then used to generate covariance matrices in multipole space, for multipoles of

00, 02, 20 (transpose) and 22:

il1+l2 Z ∞ Cξ (s , s ) = k2σ2 (k)¯j (ks )¯j (ks ) dk, (2.17) l1,l2 i j 2 l1l2 l1 i l2 j 2π 0 where the multipole-weighted variance integral is

Z 1 2 2 (2l1 + 1)(2l2 + 1) 1 σl1l2 (k) = P (k, µ) + Ll1 (µ)Ll2 (µ) dµ, (2.18) Vs −1 n¯ and the bin-averaged spherical Bessel function is

Z si+∆s/2 4π 2 ¯jl(ksi) = s jl(ks) ds. (2.19) Vsi si−∆s/2

3 3 Here Vsi = 4π[(si + ∆s/2) − (si − ∆s/2) ]/3, Vs is the volume of the entire sample, jl

is the spherical Bessel function of the ﬁrst kind, k is the wavenumber, s is the distance

53 in redshift space, andn ¯ is the number density of galaxies for the sample in question.

In this case, since there is shot noise from both the data and the random catalogs, we

make the replacement 1/n¯ → 1/n¯ + 1/n¯R in Eq. (2.18); this increases the shot noise

4 by a factor of 3 for nR/nD = 3. Our simulations are not volume-limited, and have a galaxy number density,n ¯,

which is implicitly dependent on redshift, while our theoretical method to calculate

the covariance matrices assumes a constant galaxy number density. To account for

this, we further divided each of our ZH bins into three sub-bins by redshift. The

covariance matrix of each sub-bin was calculated using the volume and number density

of galaxies within that speciﬁc sub-bin. The covariance matrix of the entire ZH bin

was calculated by: X Cξ (s , s ) = w2Cξ (s , s ), (2.20) l1,l2 i j k l1,l2,k i j k where k indicates the speciﬁc sub-bin, and:

2 vkn¯k wk = P 2 , (2.21) k0 vk0 n¯k0

For our Z2 bins, the redshift cutoﬀs were at 0.9 and 1.1, and for Z3, they were 1.55

and 1.75. The volume of our survey area over the redshift-range of the subset k is

designated as vk.

2.5.2 RSD parameter ﬁtting

The ﬁt on small scales follows the procedure in Martens et al. (2018), focusing

on the redshift space distortion parameters. We ﬁt on scales of 42 to 200 Mpc, with

a factor of 4 lower spatial resolution than for the BAO ﬁts; it was reduced in order

to prevent small-scale ﬂuctuations in the correlation function dominating the best-ﬁt

values.

54 To calculate the theoretical ﬁt, we use Convolution Lagrangian Perturbation The-

ory (CLPT), modiﬁed on small scales by Gaussian Streaming Redshift Space Dis-

tortions (GSRSD). In order to ﬁt a theoretically produced correlation function to

our simulations, we use an extension of Convolution Lagrangian Perturbation Theory

(CLPT) (Carlson et al., 2013). CLPT extends perturbation theory beyond linear

order to match up to quasi-linear scales of the correlation function; the Gaussian

streaming model tailors the ﬁt to behave better on small scales (Wang et al., 2014).

2 2 GSRSD takes as input σFOG, fv, and bg. We treat σFOG as a ﬁxed parameter, however

2 we run several fits over different set values of σFOG, fitting for fv and bg with each different set value. This is done to provide a more stringent test to the similarity

2 of the observed and shuﬄed catalogs, since ﬁxing σFOG shrinks the uncertainties in

the ﬁt for fv; it should be noted that the real data collected by WFIRST will be

2 simultaneously fit for fv, bg, and σFOG, although the scales over which these will be fit could differ from those presented here.

2 The observed and shuﬄed catalogs have a diﬀerent set value of σFOG than the true catalogs, due to the spreading introduced in Eq. (2.2). This additional spreading was

calculated separately for each ZH bin, and was found to be 24.222, 24.324, 21.097,

21.098, and 21.080 h−2 Mpc2 for ZH bins of Z2H2, Z2H3, Z3H1, Z3H2, and Z3H3,

2 −1 2 respectively. We run our ﬁts for σFOG values of 5, 20, and 35 (h Mpc) for the true catalogs, which is added to the additional spreading found for the observed and

shuﬄed catalogs.

In order to produce these outputs, GSRSD also takes as input the galaxy bias, and

fv, which are our primary ﬁtting parameters. The code outputs the redshift-space

55 correlation function in terms of moments, ξ0,2,4(r), which is directly comparable to

our simulated correlation functions.

2.5.3 BAO Parameters

WFIRST will perform accurate BAO measurements up to z ∼ 1.9 using Hα, and

to higher redshifts using [O iii] and [O ii] emitters, pinning down the expansion rate of the Universe, H(z) and the angular diameter distance, DA(z). However, the line blending effect studied in this paper will potentially bias the results. To quantify this bias and track down how much arises from the redshift error PDF versus its correlation with large-scale structure, we will compare the BAO parameters measured from our true, observed, and shuffled catalogs. In this subsection, we first introduce the BAO model, and then discuss about our fitting process. They do not include reconstruction,

which we leave to future analysis.

The BAO ﬁts are performed over scales from 60 to 200 Mpc, and are intended

to calculate the expansion rate of the Universe at a speciﬁc redshift, H(z), and the

angular diameter distance to that redshift, DA(z). We follow the standard BAO

convention in deﬁning

fid fid (H(z)rd) DA(z)rd αk = and α⊥ = fid , (2.22) H(z)rd DA (z)rd

where here, a superscript of “ﬁd” indicates a ﬁducial value, and rd is the sound horizon

at the kinetic decoupling epoch. Given P (k, µ), we calculate the multipole moments

2l + 1 Z 1 Pl(k) = P (k, µ)Ll(µ)dµ , (2.23) 2 −1

and then transform them into real space as

il Z dk ξ (s) = k3P (k)j (ks) , (2.24) l 2π2 k l l 56 so that the correlation function is expressed as

X ξ(s, µ) = ξl(s)Ll(µ), (2.25) l where we only sum to l = 4.

We ﬁt to the same BAO model in Ross et al. (2017),

a a ξ (s) =B ξ (s, α , α ) + 01 + 02 + a and (2.26) 0,mod 0 µ0 ⊥ k s2 s 03 5 ξ (s) = [B ξ (s, α , α ) − B ξ (s, α , α )] 2,mod 2 2 µ2 ⊥ k 0 µ0 ⊥ k a a + 21 + 22 + a , (2.27) s2 s 23

where ξµ0, ξµ0 are µ-averaged ξ(s, µ), deﬁned by Z 1 0 0 ξµ0(s, α⊥, αk) = dµ ξ(s , µ ) (2.28) 0 and Z 1 02 0 0 ξµ2(s, α⊥, αk) = dµ 3µ ξ(s , µ ), (2.29) 0 0 q 2 2 2 2 0 q 2 2 2 2 with µ = µαk/ µ αk + (1 − µ )α⊥, and s = s µ αk + (1 − µ )α⊥. Parameters Bi

and aij set the size of the BAO and the broadband feature.

We ﬁt the model by minimizing the χ2 in Eq. (2.12). Since the model depends

2 linearly on the parameters Bi and aij, the χ is a quadratic function of them. For each

2 given pair of (α⊥, αk), one can calculate the other 8 parameters where the χ takes a

(8) minimum. Let p be the vector of the 8 parameters, and pi be its i-th component; then the χ2 can be written as

8 8 2 ⊥ k (8) T −1 X X χ (α , α ; p ) = d C d − 2 piJi + pipjKij. (2.30) i=1 i,j=1 T −1 T −1 Here d is the data vector, and we deﬁne Ji = mi C d and Kij = mi C mj, where mi is the model vector if the parameter pi is set to be 1 and all other seven com-

(8) 2 (8) −1 ponents of p are set to be 0. The minimum of χ is obtained if pi = [K ]ijJj.

57 We ﬁrst use Eq. (2.24) to generate the theoretical ξ0,2,4(s) with 5000 s-values log-

arithmically spaced in [1,240] Mpc for each ZH bin. Several techniques have been

applied to improve the accuracy of the integration. Firstly, we use 50,000 k-values

logarithmically sampled from 10−4 to 100 Mpc−1. Secondly, a window function that

has continuous ﬁrst and second derivatives9 is applied to the high-k end of the k-

3 dependent function k Pl(k), to remove the high-frequency ringing appearing in ξl(s).

Finally, we multiply jl(ks) by a factor of sin(ks∆ ln k/2)/(ks∆ ln k/2) to eﬀectively

average out the contribution from the rapidly oscillating spherical Bessel functions at

0 large k. During each iteration of the minimization, ξl(s ) is calculated from the pre-

calculated theoretical ξl(s) using the cubic interpolation method. The minimization

uses the scipy.optimize.minimize routine (Jones et al., 2001) with the Nelder-Mead

method, and the initial guess of (α⊥, αk) is always (1, 1).

2.5.4 Fitting Results

Here, we show the resulting parameters that were ﬁt to each correlation function,

and discuss trends in the results. For each ZH bin, we have 6 separate regions, which

serve as 6 realizations of simulation data. We average the correlation functions from

all sectors: ξ + ξ + ξ + ξ + ξ + ξ ξ = 1 2 3 4 5 6 . (2.31) avg 6

We then ﬁt parameters to ξavg, which are denoted asp ¯ for a given parameter p. These

ﬁts provide the listed parameters in Tables 2.4 and 2.5, as well as the listed χ2 per degree of freedom.

9 −1 See Eq. (C.1) in McEwen et al. (2016). Here we have kmax = 100 Mpc and the window applies from 80 to 100 Mpc−1.

58 We ﬁnd the error bars for each parameter using the jack-knife method, where we

construct 6 separate combinations of the correlation functions, with each combination

containing all regions except for one:

6 1 X ξ¯ = ξ . (2.32) i 5 i j=1,j6=i ¯ We then separately ﬁt the correlation functions ξi for the BAO and RSD parameters.

We show error bars which are derived from the variance of the set of parameters of our combinations: v u 6 u5 X σ = t (¯p − p¯)2. (2.33) p¯ 6 i i=1 The jack-knife method, while providing an estimate of the errors in our parameters due to deviations in the correlation functions, is limited by our small sample set of only 6 realizations. Table 2.4 displays the fv and bg best-fit parameters for each ZH bin, and each of the true, observed, and shuffled catalogs. There are three columns in the table, each indicating the best-fits in cases of varying values for the Finger of

2 2 God spreading, σFOG. We list the true catalog value of σFOG for the True catalog in

2 Table 2.4. These values were chosen to represent a wide range of values for σFOG for our samples, and to show how their choice affects the differences between the observed and shuffled catalogs. The error bars here are calculated using Eq. (2.33). Table 2.4 also shows, for each ZH bin, the percent error between the fitted parameters between the observed and shuffled catalogs. This is calculated by:

p − p ∆p ≡ obs shf × 100%, (2.34) 0.5(pobs + pshf ) where p is either bg, fv, α⊥ or αk.

59 Table 2.4: Here we display the best ﬁt parameters for the growth of structure parameter fv and the galaxy bias bg, for each ZH bin and redshift catalog. We calculate % error (Eq. 2.34) to indicate the agreement between the shuﬄed and observed parameters. The uncertainties provided by the jack-knife sampling have a fractional error of p2/5, due to the sample size of 6 regions.

2 −1 2 2 −1 2 2 −1 2 σFOG = 5 (h Mpc) σFOG = 20 (h Mpc) σFOG = 35 (h Mpc)

2 2 2 bg fv χ /d.o.f bg fv χ /d.o.f bg fv χ /d.o.f

Z2H2 True 1.4817 ± 0.0139 0.7080 ± 0.0148 1.219 1.4658 ± 0.0139 0.7593 ± 0.0147 1.285 1.4487 ± 0.0137 0.8129 ± 0.0149 1.819 Observed 1.4938 ± 0.0120 0.6878 ± 0.0126 1.674 1.4729 ± 0.0119 0.7489 ± 0.0127 1.359 1.4497 ± 0.0119 0.8135 ± 0.0128 1.813 Shuffled 1.4934 ± 0.0120 0.6898 ± 0.0113 1.618 1.4724 ± 0.0118 0.7504 ± 0.0115 1.336 1.4493 ± 0.0119 0.8154 ± 0.0115 1.828 %Error 0.03 ± 0.10 % -0.28 ± 0.32 %- 0.04 ± 0.12 % -0.20 ± 0.30 %- 0.03 ± 0.11 % -0.22 ± 0.25 %- Z2H3 60 True 1.4585 ± 0.0139 0.6891 ± 0.0200 1.512 1.4429 ± 0.0138 0.7380 ± 0.0199 1.574 1.4262 ± 0.0139 0.7892 ± 0.0201 1.971 Observed 1.4643 ± 0.0107 0.6734 ± 0.0161 1.464 1.4444 ± 0.0107 0.7298 ± 0.0161 1.376 1.4227 ± 0.0107 0.7898 ± 0.0164 1.803 Shuffled 1.4676 ± 0.0127 0.6707 ± 0.0178 1.617 1.4479 ± 0.0125 0.7271 ± 0.0179 1.478 1.4262 ± 0.0125 0.7871 ± 0.0180 1.847 %Error -0.22 ± 0.37 % 0.40 ± 0.51 %- -0.25 ± 0.39 % 0.36 ± 0.49 %- -0.24 ± 0.38 % 0.34 ± 0.41 %- Z3H1 True 2.1512 ± 0.0072 0.7631 ± 0.0065 2.373 2.1294 ± 0.0074 0.8287 ± 0.0066 2.112 2.1056 ± 0.0073 0.8974 ± 0.0066 2.949 Observed 2.1531 ± 0.0054 0.7561 ± 0.0076 2.525 2.1256 ± 0.0052 0.8305 ± 0.0078 1.956 2.0956 ± 0.0054 0.9090 ± 0.0079 2.992 Shuffled 2.1539 ± 0.0060 0.7574 ± 0.0064 2.362 2.1268 ± 0.0059 0.8317 ± 0.0064 1.952 2.0967 ± 0.0058 0.9104 ± 0.0065 3.156 %Error -0.04 ± 0.09 % -0.18 ± 0.41 %- -0.06 ± 0.09 % -0.14 ± 0.41 %- -0.05 ± 0.07 % -0.15 ± 0.36 %- Z3H2 True 2.1307 ± 0.0066 0.7476 ± 0.0070 2.149 2.1095 ± 0.0066 0.8093 ± 0.0073 1.939 2.0867 ± 0.0063 0.8740 ± 0.0073 2.593 Observed 2.1335 ± 0.0045 0.7483 ± 0.0087 2.095 2.1079 ± 0.0045 0.8170 ± 0.0088 1.939 2.0798 ± 0.0043 0.8894 ± 0.0092 2.980 Shuffled 2.1350 ± 0.0039 0.7467 ± 0.0097 2.211 2.1095 ± 0.0039 0.8156 ± 0.0100 2.069 2.0812 ± 0.0038 0.8881 ± 0.0100 3.121 %Error -0.07 ± 0.10 % 0.21 ± 0.34 %- -0.08 ± 0.11 % 0.17 ± 0.33 %- -0.07 ± 0.10 % 0.15 ± 0.29 %- Z3H3 True 2.1535 ± 0.0121 0.7217 ± 0.0112 0.946 2.1332 ± 0.0124 0.7737 ± 0.0112 1.260 2.1112 ± 0.0123 0.8276 ± 0.0114 1.945 Observed 2.1650 ± 0.0105 0.7152 ± 0.0116 1.425 2.1420 ± 0.0104 0.7710 ± 0.0117 1.762 2.1171 ± 0.0103 0.8290 ± 0.0120 2.543 Shuffled 2.1616 ± 0.0126 0.7079 ± 0.0109 1.365 2.1390 ± 0.0126 0.7632 ± 0.0111 1.522 2.1143 ± 0.0127 0.8214 ± 0.0114 2.118 %Error 0.15 ± 0.30 % 1.03 ± 1.21 %- 0.14 ± 0.30 % 1.01 ± 1.11 %- 0.13 ± 0.31 % 0.92 ± 1.09 %- In Figure 2.9, we plot the best-fit parameters for a visual comparison of the different RSD results, we show the full RSD best-fit parameters in Table 2.4, and the best-fit BAO parameters are displayed in Table 2.5.

Systematic errors in the mean parameter ﬁts could arise from a few assumptions made in the modeling. Mean redshifts were calculated for each ZH bin, and were used to determine the parameters Σ⊥, Σk, Σs, and generate the matter power spectra. Since our samples are not volume-limited, the redshift dependence of the galaxy number densities in each bin may introduce errors in the modeling.

Some of the large χ2/d.o.f. values result from the combination of the bin-to-bin scatter in ξl(s) and the strong correlations between neighboring radial bins (i.e. the oﬀ-diagonal terms of the covariance matrix). We suspect that the analytic covariance matrix may not be adequate for describing these rapidly oscillating modes in ξl(s).

Although we attempt to generalize it to a non-volume-limited sample, we still must estimate a galaxy bias for the input power spectrum, and the rapidly oscillating modes correspond to high k where non-linear biasing may be important. The covariance matrix uses a Gaussian approximation, neglecting the galaxy trispectrum and super- sample variance eﬀects, which may also break down for these modes. In any case, these highly oscillating modes are orthogonal to the broadband modes that dominate fv, and changes in our treatment of the covariance matrix in the preparation of this paper had a small impact on changes in the parameter shifts.

2.6 Measuring and Correcting Error Terms

As discussed in Sec. 2.2, WFIRST has a requirement to meet, and preferably exceed, the observational systematic error requirements in the SRD for measured

61 Table 2.5: The best-fit parameters for the BAO shift parameters αk and α⊥, and their χ2 values. We calculate % error (Eq. 2.34) to indicate the agreement between the shuffled and observed parameters. The uncertainties provided by the jack-knife sampling have a fractional error of p2/5, due to the sample size of 6 regions. BAO Parameters 2 αk α⊥ χ /d.o.f Z2H2 True 0.9858 ± 0.0111 0.9800 ± 0.0066 0.484 Observed 0.9920 ± 0.0092 0.9780 ± 0.0062 0.678 Shuffled 0.9895 ± 0.0098 0.9792 ± 0.0054 0.807 %Error 0.26 ± 0.17 % -0.12 ± 0.11 %- Z2H3 True 1.0045 ± 0.0166 0.9739 ± 0.0088 0.486 Observed 0.9977 ± 0.0154 0.9708 ± 0.0076 0.826 Shuffled 0.9986 ± 0.0153 0.9726 ± 0.0078 0.871 %Error -0.08 ± 0.24 % -0.19 ± 0.11 %- Z3H1 True 0.9614 ± 0.0048 0.9763 ± 0.0038 1.537 Observed 0.9598 ± 0.0044 0.9799 ± 0.0039 1.362 Shuffled 0.9586 ± 0.0051 0.9793 ± 0.0041 1.241 %Error 0.13 ± 0.19 % 0.07 ± 0.05 %- Z3H2 True 0.9592 ± 0.0025 0.9753 ± 0.0041 1.668 Observed 0.9668 ± 0.0042 0.9690 ± 0.0035 1.427 Shuffled 0.9638 ± 0.0025 0.9699 ± 0.0037 1.468 %Error 0.31 ± 0.23 % -0.10 ± 0.10 %- Z3H3 True 0.9625 ± 0.0110 0.9757 ± 0.0033 1.286 Observed 0.9488 ± 0.0088 0.9702 ± 0.0064 1.109 Shuffled 0.9504 ± 0.0080 0.9721 ± 0.0060 1.189 %Error -0.17 ± 0.55 % -0.19 ± 0.15 %-

62 Figure 2.9: For each ZH bin and Observed, True, and Shuffled catalog, we plot the fitted values for fv and bg. Circles indicate the True catalog, V’s indicate the Observed 2 catalog, and triangles indicate the Shuffled catalog. For this fitting calculation, σFOG was set to 20 [h−1Mpc]2 for the True catalog.

63 parameters; speciﬁcally, the parameters measured in this work are the BAO shift

parameters αk and α⊥, and the growth of structure parameter fv. The error limits are 0.41% for α⊥, 0.74% for αk, and 0.74% for fv (there is no top-level requirement on the systematic errors to the galaxy bias bg, as we will marginalize over it in estimating cosmological parameters). In this section, we discuss the breakdown of line blending errors within the structure of our catalogs, compare our results to the error budget of the SRD, and briefly introduce potential mitigation techniques that could limit the negative effects of line blending on the observed redshift. Note that in our discussions of “percent of error budget” used, we will use root-sum-square (RSS) error budgeting, so that an effect whose amplitude is 50% of the maximum allowed error is considered to use 25% of the total budget.

In this paper, we have presented the ﬁt parameters for three separate catalogs of redshift: true, observed, and shuﬄed. Their derivation is described in detail in Sec.

2.4.3. The purpose of these catalogs is to separate the redshift errors introduced by line blending into two distinct sources. The diﬀerence between the parameters of the observed and true catalogs (which we wish to eventually mitigate) can be written as:

pobs − ptrue = (pobs − pshuﬄed) + (pshuﬄed − ptrue). (2.35)

The ﬁrst part of Eq. (2.35) is comprised of the diﬀerence between the parameters of

the observed and shuﬄed catalogs. By construction, these two catalogs have identical

distributions of redshift errors; however, in the shuffled catalog, the line-blending redshift difference has been uncoupled from the specific galaxy that generated it, destroying the correlation between [N ii]/Hα and galaxy environment. Therefore, this term describes the effect of this correlation. It is this term that – although small

– is not amenable to mitigation by measuring the one-point PDF of galaxy properties.

64 In contrast, the second part of Eq. (2.35) depends only on the one-point properties of galaxies, such as their metallicity distribution. Because there is no large-scale structure information encoded in this term, it is more straightforward to mitigate via detailed observations of a small number of galaxies. We will analyze the error encompassed by these terms separately, in order to discuss the necessary mitigation techniques.

2.6.1 Eﬀect of correlations between [N ii]/Hα and large-scale environment

We first analyze the difference between the shuffled and observed catalogs. Al- though we have fit a variety of redshift and luminosity bins, we will confine our numerical analysis here to the Z2H2 and Z3H2 bins, since the limiting flux of 1.3×10−16 erg cm−2 s−1 is closest to the planned WFIRST flux limit.10 For the BAO parameters

α⊥ and αk, the best-fit values are listed in Table 2.5, along with the percentage difference between the observed and shuffled best-fit values. These differences are also shown graphically in Fig. 2.10. The magnitude of the differences in αk fall generally between 0.1 − 0.3%, while for α⊥ they range from 0.1 − 0.2%. The errors on these differences are provided by the jack-knife method for all 6 regions we fit for. It is important to note that the jack-knife errors are based on a small sample size of 6 regions (for which the expected fluctuations in error bars are ±1/p2(6 − 1) ≈ 32%); this could explain why in some cases we have a larger error bar for a ZH bin with a smaller sample size.

10This ﬂux limit varies somewhat as a function of wavelength and ecliptic latitude, and is subject to change during future optimization.

65 Speciﬁcally, we ﬁnd that ∆αk = 0.31 ± 0.23% (0.26 ± 0.17%) for Z3H2 (Z2H2), and ∆α⊥ = −0.10±0.10% (−0.12±0.11%) for Z3H2 (Z2H2). Using the largest value of each, this corresponds to ∆αk using 18% of the systematic error budget, while

∆α⊥ uses 9%. This is a small percentage of the systematic error budget, especially considering that it is for the unmitigated result. We can also compare these errors to a previous estimate of the line blending eﬀect: Faisst et al. (2017) looked at the overall redshift errors due to line blending, and propagated it the calculation of the

BAO parameters; they predicted an upper estimate of these errors to be in the range of 0.5 − 1.6% for the αk and α⊥ parameters. Our clustering analysis has shown that this estimate is indeed an upper limit, as the values we have found due to clustering are signiﬁcantly lower.

The error from this portion of Eq. (2.35) can be potentially reduced if the [N ii]/Hα ratio can be predicted from other available data. For example, the broadband LSST+WFIRST

photometry (which extends into the rest-frame optical) allows estimates of the stellar

mass M?, and observations of a small fraction of the WFIRST sources with high- resolution ground-based NIR spectrographs would enable the correlation between

[N ii]/Hα ratio to be determined as a function of redshift z and inferred stellar mass

M?. After applying this correction, the remaining systematic error from line blending would be associated only with the residuals from the [N ii]/Hα vs. (z, M?) ﬁt and their correlation with large-scale environment. Future work will be required to determine how much the systematic errors can be mitigated by this method.

We display the best-fit parameters for the RSD fits in Table 2.4, with the percentage difference between the observed and shuffled best-fit values also displayed in

66 2 Fig. 2.11. We have fit fv given several different values of σFOG, and found consistency in the percentage errors in each case; here we use the values from the fit for

2 −2 2 the σFOG = 20h Mpc bin. The diﬀerences in fv range between 0.14 − 0.36%, with the exception of the Z3H3 bin, which has an error of 1.01 ± 1.11% (the error bar is

large due to the high ﬂux limit and corresponding small sample size). For purposes of

comparing to the SRD, we find that the error for fv is 0.17 ± 0.33% (−0.20 ± 0.30%) for Z3H2 (Z2H2). For Z2H2, the larger difference, this corresponds to fv using 7% of the systematic error budget, which again is a small enough offset that mitigations may not be required. If necessary, the aforementioned mitigations based on measurement of the [N ii]/Hα vs. (z, M?) relation could reduce it even further.

2.6.2 Eﬀects of the one-point redshift error PDF

Next, we evaluate the differences between the true and shuffled catalogs, which constitutes the second term in Eq. (2.35). We find that the difference for αk is approximately −0.48% (−0.37%) for the Z3H2 (Z2H2) bin, which constitutes 42% of the systematic error (for the largest case). This error is similar for α⊥ at 0.56%

(0.08%) for Z3H2 (Z2H2), but the lower error budget means it constitutes 187% of the total allotment (again for the largest case). In order to move these errors below 25% of the SRD limits, this would require a reduction in the errors by factors of approximately 1.3 and 2.8 for αk and α⊥, respectively. For fv, the diﬀerence is

−0.78% (1.18%) for Z3H2 (Z2H2), resulting in a maximum value of 254% of the total allotment. This requires a reduction in the errors by a factor of about 3.2 to get within this 25% limit.

67 This could be achieved with a spectroscopic re-observation of some subset of galax-

ies already observed by WFIRST with a high-resolution ground-based spectrograph

that completely resolves Hα from the [N ii] doublet. Only a tiny fraction of WFIRST

source could be followed up this way, but it would provide a clean measurement of

the redshift error PDF. In this way, we gain knowledge of each re-observed object’s

speciﬁc ztrue, and can construct a probability distribution Pest(δz|zobs) based on these

re-observed objects. The more objects we are able to re-observe, the more our subsam-

ple approaches the full sample, and the more our probability distribution Pest(δz|zobs)

approaches the true distribution P (δz|zobs).

In the case of fv, we predict how many galaxies would need to be re-observed in

order to make this reduction. From Table 2.4 it can be seen that (for both Z2H2 and

Z3H2):

∂lnfv −1 −2 2 ≈ 0.005 (h Mpc) . (2.36) ∂σFOG

This quantiﬁes the dependence of fv on the Finger of God length; if we place limits

2 on fv, we need to know σFOG to some certainty as well, since they are correlated in

2 their ﬁt values. Therefore, to ﬁt within 25% of the error budget, we need σFOG to be known to:

∂lnf −1 √ σ = v × f 0.25 F OG,err ∂σ2 v,err FOG√ 0.0074 0.25 = = 0.75h−2 Mpc2. (2.37) 0.005

Given the variance of the redshift oﬀset δz, which corresponds to 24.2 h−2Mpc2 for

−2 2 2 Z2H2 and 21.1 h Mpc for Z3H2, then we need to know σFOG to about 0.75/24.2 = 3.1% for Z2H2 and 0.75/21.1 = 3.6% for Z3H2. If assuming Gaussian errors, this would require an observation approximately in the range of 2/0.0312 ≈ 2100 galaxies.

68 In practice, the number of re-observations would be larger for a realistic non-Gaussian error distribution, and a few redshift bins would be necessary to track the redshift dependence of the error PDF.

Additional work will be necessary to determine the optimal strategy for measuring the redshift error PDF. Following up thousands of emission-line galaxies with ground- based NIR spectroscopy at the ∼ 10−16 erg cm−2 s−1 depth is certainly possible, especially if the targets can be pre-selected from WFIRST to have lines that will not collide with atmospheric OH features. However, since most of the redshift error is statistical error due to photon noise, it may be more efficient to use repeat observations in the WFIRST deep fields to measure the purely statistical scatter, and then high- resolution ground-based NIR spectra to constrain the specific contribution from line blending. These possiblities should be explored in future work.

2.7 Discussion

In this paper, we have examined the eﬀects of the grism resolution proposed for

WFIRST on the observed redshifts of galaxies, and the resultant changes in the fitted cosmological parameters αk, α⊥, and fv. We have used the Buzzard-v1.1 mock galaxy catalog to probe these parameter differences by simulating the observation of line-blended galaxies and compared them to the true redshift distributions. We then created a“shuffled”catalog; this catalog uses the same distribution of δz values (see Eq.

2.5) as the observed catalog, but with these values randomly shuffled between different galaxies. This results in all correlations between galaxy location and metallicity being erased. By analyzing the differences between the parameters fit to these catalogs, we can gain a sense of the potential parameter errors due to the line blending effect,

69 1.0 Z2H2 Z2H3 Z3H1 0.5 Z3H2 Z3H3 α ¯ α

∆ 0.0 0 0 1

0.5

1.0 0.4 0.2 0.0 0.2 0.4 ∆α 100 α¯

Figure 2.10: For each ZH bin, we show the percent difference of α⊥ and αk relative to the systematic error budget (red box) of WFIRST for each parameter. The contours show the spread of fits for all 6 jack-knife combinations for that specific ZH bin, as referenced in Eq. 2.33, while the central values are calculated from the fits of the average of all regions.

70 Z2H2 2 Z2H3 Z3H1 Z3H2 1 Z3H3 g g b ¯ b ∆ 0 0 0 1

2 1 0 1 2 ∆fv 100 ¯ fv

Figure 2.11: For each ZH bin, we show the percent difference of fv and bg relative to the systematic error budget of WFIRST for fv, represented by the solid red lines. 2 −1 2 For this fitting calculation, σFOG was set to 20 [h Mpc] for the True catalog. The contours show the spread of fits for all 6 jack-knife combinations for that specific ZH bin, as referenced in Eq. 2.33, while the central values are calculated from the fits of the average of all regions.

71 and to what extent they may require mitigation in order to meet the systematic requirements for WFIRST.

We found that errors dependent on the large-scale structure, i.e. the diﬀerence between the shuﬄed and observed catalogs, were ∆αk = 0.31±0.23%, ∆α⊥ = −0.10±

0.10%, and ∆fv = 0.17 ± 0.33% (1.355 ≤ z < 1.994), ∆αk = 0.26 ± 0.17%, ∆α⊥ =

−0.12 ± 0.11%, and ∆fv = −0.20 ± 0.30% (0.705 ≤ z < 1.345), all quoted at the

1.3 × 10−16 erg cm−2 s−1 flux limit. This uses approximately 18%, 9%, and 7% of their respective error budgets, in an RSS sense. These errors are small – and in particular are smaller than the upper limits presented by other recent analyses (Faisst et al., 2017) – and can be made smaller still through the use of mitigation techniques described in Sec. 2.6. Errors which are dependent on the knowledge of the distribution of galaxy parameters, i.e. the difference between the shuffled and true catalogs, were larger; however these errors are more easily mitigated since the redshift error PDF can be measured by re-observing a small fraction of the sample with high-resolution spectrographs. We estimated that direct mitigation would require re-observation of thousands of galaxies, but recommend more work to refine this estimate and define the optimal strategy.

It is important to note that our results are dependent on the accuracy of the

[N ii]/Hα as a function of redshift and environment. In this work, we have treated the environmental dependence as coming only from the mass-metallicity relation. We expect this to be the dominant eﬀect, both because the additional dependence on star formation rate (at ﬁxed M? and z) is observed to be weak, and because there is a strong relation between stellar mass and clustering strength. Nevertheless, investigation of these other dependences in future work is warranted.

72 There are a variety of additional steps that could be taken to further reﬁne the re-

sults presented. The software and analysis tools used in this project could be improved

upon. Ideally we would have enough simulated realizations to build a mock-based co-

variance matrix (and thus avoid issues related to variable galaxy number density as a

function of redshift and non-Gaussianity of the galaxy density ﬂuctuations). Larger

volumes would also reduce the uncertainties themselves; at present, with a simulated

area of π sr (only 5 times the WFIRST reference survey footprint), it is hard to measure biases that are small compared to WFIRST statistical errors, even though the diﬀerences of true/observed/shuﬄed catalogs cancel some of the sampling variance.

It would also be interesting to incorporate reconstruction into the BAO fits, which we did not do in this paper. Finally, future work should explore the robustness of these results under different semi-analytic model assumptions, and different prescriptions for the galaxy SEDs and Hα emission line properties.

In summary, we have presented a simulation-based analysis of the eﬀects of [N ii] and Hα line blending on BAO and RSD parameter ﬁtting. We conclude that the errors due to large-scale environment are small compared to WFIRST requirements even without mitigation, and with mitigation should not be a concern for these applications of the WFIRST galaxy redshift survey. We have also concluded that the redshift error probability distribution function will need to be measured accurately; while a brute- force approach seems feasible, we recommend further study of the optimal approach.

73 Chapter 3: A Radial Measurement of the Galaxy Tidal Alignment Magnitude with BOSS Data

3.1 Introduction

Galaxy peculiar velocities have long been known as a source of noise in using the redshift of a galaxy to infer its distance from the observer11. When redshift surveys are used to map the large-scale structure of the Universe, these peculiar velocities leave artifacts known as redshift space distortions (RSDs). On small scales, RSDs lead to “fingers of God” – structures that are smeared in redshift space by their internal velocity dispersion rather than by their physical size, and hence appear to be pointed at the observer (Jackson, 1972). This smearing increases the difficulty of making precise measurements in redshift space. However, on large scales, RSDs are also on a short list of invaluable cosmological probes. Matter overdensities cause infall of galaxies on these scales, which results in a “squashing” effect when viewed in redshift space. Measurements on the magnitude of this large-scale infall can reveal information on the clustering of matter in the Universe, even out to linear scales.

Early galaxy surveys (Kirshner et al., 1981; Bean et al., 1983) lacked the number density of sources to well quantify the eﬀects of RSDs, as they were limited to

11This chapter has been submitted for publishing to the Monthly Notices of the Royal Astronomical Society. The authors are Daniel Martens, Christopher M. Hirata, Ashley J. Ross, and Xiao Fang.

74 a few thousand galaxies, although attempts were made to model them (Davis and

Peebles, 1983). By the 1990’s, however, improved spectroscopic techniques led to a

higher number density in galaxy surveys, which provided greater empirical evidence

for mapping nonlinear eﬀects (Kaiser, 1986). The uniformity of the Universe on scales

above approximately 100h−1 Mpc was seen in tandem with a complicated network

of non-linear behaviors on smaller scales. While on small scales, RSDs trace the

velocity dispersion of the galaxies, the RSD signal on scales large compared to the

Finger of God length does not become negligible; instead it traces linear-regime infall

into potential wells. The distortions on these larger scales can measure the matter

density of the Universe, Ωm, as described in detail by Kaiser (1987). Following this,

Hamilton (1992) derived formulas to measure Ωm given the characteristic anisotropic quadrupole of the correlation function.

These methods have been applied to increasingly larger surveys, primarily pro- ducing measurements for the RSD parameters themselves. Hamilton (1993) applied these to the Infrared Astronomical Satellite (IRAS) 2 Jy, observing 2,658 galaxies,

+0.5 and ﬁnding that Ωm = 0.5−0.25. Cole et al. (1995) measured redshift-space distortions for 1.2-Jy and QDOT surveys, and compared their results to N-body simulations to

ﬁnd points of breakdown with linear theory, further studied by Loveday et al. (1996)

on the Stromlo-APM redshift survey. The optically selected Durham/UKST Galaxy

Redshift Survey pursued similar measurements (Ratcliﬀe et al., 1998). Within recent

years, galaxy samples have become larger, and errors on measured parameters have

become correspondingly smaller. Peacock et al. (2001) used the 2dF Galaxy Redshift

Survey to measure redshift-space distortions from 141,000 galaxies, and detected a

large-scale quadrupole moment at greater than 5-sigma signiﬁcance. This result was

75 expanded by Verde et al. (2002) through incorporating the galaxy bispectrum to place a measurement on the matter density, ﬁnding Ωm = 0.27±0.06. Currently, RSD clustering measurements serve to test the standard models of the growth of structure. In this endeavor, larger samples of galaxies have been analyzed using higher redshift surveys (Ross et al., 2007; Guzzo et al., 2008), 6dFGS (Beutler et al., 2012), WiggleZ

(Blake et al., 2012), and SDSS and BOSS (Tegmark et al., 2004; Okumura and Jing,

2009; Dawson et al., 2013; Chuang et al., 2017; Satpathy et al., 2017; Ross et al., 2017;

Gil-Mar´ın et al., 2016; Beutler et al., 2017). RSD measurements have also been con- ducted for high-redshift Lyman break galaxies (da Angelaˆ et al., 2005; Mountrichas et al., 2009; Bielby et al., 2013). Finally, results from these analyses have been combined with CMB observations (Tegmark et al., 2006; Alam et al., 2017) to measure the full range of cosmological parameters.

As galaxy survey sizes grow and the precision on RSD measurements increases, it is important to keep track of and mitigate potential sources of error. One such source of error, discussed initially by Hirata (2009), is caused by the intrinsic alignment of galaxies due to large-scale tidal fields. Luminous red galaxies (LRGs), which have been targeted extensively by recent surveys, are preferentially aligned along the stretching axis of the local tidal field, which when combined with observational selection effects based on orientation, results in differences in the observed density modes, depending on if the mode is parallel, or perpendicular to, the line of sight. Specifically, a viewing dependent selection effect of observing more galaxies which are pointed toward us can result in an increase in perpendicular k-modes and decrease of parallel k-modes. Hirata (2009) estimates that this effect could result in the contamination of

RSD measurements by 5–10 percent, depending on speciﬁcs of the redshift and mass

76 distribution of the galaxy sample in question. Related eﬀects have been studied in the context of other galaxy samples (e.g. Lyman-α emitters, Zheng et al. 2011; Behrens and Braun 2014) and of higher-order statistics (Krause and Hirata, 2011).

Intrinsic alignments of luminous red galaxies are correlations between their respective orientations, shapes, and physical positions. The development of these alignments is a complicated, non-linear process associated with the history of galaxy formation, but the payoﬀ for cosmology if they are understood can be huge (Chisari and Dvorkin,

2013). Proposed alignment mechanisms include linear alignment by large-scale tidal

ﬁelds (Catelan et al., 2001; Hirata and Seljak, 2004), large clusters of galaxies (Thomp- son, 1976; Ciotti and Dutta, 1994), or tidal torque contributions to galaxy angular momentum (Peebles, 1969). LRGs speciﬁcally have had their intrinsic alignment amplitudes analyzed both in observations (Hirata et al., 2007; Bridle and King, 2007;

Okumura and Jing, 2009; Blazek et al., 2011; Chisari et al., 2014) and in simulations

(Velliscig et al., 2015; Chisari et al., 2015; Tenneti et al., 2016; Hilbert et al., 2017).

Analytical models for describing these processes have been developed and expanded upon (Blazek et al., 2015, 2017). Their eﬀects on weak lensing has been, so far, the greatest motivation for their study (Troxel and Ishak, 2015). Intrinsic alignments have been measured using galaxy-galaxy lensing (Mandelbaum et al., 2006; Joachimi et al., 2011; Huang et al., 2016; van Uitert and Joachimi, 2017; Huang et al., 2018;

Tonegawa et al., 2017) for the LOWZ sample (Singh et al., 2015) which we compare to in this work.

In this chapter, we use the fundamental plane of elliptical galaxies to determine, in a statistical way, which galaxies are intrinsically aligned along or across our line

77 of sight. We then divide the BOSS catalogs into subsamples of galaxies in each orientation classiﬁcation. Our aim is to obtain a measurement of the linear alignment amplitude B using radial alignment measurements, to complement existing shear measurements of the same quantity. We perform this analysis using data from the Baryon

Oscillation Spectroscopic Survey (BOSS; Dawson et al. 2013), which was observed as part of the Sloan Digital Sky Survey - III (Eisenstein et al., 2011). Speciﬁcally, we use both the “LOWZ” and “CMASS” catalogs (Reid et al., 2016) from Data Release

12 (DR12; Alam et al. 2015). We divide these catalogs into two separate populations depending on each galaxy’s estimated orientation relative to us, as assessed using the oﬀset from the Fundamental Plane. Next, we perform clustering measurements to determine RSD parameters for each group, testing the predictions of this eﬀect by Hirata (2009), and comparing our radial measurement of B to a perpendicular measurement using galaxy ellipticities.

This chapter is organized as follows. In Section 3.2, we discuss the theory behind the galaxy intrinsic alignments and how they mimic redshift space distortions.

Speciﬁcally, in Section 3.2.3, we discuss the Fundamental Plane, and how we will use it to determine each galaxy’s orientation relative to our line of sight. Section 3.3 describes the BOSS DR12 dataset – both the catalogs themselves, the selection choices we have used for our sample, and our methods of blinding and systematics tests. We discuss in Section 3.4 the methods by which we calculate the clustering statistics on our samples, as well as how we generate our covariance matrices. Finally, in Section

3.5 we discuss how we ﬁt RSD parameters to our samples, and analyze the results.

We conclude in Section 3.6.

78 Unless noted otherwise, we use the following cosmological parameters: ΩΛ = 0.693, h = 0.68, Ωb = 0.048, Ωm = 0.307, ns = 0.96, σ8 = 0.83, and Tcmb = 2.728 K (Planck

Collaboration et al., 2014).

3.2 Theory

3.2.1 Redshift space distortions

There are two main effects which change the na¨ıve linear power spectrum in redshift space. Both of these are due to changes in radial velocities of observed galaxies, although the effects differ in both their sources and their end results. First, peculiar velocities of galaxies cause a random shift in their observed radial velocity, with some galaxies moving toward us and others moving away. This causes a spread in the observed distribution of galaxies on small scales, creating an incoherent “Finger of God” effect, where groups of galaxies appear to be pointing toward the observer, when viewed in redshift space. Although this effect has long been understood, a second effect detailing a coherent redshift space distortion was described by Kaiser (1987).

On larger scales, galaxies are pulled into local gravitational overdensities, leading to a squashing eﬀect in redshift space, which is a signiﬁcant correction to the redshift space galaxy correlation function, even on linear scales.

These distortions aﬀect measurements along the line of sight, and so changes to the linear power spectrum are a function of µ, the cosine of the angle between a given direction and the line of sight. We can write the observed overdensities of galaxies in

Fourier space as:

2 δg(k) = (b + fµ )δm(k), (3.1)

79 where δg and δm refer to perturbations measured by galaxy tracers and matter, respectively, b is the linear galaxy bias, and f = d ln G/d ln a, where G is the growth function. It can be seen that we recover the matter power spectrum multiplied by the tracer bias if we are looking at modes transverse to the line of sight. This equation can also be expressed as the relationships between the matter and galaxy tracer power spectra:

2 2 Pg(k) = (b + fµ ) Pm(k) (3.2)

It is common to measure the amount of anisotropy by the parameter β = f/b.

3.2.2 Intrinsic alignment eﬀects on RSD measurements

Hirata (2009, hereafter H09) showed that systematic errors to redshift space distortion measurements can occur through the intrinsic alignment of galaxies by large scale tidal ﬁelds. It requires two conditions to be met for the galaxy sample in question. First, galaxies must be intrinsically aligned along the stretching tidal ﬁeld axis

(“linear alignment” in the nomenclature of intrinsic alignment theory). Second, the sample must have a selection effect that depends on the galaxy orientation relative to the line of sight. The systematic error in f depends on the product of these effects. In this work, we are deliberately enhancing the selection effects by splitting the BOSS sample into subsamples using a proxy for orientation. This enables us to use the difference in inferred f from the subsamples to probe intrinsic alignments.

It can be seen in a physical sense how alignments can lead to these systematic effects by understanding their resulting changes to observed Fourier density modes, as shown qualitatively in Figure 3.1. The anisotropic selection has a different effect on density modes depending on the angle of the mode with the line of sight directional

80 W33 Positive

Observer W33 Negative Observer

Figure 3.1: Here we show an exaggerated situation demonstrating a selection bias having an anisotropic effect on the resulting k-modes. If we assume that our observations are less likely to observe galaxies which are aligned perpendicularly to our line of sight, then some of the galaxies visualized above will not appear to the observer (the crossed-out galaxies). Since galaxies are more likely to be aligned with the higher density modes of the gravitational field, this changes the measured k-modes of the observed galaxies. Fourier modes perpendicular to the line of sight will have deeper troughs, and thus a higher inferred amplitude, while k-modes parallel to the line of sight will have shallower peaks, and thus a lower inferred amplitude. In this work, we intentionally group galaxies by their orientation, shown here by the color of green or blue. If we were to view all of these galaxies, but were less likely to see galaxies of the ‘’‘blue” type, then these distortions would dampen the linear Kaiser effect of clustering in our observation. (Note: this figure is similar to Figure 1 of Hirata (2009), but adapted to the situation with two sub-samples.)

81 vector. Selection of galaxies which are aligned perpendicular to the line of sight of the observer leads to a further decrease in the amplitude of troughs in k-modes perpendicular to the line of sight, which serves to amplify those k-modes. Conversely, this selection will lead to a decrease in the amplitude of peaks in k-modes parallel to the line of sight, resulting in a smoothing of these k-modes. This anisotropic effect mimics clustering signals by amplifying modes in different ways along the line of sight than transverse to it. This is the “physical picture” which serves to illustrate how redshift-space distortion measurements can be biased high or low, depending on the galaxy selection effect in question.

To describe this eﬀect quantitatively, we give a quick synopsis of the analysis of

H09 and highlight the results needed for understanding LRG clustering (we refer the reader to H09 for additional details on the formalism). We denote the contribution of anisotropic selection eﬀects to the galaxy overdensity by – that is, the observed number density of galaxies at point x as seen from direction nˆ is taken to be 1+(nˆ|x) times the number density that would be observed if the galaxy orientations were randomized. At linear order,

2 δg(k) = (b + fµ )δm(k) + (eˆ3|k), (3.3) where we take the convention that the observer’s line of sight is the 3-axis.

In the linear tidal alignment model (relevant for LRGs), the intrinsic alignments trace the large-scale tidal ﬁeld:

(nˆ|x) = Asij(x)ˆninˆj, (3.4)

Here A is a biasing parameter, which depends on both the intrinsic alignments and the observational selection eﬀects (we explore this in more detail below), and sij is

82 the dimensionless tidal ﬁeld:12

1 s (x) = ∇ ∇ ∇−2 − δ δ (x). (3.5) ij i j 3 ij m

This leads to a modiﬁed equation for the power spectrum:

A 2 P (k) = b − + (f + A)µ2 P (k). (3.6) g 3 m

This equation, although within the confines of the assumptions we have made, is powerful. We have introduced a single parameter, A, to describe the effects of intrinsic alignment upon redshift space distortions. Note that the functional behavior of the distortions remains the same, but the intrinsic alignments have shifted the usual parameters in the model. Therefore, any systematics tests aiming to filter out effects of a different functional form will not be able to sift out this specific type of contamination. In the presence of intrinsic alignment effects, the usual RSD measurement of f can be re-interpreted as a measurement of f + A, with the “standard” assumption being that A is small.

In order to extract specifically the tidal alignment effect, we need to factor A into pieces that depend on the observational selection effects (over which we have some control) and pieces that depend on intrinsic alignments (which we seek to measure).

This means we need a mathematical description of the intrinsic alignments. Following

H09, we treat elliptical galaxies as a triaxial system which is optically thin. For any position s from the center of the galaxy, the volume emissivity is j(s) = J(ρ), where

ρ is the ellipsoidal radius, and is deﬁned by:

ρ2 = s · exp(−W )s. (3.7)

12 Since sij is traceless-symmetric, it follows mathematically that (nˆ|x) averages to zero if we average over viewing directions nˆ.

83 Here W is a 3 × 3 traceless-symmetric matrix that contains information on the

anisotropy of the galaxy, and J(ρ) speciﬁes the radial proﬁle.13 For a spherical galaxy,

W = 0. The radial alignment is encoded in W33 (with positive W33 for a prolate

galaxy pointed at the observer), whereas the sky-projected alignments relevant for

weak lensing are encoded in W11 − W22 and 2W12.

Note that real elliptical galaxies cannot generally have homologous iso-emissivity

contours since this does not explain the well-known isophote twist (see §§4.2.3 and

4.3 of Binney and Merrifield 1998 for an overview). This could lead to a difference of the alignment parameters measured by different techniques (e.g. Fundamental Plane offset versus ellipticity). However we expect the effect of intrinsic alignments to still be present, and qualitatively similar to the homologous case.

In the linear tidal alignment model, we take the ﬁrst order approximation in the

Taylor expansion of hWiji in the tidal ﬁeld:

hWiji = 2Bsij. (3.8)

This leads to

A = 2(ηχ)eﬀ B, (3.9) where

∂(eˆ3|x) ∂ ln Ngal (ηχ)eﬀ = = (3.10) ∂W33(x) ∂W33

is the selection-dependent conversion factor from W33 (radial galaxy orientation) to observed number of galaxies.14 Here again it is seen that A is only non-zero if we have

13H09 worked only to linear order in W , and Sec. 5.1 of H09 defined ρ with (I + W )−1 instead of exp(−W ). For an observational project working with real galaxies, we need to handle higher orders in W ; the definition in Eq. (3.7) is more convenient as the total luminosity of the galaxy extrapolated to ρ = ∞ is independent of W . 14In H09, a simple flux threshold was assumed, with various definitions for the flux of an extended object considered. In this case, one could write A = 2ηχB, where η is the slope of the luminosity

84 both non-random orientations B 6= 0 and an orientation-dependent selection eﬀect

(ηχ)eﬀ 6= 0.

The parameter B can be compared to other parameters for the linear intrinsic

alignment model used in the literature, such as bκ (Bernstein, 2009), where

B b = . (3.11) κ 1.74

We can also compare to the lensing measurements AI of Singh et al. (2015) (derived in Appendix A.2): Ω B = −0.0233 m A (3.12) G(z) I

where Ωm is the matter density and G(z) is the redshift-dependent growth function.

3.2.3 Sub-samples using the Fundamental Plane oﬀset

The Fundamental Plane (FP) relation for elliptical galaxies was observed by Djor-

govski and Davis (1987), who noticed an empirical relationship between a galaxy’s

luminosity, radius, and projected velocity dispersion. Increasingly detailed measure-

ments of these correlations have been made (Jø rgensen et al., 1996; Bernardi et al.,

2003a; Hyde and Bernardi, 2009) and there are good theoretical arguments for its ex-

istence (Bernardi et al., 2003b). We use the redshift and luminosity of our observed

function and χ depends on how ﬂuxes area measured for non-spherical objects. In this work, we implement a more complicated selection algorithm, but we can still think of the conversion factor as an “eﬀective” ηχ.

85 galaxies in tandem with the FP relation to estimate each galaxy’s radius15. This radius can then be compared to the radius ﬁt by the SDSS imaging pipeline to estimate the galaxy’s orientation, through use of the estimator W33 (referenced in Section 3.2).

The key idea is that from our vantage point we do not observe the full 3-dimensional galaxy, but a 2-dimensional projection. In the case of an optically thin galaxy (as for most LRGs), this is simply

Z I(s⊥) = j(s⊥ + s3eˆ3) ds3. (3.13)

The projection of a homologous triaxial galaxy gives an image with an eﬀective ra- dius16: 1 1 r = exp − W − (W 2 + W 2 ) + ... r , (3.14) e 4 33 4 13 23 es where res is the eﬀective radius for the spherical galaxy (W = 0). With this knowledge, we can use Eq. (3.14) to construct an estimator for W33, accurate to linear order:

obs re W33 = −4 ln . (3.15) re0 where we have distinguished the true value of W33 from the linear-order approxi-

obs mation, which we deﬁne as W33 . For the rest of the chapter, we will refer to the

15We do not use the velocity dispersion information, for several reasons. The velocity dispersion is dependent upon what axis our line of sight takes through the given galaxy, which complicates our assessment of orientation-dependent offsets from the Fundamental Plane. Moreover, the BOSS fibers do not sample the whole galaxy, and we have not explored the subtle correlations that might be imprinted when this effect couples to orientation effects and observational conditions. We avoid these issues entirely, at the cost of some extra statistical error, by fitting for only the redshift and luminosity. 16For an elliptical image, we define the effective radius as the geometric mean of the semi-major 2 and semi-minor axes of the half-light ellipse. This ellipse has a second moment matrix res[exp W ]proj, where “proj” denotes the 2×2 sub-block of a 3×3 matrix; see the appendix of H09 for a discussion of projections. Equation (3.14) is the Taylor expansion of the square-root-determinant of this second moment matrix.

86 linear-order quantity as W33 in order to simplify notation; however, it must be understood that it contains both measurement noise and noise from intrinsic scatter in the FP relationship. Here, re is the eﬀective radius as measured by BOSS, and re0 is the value of the radius found using the FP. The combination of these measurements gives us an estimate for each galaxy’s W33 value, and hence each galaxy’s orientation.

Speciﬁcally, we ﬁnd an estimate of the isotropic average size from each galaxy by

ﬁtting a simple linear model over all galaxies in the survey for the relation between the logarithmic size and the intrinsic magnitude, as well as the galaxy’s redshift:

log10(re0) = C1Mrs + C2z + C3, (3.16)

where re0 is the isotropic average size, Mrs is the rescaled galaxy absolute magnitude

(to be defined in Eq. 3.26), and z is the redshift, while the C values are the parameters fit to the model. We fit this model using the Levenberg-Marquardt algorithm

(Levenberg, 1944) from the Scipy curve-ﬁt algorithm (Jones et al., 2001). With this model, we estimate each galaxy’s angle-averaged size, and hence its orientation, which allows us to separate our catalogs into orientation-divided subsamples. For the

CMASS sample, it was found that this fitting procedure resulted in a redshift-biased result for W33, as shown in Fig. 3.2. Specifically, higher redshift galaxies were more likely to have fitted radius re0 > re, and therefore a more negative W33 value. This

implies a change in the slope of the fundamental plane relation with respect to red-

shift for our CMASS sample. We dealt with this by using four redshift bins in the

CMASS sample: 0.43 < z < 0.5, 0.5 < z < 0.6, 0.6 < z < 0.65, and 0.65 < z < 0.7.

Within each redshift bin, we ﬁt to Eq. (3.16), which helped to mitigate the redshift

bias. For the LOWZ sample, we also ﬁt to Eq. (3.16), but for the entire redshift

range, 0.15 < z < 0.43. Once all W33 values were set, we then subtracted the median

87 value for both the LOWZ and CMASS samples, and identiﬁed a galaxy’s inferred orientation by the sign of its W33 value.

3.2.4 Redshift space distortions of sub-samples

In this work, we primarily fit for the RSD parameter f+A that is present in our two sub-samples (split based on a proxy for orientation); the difference ∆(f + A) = ∆A is independent of the linear rate of growth of structure f, and isolates the anisotropic selection effects. This is the key parameter of our study; however, we also account

2 for two other parameters, bg and σFOG (a real-space clustering amplitude and Finger of God length parameter), in order to accurately reproduce the correlation function.

These are described in more detail in Sections 3.5.1 and 3.5.2.

We now outline the central idea of this project. We split a parent galaxy sample into two sub-samples, using a proxy for orientation (in this case, offset from the fundamental plane); we expect the different subsamples to have different values of

(ηχ)eff . We next compute the difference between the redshift space distortion amplitudes fv = f + A measured from the two sub-samples. This difference should cancel out the dependence on the true rate of structure growth (via f); there should also be some cancellation of the non-linear effects (though this may not be perfect if the sub-sample splitting is sensitive to galaxy properties other than orientation, and these properties are correlated with large-scale environment).

Using Eq. (3.6), we can see the eﬀect that galaxy samples with diﬀerent values of

A will have on the power spectrum. Specifically, separate samples of galaxies, with different values of W33, will theoretically have different values of A, but the same value of the growth function, and thus the same fv. We can therefore find the difference in

88 Figure 3.2: For the CMASS samples, we chose to ﬁt Eq. 3.16 separately to galaxies in ranges of 0.43 < z < 0.5, 0.5 < z < 0.6, 0.6 < z < 0.65, and 0.65 < z < 0.7. This helped mitigate an observed correlation between galaxy redshift and median W33 for the CMASS sample. In this plot we compare to the same equation ﬁt over the full redshift range of 0.43 < z < 0.7 (‘single redshift bin’). Here we show the resulting change in median W33 value by binned redshift for the CMASS SGC sample.

89 A between samples as

∆A ≡ A2 − A1 = (f + A2) − (f + A1) = fv,2 − fv,1, (3.17)

where fv,X indicates the measurement of fv for sample X. Note that the “measured” value of the bias (from a ﬁt to the correlation function) is actually

A b = b − . (3.18) meas 3

From our measurement of ∆A we can then calculate B:

1 ∂Ngal− 1 ∂Ngal+ ∆A = 2∆(ηχ)eﬀ B = 2 − B, (3.19) Ngal− ∂W33 Ngal+ ∂W33

where a subscript of “+” indicates the group where W33 is greater than the median

value (galaxies are brighter/smaller than predicted by the fundamental plane) and a

subscript of “−” indicates the group where W33 is less than the median value (galax-

ies are dimmer/larger than predicted by the Fundamental Plane). Throughout this

chapter, any sign of the diﬀerence in parameters, such as ∆fv, indicates the value

in the “−” group minus the value in the “+” group. While A depends on how well

our sample splitting procedure traces orientation, this dependence is removed when

considering B, which depends only on galaxy type and redshift.17 For our sample

splits, we have calculated the values of ∆(ηχ)eﬀ to be −1.111, −1.100, −1.515, and

−1.438 for CMASS NGC, CMASS SGC, LOWZ NGC, and LOWZ SGC respectively.

3.2.5 Theoretical expectations

One of the foundational principles in cosmology is statistical homogeneity and

isotropy. In the context of intrinsic alignments, this means that the intrinsic alignment

17As discussed in Sec. 3.2.2, there are other conventions for the intrinsic alignment amplitude that are related to B.

90 parameters that relate the radial tidal ﬁeld s33 to the radial alignment W33 are the same as those that relate the plane-of-sky tidal ﬁelds (s11 − s22, 2s12) to the plane-of-

sky alignment (W11−W22, 2W12). That is, the measurement of how strongly a galaxy’s

alignment is aﬀected by its tidal ﬁeld should not depend on whether we measure it

radially, as in this work, or in the plane of the sky, as is done in lensing studies. We

explore this concept in our comparison to theory, and previous literature. The caveat,

of course, is that given practical observational limitations the radial and plane-of-sky

measurements may not be measuring exactly the same thing. Possible issues include

non-homologous isophotes (but with a pipeline that assumes a homologous proﬁle);

dust extinction; selection eﬀects for the parent samples; and non-linear response (i.e.

higher-order terms in Eq. 3.14). Plane-of-sky measurements may depend on their own

set of assumptions; see Singh et al. (2015); Troxel and Ishak (2015) for details.

We directly compare our CMASS and LOWZ measurements of B to that obtained

by Singh et al. (2015), who use correlations of galaxy shapes in the plane of the sky.

Singh et al. (2015) performs this measurement for the SDSS DR11 LOWZ sample,

which we can directly compare to our DR12 LOWZ sample.

As a further comparison, we can calculate the theoretical intrinsic alignment

strength as predicted in Hirata (2009), given a luminosity value for the sample in

question. H09 ﬁnds that, using catalogs of luminous red galaxies:

L 1.48±0.64 B = (1.74)(−0.018 ± 0.006) , (3.20) L0 where L is K + e-corrected to z = 0 and L0 is the corrected r-band absolute magnitude of −22. Our r-band absolute magnitudes are listed in Table 3.2. Using the tables provided by Wake et al. (2006), who formulate star evolution on models from

Bruzual and Charlot (2003), we ﬁnd that our K +e-corrected r-band median absolute

91 magnitudes are −22.17 for LOWZ and −22.295 for CMASS. We then ﬁnd that the

ratio L/L0 referenced in Eq. (3.20) is 1.170 for LOWZ, and 1.312 for CMASS. This

results in a prediction of B equal to −0.039 ± 0.010 for LOWZ, and −0.046 ± 0.011 for CMASS. With these values, we can compare our results to that expected by H09.

We can also compare the ellipticity-based measurement of Singh et al. (2015) to that predicted by H09 for their measurement of L/L0. We will use the following

equation (derived in Appendix A.2) to compare values from Singh et al. (2015), who

measure the linear alignment parameter AI , to our measurement:

Ω B = −0.0233 m A , (3.21) G(z) I where Ωm and G(z) are the matter density and growth function at a given redshift for their speciﬁc parameter choices, which are: Ωm,Singh = 0.282 and G(0.32) = 0.869.

Therefore we can calculate that the lensed measurement for the intrinsic alignment magnitude of the LOWZ sample is:

Bpred = −0.0348 ± 0.0038. (3.22)

Using the same method, we calculate H09’s prediction for the L/L0 values found in

Singh et al. (2015), which is 0.95. This leads to a theoretical prediction of −0.029 ±

0.010. All of these predictions are compared to our ﬁnal results in Section 3.5.

3.3 Survey data

3.3.1 Sample deﬁnition and characteristics

Our data is taken from the completed and publicly available SDSS-III (Eisenstein et al., 2011) BOSS (Dawson et al., 2013) Data Release 12 (DR12; Alam et al. 2015)

LOWZ and CMASS catalogs. The purpose of BOSS was, through measurements of

92 large scale galaxy clustering, to determine the scale of baryon acoustic oscillations and

hence constrain the cosmic distance scale. Using spectroscopic redshift measurements,

it recovered the three-dimensional distribution of approximately 1.4 × 106 galaxies,

with an eﬀective area of 10, 000 square degrees. Here, we describe the observations

that produced the BOSS data and the general properties of BOSS galaxies.

BOSS objects were selected for spectroscopic observation based on SDSS-I/II/III

imaging data in ﬁve broad bands (ugriz) (Gunn et al., 2006; Doi et al., 2010). SDSS

I/II (York et al., 2000) made use of photometric pass bands (Fukugita et al., 1996;

Smith et al., 2002; Doi et al., 2010), imaging roughly 7606 deg2 in the Northern galactic cap (NGC) and 600 deg2 in the Southern galactic cap (SGC), data of which was released in DR7 (Abazajian et al., 2009). For DR8, an increased area of the

SGC was surveyed (3172 deg2) (Aihara et al., 2011). The full SDSS DR8 dataset

includes “uber-calibration”, which updates the photometric calibrations within SDSS

(Padmanabhan et al., 2008).

Spectroscopy was obtained for BOSS targets using the BOSS spectrograph (Smee

et al., 2013). Still in operation, it simultaneously obtains 1000 spectra in a given

observation. Using the reduction pipeline described in Bolton et al. (2012), redshifts

were classiﬁed from BOSS spectra with statistical errors from photon noise at a few

tens of km s−1.

The LOWZ sample selects bright, red objects using the specific color-cuts and r- band flux limit defined in Reid et al. (2016). Over the redshift range 0.2 < z < 0.4, the sample is nearly volume-limited, with a constant space density of ≈ 3×10−4h3 Mpc−3.

13 −1 LOWZ galaxies reside in massive halos, with a mean halo mass of 5.2 × 10 h M

(Parejko et al., 2013). In our sample, we applied a cut to galaxies outside of the range

93 0.15 < z < 0.43, since galaxies suﬃciently far outside of the target redshift range are more likely to be mis-assigned objects. This cut also follows the procedure adopted

ﬁrst in Anderson et al. (2012) and subsequently in other BOSS analyses studying the

CMASS and LOWZ sample separately. This makes the two samples more independent by avoiding overlaps in redshift range. Further details on the properties of the LOWZ sample can be found in Parejko et al. (2013); Tojeiro et al. (2014); Kitaura et al.

(2016).

The CMASS sample applies the specific color cuts and i-band flux limit defined in Reid et al. (2016) in order to provide a sample with a number density that peaks at 4.3 × 10−4h3 Mpc−3 at z = 0.5 and is greater than 10−4h3 Mpc−3 within the range

0.43 < z < 0.65. The selection was designed to obtain a stellar-mass limited sample, regardless of color (Reid et al., 2016). This results in a sample of galaxies that are approximately 75% red in color (e.g., Ross et al. 2014 and references there-in) and elliptical in morphology (Masters et al., 2011). The majority (∼ 90%) of the galaxies

13 −1 are centrals, living in halos of mass ∼ 10 h M (White et al., 2011). In our sample, we applied a redshift cut to galaxies outside the range 0.43 < z < 0.7, again following the procedure of Anderson et al. (2012), and again to make the samples independent.

The BOSS footprint is divided into two distinct North and South Galactic cap regions, which we refer to as NGC and SGC. These regions have slightly diﬀerent properties, as described in the appendix of Alam et al. (2017) and references there-in.

Thus, we analyze the NGC and SGC separately throughout our pipeline, for both

LOWZ and CMASS. This yields a total of 4 samples for which we will obtain results.

94 3.3.2 Data preprocessing

Separate from our cosmological parameters are the set of galaxy parameters which

describe measured quantities of each speciﬁc galaxy. The galaxy parameters used in

our pipeline fall into two general categories. The “science” parameters, speciﬁcally

right ascension (RA), declination (Dec), redshift (z), and de Vaucouleurs ﬁt param-

eters (magnitudes and radii), are used both in calculating the orientation parameter

W33 and in ﬁnding the multipoles of the correlation function. These parameters are critical to the purpose and results of this project. All other parameters we refer to as

“null” parameters. These parameters are not used directly for science, but are used to trace possible observational systematics that could impact the clustering results, and are not related to the intrinsic parameters of any speciﬁc galaxy. They are the airmass, galactic reddening E(B − V ), sky ﬂux, and point-spread function (PSF)

full-width half-maximum (FWHM). We will describe our use of these null parameters

in the Section 3.3.4.

This section details our use of the science parameters to calculate the value of

W33 for each galaxy. It is important to note that for many of our science parameters

(and null parameters), we are able to use values speciﬁc to each band, whether it be u, g, r, i or z. We found that the fundamental plane method had the best-ﬁt results within the i-band, and that the i and r bands both had the smallest deviation from

the FP ﬁts. Thus, we used the i-band for all variables in our analysis. Beyond band

choice, there is also the choice of which radius estimator to use. The de Vaucouleurs

radius r of a galaxy is found by ﬁtting the surface brightness, I, to:

−7.669[(R/r)1/4−1] I(R) = I0e , (3.23)

95 where I0 is the surface brightness at the de Vaucouleurs radius (de Vaucouleurs, 1948).

This law is a specific case of a general Sérsic profile, with a Sérsic index of n = 4. We modify the measured radius by incorporating the asymmetry of elliptical galaxies; we convert from the fitted de Vaucouleurs radius to an effective radius, using:

p reﬀ = rdev abDev, (3.24)

where rdev is the de Vaucouleurs radius and abDev is the ratio of the short elliptical axis to the long elliptical axis, again measured using the de Vaucouleurs ﬁt. A ratio value of 1.0 returns the de Vaucouleurs radius, while a ratio smaller than one appropriately scales our ﬁt radius to treat highly elliptical galaxies. The values of abDev for our samples are listed in Table 3.2. We next use the spectroscopic redshift from the

CMASS and LOWZ tables to calculate the comoving angular diameter distance D to each galaxy. With these distances, we found the size of the galaxy, parameterized by the physical half-light radius:

s = Dreﬀ . (3.25)

There is a separate size for each galaxy in each band; our analysis uses the i-band sizes, and perform our ﬁts using log10 size. The SDSS database does not include extinction correction, so we apply that here, following the maps of Schlegel et al. (1998), as provided by the BOSS catalog; we do not perform any K-corrections (Hogg et al.,

2002), except when we compare our results to theoretical expectations in Section 3.5.

We calculate the rescaled absolute magnitudes using:

Mrs = m + 5 − 5 log10 DA = M + 10 log10(1 + z), (3.26)

96 where Mrs is the rescaled absolute magnitude, M is the absolute magnitude, m is the

18 observed magnitude, and DA is the angular diameter distance . We perform redshift

cuts speciﬁc to each survey, restricting LOWZ to 0.15 ≤ z ≤ 0.43 and CMASS to

0.43 ≤ z ≤ 0.70. There is a clear cutoﬀ in the redshift space distribution of each

survey where the color cuts creating the survey were designed to limit the galaxies.

However, speciﬁcally with the low side of the CMASS survey, there are “trailing”

galaxies which have escaped the color cuts. We make these redshift cuts for two

reasons. First, eliminating these galaxies removes galaxies that were not intended to

be included by the color cuts, and prevents outliers in redshift space from dominating

our ﬁt. Galaxies on the fringes of the redshift distribution could have properties

which tend to group them within the same orientation subdivision, which would then

represent a potential systematic error in the sub-sample clustering analyses. Second,

the redshift cuts we perform are generally made across the literature for clustering

analyses of CMASS and LOWZ to which we compare (Ross et al., 2017; Gil-Mar´ın

et al., 2016; Chuang et al., 2017).

At this point, we have values for the size, rescaled absolute magnitude, and redshift

of each galaxy. We can then begin the processes described in Section 3.2.3 for ﬁnding

galaxy orientations for a speciﬁc galaxy, with respect to our point of observation.

Table 3.1 shows the number of galaxies in each subsample and the number removed

18We originally intended to use the absolute magnitude M here, but at a late stage discovered that this is what was implemented within the pipeline, which we have called the rescaled absolute magnitude. The rescaled absolute magnitude differs from that using the absolute magnitude M by an added factor of 10 log10(1 + z). This should not make any substantial difference in our group selection; the difference in the magnitude will be primarily absorbed into the fit coefficients, with a 1.4% (0.5%) z-dependent change in the value of C3 for CMASS (LOWZ) in Eq. (3.16). We tested this change by comparing groups with a radial magnitude assignment to those constructed with radial distance in Eq. (3.26), and found less than 0.3% difference between the definitions of the groups. Note that in the absence of a K-correction, the multiples of log10(1 + z) are essentially arbitrary as they correspond to a different slope of the spectral energy distribution that defines the zero-point.

97 Table 3.1: Redshift cut statistics for the BOSS CMASS/LOWZ subsamples. Category CMASS NGC CMASS SGC LOWZ NGC LOWZ SGC Initial Galaxy Count 618,806 230,831 317,780 145,264 Redshift Cut (%) 8.08 9.71 21.88 21.85 Remaining Galaxies 568,776 208,426 248,237 113,525 S/N 30.389 26.679 83.583 74.306

Table 3.2: CMASS and LOWZ medians and standard deviations for both the W33– divided and total samples. Note that the median of W33 for the full samples is zero by construction.

Statistic W33 > 0 W33 < 0 Total Sample CMASS sample Redshift 0.537 ± 0.065 0.546 ± 0.066 0.541 ± 0.063 Absolute Magnitude (r-mag) −22.184 ± 0.317 −22.180 ± 0.307 −22.182 ± 0.310 Apparent Size (log10 kpc) 1.089 ± 0.168 1.328 ± 0.169 1.213 ± 0.206 abDev 0.669 ± 0.178 0.698 ± 0.175 0.684 ± 0.175 W33 0.956 ±0.843 −1.000 ±0.905 0.000 ±1.438 LOWZ sample Redshift 0.315 ± 0.073 0.319 ± 0.072 0.317 ± 0.075 Absolute Magnitude (r-mag) −22.520 ± 0.385 −22.516 ± 0.337 −22.518 ± 0.399 Apparent Size (log10kpc) 1.117 ± 0.141 1.300 ± 0.163 1.206 ± 0.172 abDev 0.735 ± 0.169 0.734 ± 0.183 0.747 ± 0.166 W33 0.700 ±0.865 −0.793 ±0.637 0.000 ±1.217

due to redshift cuts. The medians and standard deviations of the galaxy parameters

for each sample, once cut by redshift, are shown in Table 3.2.

3.3.3 Sample splitting by orientation

In order to reduce systematic errors in evaluating the W33 component of each galaxy, it is necessary to accurately reﬂect the redshift evolution of the fundamental

98 plane within our sample. We ﬁt all the LOWZ galaxies with a single ﬁt as described

in Eq. (3.16). For the CMASS samples, in order to better adjust for changes in the FP

due to redshift, we ﬁt this model separately by several bins of redshift. We used four

redshift bins of 0.43 < z < 0.5, 0.5 < z < 0.6, 0.6 < z < 0.65, and 0.65 < z < 0.7.

We found that this scheme resulted in galaxy samples with orientations as seen in

Fig. 3.3. We plot the distributions of W33 in Fig. 3.4.

Once values of W33 were calculated for each galaxy in our catalog, we then subtracted the median W33 from each galaxy, and divided our catalogs into two separate samples, depending on the sign of their W33 value. Galaxies with negative W33 are larger than we would predict by using the FP method, and we presume that statistically these galaxies are aligned perpendicular to our line of sight. Conversely, galaxies with positive W33 are smaller than we would estimate from the FP method, and we presume that statistically these galaxies are aligned parallel with our line of sight. Although on a galaxy-by-galaxy basis there are other eﬀects which could cause a galaxy to appear bigger or smaller, these eﬀects should cancel out when measuring

∆fv for the full sample, assuming that these eﬀects are not correlated with the true value of W33 and that all galaxy sub-samples have the same true value of f.

In Figure 3.5, we compare the photometrically (i-band) measured size of each galaxy to predicted size based on each galaxy’s redshift and luminosity, using the

FP relationship. We plot 25, 000 randomly chosen galaxies from each sample, with a line displaying an exact match between the predicted and observed galaxy size. Our samples of positive and negative W33 are deﬁned based on the relationship of these

parameters.

99 Figure 3.3: For each sample, we show the fraction of positive W33 as a function of binned redshift. Note the good agreement of each distribution across the entire redshift range. For the LOWZ samples, this was achieved by using the model within Equation 3.16. For the CMASS samples, it was necessary to ﬁt this model separately to galaxies in ranges of 0.43 < z < 0.5, 0.5 < z < 0.6, 0.6 < z < 0.65, and 0.65 < z < 0.7.

100 Figure 3.4: Here, we plot the histograms for each survey’s distribution of W33.

101 Figure 3.5: For each sample, we use relations of the Fundamental Plane to ﬁt an assumed galaxy radius, based on galaxy luminosity and redshift. Here we plot this assumed radius to the ’true’ galaxy radius which has been measured photometrically in the i-band. Galaxies are assigned into bins of positive and negative W33, based on whether their ﬁt radius is greater or less than their measured radius.

102 Our primary concern with systematic errors is that some parameter related to

observation will leak into our estimates of W33, and then imprint the spatial structure of the observational effect onto our sub-samples. For example, if galaxies observed at higher airmass have fainter measured magnitudes, then this would offset them from the fundamental plane (they would look “too large” for their magnitude and their measured W33 would be biased low). In Fig. 3.6 we plot the mean W33 value in bins of airmass. For an unbiased measurement of W33, we would expect this to fluctuate around zero, as it should be uncorrelated with the airmass. The effect of seeing on the observed galaxy number density has been previously documented in Ross et al. (2017), and could lead to a systematic effect in our measurement. Specifically, in the target selection algorithm, the distinction between stars and galaxies can be more blurry in situations of bad seeing. As seeing gets worse, the smallest galaxies (large W33 values) would be more likely to be labeled as stars, and less likely to appear in the galactic sample. This cut is related to the comparison of the model and PSF magnitudes of the object in question, and is explained in detail in Reid et al. (2016); Ross et al.

(2011). In Fig. 3.7, we plot the fraction of galaxies with positive W33 value in bins of sky flux. In Table 3.3 we display the slope of a linear fit between the W33 value and four potential systematic effects: airmass, extinction, sky flux and the FWHM of the point spread function. A full quantitative treatment of these systematics and how they influence our results is given in Section 3.3.4.

3.3.4 Systematics tests, random splits, and blinding procedures

The treatment of systematics is an important aspect of this project. We aim to split our sample by the W33 values of each galaxy, and to attribute diﬀerences among

103 Figure 3.6: Here we show the average W33 value, binned by the PSF FWHM. There is a clear correlation between the PSF FWHM value and a galaxy’s calculated W33 value. This will be analyzed in more detail in Section 3.3.4. Further evidence of this can be seen in Table 3.3.

Figure 3.7: For each sample, we show the fraction of galaxies with positive W33 as binned by the sky flux at the time of observation. Note that there is no obvious correlation between the sky flux on the night of observation, and the calculated W33 value. Toward the edges of the range we can larger deviations from the center, which is due to smaller numbers of galaxies observed with those values of sky flux.

104 samples as evidence for a radial measurement of intrinsic alignments. However, we want to be sure that the differences in clustering properties of the sub-samples are associated with the targets galaxies themselves, and not due to excess clustering imprinted by observational systematics. We do this by intentionally injecting our orientation measurement with a dependence on different systematic effects, and compare the resulting parameters with the intrinsic uncertainty in our true signal. This section discusses these tests, as well as associated blinding procedures.

First, for what we call Phase I, in Section 3.3.4 we explain our random division of samples in order to estimate the amount parameters can vary in samples that have no diﬀerences in W33. There is no speciﬁc blinding procedure required at this stage, since we are not computing any clustering properties based on the real sub-sample split (based on W33).

Next, in Section 3.3.4 we describe our injection of systematic effects into the splitting criteria for W33. This is still considered a part of Phase I, since there is again no need to blind the output fit parameters, because the values of W33 are internally shuffled before being placed in a group. In order to begin Phase II, we require consistency in the χ2 of the fits with our full samples and null samples, as well as final parameter offsets that are consistent with zero, to indicate that systematics are not imitating the effects of W33 splitting.

We enter Phase II in Section 3.3.4 by discussing the fitting of our true W33-based sample division, while blinding ourselves to the true difference in fv. We must be careful here, when checking our samples for errors, to only display combinations of parameters that are independent of the final ∆fv measurement. The parameter combination p(∆bg, ∆fv) which achieves this requirement is described in Section 3.3.4.

105 We also carry out several tests in Phase II for whether diﬀerences in the Finger of

God properties of the subsamples could bias our measurement of ∆fv; in these tests, only absolute values of relevant parameters are revealed, so that we remain blind to the direction of any possible correction. In order to ﬁnish our calculation and fully un-blind ourselves for Phase III, we require consistency in both samples of this parameter combination, as well as consistency in the χ2/dof of the ﬁts.

Phase I: Random Splitting

As described in Section 3.2, we want to accurately measure B, and therefore ∆A, for our samples which are separated based on galaxy alignment. We ﬁrst need to estimate the statistical error in ∆A, by measuring the diﬀerences in parameters that can occur due to random sample selection alone. In order to achieve this, we create

500 random splits of each survey, and measure the standard deviation of ∆A, that is:

σ∆A. We then compare parameter differences for true samples to this value, to decide if true results are significant or if systematic effects need to be mitigated. Blinding is not necessary in Phase I, as we have not divided the sample in any meaningful way, and thus the parameters we are measuring are not affected by intrinsic alignments.

Null parameter scrambling

We continue Phase I by examining potential systematic eﬀects in more detail. For this step, we do not inject any random parameter values for blinding purposes; however, at one point during this test, we scramble the W33 measurements so that they are uncorrelated with their original galaxies, thus destroying any relations between

W33 and the local tidal ﬁeld for any particular galaxy; therefore, additional blinding

106 Table 3.3: Values of mθ, for each combination of survey and systematic. Survey Airmass Extinction Sky Flux PSF FWHM CMASS South −0.039 0.409 −0.007 −0.331 CMASS South −0.103 0.359 −0.011 −0.255 LOWZ North 0.011 0.232 0.001 −0.261 LOWZ South −0.069 0.318 −0.004 −0.219

protocols at this stage are not necessary. As discussed above, our list of null param-

eters includes measurements of airmass, extinction, sky ﬂux and the FWHM of the

point spread function.

For each survey and null parameter, before we perform any scrambling, we ﬁt a line

to a scatter plot between W33 and the specific null parameter θ. From this linear fit, we find the slope of the line, mθ. Ideally, if there is no dependence of W33 on this null

parameter, then mθ ≈ 0. The values of mθ are listed in Table 3.3. Next, we scramble the W33 values randomly among galaxies, destroying any real signal information for blinding purposes. To each galaxy, we modify the scrambled W33 value to become:

¯ W33 ⇒ W33 + mθ (θparam − θparam), (3.27)

¯ where θparam is the median value of that parameter within the full sample. This eﬀectively injects the correlation signal with the null parameter into the W33 signal.

By fitting each sample set for clustering parameters, we find a measurement of ∆A, which represents the potential spurious value if our estimate of W33 is contaminated by the null parameter in question, θ. Ideally, we want this difference to be much less than the statistical errors on our test measurements of ∆A.

107 However, even this method is prone to statistical errors. The scrambling step is

equivalent to a random assignment of W33 values to galaxies, and this random assign-

ment step could result in a parameter shift that has nothing to do with our injected

signal, instead dominated by random ﬂuctuations. To beat down these statistical

errors, we take two steps. First, we perform the random scrambling step multiple

times, so that a true systematic error will become more evident, while statistical ﬂuc-

tuations will tend to cancel out. Second, for each random assignment realization, we

perform a “mirror” realization where the random groups are the same, but we use the

conversion W33 → −W33 before using Eq. (3.27). This helps to beat down statistical errors even more quickly, as a given realization and its mirror will mostly cancel each other’s statistical ﬂuctuations without aﬀecting each other’s biased parameter shift.

We perform 20 realizations for each null-test, each with their own mirror realization, resulting in 40 total realizations.

Phase II: fv Blinding

As a ﬁnal test, we disable the scrambling mechanism above, and ﬁt for the true divided samples. However, we blind ourselves to the sample values of fv. Instead, we will view the parameter combination which we call p(∆bg, ∆fv):

1 p(∆b , ∆f ) ≡ ∆b + ∆f , (3.28) g v g 3 v

where ∆fv and ∆bg indicate the differences in the growth parameters and biases for a sample pair. As can be seen from Eq. (3.6), this parameter is independent of our offset due to orientation (it does not depend on ∆A). We also would like this parameter to be uncorrelated with the fv fit for the sample. To test this, we calculated this correlation for all random separations we performed. The results are displayed in

108 Fig. 3.8. We can see that there is a non-zero correlation value, varying (depending on the sample) between −0.1 and −0.16. However, as this correlation results in small changes in fv when compared to the random sample separations (see Section 3.5.4), and this step is only necessary to blind ourselves, not find a final uncorrelated result, we believe that this parameter will be effective as a check at this blinding stage.

During Phase II, we also run several tests to establish that the Finger of God length is not diﬀerent enough between the subsamples to aﬀect our measurements of

∆fv. These tests consist ﬁtting our ﬁnal samples for a smaller range of scales at 20–40

2 2 Mpc and reporting |∆(σFOG)|, and also attempting to measure |∆(σFOG)| via direct integrals of the correlation function. They are described in detail as part of Section

3.5.2. Note that only absolute values of sub-sample diﬀerences are revealed in these

Finger of God tests, so that the direction of any correction (were we to attempt one) is not revealed.

In order to pass this stage, and move on to the unblinded final results of Phase III, we require consistency in the samples for this parameter combination p(∆bg, ∆fv), and consistency in the χ2 of the fits. A discrepancy could indicate that our two groups are preferentially selecting galaxy groups of different biases. In Appendix A.1, we calculate how this could influence the size of our error bars. Based on our random separation tests, p(∆bg, ∆fv) has a standard deviation of approximately 0.04 for LOWZ, and 0.08 for CMASS. If any of our subsamples have a value of p > 0.2, where bg is the galaxy bias of the full subsample, we will investigate further into the sources of this bias difference, although this still should not affect our measurement

2 of ∆fv. Furthermore, we require our χ values, for each subsample, to be within 3σ

109 Figure 3.8: Here, we have created 500 random sample separations for each survey. We have compared the parameter p(∆bg, ∆fv) to fv (Eq. 3.28). The Pearson correlation coeﬃcient is listed above each plot. We can see that although the coeﬃcient is non- zero, we believe that it is small enough to serve our purpose here as a blinding parameter which is independent of the change due to galaxy orientation.

of the mean; speciﬁcally, the χ2/dof values must be below 1.50, 1.60, 1.82, and 1.88 for CMASS North, CMASS South, LOWZ North, and LOWZ South respectively.

Phase III: Final Results

Before entering Phase III and unblinding ourselves to the ﬁnal results, we deﬁne what will constitute as a detection, consistency between samples, and consistency between the measurements and theoretical expectations. These requirements are made before unblinding so that we can be as objective as possible in stating the results of our tests.

110 We will check sample consistency by comparing the measured values of B for the north and south subsamples of both CMASS and LOWZ. If these measurements are within 3σ of each other, then we will consider them to be consistent. Any discrepancy larger than this will merit a look into what could be causing a diﬀerence between the north and south observations. Similarly, if our samples measure B to be greater than

3σ from zero, we will consider this a “detection” of the intrinsic alignment magnitude.

We will consider a measurement of B between 2 and 3σ from zero to be “evidence”

for non-zero intrinsic alignment magnitude.

Next, we want to analyze how well our results agree with the theoretical expec-

tations. In order to evaluate our results as a whole, we will combine them and their

uncertainties with the predictions from Hirata (2009), as listed in Section 3.3.4. We

will evaluate the ratios of the observed values to the theoretical values:

h th i2 P Bi Bi th Obs i σi B 1 = i ± , (3.29) h th i2 r 2 Theory P Bi h th i P Bi i σi i σi

th where Bi is our measurement for a speciﬁc subsample, Bi is the theoretical prediction for that subsample, and σi is the error on our measurement. A value consistent with

1 indicates that our results are compatible with the theory. Speciﬁcally, we will calculate Eq. 3.29 for the LOWZ and CMASS subsamples individually, as well as for all 4 samples in total. We have decided to label a diﬀerence from 1 that is less than

2σ as ‘consistent’, between 2-3σ as in ‘tension’, and as 3+σ ‘incompatible’. We have no specific requirement for the closeness to results from Singh et al. (2015). This is because H09’s results are in agreement with that found by Singh et al. (2015), and differences in L/L0 will not be a factor in comparing to H09, while they could be a factor in comparison with the specific sample used by Singh et al. (2015).

111 3.4 Clustering statistics

3.4.1 Correlation functions

For each of our divided catalogs, we use the random galaxy catalogs provided

by the BOSS DR12 collaboration in order to normalize geometric sampling eﬀects.

Randomly oriented galaxies were given redshifts pulled from the redshift distribution

of the matching survey. The random galaxy count is equal to 10 times the real

galaxy count. Pair counts were done on the random catalogs in order to construct

the correlation function using the Landy-Szalay method (Landy and Szalay, 1993;

Peebles and Hauser, 1974). The correlations are calculated as a function of redshift-

space separation s and µ = cos θ, where θ is the angle with respect to the line of sight.

Speciﬁcally, we use:

DD(s, µ) − 2DR(s, µ) + RR(s, µ) ξ(s, µ) = , (3.30) RR(s, µ) where DD refers to the number of pairs of galaxies in the data sample within a speciﬁc

1 1 distance shell, s ± 2 ∆s, and within a speciﬁc angular range µ ± 2 ∆µ. RR refers to the same, but for the random sample, while DR refers to the counts of pairs between

one data galaxy and one random galaxy. We use 50 logarithmically spaced radial bins

from s = 43 Mpc to s = 100 Mpc. Both DD and RR counts are normalized by the total number of galaxies in that bin, speciﬁcally:

DD RR DD → and RR → , (3.31) nD(nD − 1) nR(nR − 1) while the DR counts are normalized by:

DR DR → . (3.32) nD × nR

112 Our correlation function code calculates pairs in 20 µ-bins, from −1 to +1, with a separation of ∆µ = 0.1. We do not use a galaxy weighting scheme, for three reasons. Primarily, the importance in our measurement is through the difference in parameters between pairs of samples. As long as the treatment of each sample is the same, the difference in parameters we care about should be unaffected. Second, galaxy weighting has a higher impact on large scales, where our fitting is focused on small to medium scales. Finally, weights are given in BOSS to offset effects such as close pairs of galaxies, where multiple fibers cannot successfully observe both spectra.

It is not immediately clear how to treat a weighting scheme where we have divided the samples in a way that does not preserve these close pairs.

For completeness, we tested a subset of 30 galaxy separations (60 total samples) for both CMASS and LOWZ, finding the correlation functions both with and without weights, and fitting parameters to each, with the weights defined as

wtot = wsystot(wcp + wnoz − 1), (3.33)

where wsystot accounts for ﬂuctuations in target density with seeing and stellar density, wcp deals with ﬁber collisions, and wnoz counteracts redshift dependent bias due to redshift failures. For details on the weighting schemes, see Reid et al. (2016).

Within our 30 galaxy testing subsets, we found that for CMASS (LOWZ) the errors in measuring ∆fv were 0.058 (0.070) for weighted and 0.052 (0.074) for non-weighted, indicating the weighting scheme did not substantially change our ﬁnal error bars19

(see Sec. 3.5). The changes in bg and fv due to inclusion of weights are shown in

Table 3.4. It is apparent that the largest change between weighted and non-weighted

19 The mean value of ∆fv is of course zero for random splits by construction; there is no additional systematic information in the mean value, since the BOSS weights do not know about our sub-sample splits.

113 Table 3.4: The parameter diﬀerences between identical samples which we counted using both weighted and non-weighted methods. See Sec. 3.4 for details.

bg fv Weighted Non-Weighted Weighted Non-Weighted

CMASS North 1.882 ± 0.029 1.898 ± 0.028 0.617 ± 0.025 0.619 ± 0.030 CMASS South 1.942 ± 0.036 1.986 ± 0.034 0.627 ± 0.042 0.614 ± 0.034 LOWZ North 1.773 ± 0.030 1.774 ± 0.027 0.642 ± 0.038 0.641 ± 0.037 LOWZ South 1.790 ± 0.047 1.785 ± 0.045 0.597 ± 0.048 0.593 ± 0.049

is in the bias measurements of the CMASS samples. The value of the bias should be

independent of the clustering value fv, although it may have an eﬀect on error bar size, which we discuss in Appendix A.1.

The correlation functions were converted from µ-binned“wedge”space to multipole space, for easier plotting and parameter ﬁtting. The formula for conversion is the same as that used in SDSS BOSS analyses (e.g. Ross et al., 2017):

i 2l + 1 Xmax 2 ξ (r) = ξ(r, µ )L (µ ), (3.34) l 2 i i l i i=1 max

1 where µi = (i − 2 )/imax, Ll is the Legendre polynomial of order l, ξ is the µ-binned

correlation function, and imax = 10 is the number of positive µ-bins (we simply

double the result, since auto-correlations are symmetric under µ ↔ −µ). We use the

monopole (l = 0) and quadrupole (l = 2) of galaxy clustering in our analyses, since

they carry most of the information on large-scale redshift space distortions.

114 3.4.2 Covariance matrices

To provide a best-ﬁt to the observed correlation functions, we assume a Gaussian- distributed likelihood for our vector of measured correlation functions:

L(p) ∝ e−χ2(p)/2, (3.35) where χ2 is given by:

2 X X ˆi i −1 ˆj j χ (p) = (ξ`(p) − ξ`)[C ]``0ij(ξ`0 (p) − ξ`0 ). (3.36) `,`0 i,j Here p is a vector of parameters; ` and `0 are the moments of the correlation function

(here equal to 0 or 2); i and j refer to the separation bins; ξˆ is the model correlation function; and C is the covariance matrix (Sánchez et al., 2008; Cohn, 2006), which we calculate using the method from Grieb et al. (2016), who introduce a theoretical model for the linear covariance of anisotropic galaxy clustering observations, making use of synthetic catalogs. The calculation of the covariances are based on an input linear galaxy power spectrum dependent on both the wavevector and the angle with the line of sight, P (k, µ). In order to calculate this, we first calculate the linear matter power spectrum using CLASS (Blas et al., 2011) and then compute the no- wiggle power spectrum from the formulae listed in Eisenstein and Hu (1998). This is done at the median redshift of each sample. We next account for redshift space distortion effects to the power spectrum using the procedure outlined in Ross et al.

(2017). First we take

h 2 2 i 2 −k σv P (k, µ) = C (k, µ, Σs) (Plin − Pnw)e + Pnw , (3.37) where we have used

2 2 2 2 2 σv = (1 − µ )Σ⊥/2 + µ Σk/2 (3.38)

115 and 1 + µ2β C(k, µ, Σs) = 2 2 2 . (3.39) 1 + k µ Σs/2

−1 −1 The parameters used are β = 0.4, Σs = 4h Mpc, Σk = 10h Mpc, and Σ⊥ =

6h−1 Mpc, which gives us the the linear matter power spectrum. In order to be

used in our covariance matrix calculation, this must be converted to a galaxy power

spectrum using a bias appropriate for our speciﬁc matter tracers. We follow the

results of Gil-Mar´ın et al. (2016), who have previously measured full-sample clustering

parameters for the LOWZ and CMASS samples, by deﬁning bg = 1.921 for our LOWZ

sample, and bg = 1.993 for our CMASS sample. These power spectra are then used

to generate covariance matrices in multipole space, creating covariance matrices for

ξl(r) for l = 0, 2:

il1+l2 Z ∞ Cξ (s , s ) = k2σ2 (k)¯j (ks )¯j (ks ) dk, (3.40) l1,l2 i j 2 l1l2 l1 i l2 j 2π 0 where the multipole-weighted variance integral is

Z 1 2 2 (2l1 + 1)(2l2 + 1) 1 σl1l2 (k) = P (k, µ) + Ll1 (µ)Ll2 (µ) dµ, (3.41) Vs −1 n¯ and the bin-averaged spherical Bessel function is

Z si+∆s/2 4π 2 ¯jl(ksi) = s jl(ks) ds. (3.42) Vsi si−∆s/2

Here Vsi is the volume of the bin for that iteration of ∆s, Vs is the volume of the entire sample, jl is the spherical Bessel function of the ﬁrst kind, k is the wavenumber, s is

the distance in redshift space, andn ¯ is the comoving number density of galaxies for

the sample in question.

116 3.5 Analysis

In this section, we describe both our ﬁtting methods and the results for the mul-

tiple Phases. In Sec. 3.5.1, we explain which parameters we are ﬁtting for, and how

the ﬁts are performed. In Sec. 3.5.3, we begin the Phase I results. We describe the

parameter ﬁts to the full samples of our catalog, without any splitting, and compare

them to previous clustering measurements in the literature. In Sec. 3.5.4, we show

the results of our random splitting analysis and explain its signiﬁcance. We explain

the results of our quantitative systematics testing (Phase II) in Sec. 3.5.5, and in

Sec. 3.5.7 we show our measurements for the true splitting of our samples based on

orientation, and describe the result for intrinsic alignments (Phase III).

3.5.1 Parameter ﬁtting methods

For each subsample, regardless of whether it is a full sample or it is split in some

way, we ﬁt with an identical procedure. We calculate the correlation function on

scales of 43 to 100 Mpc in 50 radial bins. We ﬁt for the rate of growth fv and the

galaxy bias bg.

It is important to note that the critical measurements from this project are not

the values of fv and bg for our samples. Although we have attempted to measure these as accurately as possible, the systematics tests and blinding protocols in this work are instead aimed at ensuring an accurate measurement of ∆A. Our values of

fv and bg should not be used as references for these ﬁts; instead see papers such as

Gil-Mar´ın et al. (2016) or Chuang et al. (2017) for a reference to the sample clustering

parameters.

117 The ﬁt on small scales focuses primarily on parameters due to redshift space

distortions; it does not include the BAO peak, which would have little information

on the amplitude of intrinsic alignments (but see Chisari and Dvorkin 2013). To

calculate the theoretical ﬁt, we use Convolution Lagrangian Perturbation Theory

(CLPT; Carlson et al. 2013), modiﬁed on small scales by Gaussian Streaming Redshift

Space Distortions (GSRSD). The code for this program was featured in Wang et al.

(2014), and is publicly available.20 Note that this code uses h−1 Mpc as its native units

whereas this work uses Mpc; we have checked that our interface contains the correct

conversions (but note that some explanatory material in this section uses h−1 Mpc).

We modiﬁed the GSRSD code to match the speciﬁc spatial binning scheme used when

we performed the counting algorithms. The code uses CLPT to produce the real-space

galaxy correlation function ξ(r), the real-space galaxy pairwise in-fall velocity v12(r),

2 and the real-space galaxy pairwise velocity dispersion σ12(r, µ). CLPT takes as input a linear matter power spectrum at the redshift of the sample being ﬁtted, which we

generate using CLASS (Blas et al., 2011) set with the cosmological parameters listed

in Section 3.1. These results are then used by GSRSD to calculate the redshift-space

correlation function: Z s dy 1 + ξ (s⊥, sk) = [1 + ξ(r)] q 2 2 2π[σ1,2(r, µ) + σFOG] ( 2 ) sk − y − µv12(r) × exp − 2 2 . (3.43) 2σ1,2(r, µ) + 2σFOG Here y is the real-space pair separation along the line of sight, and µ = y/r. The

2 parameter σFOG is the increase in variance due to the “Finger of God” eﬀect. GSRSD uses a default spacing of ∆y = 0.5h−1 Mpc in the numerical integral, and a default

20GitHub link: https://github.com/wll745881210/CLPT_GSRSD. We started from the version that was most recently edited on 29 May 2015.

118 integration of ±200h−1 Mpc around the peak value. Ideally, we require a high enough

resolution and range to accurately reproduce the correct correlation functions, but

minimize the computation time when running a ﬁt. We found that, in our range of

scales and parameter values, a step size ∆y = 0.8h−1 Mpc and a default integration of ±45h−1 Mpc resulted in the same correlation function, but signiﬁcantly reduced computation time. We made a slight modiﬁcation to allow the code to accurately

2 21 2 treat cases where σFOG < 0. This change allowed the parameter space for σFOG to be extended to include physical situations of a decrease in the spread of red-shift space distortions. In order to produce the redshift-space correlation functions, GSRSD also takes as input the galaxy bias, and the structure growth rate, fv. The code outputs the redshift-space correlation function in terms of multipole moments, ξ0,2(r). It is these correlation functions we compare to our measured correlations from the BOSS datasets.

We use the Scipy minimize function (Jones et al., 2001), speciﬁcally the Nelder-

Mead method (Powell, 1973), to ﬁnd the best-ﬁt parameters by minimizing the χ2

(see Eq. 3.36). Each iteration of the method uses CLPT and GSRSD to create a

fitting attempt of the monopole and quadrupole correlation functions, which are then compared to the BOSS values. We find that convergence always occurs within a few hundred iterations, assuming starting values of fv = 0.7 and bg = 1.8. In Fig. 3.13 we show an example fit overlaid on the data for a random separation in the CMASS

NGC sample.

21 2 2 Previously, GSRSD set the value of σ12 + σFOG to zero, if it were to extend to negative values. This created a hard cutoﬀ in the χ2 of our ﬁts, since these values created unrealistic correlation 2 2 2 functions. We analytically extended Eq. (3.43) to negative values of σ12 + σFOG, where f(x|σ ) → 2 2 2f(x) − √ 1 R dyf(x + y)ey /2σ . −2πσ2

119 3.5.2 Finger of God Eﬀects

The Finger of God eﬀect, as explained in Sec. 3.2, results in a smearing of galaxy velocities along the line of sight for small scales. It can be seen represented in

2 Eq. (3.43) as σFOG. This parameter can have an important eﬀect on the quadrupole of our correlation function, and ideally should be taken into account as a nuisance parameter. While running our ﬁts over the divided samples, as discussed in Section

2 3.3.4, we found that the value of σFOG was not well constrained, and likely to vary highly across the parameter space. This indicated a very shallow χ2 surface, relative

2 to this speciﬁc parameter choice. We decided then to set σFOG to a value consistent with our model ﬁts, which also matched the literature. Gil-Mar´ın et al. (2016) found,

2 −2 2 depending on the method, a value of σFOG = 11.4 to 22.9h Mpc for LOWZ, and

2 −2 2 a value of σFOG = 9.2 to 13.9h Mpc for CMASS. We re-ﬁt 50 subsamples to just two parameters, and found that for LOWZ, the average χ2/dof changed from 1.3745

2 to 1.3695, and for CMASS, from 1.385 to 1.3695. This indicated that ﬁtting for σFOG was not improving our ﬁts, and could be set identically for all subsamples taken from the total data.

Ideally, we would ﬁt for all three parameters in each subsample; however, for a

2 variety of reasons we felt that it was better to set a value for σFOG here. First, as stated above, the addition of this parameter did not improve the quality of the ﬁts

2 2 - in fact, the χ /dof decreased when σFOG was ﬁtted for. Second, the important measurement we are conducting is in the diﬀerence of fv between two otherwise

2 identical samples. By treating each sample with the same value of σFOG, we reduce the scatter in the resulting values for fv, and increase the signiﬁcance of our measurement of the orientation-based splitting. Finally, we only ﬁt large scales where the Finger

120 2 of God eﬀect has little impact; hence even large changes in the value of σFOG result in a small change of χ2.

A possible concern is that when we split into subsamples based on the observed

W33, the two subsamples may have diﬀerent satellite fractions and hence diﬀerent

2 2 Finger of God lengths. In a ﬁt with ﬁxed σFOG (hence ∆σFOG forced to be zero), this

2 could leak into the measurement of ∆fv due to the degeneracy of σFOG and fv. To estimate the potential impact of this eﬀect, we measured the covariance between the

2 ﬁt values of ∆fv and ∆σFOG, and found that

2 −3 Cov(∆σFOG, ∆fv) 3.7 × 10 h/Mpc (CMASS) 2 = −3 . (3.44) Var(∆σFOG) 3.3 × 10 h/Mpc (LOWZ)

2 2 This means that an error of ∆σFOG = 1 (Mpc/h) propagates into an error of 0.0037

(0.0033) in ∆fv for CMASS (LOWZ). For a “nightmare” scenario where our samples

2 were split within groups where one group had a value of σFOG = 0, and the other of

2 2 σFOG = 22 (Mpc/h) , this would constitute a change of ∆fv ≈ 0.081(0.072) between the two samples for CMASS (LOWZ), which is comparable to the error bars in ∆fv of approximately 0.05 (0.08).

We address this concern by calculating a value I for both the full samples and a selection of 15 randomly split samples, where:

X 3 I = r ξ(r, µ)i. (3.45) i This value represents an integral over the correlation function. We sum over a range

2 2 2 of 1 ≤ r⊥ ≤ 4 Mpc and 3 ≤ rk ≤ 22 Mpc, where r⊥ = r (1 − µ ), and rk = rµ. The index ‘i’ refers to all eligible µ and r bin combinations that meet these criteria within our correlation calculation. The purpose of this is to gain a sense of the FOG size of each subsample, and to understand the variance of this value for a random separation

121 by calculating:

I2 − I1 Isplit ≡ (3.46) Ifull where Ii is calculated for the ith subsample split, and Ifull is calculated for the full sample.

We ﬁnd that for the random splits the means and standard deviations of Eq. (3.45) are 0.017 ± 0.026 for LOWZ South, 0.001 ± 0.018 for LOWZ North, 0.003 ± 0.014 for CMASS South, and −0.002 ± 0.016 for CMASS North. The means are consistent with zero, as expected for a random split, and the standard deviations decrease for the subsamples with larger numbers of galaxies. With this information, we can test our

2 true subsample splits for a diﬀerence in their σFOG value without unblinding ourselves

22 to their true fit parameters. From the values of |Isplit| for our true subsamples we can calculate the prospective effects on ∆fv, and determine whether or not this issue will affect our measurements. We reveal our results for this test in Section 3.5.6.

Finally, we also measured 15 randomly split samples over the ranges of 20–40

2 Mpc, ﬁtting for fv, bg, and σFOG. We found that the standard deviations for each

2 2 2 sample pair of σ1 −σ2 were 10.4, 8.4, 11.0 and 7.6 (Mpc/h) for LOWZ South, LOWZ North, CMASS South and CMASS North respectively. In Section 3.5.6 we reveal the absolute value of same parameter diﬀerences for our true samples, to further analyze

2 if our measurements will be biased by the value of σFOG. (The use of the absolute value ensures that we do not know which direction the potential bias would go.)

22As noted in Sec. 3.3.4, only the absolute value is revealed so that if we decided a correction to ∆fv were needed, we would not know what direction it would go.

122 3.5.3 Full sample results

In order to establish that our fitting process is accurately finding the best-fit pa-

rameters, we compare our correlation functions, for the scales of interest, to those

calculated by Chuang et al. (2017). We used 2.4 million random galaxies to pro-

duce these correlation functions. In Figures 3.9 and 3.10, we plot the monopole and

quadrupole of our correlation functions together with those from Chuang et al. (2017),

for both the LOWZ and CMASS subsamples, for the range of separations where we ﬁt

RSD parameters. We also show the percent diﬀerence between the two comparisons.

Diﬀerences at this stage are primarily due to weighting schemes in the pair counting,

which were used in Chuang et al. (2017), while our pair-counts were unweighted (all

weights equal to 1). Note that we plot in Mpc, as opposed to the h−1 Mpc used in

Chuang et al. (2017).

Next, we used the correlation functions from Chuang et al. (2017) in our own pipeline to find best-fit values for bg and fv, and compared them to our own full sample fits, as well as the results listed within Chuang et al. (2017) and Gil-Mar´ın et al.

(2016). We plot the results in Fig. 3.14. Here the we can see how the differences in our results break down between differences in the correlation function, and differences in the fitting process. Green points indicate our results for the full samples, while red points indicate running the correlation function of Chuang et al. (2017) through our pipeline. For both samples, these points agree well, which indicate that differences in the calculation of the correlation function are not a major factor when fitting our parameters. The black and blue points refer to the fits presented in Chuang et al.

(2017) and Gil-Mar´ın et al. (2016), respectively. The differences between these points and our own are primarily due to scale differences; we fit on scales of approximately 40

123 Figure 3.9: Top panel: We plot the correlation function monopoles for the CMASS and LOWZ. We compare our calculation, with normal lines, to the calculations of Chuang et al. (2017), in dotted lines. Bottom panel: The percent diﬀerence between the sample diﬀerences.

to 100 Mpc, while they ﬁt on a much larger range of scales, between approximately 60 and 260 Mpc. Furthermore, they are ﬁtting and marginalizing over a larger parameter set.

Our point of comparison here is to highlight the consistency of the pipeline itself; the purpose of our project is a differential measurement between sub-divisions of galaxies. The actual values of the parameters are not important for our final results, as long as the sub-divisions are taken from the same sample. We do not recommend using our best-fit full sample parameter values for reference outside of this work; instead, use values from Chuang et al. (2017) or Gil-Mar´ın et al. (2016) to exemplify the best clustering parameters to describe the LOWZ and CMASS samples.

124 Figure 3.10: Top panel: We plot the correlation function quadrupoles for the CMASS NGC and LOWZ SGC. We compare our calculation, with normal lines, to the calculations of Chuang et al. (2017), in dotted lines. Bottom panel: The percent diﬀerence between the sample diﬀerences.

3.5.4 Phase I: Statistical uncertainties

Random splittings of the galaxy sample will have a difference in fitted parameters, if only due to statistical fluctuations. In order to evaluate the significance of our orientation-based splitting, we calculate the standard deviation of a random splitting,

σ∆A, by dividing each sample 500 times and calculating the distributions of ∆A. This is assuming that the parameters fitted for the difference between two random sub- selections will be Gaussian-distributed. We test for this by calculating the kurtosis of both the ∆bg and ∆fv parameters, for each subset. We find that the kurtosis for

∆bg (∆fv) are −0.25, 0.05, −0.16, and −0.15 (−0.17, −0.25, −0.07, and −0.24) for

CMASS North, CMASS South, LOWZ North, and LOWZ South, respectively; for

125 Table 3.5: Fitting statistics for the 1000 random splits of each survey. 2 Survey σδA σB <χ /dof> CMASS North 0.605 ± 0.022 1.852 ± 0.022 0.040 0.018 1.109 CMASS South 0.610 ± 0.034 1.960 ± 0.035 0.063 0.029 1.146 LOWZ North 0.604 ± 0.032 1.725 ± 0.031 0.059 0.020 1.281 LOWZ South 0.580 ± 0.048 1.737 ± 0.046 0.089 0.031 1.323

500 points, the expected error in the kurtosis is ≈ p24/500 = 0.22, so these are consistent with zero.

In Fig. 3.11, we show the results of fitting clustering parameters of bg and fv to each half of 500 random splits of each survey catalog, resulting in 1000 plotted points. We show in Fig. 3.12 the differences in parameters fit for each random split.

For each sample, the mean of these diﬀerences is centered at zero, as it must be.23

We are interested in the inferred error bars, speciﬁcally the standard deviation of the diﬀerences in fv, which is equivalent to the standard deviation for our value of ∆A.

These values are listed in Table 3.5. We discuss in Appendix A.1 how large of an eﬀect ∆bg could have on error bars.

3.5.5 Systematics-biased subsamples

Here we show the results for subsamples that have been randomized (to eliminate any real signal) but subsequently biased to reﬂect selected observational systematics.

As mentioned in Section 3.3, some of our variables describing the data are used solely for systematics checking – speciﬁcally, making sure that observation elements are not leaking into our W33 estimator and hence imprinting the systematics maps

23This is more of a code check then a systematics test, since there is no way for a truly random split to give a non-zero average ∆bg or ∆fv.

126 Figure 3.11: We show the results of ﬁtting clustering parameters of bg and fv to each sample of 500 random splits of each survey catalog, resulting in 1000 points.

127 Figure 3.12: The diﬀerences in parameter values for 500 random splits of the data.

128 Figure 3.13: Correlation functions from the best-fit parameters, found by the fitting methods in section 3.5.The shaded areas show the correlation functions from the data with error bars. The dark lines are produced by the best-fit parameters. This is from a random separation of the CMASS NGC catalog. This fit has a χ2 per dof of 1.034. The best-fit parameters are fv = 0.3712 and bg = 1.9486.

129 Figure 3.14: Here, we plot the final parameters of bg and fv for a variety of samples. Triangles indicate results from CMASS, and circles indicate results from LOWZ. The green points are the results from our full samples for CMASS and LOWZ. The red points are the results from the correlation functions from Chuang et al. (2017) being run through our pipeline. (Note that for LOWZ, the green and red points lie directly on top of each other). The black points are the full results from Chuang et al. (2017). Finally, the blue points are the results from Gil-Mar´ın et al. (2016), where we have shown several of their final parameters, which depend on details of the mocks used for fitting.

130 diﬀerently on the two subsamples. We run systematics testing against sky ﬂux, air

mass, extinction, and the point-spread function at full-width half-maximum (PSF

FWHM). These tests are described in detail in Section 3.3.4. Here, we show the

resulting ﬁt parameters from these tests, and discuss their relevance with respect to

the value of σ∆A calculated above.

Given a speciﬁc survey and systematic, we create 40 separations of a positive and negative W33 group. For each separation, we calculate the diﬀerence in their parameters for bias, bg,− − bg,+, and for the growth parameter, fv,− − fv,+. In Table

3.6 we show the means of these parameter diﬀerences for all 40 realizations, as well as the standard deviation of our realizations. In this setting, the mean parameter diﬀerence represent the systematic bias potentially added to our signal, while the standard deviations indicate whether or not these biases are consistent with zero.

In Fig. 3.15 we visually display all 40 realizations, as well as the mean parameter diﬀerences, for the four systematics templates (airmass, extinction, sky ﬂux, and PSF

FWHM).

The mean offsets and deviations are also displayed in Table 3.6. It can be seen here that most offsets are consistent with zero. Our largest offsets, within the systematic of the PSF FWHM, reach as large as −0.022 for bias and 0.013 for fv in the LOWZ

South sample. When we compare this to the intrinsic uncertainty in the splitting of the LOWZ South sample, at −0.031 for bias and 0.089 for fv, we can see that these oﬀsets are well within the standard deviation simply due to random ﬂuctuations.

131 Table 3.6: The average of parameter diﬀerences for our systematic-injected subsamples. Consistency with zero implies that the systematic in question will not disguise itself as galaxy orientation for our sample splitting. Here, the σ∆f are the uncertainties in the systematic error from each respective systematic, not the uncertainty in our measurement of ∆fv. Bias Growth

< bg,− − bg,+ > σ∆b < fv,− − fv,+ > σ∆f Extinction CMASS North 0.006 0.002 0.000 0.002 CMASS South 0.016 0.007 0.004 0.006 LOWZ North -0.002 0.004 0.007 0.003 LOWZ South 0.02 0.008 -0.007 0.007 Airmass CMASS North 0.000 0.002 0.000 0.002 CMASS South 0.005 0.006 -0.004 0.005 LOWZ North -0.000 0.001 0.001 0.001 LOWZ South -0.006 0.006 -0.001 0.010 Sky Flux CMASS North 0.001 0.003 0.000 0.003 CMASS South -0.002 0.008 -0.002 0.006 LOWZ North 0.000 0.002 -0.000 0.002 LOWZ South -0.004 0.006 0.004 0.005 PSF FWHM CMASS North -0.015 0.007 0.013 0.006 CMASS South -0.023 0.009 0.005 0.008 LOWZ North 0.005 0.013 -0.007 0.010 LOWZ South -0.022 0.012 0.006 0.013

132 Airmass Extinction

Sky ﬂux PSF FWHM

Figure 3.15: For each survey, we show the difference in clustering parameters for each pair of systematics-biased subsamples (i.e. with W33 scrambled but then each systematic template added in with the best-fit coefficient mθ). A non-biased result should be consistent with zero, for both parameters. Plotted are the differences in measured bias and measured fv for each survey, where the shaded regions represent 1 and 2 standard deviations. We see here that the systematics due to airmass, extinction, sky flux, and PSF FWHM are consistent with zero.

133 3.5.6 Phase II

p(∆bg, ∆fv) Measurement

At this point, we move into the second stage of the blinding procedure. In this

stage, clustering statistics are computed on the true W33-split subsamples, but we

blind ourselves to the value of ∆fv, and thus ∆A. The only information from the

ﬁts that is revealed in this stage is the parameter p(∆bg, ∆fv) (which is described in

Eq. 3.28 and contains no dependence on ∆A), and the χ2/dof of the resulting ﬁts.

2 In Fig. 3.16 we display both the χ /dof of each survey ﬁt, as well as ∆bg +

2 ∆fv/3. All of the ﬁts have χ values that are well within the range of acceptability.

However, we do note a consistent positive oﬀset for our blinded parameter, p(∆b, ∆f).

Speciﬁcally, we ﬁnd that p(∆bg, ∆fv) = 0.112 ± 0.031 (0.153 ± 0.042) for CMASS

(LOWZ).

It is possible that we are seeing higher clustering in the groups above and below the

fundamental plane, which correspond to diﬀerent measurements of bg. As explained

in Section 3.2, W33 is a proxy for orientation, but contains other information as well, which could lead to a subgroup selection eﬀect that is bias-dependent. For instance, the fundamental plane estimation may have preferentially selected central or satellite galaxies, depending on the subgroup, which could explain a bias diﬀerence.

The investigation of this bias diﬀerence could motivate future work, although it is outside the scope of this work to explore. A promising route would be Halo

Occupation Distribution (HOD) modeling of the subsamples, to identify where the diﬀerence arises (including whether it is associated with satellites or centrals). In any case, this result does not change our expected values of ∆fv, and therefore our measurement of intrinsic alignments should remain uncorrupted. As explained in

134 Appendix A.1, there could be an eﬀect on the measured size of the error bars, but we

estimate this to be below 5% in all realistic scenarios.

2 σFOG Measurement

As referenced in Section 3.5.2, we are interested in a ﬁnal test of our subsamples

before fully unblinding them. We measured the values of |Isplit| (see Eq. 3.46) for

each of our truly divided subsamples. Note that we measure and list here the absolute

value of Isplit only; we are still blinded to the parameter diﬀerences of fv and bg, and anything indicating the sign of those values. We ﬁnd that |Isplit| is equal to 0.07001

± 0.0257 for LOWZ South, 0.04749 ± 0.0179 for LOWZ North, 0.01924 ± 0.0143 for

CMASS South, and 0.00932 ± 0.0163 for CMASS North, where the uncertainties are

provided by the standard deviations on the random split samples discussed in Section

3.5.2.

It can be seen that the LOWZ values are inconsistent with a measurement of zero,

while our CMASS values are consistent with a measurement of zero between 1-2σ.

This is not unexpected; given our results for the values of p(∆bg, ∆fv) for these sub-

samples, we expect there to be some level of a potential diﬀerence in satellite/central

2 galaxy subscription, which would change the measurement of σFOG for each sample. However, we want to highlight the magnitude of this discrepancy, which is the im-

portant point for our measurement. Our “nightmare” scenario discussed a diﬀerence

2 2 of 22 (Mpc/h) in ∆σFOG, which corresponds to:

2 2 |σF OG,1 − σF OG,2| 2 ≈ 2. σF OG,full

2 where we have assumed that we have one subsample of galaxies with σFOG ≈ 0 (no

2 2 Finger of God) and another with σFOG ≈ 22 (Mpc/h) , which when averaged over the

135 full sample appear to be 11 (Mpc/h)2. From our measurements, we have found:

2 2 |σF OG,1 − σF OG,2| 2 ≈ 0.07. σF OG,full

2 This corresponds to a diﬀerence in σFOG of roughly 0.77, which propagates to an uncertainty in ∆fv of approximately 0.003. This is much less than our statistical uncertainty in ∆fv; in fact, if we are somehow underestimating the measurement of

2 σFOG with this exercise by a factor of 10, we still have a change in ∆fv less than our statistical errors in the worst case listed above.

2 As discussed in Sec. 3.5.2, we ﬁt our true samples for fv, bg, and σFOG on scales of 20–40 Mpc. It can be seen in Table 3.7 that in all cases, our resultant dispersions are consistent with zero given the errors provided by the random ﬁts. Furthermore, we derive the propagated error on ∆fv, and determine that in all cases this error is less than our statistical error on fv.

2 These two tests (measuring |Isplit| and measuring σFOG on small scales) comple-

2 ment each other in determining the effects of setting the value for σFOG. When testing the value of |Isplit| on the subsamples, we have a physical interpretation of what we are measuring, with small enough statistical errors that we have a detection of the differences of this measurement between the subsamples. This difference is at its largest value for LOWZ South, and propagates to an error in ∆fv of 0.003. However, this measurement specifically probes small scales, and it could be the case that our

2 model is collecting larger-scale eﬀects into its parameter value for σFOG. This issue

2 is resolved by our second test, which is a direct measurement of σFOG and therefore

2 picks up any and all quasilinear eﬀects that may be contained in the σFOG parameter.

2 2 The second test, in the worst case, resulted in ∆σFOG of 7.14 (Mpc/h) for CMASS North, which is consistent with zero, as the standard deviation was 7.56 (Mpc/h)2

136 2 Table 3.7: Errors on fv due to diﬀerences in σFOG for the true divided subsamples. Category CMASS NGC CMASS SGC LOWZ NGC LOWZ SGC 2 2 |∆σFOG| [Mpc/h] 7.14 4.30 6.32 3.58 2 Statistical error on ∆σFOG 7.56 11.00 8.40 10.41 Propagated Error to ∆fv 0.026 0.016 0.023 0.013 Statistical Error on ∆fv 0.040 0.063 0.059 0.089 Propagated Error / Statistical Error 0.65 0.25 0.39 0.15

for our random separations. Given the results from the ﬁrst test (a propagated error

2 on ∆fv much smaller than statistical errors) and the second test (a diﬀerence in σFOG that is consistent with zero) we have determined that setting the value of the Fin- ger of God when measuring fv and bg will not signiﬁcantly bias our results for these parameters.

3.5.7 Phase III: Final results

Throughout this research project, we have been careful to notate and adhere to our specific blinding procedures (Section 3.3.4). To be clear, all parts of this chapter were written before unblinding ourselves to the final results of our parameter fits for the true subsamples divided by orientation, except for the specific parts/changes listed below:

• Errors in the plotting scripts for Figures 3.14 and 3.17.

• Minor changes to the historical discussion in the introduction.

• A typo was ﬁxed in Eq. (A.9).

• The chapter conclusion.

137 Figure 3.16: Here, we plot p(∆b, ∆f) by the χ2 per degree of freedom of the fits. The solid colored lines indicate the limits of acceptable χ2 for our fits; it can be seen that all of our fits meet this criteria. The ’X’ markers indicate the average χ2 for each survey, while the ’O’ markers indicate the the χ2 for each subsample. All fits result in p(∆b, ∆f) < 0.2, the limit discussed in Sec. 3.3.4. However, they all show a consistent offset of approximately 0.11 for LOWZ and 0.15 for CMASS.

138 • This section of the chapter.

• The acknowledgements.

• Grammatical corrections (e.g. punctuation).

Now we discuss the final unblinded results of our split samples. In Fig. 3.18, we show the values of ∆fv and ∆bg for the separate subsamples. We see a consistent offset in both the bias parameter (which was expected given our results in Sec. 3.5.6) and in the growth parameter, which was the desired result. We display the final result for B in Fig. 3.17, and compare it to the theoretical predictions by Hirata (2009).

Our ﬁnal results for the intrinsic alignment values for B are −0.016 ± 0.018 for

CMASS North, −0.043 ± 0.029 for CMASS South, −0.033 ± 0.019 for LOWZ North,

and −0.022 ± 0.031 for LOWZ South. It can be seen that each individual result

for the SGC and NGC groups are consistent with each other, as they are less than

3σ apart. We combined our results into a total estimate of B for the LOWZ and

CMASS groups individually, and as a total value for both, and compared them to

theory, as described in Eq. 3.29. We ﬁnd an Obs/Theory ratio of 0.51 ± 0.33 for

CMASS, 0.76 ± 0.42 for LOWZ, and 0.61 ± 0.26 for the total sample. In all cases,

we see that this ratio is in between 1–2σ from the value of 1, and we conclude that

our results are consistent with the theoretical expectations. Finally, our combined

results for B are −0.024 ± 0.015 for CMASS, and −0.030 ± 0.016 for LOWZ, with a

total combined signal of −0.026 ± 0.011, which meets our criteria as evidence for the

intrinsic alignment magnitude.

139 Figure 3.17: Here, we display our ﬁnal results for the value of B along with theoretical predictions. The data point from Singh et al. (2015) was measured from the LOWZ sample using galaxy-galaxy lensing, and is labeled ‘Lensing.’ For each measurement we also show the theoretical prediction of a radial intrinsic alignment measurement at the calculated median luminosities of each respective sample, using the machinery of Hirata (2009). These predictions are shown as red ranges on the plot, overlapping the true measurement with the same sample luminosity.

140 Figure 3.18: For each subsample split, we show the values of ∆fv and ∆bg.

141 3.6 Conclusion

In this work, we have used the Fundamental Plane to divide galaxies from the

CMASS and LOWZ catalogs within SDSS BOSS DR12 into groups depending on their orientation seen by an observer on Earth. By measuring the diﬀerences in the clustering parameter fv of these subsamples, we have attempted to quantify the intrinsic alignment magnitude by measuring the dimensionless parameter B. We found that for CMASS (LOWZ), the measured B was −0.024±0.015 (−0.030±0.016).

We take their ratio relative to our theoretical prediction and combine the results, based on Eq. (3.29). We ﬁnd Obs/Theory = 0.61 ± 0.26; this constitutes evidence

(between 2 and 3σ) for a measurement of the radial intrinsic alignments, and is consistent with expectations (< 2σ diﬀerence).

Our result gives additional motivation for searches of similar intrinsic alignment eﬀects in samples other than LRGs, such as Emission Line Galaxies (ELGs) or Lyman-

Alpha Emitters (LAEs). It is important to note that in those samples, the analysis could become more complicated due to radiative transfer eﬀects inﬂuencing the relationship between ∆fv and the ellipticity of the galaxies.

Given our result, future RSD measurements should aim to minimize orientation- dependent biases in target selection to the extent possible, and understand the possible impact of intrinsic alignments on the full analysis chain. Even “null tests” in which the clustering statistics from various sub-samples are compared can be aﬀected, if the method of splitting into sub-samples is correlated with galaxy orientation. While in this project such a split was done deliberately to probe intrinsic alignments, in a cosmological parameter analysis using RSDs it is possible that an orientation-dependent

142 split could arise unintentionally, and thus cause a null test to “fail” even if selection eﬀects in the full parent sample are well understood.

Several choices were made in our analysis that could be improved for future studies.

The scatter in the Fundamental Plane could be reduced by incorporating velocity dispersion information, although the modeling of orientation-dependent eﬀects would be more complex. Multiple bins could be analyzed instead of the two-bin orientation system implemented here, which could improve the aggregate signal-to-noise ratio.

Our work could be extended to future surveys: for example the DESI experiment will produce an LRG sample that is 3 times larger than the BOSS sample used in this work (DESI Collaboration, 2016). Increases in both the sample size and the redshift range of target galaxies should allow for more robust measurements of the clustering parameters, and decrease error bar sizes in the resulting measurements of B.

Using radial methods to measure intrinsic alignments can complement the traditional ellipticity-based approach, in order to gain a better understanding of the behaviour of intrinsic alignments.

143 Chapter 4: The Eﬀects of Resonant Line Scattering During Recombination on the CMB Power Spectrum

4.1 Introduction

The hot, early universe consisted of a tightly coupled photon-baryon fluid. The interactions within this fluid were dominated by Thomson scattering between electrons and photons. However, as the temperature of the Universe fell, protons and electrons in this fluid were able to begin to permanently recombine into neutral hydrogen, in an event called recombination. With the onset of recombination, the free electron fraction fell, and the number of neutral hydrogen, and eventually neutral helium, particles increased (see Fig. 4.1). The cross section for photons to interact with these neutral particles is much smaller than with electrons, and so the photons then began to free stream outwards as the Cosmic Microwave Background (CMB).

Analysis of the CMB can give insights into the very early conditions of the Uni- verse. Due to the high amount of scattering between photons and electrons, the frequency proﬁle of the CMB is almost a perfect blackbody. However, there are spatial ﬂuctuations in the CMB temperature at the level of δT/T ≈ 10−5. These temperature anisotropies can be analyzed using the angular power spectrum to constrain parameters of cosmological models, such as Ωm and Ωb. The anisotropies

144 themselves are frequency-independent, since they are produced by Thomson scattering, a frequency-independent process. However, the appearance of neutral hydrogen during recombination resulted in resonant line scattering between photons and hydrogen, which produces frequency-dependent spectral distortions in the CMB blackbody spectrum, as well as frequency-dependent changes in the temperature anisotropies.

The imprint of these cosmological recombination lines on the CMB power spectrum was analyzed by Rubi˜no-Mart´ın et al. (2005), who focused on the three strongest lines of the Balmer, Paschen, and Brackett series, ﬁnding a maximum change in the

CMB power spectrum of 0.3 µKelvin for the H-alpha line. The eﬀects on the spectral changes in the CMB were computed by Rubino-Martin et al. (2006), tracking all Hydrogen states up to n = 30, with individual treatment for all angular momentum sub-states. This was extended in Chluba and Sunyaev (2007) to include particle collisions, and tracking Hydrogen states up to n = 100.

General CMB spectral distortions can have a variety of sources, and have been studied extensively (Sunyaev and Zeldovich, 1970; Burigana et al., 1991; Hu and Silk,

1993); see Sunyaev and Khatri (2013) for a review. In our speciﬁc case, these distortions would be due to the radiation emitted from the recombination process (Sunyaev and Zeldovich, 1969; Peebles, 1968; Dubrovich, 1975). Although the expected amplitudes of these distortions are too small to be seen with the current generation of CMB experiments, in principle they could be seen by the next generation, including Pixie

(Kogut et al., 2011), Prism (Andr´eet al., 2014) and Litebird (Suzuki et al., 2018;

Matsumura et al., 2014). Furthermore, Sathyanarayana Rao et al. (2015) studied what type of ground-based experiment would be needed to measure recombination lines in the CMB power spectrum, and Abitbol et al. (2017) looked at the prospects of

145 these measurements in the presence of foregrounds. The analysis of how uncertainties around recombination calculations could propagate to the CMB parameter estimates was studied by Rubino-Martin et al. (2010).

Resonant line scattering during recombination also eﬀects the polarization of the

CMB. The CMB anisotropies become ‘E-mode’ polarized primarily due to temperature quadrupoles present at the ‘last scattering’ of the photons, and as such provides a snapshot into the exact conditions at last scattering. This can be contrasted to so-called ‘B-mode’ anisotropies which are a signature of inﬂation in the primordial

Universe.

The change in CMB polarization anisotropies from cosmological recombination lines was studied by Hernandez-Monteagudo et al. (2007) who calculated the main transitions of the Balmer and Paschen series only. They found that the polarization power spectra were distorted in the order of 10−3 µKelvin2 which although small, would require a similar measurement precision as that needed for a detection of B- mode polarizations signaling gravitational waves from inﬂation.

In this chapter, I lay the mathematical groundwork for a general extension of the research by Hernandez-Monteagudo et al. (2007) with a formalism that allows all transitions for hydrogen to be accounted for. These distortions provide a direct determination of the redshift of recombination, as well as an independent measurement of the baryon density, which can be compared to other observations. We do this via a theoretical calculation of the time rate of change of atomic excited states in an anisotropic radiation ﬁeld. The results provide ﬁrst order corrections to the

Boltzmann equations of recombination, which result in second order corrections to the observed CMB power spectrum. The machinery developed in this work could also

146 serve to provide predictions for observed atomic lines in future CMB observations.

Although the distortions are small, they are frequency-dependent, which may provide a tool to use to extract their information. Because of the strong interest in constraining inflationary theories, efforts are being taken to measure CMB anisotropies to the level capable of detecting a B-mode, which is also the level needed for this work. Full calculations of resonant scattering’s effects on the polarization power spectra should therefore be complete by the time we are experimentally able to reach these precision ranges.

This chapter is outlined as follows. In Section 4.2, we discuss the formalism and notation behind our calculations. We evaluate changes to the atomic population in Section 4.3, while in Section 4.4 we ﬁnd the changes to the photon phase space distribution. In Section 4.5 we combine our photon and atomic changes to write the full equation sets together, in a way which is consistent with the notation used in

CLASS (Blas et al., 2011). We end in Section 4.6, where we discuss the possible uses for this work.

4.2 Formalism

Although it is common in the literature to track single atomic transitions independently, since we are analyzing spectral distortions of the perturbations to the atomic populations, we will need to independently track all of the populations of the atomic substates, not just the overall population of the n-state itself. For this purpose, we will use density matrix formalism, which allows us to track a statistical ensemble of quantum states. In general, for a group of atoms with a mixture of states |ψαi which are weighted by Wα, then we deﬁne the density operator:

147 Boltzmann Equations Dark Photons Matter Neutrinos

Compton Scattering Metric

ElectronsElectrons Protons Coulomb Scattering

Boltzmann Equations Dark Photons Matter Neutrinos

Thomson Scattering Metric Resonant Line Scattering

Electrons /Protons Neutral Recombination Hydrogen

Figure 4.1: Charts indicating particle interactions before and after recombination. Top panel: Protons and electrons interact via Coulomb scattering, while photons and electrons interact via Compton scattering. Neutrinos and Dark Matter (essentially) do not interact with any of the other forms of matter. However, all forms of matter interact to some extent gravitationally, referenced by their relationship with the metric. Bottom panel: Protons and electrons recombine to form neutral hydrogen, while still Thomson scattering with the photons. Neutrinos and Dark matter are still essentially ignoring all other forms of matter. All forms of matter interact gravitationally, referenced by the relationship with the metric. The red arrow indicates the resonant line scattering interaction between neutral hydrogen and photons, which is the focus of this Chapter.

148 X ρ ≡ Wα |ψαi hψα| (4.1) α which is summed over all possible states. The expectation value of an operator upon this weighted ensemble can be found with:

hMi = T r [ρM] (4.2)

Following Venumadhav et al. (2017), we choose basis states which are both orthonor-

mal and complete:

hΦI |ΦJ i = δIJ (4.3)

X |ΦI i hΦI | = 1 (4.4) I So the matrix elements of the density operator ρ are:

X ρIJ = h|ΦJ i hΦI |i = Wα hΦI |Ψαi hΨα|ΦJ i (4.5) α

The number of matrix elements in our operator depends on how many atomic transitions we want to take into account. For example, if we look speciﬁcally at the rate at which the hydrogen ground state changes, then we want the density matrix for

g the hydrogen ground state, ρIJ . This number can become as high as needed, and by leaving it in operator form, we keep the general solution. The density operator allows us to track, in matrix form, the population of each speciﬁc sub-state of our entire system, in order to accurately compute transitions between diﬀerent sub-states.

To account for resonant line scattering, we need to calculate the change in a given state’s population due to all possible interactions with radiation; this includes absorptions resulting in departures from that state, absorptions leading to repopulation of that state, and emissions leading to depopulation and repopulation as well. We

149 will calculate these four terms separately, and then recombine the results, ﬁnding the

changes to the Boltzmann equation beyond thermodynamic equilibrium.

The general equation we will be using to describe changes in the atomic density

matrix due to photon absorption is given by Happer (1972). We know that for

“depopulation pumping”, the time rate of change of the density matrix is:

∗ dρµν 1 X hµ| E · D |mi hm| E · D |αi = ραν ~Γ dt ~ i(Eγ − Emα) − αm 2 (4.6) 1 X hα| E · D |mi hm| E∗ · D |νi − ρµα i(E − E ) + ~Γ ~ αm γ mν 2 Here, E is the electric ﬁeld vector of the incident radiation, D is the dipole moment operator, Γ is the width of the excited state, Eγ is the energy of the absorbed photon, and Eab is the energy diﬀerence between any two states, Ea − Eb.

The subscripts µ, m, α, and ν refer to different matrix elements within the density matrix. For the specific case of the ground state of hydrogen we are summing over all possible destinations of that excitation- i.e., we are summing over all the substates of the first excited state of hydrogen. So our basis states are:

|ΦI i = (|2si , |2pxi , |2pyi , |2pzi) (4.7)

When we then look at the density matrix, we can write:

ρaa ρam ρIJ = (4.8) ρma ρmn

Where ρaa is a sub-matrix describing the probability of ﬁnding a hydrogen atom in

the 2s state, and ρmn is a 3x3 matrix describing the populations of each state within

the triplet 2p state. The oﬀ-diagonal elements describe interference between the 2s

and 2p states. We assume here that these oﬀ-diagonal elements vanish, since the time

150 of existence is inversely proportional to the energy diﬀerence of the two elements in

question. For large populations, these terms should be statistically zero and not have

a noticeable eﬀect on the changes in sub-state populations.

In Eq. 4.6, when we look at states like |αi, what we really need to specify is the

full description of that state using quantum numbers:

|αi = |n, l, j, mji (4.9)

Where n is the principal quantum number, l is the angular quantum number, j is the

sum of the orbital and spin angular momentum, and mj is the magnetic quantum

number. For the 2s states,

|αi = |2, 0, 1/2, ±1/2i (4.10)

While for the 2p states:

|αi = |2, 1, 3/2, [±1/2, ±3/2]i (4.11)

So we can write our sub-matrices from Eq. 4.8 as: s1 s2 ρaa = (4.12) s3 s4

and   ρ11 ρ12 ρ13 ρ14  ρ21 ρ22 ρ23 ρ24  ρmn =   (4.13)  ρ31 ρ32 ρ33 ρ34  ρ41 ρ42 ρ43 ρ44 Here, the dimensionality of the submatrices for a given state is 2j + 1. In general, we can write the matrix elements of ρmn in terms as irreducible tensor moments, with the following equation:

j p X j−m2 Psm = (2j + 1) (2s + 1) (−1) m1,m2 (4.14) j s j j × ρm1m2 −m2 m m1

151 where the superscript, j, describes the total angular momentum of the state being described. For each state, the ms values can be described by either our density matrix

ρmn, or through a separate basis, Psm. We can write the inverse equation to ﬁnd the

initial matrix elements from the irreducible ones: r j X 2s + 1 j−m2 j s j j ρm1m2 = (−1) Psm (4.15) 2j + 1 −m2 m m1 sm In the appendix we list the irreducible tensor moments for speciﬁc states in this simple example, where a superscript p indicates the ﬁrst excited state submatrix, and a superscript s indicates the ground state.

4.3 Atomic State Equations

Taking resonant line-scattering into account during recombination will change both the sub-state populations of hydrogen atoms as well as the energy distribution of the photons. This section specifically focuses on the changes on the atomic populations, due to different processes (Fig. 4.3). For each subsection below, we calculate the rate of a specific transition from n to n0, where we later sum these transitions to obtain the net effect; we also use the density matrix formalism described in Sec. 4.2, so that we can track the full statistical ensemble of each atomic population.

For a given state n, there are four processes to consider. The effect of absorption in both bringing atoms up to state n from some state m, absorption from state n to some other state m, the effect of emission in decaying from state n to some state m and decaying to state n from some state m. All four terms will affect the rate of change within the state n. In Section 4.3.1, we derive the equations for a change in the ground state of a transition, due to the absorption of a photon. In Section

4.3.2 we discuss the change in the excited state of a transition for the same photon

152 Spontaneous Stimulated Absorption Emission Emission

Figure 4.2: The ways in which light and matter interact for resonant line scattering. In absorption, a photon (red squiggly line) is absorbed by an atom. The atom increases in energy, jumping to a higher state. This process is discussed in Sections 4.3.1 and 4.3.2. For spontaneous and stimulated emission, the atom loses energy by photon emission and drops to a lower energy state. These processes are relevant for Sections 4.3.3 and 4.3.4.

absorption process. In Sections 4.3.3 and 4.3.4 we ﬁnd the equations for the changes in the ground and excited states due to the emission of a photon. These eﬀects are combined and summarized in Section 4.3.5.

4.3.1 Ground State Change Due to Absorption

First, we will describe the eﬀect of a photon absorptions causing a lower-energy state to become a higher-energy state. For this, we must return to Eq. 4.6, which we will simplify by tailoring it to match our speciﬁc situation. First, we take a closer look at the matrix elements:

hm| E · D |αi

153 The dipole operator D is equal to the charge of the atom multiplied by the position

operator for each electron: N Xe D = Q ri = Qr (4.16) i=1 Where here we have simpliﬁed to the case of the hydrogen atom in the second equality. Therefore we can expand:

1 X q hm| E · D |αi = QE · hm| r |αi = QE−q hm| rq |αi (−1) (4.17) q=−1 where in the last step we have expanded our vector dot product into a sum within the spherical basis. The spherical basis will allow us to more easily perform calculations because of the inherent symmetries it contains. A brief description of the spherical basis is given in Section B.2. Now, we can further expand our expression:

0 0 0 0 hm| rq |αi = hn , l , j , mj| rq |n, l, j, mji 0 0 0 j k j (4.18) j −mj 0 0 0 = (−1) hn , l , j | |r| |n, l, ji 0 −mj q mj Where the 3 by 2 matrix is a Wigner 3-j symbol, with k = 1 since we are dealing with a 3-vector in rq. The 3-j symbol is an alternative method of describing angular momentum addition in quantum mechanics, and an introduction can be found in

Section B.4. The second element in our expression is a reduced matrix element, and in general it can be expanded as:

where we have ignored the nuclear spin eﬀects. The 3 by 2 matrix within the curly

brackets is a 6-j symbol, introduced in Section B.5. We can combine these expressions

154 Table 4.1: A reference for the labelling of states throughout this chapter. Although we combine all of these effects with a unified nomenclature in Sec. 4.3.5, each specific calculation requires names for different numbers of states in the excited and ground configurations.

Sec. 4.3.1 Sec. 4.3.2 Sec. 4.3.3 Sec. 4.3.4 Absorption, Ground Absorption, Excited Emission, Excited Emission, Ground

to get the full equation for the matrix element:

It is important to be clear of our notation. Here, we have:

0 0 0 0 0 |mi = |n , l , j ,F , mF i (4.21)

|αi = |n, l, j, F, mF i (4.22)

00 00 00 00 00 |µi = |n , l , j ,F , mF i (4.23)

000 000 000 000 000 |νi = |n , l , j ,F , mF i (4.24)

Each of these variables represents a speciﬁc ‘state’ of the hydrogen atom, where we n, l, and j are the principal, angular, and magnetic quantum numbers of that state, and

F and mf refer to the nuclear quantum numbers. However, we have already stated we are ignoring nuclear spin, and since our sums over the states indicated by Greek

155 indices are really sums over the same n, l, and j values, then the only difference is in the sub-level state, defined by mj. Furthermore, there are only three possibilities of our ground sub-state, which differ in their mj values - they share the other quantum numbers. So, we can simplify the above to write:

0 0 0 0 |mi ≡ |n , l , j , mji (4.25)

|αi ≡ |n, l, j, mji (4.26)

00 |µi ≡ |n, l, j, mj i (4.27)

000 |νi ≡ |n, l, j, mj i (4.28)

The m state is the state to which we are transitioning, and the greek indices refer to diﬀerent sub-states which we may be transitioning from. The labels for these states, which are unique for each transition we analyze here, are shown in Table 4.1.

At this point, we can calculate the second matrix element from the ﬁrst term in Eq.

4.6, using the same strategies as above. It is important to note that when expanding the imaginary components of E in the spherical basis, we have:

∗ q ∗ q ∗ (E )q = (−1) (E−q) = (−1) E−q (4.29)

The second matrix element then becomes:

156 By using the symmetries of 6-j symbols (as described in Section B.5), we can now write the numerator of the ﬁrst term in our depopulation expression (Eq. 4.6):

∗ 2 X q+q0 ∗ hµ| E · D |mi hm| E · D |αi = Q (−1) Eq E−q0 q,q0 −m00−m0 × (−1) j j (2j + 1) (2j0 + 1) j0 1 j j 1 j0 (4.31) × 0 0 00 0 −mj q mj −mj q mj l0 j0 1/2 2 × hn, l| |r| |n0, l0i hn0, l0| |r| |n, li j l 1 At this point, we have all the tools necessary to fully describe the change in the ground state due to photon absorption. The second term in Eq. 4.6 follows the same decomposition as the ﬁrst, with the only changes being the speciﬁc states from which we are transitioning. By incorporating the use of irreducible tensor moments, our expression for the change in the ground state becomes:

d 000 g p 0 X j−mj P 0 0 = (2j + 1) (2J + 1) (−1) dt J M µν (4.32) 0 j J j d g × 000 0 00 ρµν −mj M mj dt where: d ρg = T − T (4.33) dt µν 1 2

2 0 0 2 Q ∗ l−m00+q0−m0 +l0 l j 1/2 T = E E 0 (−1) j j 1 B q −q j l 1 0 0 0 0 j 1 j j 1 j hn, l| |r| |n , l i 0 0 00 0 −mj q mj −mj q mj (4.34) p (2J + 1) 000 j J j 0 0 j−mj × hn , l | |r| |n, li p (−1) 000 (2j + 1) −mj M mj g 0 × PJM (2j + 1) (2j + 1) 157 and:

2 0 0 2 Q 0 0 ∗ l−mj +q−mj +l l j 1/2 T = E 0 E (−1) 2 C q −q j l 1 0 0 0 0 j 1 j j 1 j hn , l | |r| |n, li 0 0 000 0 −mj q mj −mj q mj (4.35) p 0 0 (2J + 1) j−mj j J j × hn, l| |r| |n , l i p (−1) 00 (2j + 1) −mj M mj g 0 × PJM (2j + 1) (2j + 1) where we have used the deﬁnitions:

B ≡ ~[i (Eγ − Emα) − ~Γ/2] (4.36)

C ≡ ~[i (Eγ − Emα) + ~Γ/2] (4.37)

This equation is accurate for an electric ﬁeld E~ . We next need to generalize our

equation to be applicable where we don’t know the exact electric ﬁeld vectors of

the incoming photons, we only know the photon phase space density, which can be

anisotropic. In the expressions above, we must change: ∗ ∗ Eq E−q0 −→ Eq E−q0 = Z 3 (4.38) 4~ d k ∗ 2 3 fq (ω, nˆ) f−q0 (ω, nˆ) V 0 (2π) ∗ where fq (w, nˆ) is the q-th component of the phase space density of photons in a speciﬁc directionn ˆ and frequency ω.

This can be handled if we perform a multipole decomposition on the phase space density, in terms of the Stokes Parameters (introduced in Section B.7). We will deﬁne: f++ f+− fI + fV −fQ + ifU fαβ = = (4.39) f−+ f−− −fQ − ifU fI − fV

Which allows us to write:

Z 3 ∗ d k ~ω X ∗ Eq E−q0 = 3 fβα (ω, nˆ) α,qβ,−q0 (4.40) 0 (2π) α,β=+,−

158 Where refers to the polarization of the photons. In order to evaluate this integral,

we will write the polarizations in terms of spin-weighted spherical harmonics (intro-

duction in Section B.3), and separate out the angular components. The following

equations separate the angular dependence from the frequency dependence of the

distribution terms: r X 4π ∗ f = f Y 0 (4.41) I 2l + 1 I,lm lm lm r X 4π ∗ f = f Y 0 (4.42) V 2l + 1 V,lm lm lm r X 4π ∗ − f + if = (−f + if ) Y 2 (4.43) Q U 2l + 1 E,lm B,lm lm lm r X 4π ∗ − f − if = (−f − if ) Y −2 (4.44) Q U 2l + 1 E,lm B,lm lm lm We will also need the following expression, which describes the polarization vectors in terms of spin-weighted spherical harmonics:

r4π [ ] (ˆn) = ∓ Y ±1 (ˆn) (4.45) ± q 3 1,−q

as well as the integral identity, Equation B.20. After combining all of the above steps, we ﬁnd that:

∗ X φ+q 1 1 λ E E 0 = (−1) K (4.46) q −q q q0 −φ λφ λφ where:

Z 2 " k dk ~ω λ 1 1 λ λ 1 1 λ Kλφ = 4π 3 1 + (−1) fI,λφ − 1 − (−1) fV,λφ (2π) 0 0 1 −1 0 1 −1 # 1 1 λ 1 1 λ + (−f + if ) + (−f − if ) E,λφ B,λφ −1 −1 2 E,λφ B,λφ 1 1 −2 (4.47)

159 With these deﬁnitions, we can return to our main purpose: generalizing Eq. 4.32

for an anisotropic photon distribution. When we include Eq. 4.46, we ﬁnd that each

term has ﬁve 3-j symbols. These terms are expressions for a single state to state

transition, but to ﬁnd the resultant change for all states, we will need to sum over all possible substates. A sum over ﬁve 3-j symbols makes use of the following identity:

j4 j5 j6 j7 j8 j9 X X X X X X

m4=−j4 m5=−j5 m6=−j6 m7=−j7 m8=−j8 m9=−j9 × (−1)j4+j5+j6+j7+j8+j9 (−1)−m4−m5−m6−m7−m8−m9 j j j j j j j j j × 4 2 6 5 3 4 7 1 8 (4.48) m4 m2 −m6 m5 m3 −m4 m7 m1 −m8 j j j j j j × 8 6 9 9 5 7 m8 m6 −m9 m9 −m5 −m7 j j j j j j j j j = 1 2 3 1 2 3 1 7 8 m1 m2 m3 j4 j5 j6 j9 j6 j5

After using this relation, we ﬁnd that the second term and the ﬁrst term sum to

the same 3j and 6j symbols with only a diﬀerence of the phase factor:

0 (−1)λ+J+J (4.49)

From the triangle inequality of the 3-j symbol, we know that if one of these three

exponents is zero, then the others must be equal to each other, which will result in

this factor being equal to 1. If all three factors are present, then the eﬀect is a second

order perturbation eﬀect, which we can safely ignore. To ﬁrst order, we can assume

that this factor is equal to 1. This leaves us with our current expression for the

time rate of change of the reduced density matrix element of the ground state due to

photon absorption:

160 d g X l0+l+φ+λ+M+j+j0 P 0 0 = (−1) dt J M J,M,λ,φ,n0,l0 " # −Q2Γ × p(2J 0 + 1) (2J + 1) 2 2 2 (Eγ − Emα) + ~ Γ /4 0 g (4.50) × (2j + 1) (2j + 1) KλφPJM l j 1/2 2 × hn, l| |r| |n0, l0i hn0, l0| |r| |n, li j0 l0 1 λ J 0 J λ 1 1 λ J 0 J × j j j j0 j j φ M 0 −M This expression is accurate to first order in perturbation theory. We can be more precise by specifically writing out the equation for δP , the first-order part of this expression. We know that our term Kλφ and our term PJM can both be up to arbitrarily high order, but to our approximation, their combination cannot exceed

first order. Therefore, if we’re looking at the change in the first order value: X X λ J 0 J δP˙ 0 0 = (...) J M φ M 0 −M λφ JM (4.51) ¯ ¯ × (δKλφPJM + KλφδPJM ) Where an overbar indicates the non-perturbed value. For each term, we have simplifications we can make. For a value to be unperturbed, this means that its indices must be zero. This then simplifies the 3-j symbol, and kills the sums over all

4 indices.

We also know: Z 0 0 ∗ 2 hn , l | |r| |n, li = Rn0l0 rRnlr dr (4.52)

So that:

hn0, l0| |r| |n, li = (hn, l| |r| |n0, l0i)∗ (4.53) and

hn0, l0| |r| |n, li hn, l| |r| |n0, l0i = | hn0, l0| |r| |n, li |2 (4.54)

161 After simplifying, and renaming our J’ to J and M’ to M, we end up with the fol-

lowing expression for the change in the lower state of energy level En due to absorption

0 of a photon and a jump to En0 , where the sum is over n > n: lower 0 Γ ˙ n X 2 l +l δPJM = − Q (−1) 2 2 2 (Eγ − Emα) + Γ /4 abs n0l0 ~ l j 1/2 2 × (2j0 + 1)| hn0, l0| |r| |n, li |2 j0 l0 1 (4.55) " 0 0 2j + 1 J 1 1 × (−1)j +j+J δK P¯n ( )1/2 J−M 00 2J + 1 j0 j j # 1 ¯ n + √ K00δP 3 JM Next, we can make some more simpliﬁcations. Looking at:

Γ 2 2 2 (Eγ − Emα) + ~ Γ /4 we know that Γ is small compared to the entire spectrum, and so in the limit of small

Γ: Γ 2π 2 2 2 → δ(E − Emα) (4.56) (Eγ − Emα) + ~ Γ /4 ~ We can also simplify this delta function with the integral contained in the δK term.

We ﬁnd that:

Γ δKJ,−M 2 2 2 → (Eγ − Emα) + ~ Γ /4 3 " kmα J J 1 1 (−1) + 1 δfI,J−M (kmα) (4.57) ~0π 0 1 −1 # 1 1 J − δf (k ) −1 −1 2 E,J−M mα

Similarly, we ﬁnd:

Γ 2 ¯ √ 3 K00 2 2 2 → kmαf0(kmα) (4.58) (Eγ − Emα) + ~ Γ /4 ~ 30π 162 We can put these into our ﬁnal expression. In order to make it a little easier to visualize, we’re deﬁning values ∆ which are dependent upon n, l, j, n’, l’, j’, J, and

M. Their exact deﬁnitions are displayed in Section B.8. These variables will make it easier to understand physically what is happening in these equations. Furthermore, we will later re-write our total eﬀect in a way that will not be dependent on the ∆ values, so their use is temporary. Thus we have, for the change in lower state of energy level n due to photon absorption, jumping to energy n’:

lower " #

˙ n X 1 2 3 n (4.59) δPJM = ∆ δfI,J−M (kmα) + ∆ δfE,J−M (kmα) + ∆ δPJM abs n0>n

4.3.2 Excited State Change Due to Absorption

In Section 4.3.1, we have calculated the eﬀect for an absorption bringing an atom from state n to some upper state n0 > n. We examined how the population of n changed, given that it lost population due to photon absorptions. Now, we look at the inverse absorption eﬀect; how does the population of n change due to states below it absorbing photons, and being introduced into the state n? We begin with the general expression from Barrat, J.P. and Cohen-Tannoudji, C. (1961):

n dραβ 1 X 00 = hα| e∗ · D |µi ρn hν| e · D |βi (4.60) dt T λ µν λ p µν

Where Tp is set by: 1 π = 2 δ(k − kγ) (4.61) Tp ~ c Because of the finite amount of greek letters available to specify states, we will redefine our state quantum numbers, specifically for this subsection, here, as shown in Table

4.1:

|αi ≡ |n, l, j, mji (4.62)

163 0 |βi ≡ |n, l, j, mji (4.63)

00 00 00 00 |µi ≡ |n , l , j , mj i (4.64)

00 00 00 000 |νi ≡ |n , l , j , mj i (4.65)

With these deﬁnitions in place, we move through the same steps as for those in

Section 4.3.1. The ﬁnal expression is very similar to our previous one, with one main

diﬀerence- the density matrix of state n is dependent upon a sum over the matrices

of the states n00 which are below n. This makes sense, because the absorption of

light by those states is proportional to the number of atoms in those states. In

contrast, our previous expression was proportional to its own density matrix, because

the absorption was occurring from that state. Our expression for the population

change due to lower states absorbing photons is:

d n X l00+l+1+λ+M+J P 0 0 = (−1) dt J M J,M,λ,φ,n0,l0 (2J 0 + 1)(2J + 1)(2j + 1)1/2 × (2j00 + 1) 2 (4.66) Q 00 n00 00 00 2 × (2j + 1)(2j + 1)KλφPJM | hn , l | |r| |n, li | Tp  0  2 J λ J l00 j00 1/2 J 0 λ J   × j 1 j00 j l 1 −M 0 −φ M  j 1 j00 

Where we have used the same Kλφ as we have used in the previous term. Again, our next step is to look at the perturbations. We have the same expansion rules as in the ﬁrst term, except here the simpliﬁcation involves a 9-j symbol (see Section

B.6). The ﬁnal result, which is the change in the upper state En as a result of photon

00 absorption from the lower state En00 , where n > n , is:

164 upper 2 00 00 ˙ n X Q 3/2 l+l +j+j +J δPJM = (2j + 1) (−1) Tp abs n00l00 l00 j00 1/2 2 × | hn00, l00| |r| |n, li |2 j l 1 (4.67) " 00 00 j 1 j × (−1)M δK P¯n J−M 00 1 j J 0

00 00 # 1 00 ¯ n p 00 j j J + √ K00δP 2j + 1 3 JM j j 1

With the deﬁnition of Tp (Eq. 4.61), we can combine the above with the integrals in K to get simpliﬁcations such as:

k3 ¯ 1 γ K00 = √ f0(kγ) (4.68) Tp 3~0π

Deﬁning constants similar to the previous term, we can write the change in upper state of energy En as a result of absorption from En0 :

upper " # 0 ˙ n X 7 8 9 n δPJM = ∆ δfI,J−M (kγ) + ∆ δfE,J−M (kγ) + ∆ δPJM (4.69) abs n0

4.3.3 Excited State Change Due to Emission

After solving for the absorption terms, we next move to the emission terms - how do populations change due to the emissions of photons? We can save time here by noting that the emission process is the time reversal of the absorption process.

Therefore, if we time reverse the equation for the change in ground state due to absorption, we can obtain the change in the excited state due to emission.

165 The interacting Hamiltonian for resonant scattering can be written as:

X q 0 q ∗ + 0 Hint ∝ (−1) qδ hu| dq |li |ui hl| aδ(k ) − (−1) qδ hl| dq |ui |li hu| aδ (k ) k0,δ,q,u,l (4.70)

where l indicates the lower state, u indicates the upper state, and a+ and a are the

creation and annihilation operators. To ﬁnd the time reversal of this Hamiltonian, we

need to ﬁrst exchange the upper and lower states in the Hamiltonian: l ↔ u. This

change could also be achieved with a switch between the creation and annihilation

operators:

+ aδ ↔ aδ (4.71)

and the electric ﬁeld vector on which the complex conjugate is acting:

∗ (δ)q ↔ (δ)q. (4.72)

However, we must also add the eﬀect of spontaneous emission, which is not initially

present as there is no “spontaneous absorption” term. This, when combined with the

change in the annihilation and creation operators, amounts to changing the phase

space density matrix:

fαβ → fβα + δαβ (4.73)

From how the creation and annihilation operators act with the Hamiltonian H0:

[H0, aˆδ] = −~ωaδ

+ + H0, aˆδ = ~ωaδ (4.74) we can see that we must change the frequency by a sign.

ωk ↔ −ωk (4.75)

166 For the terms due to photon emission, our photon phase space density becomes:

f f f + 1 f f + f + 1 −f − if ++ +− → ++ −+ = I V Q U (4.76) f−+ f−− f+− f−− + 1 −fQ + ifU fI − fV + 1

We then move through the same calculation as for the absorption terms, with the altered representation of the Stokes Parameters. This results in a new expression for the expectation of electric ﬁeld vectors:

q+q0 ∗ q+q0 (−1) E−qEq0 = (−1) Aδq−q0 X 0 1 1 λ (4.77) + (−1)q +φ K q q0 −φ λφ λφ

Where we have deﬁned: Z 2 k dk ~ω 8π A ≡ 3 (4.78) (2π) 0 3 which speciﬁcally accounts for spontaneous emission. Although A appears to be

inﬁnite, it will only be present in our ﬁnal expression when combined with a delta

function. From here, the calculation is identical to those in Sections 4.3.1 and 4.3.2,

with the addition of the term involving the Kronecker delta. Again, we take a look

at the perturbations to K and P , and keep only ﬁrst order terms. The result, which

is the change in excited state n due to emissions to lower state n0, is: upper −ΓQ2 ˙ n X 0 δPJM = 2 2 2 (2j + 1) (Eγ − Emα) + Γ /4 em n0 ~ l j 1/2 2 × | hn0, l0| |r| |n, li |2 j0 l0 1 " Z dkk2 ω 8π ~ n (4.79) × 3 δPJM (2π) 0 3 0 1/2 0 0 2j + 1 J 1 1 + (−1)l+l +j+j +J+M δK P¯n J−M 00 2J + 1 j0 j j # l+l0 1 ¯ n + (−1) √ K00δP 3 JM

167 The states used here are referenced in Table 4.1 for clarification. Note that here, we must perform a similar simplification to that shown in Eq. 4.57. The δKJ−M term contains an integral over k, which integrates over the Lorentzian at the beginning of the above equation. After performing similar simplifications to that done in previous sections, we define some convenient constants to write the change in excited state n, due to emissions to lower state n’ as: upper " #

˙ n X 5 6 4 n δPJM = ∆ δfI,J−M (kmα) + ∆ δfE,J−M (kmα) + ∆ δPJM (4.80) em n0

4.3.4 Ground State Change Due to Emission

Finally, we calculate the fourth possible transition: the change in the ground state as a result of photon emission. We are looking at the time reversal of the change in upper state as a result of absorption, therefore we make the same switches that we did in the previous section (see Table 4.1 for clarity). Therefore, the change in lower state n as a result of photon emission from upper state n00 is: lower 2 ˙ n Q 00 00 2 δPJM = | hn , l | |r| |n, li | Tp(ω → −ω) em " l00 j00 1/2 2 × (2j + 1)3/2 (2j00 + 1)1/2 j l 1 Z 2 j+l+j00+l00 dkk ~ω 8π n00 1+J j j J (4.81) × (−1) 3 δPJM (−1) 00 00 (2π) 0 3 j j 1 00 00 00 00 h 00 l+l +j+j +J 1 00 1/2 ¯ n j j J + (−1) √ (2j + 1) K00δP 3 JM j j 1 00 # 00 j 1 j i + (−1)M δK P¯n J−M 00 1 j J

Using constants deﬁned in the Section B.8:

lower " # 0 ˙ n X 11 12 10 n (4.82) δPJM = ∆ δfI,J−M (kγ) + ∆ δfE,J−M (kγ) + ∆ δPJM em n0>n 168 4.3.5 Total Atomic Equation

Now that we have the equations for the changes in the upper and lower states, due to both absorption and emission, we can combine them all into a single equation, for a given state n, due to all possible absorptions and emissions.

" upper upper#  lower lower

˙ n X ˙ n ˙ n X ˙ n ˙ n δPJM = δPJM + δPJM + δPJM + δPJM  (4.83) n0n em abs

Where here, “upper” (“lower”) refers to the interactions of a level n with all those that are below (above) it- in this sense, it is the “upper” (“lower”) state in the interaction. Also, where necessary, we must change any reference from n00 to n0, since it is only a dummy variable, indicating the level we are summing over. For notational clarity, we do not explicitly write the summation over all l states within each n state.

Note also an implicit dependence of each P on n, l and j. Only the n is explicitly written.

We can set the total time rate of change to zero, since the rate of expansion of the Universe is slow when compared to the atomic interaction rate at recombination.

In this way, we can write the equation for the atomic densities completely in terms of the photon densities. Here, I will also make use of the substitutions to Fγ and Gγ as used in the CLASS program. We end up with:

n X −1 0 δPJ0 = − N nn0 θn (4.84) n0 where:

169 " J X 1+5+7+11 (2J + 1)(−i) θ 0 = ∆ 0 A(ck 0 ,T )F (k 0 ) n n n 4 n n γJ n n n # (4.85) 5 2+6+8+12 + √ ∆ 0 A(ckn0n,T )(Gγ0 + Gγ2) 4 6 n n

9+10 X 3+4 Nnn0 = ∆nn0 + δnn0 ∆nn00 (4.86) n00

r 2J + 1 2πω eω/T A(ω, T ) ≡ . (4.87) 4π 4T (eω/T − 1)2 To simplify the notation, we have deﬁned:

∆1+2 ≡ ∆1 + ∆2 (4.88)

This is ﬁnal result of the changes to the atomic populations due to resonant line scattering.

4.4 Photon Phase Space Density

In Section 4.3 we have calculated how including resonant line scattering will change the atomic populations during recombination. However, we also need to consider how the absorptions and emissions will aﬀect the photon phase space density. We begin with the deﬁnition of the photon phase space density matrix:

~ D + ~ ~ E fαβ(k) = aβ (k)aα(k) (4.89) where f is composed of the creation and annihilation operators of a speciﬁc wavenumber k. To analyze how f is changing, we look at what is causing the change in f. We can write the time rate of change in terms of the commutator:

d i fαβ = [Hint, fαβ] (4.90) dt ~ 170 The interacting Hamiltonian is:

X q Hint = diEi = (−1) dqE−q (4.91) q where we have written the sum in terms of spherical components, with q equal to 0,

1, or −1. We can deﬁne the electric ﬁeld component q as:

1/2 " # 2π~ X √ 0 0 ik~0·~r ∗ 0 q + 0 −ik~0·~r E = i w 0 (k )a (k )e − (k )(−1) a (k )e (4.92) q V k qδ δ −qδ δ k0 Notice here that the epsilons are unit vectors with two indices. The first indicates the spatial component of the vector, while the second indicates its polarization. When we take the commutator of the electric field with the density matrix, and use the commutation relations between the creation and annihilation operators, we find:

1/2 d 2πωk X q h q −i~k·~r D ∗ ~ + ~ E i fαβ = − (−1) dq(−1) e (k)a (k) + h.c. (4.93) dt V qα β ~ q where h.c. stands for the hermitian conjugate of the ﬁrst term in brackets. Explicitly, we switch the indices α and β and take the complex conjugate. We can also expand the dipole operator.

X q dq = |µi hν| dµν (4.94) µν This states that the dipole operator is a weighted sum over all pairs of atomic states,

µ and ν, with an appropriate coeﬃcient. Our convention here is that µ is the higher

energy state. These two states can have any quantum numbers. When we insert this

expansion, we have the expression:

1/2 d 2πωk X q h q q −i~k·~r ∗ ~ D ~ E i fαβ = − (−1) d (−1) e (k) Oµνβ(k) + h.c. (4.95) dt V µν qα ~ q,µ,ν where we have deﬁned:

~ + ~ Oµνβ(k) ≡ |µi hν| aβ (k) (4.96)

171 In order evaluate our expression for the density matrix, we must ﬁnd the expectation ~ value of the Oµνβ(k) operator. To do this, we will again use Ehrenfest’s Theorem:

d ~ i D ~ E Oµνβ(k) = [H, Oµνβ(k)] (4.97) dt ~

Earlier, the required Hamiltonian was only from the perturbative interaction of the electromagnetic ﬁeld, since the contribution from the non-perturbed Hamiltonian cancelled out. Here, we need both contributions. The non-perturbative term gives us:

H0 = Hatom + Hem (4.98)

So that: i D ~ E i D ~ E [H0, Oµνβ(k)] = (Eµ − Eν + ~ωk) Oµνβ(k) (4.99) ~ ~ To ﬁnd the perturbed portion, we have the same interacting Hamiltonian as we did before:

X q0 Hint = djEj = (−1) dq0 E−q0 (4.100) q0 Again expanding the dipole operator in terms of two other states, where λ is the

higher state:

X q dq0 = |λi hφ| dλφ (4.101) λφ our interacting Hamiltonian becomes:

1/2 X X X 0 0 2π ωk H = i (−1)q |λi hφ| dq ~ int λφ V λφ q0 k0 (4.102) h 0 0 ik~0·~r ∗ 0 q0 + 0 −ik~0·~ri × −q0δ(k )aδ(k )e − q0δ(k )(−1) aδ (k )e

We then perform a calculation to ﬁnd the commutator of this Hamiltonian with the ~ operator Oµνβ(k). During the calculation, we set the two terms to zero which refer

172 to 2-photon processes, since this is a second-order eﬀect. We also make use of the

commutator deﬁnitions:

D E ~ + ~ ~0 fδβ(k)δ~kk~0 = aβ (k)aδ(k ) (4.103) D E ~ ~ + ~0 (fδβ(k) + δδβ)δ~kk~0 = aβ(k)aδ (k ) (4.104)

We will also deﬁne:

h|νi hλ|i ≡ ρλν (4.105)

There is a subtlety here. Since this expectation value is proportional to e−i∆Et/~, we will ﬁnd that it averages to zero as long as there is an energy diﬀerence, ∆E = Eλ−Eν.

Therefore, this expectation value is only non-zero if the energy diﬀerence is zero, which means that the two states λ and ν must share the same quantum numbers n, l, and j. They will be distinguished only by their diﬀering values of mj.

After performing the calculation, and combining it with the zeroeth order commutator, we ﬁnd:

d D ~ E i D ~ E Oµνβ(k) = (Eµ − Eν + ~ωk) Oµνβ(k) dt ~ 1/2 1 X X X 2π~ωk q0 i~k·~r − (−1) 0 (~k)e (4.106) V −q δ ~ levels(γ) q0 γ

h q0 q0 i × dγµρνγ(δδβ + fδβ(k)) − dνγργµfδβ(k)

Here, the sum over ”levels” is the sum over the quantum numbers n, l, and j of all possible γ values. The indices of the density matrices are referring to the mj values within those speciﬁc quantum numbers, with the γ sum being over those magnetic quantum number values. To be explicit, the γ sum is over all values; the ﬁrst term

173 will give zero unless γ is within the same level as ν, and the second term will give

zero unless γ is within the same level as µ.

By deﬁning:

Eµ − Eν ωµν ≡ (4.107) ~ and 1/2 1 X X X 2π~ωk q0 i~k·~r Ψ (~k) ≡ − (−1) 0 (~k)e µνβ V −q δ ~ levels(γ) q0 γ (4.108) h q0 q0 i × dγµρνγ(δδβ + fδβ(k)) − dνγργµfδβ(k) then our equation becomes:

d D E D E O (~k) = i(ω + ω ) O (~k) + Ψ (~k) (4.109) dt µνβ µν k µνβ µνβ

The solution to this equation: ~ D E Ψµνβ(k) ~ i(ωµν +ωk)t Oµνβ(k) = C1e − (4.110) i(ωµν + ωk) diverges when the energy diﬀerence between states is equal to the energy of the

photon. To deal with this, we deﬁne a small parameter η << 1 which will indicate

that we are ramping up the perturbation very slowly, from a time at negative inﬁnity.

Then our solution becomes:

~ ηt D E Ψµνβ(k)e ~ i(ωµν +ωk)t Oµνβ(k) = C1e − (4.111) η − i(ωµν + ωk)

To solve for C1, we note that at a time of negative infinity, we would expect our operator to have a value of zero, since in the infinite past there would be no perturbations from the collisions between atoms and photons. The second term is trivially zero, while the first term oscillates because of the i in the exponent. Therefore, we have to set C1 to zero. After some algebra we obtain:

D ~ E i ~ Oµνβ(k) = Ψµνβ(k) (4.112) ωµν + ωk + iη 174 At this point, we can return to our original goal: changes in the photon phase space

density. By plugging in the previous equation into Eq. 4.95, we have:

1/2 d 2πωk X q fαβ = − (−1) dt V ~ q,µ,ν (4.113) q q −i~k·~r ∗ ~ i ~ × (−1) dµνe qα(k) Ψµνβ(k) + h.c. ωµν + ωk + iη With: 1/2 1 X X X 2π~ωk q0 i~k·~r Ψ (~k) ≡ − (−1) 0 (~k)e µνβ V −q δ ~ levels(γ) q0 γ (4.114) h q0 q0 i × dγµρνγ(δδβ + fδβ(k)) − dνγργµfδβ(k)

In order to simplify this expression, we can look at the Hermiticity of the operators

involved. We deﬁne:

q X X q −i~k·~r ∗ Dµναβ(k) ≡ dµνe qα(k)Ψµνβ(k) (4.115) q µν

and √ − 2πω κ ≡ √ k (4.116) ~V so that:

d q i q∗ −i fαβ = κ Dµναβ(k) + Dµνβα(k) (4.117) dt ωµν + ωk + iη ωµν + ωk − iη

Let’s deﬁne the Hermitian part of the operator:

q q Aµναβ(k) ≡ Herm(Dµναβ(k)) (4.118)

And the anti-Hermitian part:

q q Bµναβ(k) ≡ AntiHerm(Dµναβ(k)) (4.119)

175 Plugging this into our deﬁnition of the change in photon phase space density, we ﬁnd: " d q q i fαβ = κ (Aµναβ(k) + Bµναβ(k)) dt ωµν + ωk + iη # (4.120) q∗ q∗ i − (Aµνβα(k) + Bµνβα(k)) ωµν + ωk − iη

We can simplify this to: d 2 fαβ = κ 2 2 dt (ωµν + ωk) + η (4.121) q q × Aµναβ(k)η + i(ωµν + ωk)Bµναβ(k) We can see that if the D operator is either totally Hermitian or anti-Hermitian in the indices of α and β, we can eliminate one of these terms. First, let’s examine this operator. We can write D out by explicitly expanding Ψ:

q X X X q0 q ∗ Dµναβ(k) = κ (−1) dµνqα−q0δ µν γ qq0 (4.122) h q0 q0 i × dγµρνγ(δδβ + fδβ) − dνγργµfδβ

Since we only need the answer to first order in perturbation theory, we can take the atomic density matrix ρ to first order, or the photon density matrix f to first order, since having both to first order creates second-order terms which we neglect. If the Hermiticity is identical in both cases, then we have a Hermitian or anti-Hermitian matrix. First, we will examine the state where the atom is unperturbed:

0 ργν → ργν = δγν (4.123)

In this case, our operator simpliﬁes greatly to:

q X ∗ q Dµναβ(k) → κ qα−qβ(−1) (4.124) q We can see that this is Hermitian, via interchange of α and β and a complex conjugate.

When expanding the sum, they are equal. Therefore we have veriﬁed: if the atom

176 is unperturbed, then the operator is Hermitian. Next, let’s look at the case of the

photons being unperturbed:

(0) fδβ → f δδβ (4.125)

So our operator becomes:

q X X X q0 q ∗ Dµναβ(k) = κ (−1) dµνqα−q0β µν γ qq0 (4.126) h q0 (0) q0 (0)i × dγµρνγ(1 + f ) − dνγργµf

Using the following identities:

q ∗ −q q (dµν) = dνµ (−1) (4.127)

and

∗ ργν = ρνγ (4.128)

So by taking the Hermitian conjugate with respect to α and β of the operator D, and letting q → −q0 and q0 → −q, and ﬁnally in the ﬁrst term letting ν and γ switch,

while letting µ and γ switch in the second term, we ﬁnd that:

q X X X q ∗ −q HermConj(Dµναβ) = κ dµνqα(−1) −q0β µν γ qq0 (4.129) h q0 (0) q0 (0)i × dγµρνγ(1 + f ) − dνγργµf

Again, this is identical to the term before we took the Hermitian conjugate. We can see, to ﬁrst order in perturbation theory, that the operator D is Hermitian. We can then write:

˙ 2η q fαβ = κ 2 2 Dµναβ(k) (4.130) (ωµν + ωk) + η Using the identity: η lim = πδ(x) (4.131) η→0 x2 + η2

177 and also making the substitution from a sum over ﬁnite k to an integral over all k,

we can write our expression for the change in photon phase space density:

1 X X q ∗ q0 f˙ = ω δ(ω + ω ) d (−1) 0 αβ 2π k µν k µν qα −q δ ~ µνγ qq0 (4.132) h q0 q0 i × dγµρνγ(δδβ + fδβ) − dνγργµfδβ

Next, we want to rewrite this in terms of the multipole moments of the stokes param-

eters, on both sides of the equation. We can write: r 1 Z X 2l + 1 f˙ = d2nˆ S f˙ (4.133) χlm 2 χlmαβ 4π αβ αβ with another similar equation for the inverse relation:

r X ∗ 4π f = S 0 0 0 f 0 0 0 (4.134) αβ χ l m αβ χ l m 2l0 + 1 χ0l0m0

Here, S is a tensor which is composed primarily of a spin-weighted spherical harmonic

and constants. The greek indices here can be either + or −. Here, I will write out

178 the deﬁnitions of S:

SIlm++ = Ylm

SIlm−− = Ylm

SIlm+− = SIlm−+ = 0

SV lm++ = Ylm

SV lm−− = −Ylm

SV lm+− = SV lm−+ = 0 (4.135) SElm++ = SElm−− = 0

2 SElm+− = −Ylm

−2 SElm−+ = −Ylm

SBlm++ = SBlm−− = 0

2 SBlm+− = −iYlm

−2 SBlm−+ = iYlm We can divide our expression into two terms, looking at the last bracket in 4.132.

The ﬁrst term contains a kronecker delta, while the second doesn’t. Starting with the second term, we will begin by replacing the dipole operators using the Wigner Eckart theorem. We ﬁnd:

0 j 1 j j 1 j dq dq = Λ(−1)−µ−ν µ ν ν γ (4.136) µν νγ −µ q ν −ν q0 γ Where Λ is a term containing two 6-j symbols, the reduced matrix elements, and some pre-factors which do not depend on the values of µ, ν or γ. Next, we expand the atomic density matrix:

X j J j ρ = G (−1)−µ µ µ (4.137) γµ JM −µ M γ JM 179 Where: s 2J + 1 jµ nµ GJM = (−1) PJM (4.138) 2jµ + 1 Next, we rewrite the our spherical unit vectors in terms of spin-weighted spherical harmonics:

∗ 4π 1+q −α δ 0 = αδ (−1) Y Y 0 (4.139) qα −q δ 3 1q 1q

Note here that the variables α, β, and δ can only be ±1. They are all summed over

both values, so there are 8 terms in reality. Next, we can rewrite the tensors S. In

general, since S is always proportional to a spin-weighted spherical harmonic, or is

zero, we can write them compactly:

α−β Sχlmαβ = C1(χ, α, β)Ylm (4.140)

∗ 0 δ−β ∗ Sχ0l0m0δβ = C1 (χ , δ, β)(Yl0m0 ) (4.141)

where the values for C1 are deﬁned in Table 4.2. Our next step is to rewrite the

two spin-weighted spherical harmonics as a sum:

1/2 −α δ X 9(2lA + 1) 1 1 lA Y1q Y1q0 = 0 4π q q mA lAmA

1 1 l × A (−1)α−δ+mA Y δ−α (4.142) α −δ δ − α lA,−mA

Now, we note that with this single spherical harmonic, and the two from the expansion of our S tensors, we have 3 spin-weighted spherical harmonics, as well as a sum over the angle that they are functions of. Therefore, using Eq: B.20 we can write:

Z 0 1/2 2 α−β δ−β ∗ α−δ ∗ (2l + 1)(2l + 1)(2lA + 1) d nYˆ (Y 0 0 ) (Y ) = lm l −m lA,mA 4π (4.143) 0 0 0 α+β+m +mA l l lA l l lA × (−1) 0 m −m −mA β − α δ − β α − δ

180 For this state speciﬁcally, we know that jγ = jµ, since the substates themselves must be within the same overall atomic level, otherwise the atomic density matrix element would be zero. We can simplify this further by performing a sum:

X j J j j 1 j j 1 j (−1)−µ−ν−γ µ µ µ ν ν µ −µ M γ −µ q ν −ν q0 γ µνγ (4.144) 0 jν +jµ+jγ +q+q 1 1 J 1 1 J = (−1) 0 q q −M jµ jµ jν Once we have these terms, we can simplify again using the following identity: X 1 1 J 1 1 lA 0 0 q q −M q q mA qq0 (4.145) 1 = δ δ lAJ mAM 2J + 1 Thus, our second term becomes:

ωk X X X f˙ (II) = χlm 4π ~ JM αβδ χ0l0m0 0 1+α+β+m +M+jν × δ(ωµν + ωk)(−1) (δα)(2l + 1)

∗ 0 × C1(χ, α, β)C1 (χ , δ, β)ΛµνγGJM (4.146) l l0 J l l0 J × m −m0 M β − α δ − β α − δ 1 1 J 1 1 J × fχ0l0m0 α −δ δ − α jµ jµ jν The ﬁrst term has the exact same simpliﬁcations as this one, with an extra part

that collapses more quickly due to the delta-function from the spontaneous emission.

Moving forward, we will examine this spontaneous emission term, and come back to

the stimulated emission and absorption terms, which are similar.

The spontaneous emission term is proportional to:

X 1 1 l f˙ ∝ αβC (χ, αβ) (4.147) χ,lm 1 α −β β − α αβ

181 Table 4.2: C1(χ, α, β) Parameter(χ) ++ – +– –+ I 1 1 0 0 V 1 -1 0 0 E 0 0 -1 -1 B 0 0 -i i

This expression can be worked out for each Stokes Parameter. For the I parameter,

since the constant is zero unless α and β are equal, we can write:

X 1 1 l f˙ ∝ α2C (I, αα) I,lm 1 α −α 0 α 1 1 l 1 1 l = + (4.148) 1 −1 0 −1 1 0 1 1 l = 1 + (−1)l 1 −1 0

We find very similar expressions for the other 3 Stokes Parameters. Now, for the stimulated emission and absorption terms, we can find simpler expressions since we are calculating to first order in perturbation theory. For each term, there are two possibilities: the photons are perturbed and the atomics are unperturbed, or the atoms are perturbed and the photons are unperturbed. Mathematically speaking:

˙ ¯ ¯ δfχlm → δPl−mfχ0l0m0 + Pl−mδfχ0l0m0 (4.149)

When we perform simpliﬁcations in each of these terms due to zeroes in the 3-j

symbols, we find our final expressions, for each stokes parameter. We find that:

182 X ωk δf˙ = δ(ω + ω )Q2(−1)lµ+lν +jµ+jν (2l + 1)(2j + 1) I,lm 4π µν k µ µν ~ 2 lµ jµ 1/2 2 × (2jν + 1) | hnµ, lµ| |r| |nν, lνi | jν lν 1 " n −δP ν 1 1 l 1 1 l × l,−m 1 + (−1)l [(2jν + 1)(2l + 1)] jν jν jµ 1 −1 0 l f0 1 1 l + 1 + (−1) √ (4.150) 2l + 1 1 −1 0 " nµ nν # 1 1 l δPl,−m 1 1 l δPl,−m × p − √ jµ jµ jν 2jµ + 1 jν jν jµ 2jν + 1 2δf + √ Ilm 3(2l + 1) " ## ¯nµ ¯nν 1 1 0 P00 1 1 0 P00 × p − √ jµ jµ jν 2jµ + 1 jν jν jµ 2jν + 1

˙ With very similar expressions for δfElm. The V and B stokes parameters are ˙ diﬀerent: Only the l = 1 term survives for the V parameter, and we ﬁnd that δfBlm =

0. The beginning sum over µ and ν indicates that we sum over all possible interactions between any permissible pairs of states. These are not the sub-states mj: those sums were cancelled out earlier in the calculation. But they are sums including the orbital angular momentum l and the energy level n.

4.5 Results

We have separately calculated the eﬀects of resonant line scattering on the atomic population and on the photon population. In order to see the theoretical results of resonant line scattering on the power spectrum of the CMB, a cosmological calcula- tor such as CLASS should be modiﬁed to contain these changes to the Boltzmann equations. In this section, we re-write our results in terms of the variables used in

183 CLASS. Speciﬁcally from Ma and Bertschinger (1995b), which follows a diﬀerent mul-

tipole decomposition than the one we have used above. The conversion involves the

temperature moments, since they are both of spin-weight zero. We can write:

(2l + 1)(−i)lω eω/T δf (~k, τ) = F (~k, τ) (4.151) I,l0 4T (eω/T − 1)2 γl

Here, Fγ is the variable used in CLASS. So, we need to plug this in for these terms in the photon distribution equations and the atomic distribution equations. The time derivative should move straight through to the variables we are changing, since if we use comoving k-bins, the average temperature of the Universe remains constant. The change to the temperature intensity gives us:

X ωk F˙ = δ(ω + ω )Q2(−1)lµ+lν +jµ+jν (2l + 1)(2j + 1) γl 4π µν k µ µν ~ 2 lµ jµ 1/2 2 × (2jν + 1) | hnµ, lµ| |r| |nν, lνi | jν lν 1 4T (eω/T − 1)2 × ω(−i)l eω/T " n −δP ν 1 1 l 1 1 l × l,0 1 + (−1)l [(2jν + 1)(2l + 1)] jν jν jµ 1 −1 0 (4.152) f 1 1 l + 1 + (−1)l √ 0 2l + 1 1 −1 0 " nµ nν # 1 1 l δPl,0 1 1 l δPl,0 × p − √ jµ jµ jν 2jµ + 1 jν jν jµ 2jν + 1 ω(−i)l eω/T + √ Fγl 2 3T (eω/T − 1)2 " ## ¯nµ ¯nν 1 1 0 P00 1 1 0 P00 × p − √ jµ jµ jν 2jµ + 1 jν jν jµ 2jν + 1

For the polarization, let us deﬁne:

184 X ωk − τ˙ ≡ δ(ω + ω )Q2(−1)lµ+lν +jµ+jν (2l + 1)(2j + 1) 4π µν k µ µν ~ 2 lµ jµ 1/2 2 × (2jν + 1) | hnµ, lµ| |r| |nν, lνi | (4.153) jν lν 1 " # ¯nµ ¯nν 2 1 1 0 P00 1 1 0 P00 × √ p − √ 3(2l + 1) jµ jµ jν 2jµ + 1 jν jν jµ 2jν + 1 And:

X ωk S ≡ δ(ω + ω )Q2(−1)lµ+lν +jµ+jν (2l + 1)(2j + 1) E,l0 4π µν k µ µν ~ 2 lµ jµ 1/2 2 × (2jν + 1) | hnµ, lµ| |r| |nν, lνi | jν lν 1 " −δP nν l,0 1 1 l 1 1 l l × 1 + (−1) (4.154) [(2jν + 1)(2l + 1)] jν jν jµ 1 1 −2 f 1 1 l + 1 + (−1)l √ 0 2l + 1 1 1 −2 " nµ nν ## 1 1 l δPl,0 1 1 l δPl,0 × p − √ jµ jµ jν 2jµ + 1 jν jν jµ 2jν + 1 So we can write:

˙ δfE,l0 = −τδf˙ E,l0 + SE,l0 (4.155)

Note here that SE,l0 is identically zero unless l = 2. The result, for the polarization term, is:

r 2 T (eω/T − 1)2 G˙ = −τG˙ + 2 S [δ + 5δ ] (4.156) γl γl 75 ω eω/T E,20 l2 l0

4.6 Future Work

The next steps forward for these equations would involve the direct implementation of our algorithms into codes such as CAMB or CLASS. In the previous section,

185 we have detailed our equations in terms of the variables used in CLASS. However, in order to apply these, there are several changes to the code which will be necessary.

First, the CLASS code must be modiﬁed so that the photon distribution function is dependent on the wavelength of light. Currently, the code only cares about the temperature of the distribution, since for Thompson scattering the interaction cross- section is independent of light frequency, and therefore the light distribution remains as a blackbody. Our introduction of resonant line scattering would require individual populations of light at each frequency to be kept. This can be done by requiring

CLASS to keep an array of values for the distribution, as opposed to a single value.

In the initialization ﬁle for CLASS, one can deﬁne Nγ, the number of photon bins, and also the resolution to which the frequency is measured.

CLASS would also need to change how it stores the populations of hydrogen atoms. To accurately follow the derived changes to the Boltzmann equations in Sec.

4.5, each time-step would require a knowledge of the populations of all substates for the hydrogen atoms. An individual density matrix for each sub-state of hydrogen would need to be tracked, as opposed to simply the ionized and neutral populations.

The ﬁnal step, although the most important computationally, is the simplest in terms of modifying the program. We would change the Boltzmann equations to include the resonant line scattering eﬀects that we have calculated, by modifying the time rate of change of the equations. Currently, CLASS follows the equations outlined in Ma and Bertschinger (1995a), which we reproduce below: 4 2 δ˙ = − θ − h˙ (4.157) γ 3 γ 3

1 θ˙ = k2( δ − σ ) + an σ (θ − θ ) (4.158) γ 4 γ γ e T b γ 186 8 3 4 8 9 1 F˙ = θ − kF + h˙ + η˙ − an σ σ + an σ (G + G ) (4.159) γ,2 15 γ 5 γ,3 15 5 5 e T γ 10 e T γ,0 γ,2

k F˙ = lF − (l + 1)F ) − an σ F (4.160) γ,l 2l + 1 γ,(l−1) γ,(l+1 e T γl

where l ≥ 3. k G˙ = lG − (l + 1)G γ,l 2l + 1 γ,(l−1) γ,(l+1) (4.161) 1 δ + an σ −G + (F + G + G ) δ + l,2 e T γ,l 2 γ,2 γ,0 γ,2 l,0 5

Here, ne is the number density of electrons, σT is the Thompson scattering cross section, h and η are the synchronous metric perturbations, σ = F2/2 is the shear

stress, and θ = 3kF1/4 is the divergence of ﬂuid velocity. These equations encompass

the changes in the photon distributions given Thompson scattering processes; they

should be modiﬁed to include the resonant line scattering processes calculated in Sec.

4.5.

The program would need to calculate over each wavelength of the distribution

function. CLASS is made to be easily-modiﬁable, however the changes this would

require are not trivial to apply. A working knowledge of the structure in CLASS

would be beneﬁcial to this endeavor.

187 Appendix A: Radial Tidal Alignment

A.1 Eﬀect of a diﬀerence in bias on the random-split error bars

In Section 3.5 we used random splits of the BOSS samples to determine the error bars on ∆bg and ∆fv. This procedure assumes that the two subsamples have the same clustering amplitude. A possible concern is that if the clustering amplitudes are diﬀerent, then the error bars on ∆fv may have additional cosmic variance contributions and hence may be larger than we calculate based on the random-split method.

The purpose of this appendix is to investigate this possibility.

When we separate our sample into two groups based on orientation, we ﬁnd the correlation functions of, and measure clustering parameters for, each of the subgroups.

Since both subgroups are pulled from the same sample, an excess of high-biased galaxies in one subgroup would imply a lack of high-biased in the other, and we would see a diﬀerence in the bias parameters measured between the samples, ∆bg.

Since the sample measurements are correlated, we must use the covariance matrix of density ﬂuctuations between the two subsamples in order to gain an understanding of the eﬀects on the variance of our measurements.

188 We consider the issue of whether error bars are increased in Fourier space; the result could be transformed to real space as done in §3.4.2 if it turned out to be important. For a given density mode k, and a diﬀerence in bias of the samples ∆b, then the covariance matrix of errors in the bias measurement between those samples can be written as:

∆b 2 1 ∆b ∆b (1 + 2b ) P (k) + n¯ (1 + 2b )(1 − 2b )P (k) C = ∆b ∆b ∆b 2 1 , (A.1) (1 + 2b )(1 − 2b )P (k) (1 − 2b ) P (k) + n¯ wheren ¯ is the number density within each sample, assuming they are the same, P (k) is the power spectrum at wavenumber k, and ∆b is the diﬀerence in the measured bias of the two samples. The variance in the diﬀerence of our estimate of the biases of the two parameters can be written as: ˆ ˆ ˆ ˆ ˆ ˆ Var(C11 − C22) = Var(C11) + Var(C22) − 2Cov(C11, C22) (A.2) 2 2 2 = 2C11 + 2C22 − 4C12. For two samples with the same bias, we can see that this variance is dependent only upon the shot noise and power in our sample:

16 16 Var(Cˆ − Cˆ )(∆b = 0) = P (k) + . (A.3) 11 22 n¯ n¯2

However, for a general diﬀerence in bias, we ﬁnd:

16 (∆b)2 8 (∆b)2 Var(Cˆ − Cˆ ) = + 8P 2(k) + P (k) 2 + . (A.4) 11 22 n¯2 b2 n¯ 2b2

An effective way to analyze the increase in the size of our error is to look at the ratio between this variance difference with a non-zero bias difference, to that with a bias difference of zero:

Var(Cˆ − Cˆ )(∆b = 0) 1 nP¯ (k)(1 + 2¯nP (k) ∆b2 11 22 = 1 + . (A.5) ˆ ˆ Var(C11 − C22) 4 1 +nP ¯ (k) b 189 With this, we can see how our error bars are affected by a given difference in bias of the two samples. For the case relevant to our interests, we state in Sec. 3.3.4 that a change in ∆b of 0.2 could have an effect on the size of our error bars, and indicate further investigation. Given that for our samples, b ≈ 2, then this offset would mean

∆b/b ≈ 0.1. From Eq. A.5 we can then see that, for values ofnP ¯ (k) ≈ 1, the modiﬁcation to the error bar would be ≈ 0.4%. Even given an extreme case where nP¯ (k) ≈ 7, we would see an increase in error of only about 3.3%.

A.2 Comparison of Intrinsic Alignment Values

In Section 3.2, we introduced Eq. (3.21) to explain how to compare our intrinsic alignment values to those from Singh et al. (2015). Here, we go through the derivation to that equation. We need to relate Eq. (53) of Hirata (2009) to Eq. (9) of Singh et al. (2015). The ﬁrst of those equations of the 2D correlation function is: Z Z ∞ AI bDC1ρcritΩm W (z) wg+(rp) = 2 dz dkz π D(z) 0 Z ∞ 3 k⊥ (A.6) × dk⊥ 2 2 P (k, z)sin(kzΠmax) −∞ (k⊥ + kz )kz 2 × J2(k⊥rp)(1 + βµ ), where AI is the intrinsic alignment measurement, bD is the galaxy bias, C1ρcrit is

fixed at a value of 0.0134, Πmax is the line of sight separation, Ωm is the matter density, D(z) is the growth function (G(z) in our notation), and k⊥ and kz are the wavenumbers perpendicular and parallel to the line of sight, respectively. β is the ratio of fv/bg, and µ is the cosine of the angle between a given direction and the line of sight. Overall, this function wg+(rp) is the 2D projected correlation function between the galaxy density field and the galaxy intrinsic shear. W (z) is defined as:

p (z)p (z) Z p (z)p (z) −1 W (z) = A B A B dz , (A.7) χ2(z)dχdz χ2(z)dχdz

190 where pA(z) and pB(z) are the redshift probability distributions for shape and density sample, respectively, and χ(z) is the comoving distance to a given redshift z. We wish

to compare Eq. (A.6) to:

Z ∞ bκ wδ+(rp) = − Pm(k)J2(krp)k dk, (A.8) 2π 0

where wδ+(rp) is the 2d projected correlation function between matter density and

galaxy intrinsic shear, Pm(k) is the matter power spectrum at wavenumber k, and bκ

is the measurement of intrinsic alignments. The notational diﬀerences of ”g” and ”δ”

are due to Eq. (A.8) citing the correlations with the matter power spectrum, which

means we must set bD = 1 when setting the equations equal. Next, we will take the

limit of large line of sight distance (Πmax). This results in a quickly oscillating sine

function within A.6, so that we can simplify (using the residue theorem) the integral

over kz: Z ∞ 3 k⊥ π dkz 2 2 sin(kzΠmax) → k⊥. (A.9) 0 (k⊥ + kz )kz 2 Next, we know that the window function W (z) integrates to 1. Therefore, Eq. (A.6)

becomes Z ∞ AI bDC1ρcritΩm 1 wg+(rp) = dk kP (k, z)J2(krp) (A.10) 2π D(z) 0 when considering a narrow redshift range. By comparing this to Eq. (A.8), we can

see that: A C ρ Ω I 1 crit m = −b . (A.11) D(z) κ

By inserting the deﬁnition of C1ρcrit, changing to our notation of the growth function,

and using Eq. (3.11), we arrive at Eq. (3.21):

Ω B = −0.0233 m A . (A.12) G(z) I

191 Appendix B: Resonant Line Scattering

B.1 Irreducible Moments

Here we list the irreducible tensor moments from Eq. 4.14 for speciﬁc states in this simple example, where a superscript p indicates the ﬁrst excited state submatrix, and a superscript s indicates the ground state.

192 p P00 = ρ11 + ρ22 + ρ33 + ρ44 √ " r r # p 3 1 1 3 P = 3 − ρ44 − √ ρ33 + √ ρ22 + ρ11 10 5 15 15 5 √ " r # p 1 2 1 P = 12 √ ρ43 + ρ32 + √ ρ21 1−1 10 15 10 √ " r # p 1 2 1 P = − 12 √ ρ34 + ρ23 + √ ρ12 11 10 15 10 √ p P2−2 = 2 [ρ42 + ρ31] √ p P2−1 = 2 [−ρ43 + ρ21]

p P20 = ρ11 − ρ22 − ρ33 + ρ44 √ p P21 = 2 [ρ34 − ρ12] √ p P22 = 2 [ρ24 + ρ13]

p P3−3 = 2ρ41 (B.1) √ p P3−2 = 2 [−ρ42 + ρ31] √ p 2 h i P = √ ρ43 − 3ρ32 + ρ21 3−1 5 p 1 P = √ [ρ11 − 3ρ22 + 3ρ33 − ρ44] 30 5 p −2 P = √ [ρ34 + ρ12] 31 5 √ p P32 = 2 [−ρ24 + ρ13]

p P33 = −2ρ14

s P00 = s1 + s4

s P10 = s1 − s4 √ s P11 = − 2s3 √ s P1−1 = 2s2

193 B.2 Spherical Basis

Here, we discuss the basics of decomposing a vector into the spherical basis. In the Cartesian basis, we could decompose a vector A~ into:

~ A = Axex + Ayey + Azez (B.2)

In the spherical basis, this expansion becomes:

~ X µ A = −A1−1 − A−11 + A00 = (−1) Aµ−µ (B.3) µ=−1,0,1 Where we can deﬁne:

(Ax + iAy) (Ax − iAy) A1 = − √ ,A0 = Az,A−1 = √ (B.4) 2 2 and:

(ex + iey) (ex − iey) 1 = − √ , 0 = ez, −1 = √ (B.5) 2 2 where the e values indicate unit vectors in the Cartesian basis and the variables indicate unit vectors in the spherical basis. We can derive from the above equations that:

∗ µ Aµ = (−1) A−µ (B.6)

For more information, see Thompson (2004).

B.3 Spin-Weighted Spherical Harmonics

Traditional spherical harmonics are sets of complete, orthogonal functions which are useful to describe functions on the surface of a sphere. While there are diﬀerent conventions for normalization, here we use the following deﬁnition: s 2l + 1 (l − m)! Y (θ, φ) = (−1)m P (cosθ)eimφ (B.7) l,m 4π (l + m)! l,m

194 where Pl,m(cos)θ are the associated Legendre polynomials. Because they are orthonor- mal, we will make good use of the following:

Z π Z 2π ∗ 0 Yl,mYl0,m0 dΩ = δl,l0 δm, m (B.8) θ=0 φ=0

We will also need:

∗ m Yl,m = (−1) Yl,−m (B.9)

Spin-weighted spherical harmonics are a generalization to traditional spherical har-

monics, with a spin-weight s applied, in the top-right of the term in our notation. In

fact, applying a spin-weight of 0 is identical to using a traditional, non-spin-weighted

spherical harmonic:

0 Yl,m = Yl,m (B.10)

They also have an orthogonality relation:

Z π Z 2π s s∗ 0 Yl,mYl0,m0 dΩ = δl,l0 δm, m (B.11) θ=0 φ=0

and a phase relation:

s∗ s+m −s Yl,m = (−1) Yl,−m (B.12)

Finally, the equations we use in this calculation are generally relations between

existing spin-weighted spherical harmonics. However, to complete this brief reference,

use the following to calculate their values from ﬁrst principles: s (l + m)!(l − m)!(2l + 1) Y s (θ, φ) = (−1)m sin2l(θ/2) l,m 4π(l + s)!(l − s)! (B.13) l−s X l − s l + s × (−1)l−r−seimφcot2r+s−m(θ/2) r r + s − m r=0 For a deeper look into the relationship between monopole spherical harmonics and

spin-weighted spherical harmonics, see Dray (1985).

195 B.4 3-j Symbols

The Wigner 3-j symbol is an alternative to Clebsch-Gordon coefficients, used when adding angular momenta in quantum mechanics. We use 3-j symbols in our calculations because their many symmetries allow for faster calculations, and easier checks for correctness. These 3-j symbols can be defined in terms of Clebsch-Gordon coefficients:

j1−j2−m3 j1 j2 j3 (−1) = √ hj1m1j2m2| |j3(−m3)i (B.14) m1 m2 m3 2j3 + 1 where each j and m is an angular momentum quantum number. The primary beneﬁt of the 3-j symbols comes in using their symmetries to quickly evaluate zero terms, or matching terms. Below, we list the required values of a 3-j symbol. If any of these conditions are not met, then that 3-j symbol is equivalent to zero.

• m1 + m2 + m3 = 0

•| j1 − j2| ≤ j3 ≤ j1 + j2

• (j1 + j2 + j3) is an integer.

• mi is within [−ji, −ji + 1, ..., ji − 1, ji]

Furthermore, we can quickly relate pairs of 3-j symbols through the following symmetries. First, even permutations of a 3-j symbol do not change its value.

j j j j j j j j j 1 2 3 = 2 3 1 = 3 1 2 (B.15) m1 m2 m3 m2 m3 m1 m3 m1 m2

Second, odd permutations can be re-written with a phase factor: j j j j j j 1 2 3 = (−1)j1+j2+j3 2 1 3 m1 m2 m3 m2 m1 m3 (B.16) j j j = (−1)j1+j2+j3 1 3 2 m1 m3 m2

196 We can change the sign of the bottom row with a phase factor:

j j j j j j 1 3 2 = (−1)j1+j2+j3 1 3 2 (B.17) −m1 −m3 −m2 m1 m3 m2

There are two orthogonality relations we make use of in our calculation. First, a sum over the m-values is deﬁned by: 0 X j1 j2 j3 j1 j2 j3 2(j3 + 1) 0 m1 m2 m3 m1 m2 m3 m1m2 (B.18)

= δ 0 δ 0 {j j j } j3j3 m3m3 1 2 3

Where {j1j2j3} is the triangular delta, equal to 1 when its contents satisfy the triangle inequality, and zero otherwise. The second orthogonality condition is: X j1 j2 j3 j1 j2 j3 (2j + 1) = δ 0 δ 0 (B.19) 3 0 0 m1m1 m2m2 m1 m2 m3 m1 m2 m3 j3m3 A common formula we will use is the following relationship between an integral of spin-weighted spherical harmonics and 3-j symbols. If the spin weights s1, s2 and s3 add to zero, then: Z dΩY s1 Y s2 Y s3 l1,m1 l2,m2 l3,m3 r (B.20) (2j + 1)(2j + 1)(2j + 1) j j j j j j = 1 2 3 1 2 3 1 2 3 4π m1 m2 m3 −s1 −s2 −s3 Finally, we also make use of the following identity:

X j j J (−1)j−m = p2j + 1δ (B.21) m −m 0 J,0 m More information can be found in Biedenharn (1965).

B.5 6-j Symbols

The 6-j symbol is deﬁned as a product over four 3-j symbols: j1 j2 j3 X P6 (j −m ) j1 j2 j3 ≡ (−1) k=1 k k j4 j5 j6 −m1 −m2 −m3 m1...m6 (B.22) j j j j j j j j j × 1 5 6 4 2 6 4 5 3 m1 −m5 m6 m4 m2 −m6 −m4 m5 m3

197 The 6-j symbols are again, quite useful because of their symmetry relations. First, any column permutation results in the same value:

j j j j j j 1 2 3 = 2 1 3 (B.23) j4 j5 j6 j5 j4 j6

Furthermore, exchanging the upper and lower arguments in any two columns leaves the value unchanged as well:

j j j j j j 1 2 3 = 4 5 3 (B.24) j4 j5 j6 j1 j2 j6

These symmetry operations result in an array of situations where the 6-j symbol evaluates to zero. The upper row of the 6-j symbol, such as j1, j2 and j3 in our starting example, must satisfy the triangle inequality. Because of all possible symmetric permutations, this results in the restriction that the following groups of values must satisfy the triangle inequality:

• (j1, j2, j3)

• (j1, j5, j6)

• (j4, j2, j6)

• (j4, j5, j3)

If any of these conditions are not met, the 6-j symbol evaluates to zero. Furthermore, we make use of two identities for 6-j symbols. First:

δ δ j1 j2 j3 j2,j4 j1,j5 j1+j2+j3 = p (−1) (B.25) j4 j5 0 (2j1) + 1(2j2 + 1) Second, they satisfy the orthogonality relation:

δ 0 X j1 j2 j3 j1 j2 j3 j6j6 (2j3 + 1) 0 = (B.26) j4 j5 j6 j4 j5 j6 2j6 + 1 j3 For more details on their derivation, see Biedenharn (1965).

198 B.6 9-j Symbols

We make use of a single 9-j symbol in our calculations, so here I provide just a few

notes on how they are calculated. They can be derived from products of 6-j symbols:   j1 j2 j3   X 2x j1 j4 j7 j4 j5 j6 = (−1) (2x + 1) j8 j9 x  j7 j8 j9  x (B.27) j j j j j j × 2 5 8 3 6 9 j4 x j6 x j1 j2 where here, x is equal to every ji. Note, however, that many of these combinations will add zero due to the triangle equality requirements on the 6-j symbols. For symmetry identities, 9-j symbols are invariant under even permutations of its rows or columns, as well as reﬂection about either of its diagonals. A phase factor is added when we apply odd permutations of rows or columns, where the phase factor is deﬁned as:

(−1)S (B.28) where: 9 X S = ji (B.29) i=1 Finally, a special case of the 9-j symbol can be reduced:   j1 j2 j3   δj3,j6 δj7,δ8 j4 j5 j6 = p (2j3 + 1)(2j7 + 1)  j7 j8 0  (B.30) j j j × (−1)j2+j3+j4+j7 1 2 3 j5 j4 j7

More information can be found in Biedenharn (1965).

199 B.7 Stokes Parameters

Stokes parameters are a convenient way of tracking the polarization of light waves.

If independent light waves are traveling in the same direction but have diﬀerent po-

larizations, we can ﬁnd their total polarization simply by summing their Stokes pa-

rameters. There are four parameters, I, Q, U, and V. They are deﬁned, for purely

polarized light, by:

2 2 Q ≡ |Ex| − |Ey |

∗ U ≡ 2Re(ExEy (B.31) ∗ V ≡ −2Im(ExEy

2 2 I ≡ |Ex| + |Ey | ‘Q’, ‘U’, and ‘V’ each describe the amounts of polarization of speciﬁc types, while

‘I’ describes the sum total amount of radiation. Speciﬁcally, ‘Q’ describes linearly polarized light in the vertical or horizontal directions. ‘U’ is the amount of light in the diagonal directions, and ‘V’ describes light which is circularly polarized. Ex and

Ey are the amplitudes of the electric ﬁeld in the x and y directions, respectively.

Unpolarized radiation would have Q = U = V = 0.

For more information, see Collett (2005).

B.8 Delta Deﬁnitions

This will probably move, just typing them up for now.

200 Each of these terms is a function of n, l, j, n0, l0, j0,J. The atomic numbers of the two parts of the transition in question, and J is the total angular momentum.

2 0 l j 1/2 ∆1 ≡ −Q2(−1)l +l(2j0 + 1)| hn, l| |r| |n0, l0i |2 j0 l0 1 1/2 0 2j + 1 J 1 1 × (−1)j+j +J P¯n (B.32) 00 2J + 1 j0 j j k3 J 1 1 × mα (−1)J + 1 ~0π 0 1 −1 2 0 l j 1/2 ∆2 ≡ −Q2(−1)l +l(2j0 + 1)| hn, l| |r| |n0, l0i |2 j0 l0 1 0 1/2 0 2j + 1 J 1 1 × (−1)j+j +J P¯n (B.33) 00 2J + 1 j0 j j k3 1 1 J × mα (−1)J + 1 ~0π −1 −1 2 2 0 l j 1/2 ∆3 ≡ −Q2(−1)l +l(2j0 + 1)| hn, l| |r| |n0, l0i |2 j0 l0 1 (B.34) 2 3 × kmαf0(kmα) 3~0π l0 j0 1/2 2 ∆4 ≡ −Q2(2j0 + 1)| hn, l| |r| |n0, l0i |2 j l 1 (B.35) 3 2kmα h l+l0 i × 1 + (−1) f0(kmα) 3~0π 0 0 2 0 l j 1/2 ∆5 ≡ −Q2(−1)l +l(2j0 + 1)| hn, l| |r| |n0, l0i |2 j l 1 0 1/2 0 2j + 1 J 1 1 × (−1)j+j +J P¯n (B.36) 00 2J + 1 j0 j j k3 J 1 1 × mα (−1)J + 1 ~0π 0 1 −1 0 0 2 0 l j 1/2 ∆6 ≡ Q2(−1)l +l(2j0 + 1)| hn, l| |r| |n0, l0i |2 j l 1 0 1/2 0 2j + 1 J 1 1 × (−1)j+j +J P¯n (B.37) 00 2J + 1 j0 j j k3 1 1 J × mα (−1)J + 1 ~0π −1 −1 2

201 0 0 2 0 l j 1/2 ∆7 ≡ Q2(−1)l +l(2j + 1)3/2| hn, l| |r| |n0, l0i |2 j l 1 0 0 0 j 1 j × (−1)j+j +J P¯n (B.38) 00 1 j J k3 J 1 1 × γ (−1)J + 1 2π~0 0 1 −1 0 0 2 0 l j 1/2 ∆8 ≡ −Q2(−1)l +l(2j + 1)3/2| hn, l| |r| |n0, l0i |2 j l 1 0 0 0 j 1 j × (−1)j+j +J P¯n (B.39) 00 1 j J k3 1 1 J × γ (−1)J + 1 2π~0 −1 −1 2 0 0 2 0 l j 1/2 ∆9 ≡ Q2(−1)l +l(2j + 1)3/2| hn, l| |r| |n0, l0i |2 j l 1 0 0 0 j j J × (−1)j+j +J p2j0 + 1 (B.40) j j 1 3 kγ × f0(kγ) 3~0π 0 0 2 0 l j 1/2 ∆10 ≡ Q2(−1)l +l(2j + 1)3/2| hn, l| |r| |n0, l0i |2 j l 1 0 j j J × (−1)j+j (B.41) j0 j0 1 3 0 1/2 kγ J × (2j + 1) (−1) [1 − f0(kγ)] 3π~0 0 0 2 0 l j 1/2 ∆11 ≡ −Q2(−1)l +l(2j + 1)3/2| hn, l| |r| |n0, l0i |2 j l 1 0 0 0 j 1 j × (−1)j+j +J P¯n (B.42) 00 1 j J k3 J 1 1 × γ (−1)J + 1 2π~0 0 1 −1 0 0 2 0 l j 1/2 ∆12 ≡ Q2(−1)l +l(2j + 1)3/2| hn, l| |r| |n0, l0i |2 j l 1 0 0 0 j 1 j × (−1)j+j +J P¯n (B.43) 00 1 j J k3 1 1 J × γ (−1)J + 1 2π~0 −1 −1 2

202 B.9 Rotations

Here we detail how rotations aﬀect the values of both the irreducible density

matrix and the phase space density of a speciﬁc Stokes parameter. We can write the

‘I’ Stokes parameter as:

r X 4π f = f (θ, φ) = f Y 0∗ (θ, φ) (B.44) I I 2J + 1 I,JM JM JM

l Where YJM is a spin-weighted spherical harmonic. This holds true at any coordinates θ and φ. Let us perform a coordinate rotation around the lab-frame z-axis, so that:

θ → θ0 = θ (B.45)

and:

φ → φ0 = φ + δφ (B.46)

So: r X 4π f (θ0, φ0) = f Y 0∗ (θ, φ + δφ) (B.47) I 2J + 1 I,JM JM JM We could also write this as a sum over the original coordinates, but with a new weighting function:

r X 4π f (θ0, φ0) = f 0 Y 0∗ (θ, φ) (B.48) I 2J + 1 I,JM JM JM

We know how spherical harmonics change with this rotation:

0∗ −iMφ 0∗ YJM (θ, φ + δφ) = e YJM (θ, φ) (B.49)

Using this, we can see that the rotated stokes parameter distribution element is:

0 −iMδφ fI,JM = fI,JM e (B.50)

203 Let us do the same calculation for the density matrix. From the deﬁnition of the

irreducible moments: J X −m2 J j J J Pjm ∝ (−1) ρm1m2 (B.51) −m2 m m1 m1,m2 This is only non-zero if:

m = m2 − m1 (B.52)

If we can see how the density matrix changes, we can relate it to how the irreducible moments change. From Eq. 4.5, and using the following deﬁnition of states:

im1φ |m1i ∝ e (B.53) and:

0 0 im1φ |m1i ∝ e (B.54) where we have isolated the parts proportional to φ, we can combine them to write:

0 im1δφ |m1i = |m1i e (B.55)

So that:

−i(m2−m1)δφ 0 0 h|m2i hm1|i = e h|m2i hm1|i (B.56)

So, we see that:

−i(m2−m1)δφ 0 ρm1,m2 = e ρm1,m2 (B.57)

And therefore, looking back to Eq. 4.14:

0 imδφ Pjm = Pjme (B.58)

Note here that these moments transform opposite to the photon phase space den-

sity moments. This is important, because in our equations, we will have reversed m

indices on either side of our equations, in order to maintain this rotation conservation.

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