Tools for Cosmology - Combining Data from Photometric and Spectroscopic Surveys

Dissertation

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

Sujeong Lee, B.S., M.S.

Graduate Program in Physics

The Ohio State University

2019

Dissertation Committee:

Klaus Honscheid, Advisor Christopher M. Hirata Samir Mathur Richard J. Furnstahl c Copyright by

Sujeong Lee

2019 Abstract

Cosmic acceleration is the most surprising discovery in modern history. While the phenomenon has been proven by a plethora of cosmological observations, the underlying mechanism is still a mystery. There have been various attempts to understand the driver of cosmic acceleration in a form of “dark energy” or “modiﬁed gravity”, but none of these has compelling evidence. This thesis contains my PhD research projects dedicated to ﬁnd the origin of cosmic acceleration. In Chapter 2, I describe the DES-CMASS (DMASS) galaxy sample constructed from images taken from the

Dark Energy Survey (DES). The sample is designed for a joint analysis of the existing BAO and RSD measurements from BOSS using the CMASS galaxy sample and a galaxy-galaxy lensing measurement from DES. We show that DMASS successfully replicates CMASS in many ways, such as by comparing galaxy bias, angular correlation functions, and redshifts. Chapter 3 describes the DES Y1 analysis for extended cosmological models focusing on modiﬁed gravity (MG), which I contributed. DES

Y1 shear measurement signiﬁcantly improves the existing MG constraints. We show that the resulting MG constraints are consistent with general relativity. In the latter part of the chapter, I forecast the detection of MG parameters for DMASS to demonstrate the capability of DMASS to achieve tighter constraints by cancelling galaxy bias. Finally, in Chapter 4, we study the information content of the three-dimensional galaxy correlation function and power spectrum when realistic scale cuts are applied.

ii We ﬁnd that two estimators are complementary to each other and combining the two yields small improvement for joint constraints.

iii Acknowledgments

To begin with, I would like to express my sincere gratitude to my advisor, Klaus

Honcheid, for his continuous support of my Ph.D study and research. His consistent encouragement always motivated me to keep moving forward, which has shaped my career path as a scientist. Klaus has always been a caring, patient and was also very helpful and supportive during my postdoc job search. As I neared the end of my thesis, he oﬀered great advice and support that helped me complete my thesis. I could not imagine having a better advisor and mentor for my Ph.D study.

I also want to thank Chris Hirata for his insightful advice on my research. His deep understanding of physics helped me overcome theoretical challenges. In addition, I would like to thank the rest of my thesis committee, Samir Mathur and Richard J.

Funstahl, for their encouragement, insightful comments, and probing questions.

I am also grateful for my collaborators - Hee-Jong Seo, Ami Choi, Niall MacCrann,

Michael Troxel, and Jack Elvin-Poole, who engaged in meaningful and inspiring dis- cussions. I am particularly in debt to Eric Huﬀ and Ashley Ross, who have dedicated many hours to helping me not only with my research, but also in maintaining my mental health.

I wish to thank my friends in Korea -Yoonkyung Hwang, Sugyeong Kim, Jeong-

Ok Lee and Sula Lee- who have remained such good friends with me for 13 years, supporting me from the other side of the globe. My thanks also go to my friends at

iv OSU, including Hui Kong, Ben Buckman, Paulo Montero, Bianca Davis, and Mathew

Digman, who helped me with English and were good oﬃce mates.

Finally, I could not have reached such a fulﬁlling life without the love and encouragement of my family, who have been trusting and supporting of me spiritually throughout my life.

v Vita

2011 ...... B.S. Kyung Hee University, Korea

2013 ...... M.S. Kyung Hee University, Korea

Publications

Research Publications

“Producing a BOSS-CMASS sample with DES imaging” Lee, S., Huﬀ, E. M., Ross, A., Choi, A., Hirata, C., Honscheid, K., MacCrann, N., Troxel, M. A., Cawthon, R., and Elvin-Poole, J. Accepted to Monthly Notices of the Royal Astronomical Society

“Dark Energy Survey Year 1 Results: Constraints on Extended Cosmological Models from Galaxy Clustering and Weak Lensing” Abbot, T. M. C. et al. (DES Collaboration) Phys. Rev. D 99, 123505

Fields of Study

Major Field: Physics

vi Table of Contents

Page

Abstract...... ii

Acknowledgments ...... iv

Vita ...... vi

ListofTables...... x

List of Figures ...... xi

1. Introduction...... 1

1.1 Overview ...... 1 1.2 History: the discovery of accelerating universe ...... 3 1.3 Modern Cosmology: the Basic ...... 5 1.3.1 FLRW metric and Friedmann’s equations ...... 5 1.3.2 Distances and Geometry ...... 8 1.3.3 Density Fluctuations and Structure Growth ...... 10 1.3.4 Correlation Function and Power Spectrum ...... 11 1.4 Dark Energy ...... 14 1.5 ModiﬁedGravity...... 16 1.6 Cosmological Probes of Cosmic Acceleration ...... 20 1.6.1 TypeIaSupernovae ...... 20 1.6.2 Cosmic Microwave Background Radiation ...... 24 1.6.3 Baryonic Acoustic Oscillation ...... 25 1.6.4 Redshift Space Distortion ...... 27 1.6.5 Weak Gravitational Lensing ...... 30

vii 2. Producing a BOSS-CMASS sample withDESimaging ...... 33

2.1 Introduction ...... 34 2.2 Data...... 40 2.2.1 BOSS DR12 CMASS Sample ...... 40 2.2.2 DES Y1 Gold Catalog ...... 42 2.2.3 Diﬀerences between the SDSS and DES photometry . . . . 46 2.3 Constructing the Model ...... 50 2.3.1 Overview of the Algorithm ...... 50 2.3.2 The Training and Validation Sets ...... 52 2.3.3 Obtaining True Distributions with the Extreme-Deconvolution Algorithm...... 53 2.3.4 Application to the Target Galaxies ...... 54 2.3.5 Excluding Low Probability Galaxies ...... 59 2.4 Systematic Error Characterization ...... 61 2.5 Comparison with the BOSS CMASS Sample ...... 68 2.5.1 Number Density ...... 68 2.5.2 Angular Correlation Function ...... 70 2.5.3 Redshift Distribution ...... 78 2.5.4 Diﬀerence in Galaxy Bias ...... 83 2.6 Conclusion ...... 88

3. Dark Energy Survey Year 1 Results: Constraints on Extended Cosmological Models from Galaxy Clustering and Weak Lensing ...... 90

3.1 Introduction ...... 91 3.2 Theory ...... 93 3.3 Data...... 94 3.3.1 DES data ...... 94 3.3.2 External data ...... 95 3.4 Analysis...... 97 3.5 Results ...... 98 3.6 Forecast for DMASS ...... 102 3.7 Conclusion ...... 107

4. Complementary Information in Fourier and Conﬁguration Space for Cos- mological Observations ...... 108

4.1 Introduction ...... 108

viii 4.2 Power spectrum and Correlation function ...... 111 4.2.1 Modeling estimator data using the streaming model . . . . . 112 4.3 Signal to Noise of estimators ...... 120 4.4 Cosmological Parameter Constraints ...... 129 4.4.1 Individual estimator projections ...... 129 4.4.2 Combining and Comparing Constraints ...... 131 4.5 Discussion and Conclusion ...... 138

5. Conclusion...... 140

Appendices 143

A. The impact of redshift tails in BOSS CMASS on galaxy bias ...... 143

B. Diﬀerences between CMASS SGC and NGC ...... 149

C. Analytic way of marginalizing high k scales in band power ﬁsher matrix withoutinverting ...... 154

Bibliography ...... 156

ix List of Tables

Table Page

3.1 Parameters and priors used to describe the measured two-point functions, as adopted from Y1KP. Flat denotes a ﬂat prior in the range given while Gauss(µ, σ) is a Gaussian prior with mean µ and width σ. 99

2 2 B.1 χ /dof of three probes calculated between two diﬀerent samples. χsys/dof in the third column are calculated with systematic weights of DMASS obtained in Section 2.4. Values in the parentheses are corresponding PTE values. SGC and FULL in bold stand for CMASS in SGC and fullCMASS...... 150

x List of Figures

Figure Page

1.1 The Hubble diagram for the Pantheon sample. The top panel shows the distance modulus for each SN; the bottom panel shows residuals to the best ﬁt cosmology. Distance modulus values are shown using G10 scatter model. Figure is from Scolnic et al. (2018)...... 23

2.1 The two-dimensional histograms of CMASS galaxies from Stripe 82 in the g r vs. r i color plane. The left panel shows SDSS, the right − − panel shows DES colors of the same galaxies. The colorbar shows the number of galaxies binned in each histogram bin. The red line is the d⊥ cut. CMASS galaxies look bluer in the DES photometry and the d⊥ cut discards almost half of the CMASS galaxies by crossing the most dense region. The grey contours in the right panel show the full distribution of DES Y1 GOLD galaxies in the color plane. The grey contours show that blindly lowering the d⊥ cut results in accepting more non-CMASS galaxies...... 45

2.2 The response functions for the griz SDSS (shaded) and DES (solid lines) filters as a function of wavelength (amstrong) with the spectral energy density (SED) distribution of an elliptical galaxy at z = 0.4 (black solid line). Near the 4000 amstrong break where the g r − transition happens, the SDSS r filter (shaded) covers slightly lower wavelength than the DES r filter (solid lines) does. This implies the same galaxy near z = 0.4 looks redder in the SDSS photometry than intheDESphotometry...... 47

xi 2.3 Accuracy of model membership probabilities assigned to the test set. Galaxies in the test set are binned based on their probability assigned by the probabilistic model. The x-axis shows 20 bins of the assigned probability, and the y-axis shows the fraction of true CMASS galaxies in each bin. If the model successfully recovers the observed distribution of CMASS in the test region, the fraction of true CMASS galaxies in each bin should be identical to the assigned model probability. The dashed diagonal line in the ﬁgure stands for this ideal case, and the grey bars are given by Poisson errors...... 55

2.4 Histograms of the d⊥ color distributions of CMASS (blue) in the training set and DMASS (solid black line) in 10 membership probability bins. d⊥ on the x-axis consists of only DES quantities (d⊥,DES = (RDET IDET) (GDET RDET)/8.0 where the subscript ‘DET’ denotes − − − DES detmodel magnitude). ftrain in the top-right corner of each panel denotes the fraction of training galaxies binned in each probability bin, deﬁned as ftrain,i = Ntrain,i/Ntrain,total for the ith probability bin. . . . 57

2.5 Histograms of color and magnitude distributions of CMASS in the training set (blue) and DMASS in SPT (black solid line). The colors and magnitudes on the x-axis are DES quantities...... 57

2.6 Number density of DMASS with respect to the probability cut computed from three diﬀrent regions - the training region (even HEALPix pixels; square), the test region (odd HEALPix pixels; circle), and the full region (training+test; black diamond). Galaxies below a given probability cut are excluded. Number densities are divided by the number density of CMASS in the corresponding regions. The extremely similar shape of curves from diﬀerent samples implies that the model tends to boost the number density of the sample in a predictable way and this tendency can be remedied by cutting out low probability galaxies below a certain threshold. The red star at P = 1% on the black diamond curve is our choice of the probability cut...... 60

2.7 The impact of systematic weights. Starting from the left, the names of the survey properties are listed on the x-axis in the order that they are corrected. The weight for the particular property is applied on top of the other weights applied earlier. The y-axis shows the χ2 measured between the correlation function with the new and old weights. ‘veto’ denotes a veto mask applied to remove regions where fwhm in r band > 4.5...... 65

xii 2.8 Galaxy number density with respect to survey properties having the top six ∆χ2 and stellar density. The solid blue lines are calculated without correction weights. The dashed black lines are calculated with weights. The error bars on the black lines are calculated assuming Poissonian statistics...... 67

2.9 Number density and its deviation in the NGC (left), CMASS in the SGC (middle), and DMASS (right). Each region is divided into Stripe 82-size (train region) patches. Red stars represent the number density of BOSS CMASS in the training region and blue stars are the total number density of DMASS. All values are divided by the mean number density of each CMASS sample. The dark-grey-shaded region is 1σ, and the light-grey-shaded region is the 2σ level deviation of the black points in each panel...... 69

2.10 The top panel shows the angular correlation function calculated with DMASS (red), DMASS corrected by the systematic weights (orange), CMASS SGC (blue), and full CMASS (black). The bottom panel shows residuals between DMASS and CMASS SGC (blue) or full CMASS 2 2 2 (black). χSGC (χFULL) is the χ of the observed diﬀerence of two point functions of DMASS and CMASS SGC (FULL) ...... 72

2.11 Cross-correlation measurements of the DMASS (red), DMASS with systematic weights (orange), CMASS SGC (blue) and full CMASS (black) samples with WISE galaxies (left) and CMB convergence map (right). We dropped the ﬁrst three data points of the cross-correlation with CMB lensing (in grey shaded region) from the measurements of χ2 to include only reliable scales where the analytic covariance matrices arevalid...... 74

2.12 Integrated auto-correlations (Equation 2.35) of the CMASS SGC sample (black points). The grey-dashed line is the redshift evolution model (1+z)γ with γ = 0 (Equation 2.40). The value of χ2 between the model and the measurement is 15.8 for 13 data points, which indicates that the measurement is well consistent with the model...... 81

2.13 Redshift distribution of DMASS (blue) and DMASS with systematic weights (orange) recovered by the clustering-z method with redMaGiC. The solid black and dashed lines show the spectroscopic redshift distribution of CMASS SGC and full CMASS...... 82

xiii 2.14 Constraints on the galaxy bias shift ∆b and redshift distribution shift ∆z from combination of the auto-angular correlation function and clustering-z. The dashed vertical and horizontal lines show the ideal case where DMASS is perfectly matched with CMASS. Orange-solid and red-dashed contours show shifts from the values of CMASS SGC. Black-solid and Blue-dashed contours show shifts from the values of full CMASS. The DMASS systematic weights are added for dashed contours. Adding systematic weights has very little impact on galaxy bias and redshift distributions...... 86

3.1 The left panel shows constraints on the modiﬁed gravity parameters (Σ0, µ0). Blue contour is from DES alone, yellow is from external data alone, and red is the combination of the two. The dashed horizontal and vertical lines show the ΛCDM case. The right panel shows the constraints on the matter density Ωm and S8 for the three cases aforementioned, with the case of ΛCDM (dashed contours)...... 98

3.2 Left: Marginalized posterior contours in the Q2 Σ2 plane measured − by KiDS (Joudaki et al., 2017) with ﬁducial angular scales in green (denoted by ‘FS’), KiDS keeping only the largest angular scales in pink (denoted by ‘LS’), and respectively combined with Planck in grey and blue. The indices represent a particular combination of MG bins, such that z < 1 and k > 0.05h−1 Mpc. The dashed lines intersect at the GR prediction (Q = Σ = 1). Right: Marginalized posterior distributions of the MG pa rameters µ and η for Planck TT,TE,EE+lowE+lensing data alone and in combination with external data (as indicated in the legend), using the late time parameterization and neglecting any scale dependence. The dashed lines show the standard ΛCDM model. The ﬁgure is from Planck Collaboration et al. (2018)...... 103

3.3 The blue contour is forecasted 68% and 95% conﬁdence contours on (Σ, µ) obtained with DMASS. The red contour shows constraints from Simpson et al. (2013), plotted for comparison with this work. . . . . 106

xiv 4.1 Cumulative SNR of PS (black dashed line) and CF (blue dot points) as a function of the maximum wave number kmax. The blue solid is −1 a smooth ﬁt to the dot points. We assume k0 = 0 h Mpc , rmin = −1 −1 0.1 h Mpc and rmax = 180 h Mpc, i.e., an almost all scale available for CF. Throughout all relevant calculations, we adopt 0.004 h Mpc−1 for k bin size dk since SNR from correlation function becomes stable where dk is equal and smaller than 0.004h Mpc−1. Convergence tests for other dk bin sizes is in Appendix...... 120

4.2 Top: cumulative SNR of CF for different rmin and rmax. Bottom: SNR contribution from individual band power defined as ∆SNR(kn) = SNR(kn) SNR(kn−1). The black dashed line in the top and bottom − panels represent the SNR of PS while the colored solid lines are for CF1. In the top panel, at low k, SNR from correlation function increases slower than power spectrum until it reaches to the maximum value. This can be seen clearer in the bottom panel; the peak of blue line falls at larger k than the peak of black dot-dashed line (for PS) without amplitude change. The colored dashed lines show ∆ SNR when only diagonal components in Fisher matrix are considered. The location of peaks is almost in agreement with the peak of PS which implies that off-diagonal components (more so for a limited range of r) are the main factor that shift information peak to smaller scales. . . 121

4.3 Maximum cumulative SNR from PS and CF as a function of rmax (red line) and rmin (blue line), relative to the maximum cumulative SNR for −1 −1 PS between 0.01h Mpc < k < 2 hMpc . When rmax is varied, rmin −1 −1 is 0h Mpc for all points; when rmin is varied, rmax is 200h Mpc for all points. One can see that the dependence on rmax (red curve going −1 from 170 to 200 h Mpc) is small while the dependence on rmin (blue −1 curve going from 0 to 30 = rmax 170h Mpc) is particularly strong. 125 −

4.4 Fractional error on b and f as a function of rmin. This plot is to be compared with figure 5 of R11. The solid line is obtained from three CF multipoles l = 0, 2, 4, being marginalized over σv, using our model based on White et al. (2009). The same calculation has been done in R11 with a configuration-space counterpart to White et al. (2009). Our calculation agrees with R11’s result within 10% by visual inspection. ∼ The dot-dashed lines are obtained from three PS multipoles plotted at rmin = 1.15π/kmax. The difference between two fractional errors from different estimators is comparable to what we observe in confidence ellipsesinFigure4.5...... 130

xv 4.5 Confidence ellipses (1σ) for b and f for various scale limits used in recent DR12 BOSS analyses. Different color stands for different estimator; PS(blue), CF(red), joint estimator (green). The purple color stands for the joint constraint with zero cross-covariance. The scales included are specified in the left corner of each panel. The left panel shows the projection when all scales are available. As expected, the two estimators give almost the same constraints and there would be no information gain by joint estimator in this case. The right panel shows the projection with realistic scale limits. In these specific choices of ranges, PS (blue) is substantially better for constraining b and f both. When two estimators are combined (green), the gain is small for b while the gain is substantial for f and σv...... 136

4.6 Confidence ellipses (1σ) for all cosmological parameters for scale limits used in recent DR12 BOSS analyses (25 < r < 160h−1 Mpc and 0 < k < 0.25h Mpc−1). Different color stands for different estimator; PS(blue), CF(red), joint estimator (green). The purple color stands for the joint constraint with zero cross-covariance. PS (blue) is substantially better for constraining b and f both. When two estimators are combined (green), the gain is small for b while the gain is substantial for f and σv...... 137

A.1 Monopole (top) and quadrupole (bottom) correlation function of the CMASS sample before (red) and after (blue) applying redshift cuts at 40h−1 Mpc < s < 180h−1 Mpc...... 145

A.2 Comparison of galaxy bias constraints from the CMASS clustering before (blue) and after (red) applying z-cuts...... 147

B.1 Diﬀerence in galaxy biases constrained by the angular correlation func- +0.031 tion of CMASS SGC and CMASS NGC (red). ∆bNGC−SGC = 0.056−0.033. The blue-dashed and black-dot dashed histograms display ∆bSGC−DMASS and ∆bFULL−DMASS obtained in Section 2.5, respectively. The redshift bin bias ∆z of DMASS is marginalized for the latter two cases. . . . 151

xvi Chapter 1: Introduction

1.1 Overview

For over a century, Einstein’s General Relativity (GR) has provided a theory of gravity that ﬁts all observations from our solar system to galactic scales. For a homogeneous universe ﬁlled with only matter and radiation, GR predicts that the expansion of the universe will slow down due to Newtonian attraction of matter.

However, two decades ago, two teams measuring properties of distant supernovae independently found that the universe’s expansion has accelerated over the last ﬁve billion years (Riess et al., 1998; Perlmutter et al., 1999). A plethora of cosmological observations have followed to conﬁrm this remarkable discovery and established the concordance cosmological model referred to as the ΛCDM model. Λ here is a so- called cosmological constant implemented in Einstein’s GR equations to account for an accelerated expansion. CDM denotes cold dark matter.

A cosmological constant Λ is viewed as a form of vaccum energy with negative pressure, unchanging in time and spatially smooth (Sahni & Krasi´nski, 2008). This is the simplest theoretical model for the accelerating universe and so far it is consistent with existing data as well. However, the exact physics behind it is unclear, and its negligibly small magnitude compared to the value predicted from quantum ﬁeld

1 calculations is still a mystery. This is known as the cosmological constant problem.

For furtuer details, see Weinberg et al. (2013) and Silvestri et al. (2013).

There have been two main attempts to explain the accelerating expansion without the cosmological constant. The first is that the accelerating cosmic expansion is driven by a new form of energy such as a scalar field φ with potential V (φ). At late times, the scalar field evolves over time and naturally mimics the behavior of the cosmological constant. The basic concepts of dark energy and relevant theories will be introduced later. The second attempt to explain the observations without a cosmological constant is to modify Einstein’s GR in a way that produces an acceleration.

Various modifications to GR have been proposed. For a detailed review for modified gravity models and theories, see Jain & Khoury (2010). A more general method is a phenomenological approach, which investigates deviations from GR through a set of the modified gravity parameters embedded in the field equations at the linear perturbation level. I will describe the details of the phenomenological approach later in this chapter.

Both dark energy and modiﬁcation to GR alter the relation between the expansion history and the growth of matter clustering from the behavior predicted in ΛCDM.

The signatures of either one can be detected through various cosmological probes - the cosmic microwave background radiation (CMB), weak gravitational lensing, galaxy surveys, distances to supernovae, and baryon acoustic oscillations (BAO) and others.

The goal of this chapter is to provide the basic knowledge of cosmic acceleration and its possible drivers, dark energy and modiﬁed gravity, to help the readers understand the more advanced topics described in the following Chapters 2-4. The rest of

2 this chapter is organized as follows: In Section 1.2, I brieﬂy recap the history of cosmic acceleration. Section 1.3 describe the theoretical background and mathematical formalism of modern cosmology. In Section 1.4, I review the basic concept of dark energy and its empirical parametrizations. Section 1.5 introduces the phonemoneolog- ical approach for probing modiﬁed gravity and popular parametrizations frequently adopted in the cosmological analyses. Section 1.6 presents the principal cosmological probes of cosmic acceleration.

1.2 History: the discovery of accelerating universe

Along with the huge success of General Relativity (GR), Einstein (1917) introduced the first modern cosmological model based on GR. He assumed that the universe is homogeneous on large scales, matter-filled, and static. In GR, the universe must contract due to the gravitational attraction of matter. To maintain the universe static, Einstein proposed a new component whose repulsive gravity could balance the attractive gravity of the matter. This modification led to the cosmological constant

Λ in Einstein’s equation.

In 1929, Hubble discovered direct evidence of the cosmic expansion by observing galaxies receding from Earth. His discovery removed the original need for a cosmological constant causing Einstein to call it his biggest blunder (Hubble, 1929). After the discovery of cosmic expansion, Einstein and de Sitter (Einstein & de Sitter, 1932) proposed an expanding, homogeneous and isotropic, spatially ﬂat, matter-dominated universe. For such a universe, GR predicts that the expansion of the universe will slow down over time.

3 By the mid-1990s, Big Bang cosmology was advocated by a number of cosmologi-

cal simulation studies and observations, while the Einstein-de Sitter model was show-

ing numerous ﬂaws. Notably, Big Bang nucleosynthesis theory and measurements of

massive galaxy clusters at high redshift (Fabian, 1991; White & Frenk, 1991; White

et al., 1993) eﬀectively ruled out the ﬂat, matter-dominated universe. At the same

time, measurements of galaxy clustering strongly favored a low-matter density uni-

verse and furthermore suggested something like a cosmological constant as a potential

source contributing to the spatially ﬂat geometry (Maddox et al., 1990; Efstathiou

et al., 1990). The measured Hubble constant (Freedman et al., 1994) and the ages

of globular clusters (Krauss & Chaboyer, 2003) also disfavored the Einstein-de Sitter

model.

In the late 1990s, supernova surveys by two independent teams provided direct

evidence for an accelerating cosmic expansion (Riess et al., 1998; Perlmutter et al.,

1999). They found that the distant supernovae were dimmer than expected in a

matter-dominated universe. Their measurements were interpreted as evidence for an

accelerating universe with a cosmological constant, consistent with a ﬂat universe

having the total energy density Ωtot = 1 with the energy density of Λ, ΩΛ 0.7. ≈ Shortly after, CMB experiments (de Bernardis et al., 2000; Hanany et al., 2000) conﬁrmed a spatially ﬂat and accelerating universe by ruling out the hypothesis of the free expansion with the matter density Ωm 1 and ΩΛ = 0. Today, the accelerated expansion of the universe is well supported by independent evidences from a plethora of cosmological measurements.

This accelerated expansion of the universe requires the presence of a new component with strongly negative pressure. This component was soon named “dark energy”

4 by cosmologist Michael Turner (Huterer & Turner, 1999). Instead of introducing a

new energy component, one can attempt to modify gravity in a way that leads to

accelerated expansion. The distinction between dark energy and a modiﬁcation to

gravity is ambiguous. Currently, we have strong observational evidence for acceler-

ated cosmic expansion, but there is no compelling evidence that either dark energy

or modiﬁed gravity is the cause of the acceleration. In the near future, more ad-

vanced cosmological surveys along with the worldwide eﬀorts in developing observing

techniques and theoretical modeling will attempt to solve this puzzling mystery.

1.3 Modern Cosmology: the Basic

1.3.1 FLRW metric and Friedmann’s equations

On large scales, the universe is homogeneous and isotropic. Under these assump-

tions, the geometry can be described by the metric of the Friedmann–Lemaˆıtre–

Robertson–Walker (FLRW) metric as follow:

dr2 ds2 = dt2 + a2(t) + r2(dθ2 + sin2 θdφ2) , (1.1) − 1 kr2 − where r, θ and φ indicate comoving spatial coordinates, the quantity k determines the intrinsic curvature of of the geometry: k = 0 correponds to a ﬂat universe, while k > 0 corresponds to positively curved universe (spherical geometry) and k < 0 to negatively curved universe (hyperbolic geometry). The FLRW metric can be rewritten as a function of a comoving coordiate χ using the relation dχ = dr/√1 kr2: − ds2 = dt2 + a2(t) dχ2 + f 2 (χ)(dθ2 + sin2 θdφ2) , (1.2) − K where sin(χ) k = +1 f (χ) = χ k = 0 (1.3) K  sinh(χ) k = 1 ,  −  5 and the comoving distance χ between us and a galaxy at redshift z = 1/a 1 is − deﬁned as

t0 dt z dz χ(a) = = . (1.4) a(t) H(z) Zt1 Z0 The scale factor a(t) is an unknown function that is obtained by solving the

Einstein’s ﬁeld equation for the FLRW metric. The Einstein’s ﬁeld equation is given

Gµν + Λgµν = 8πGTµν (1.5)

1 where Gµν = Rµν Rgµν is the Einstein tensor, Rµν is Ricci tensor, R is Ricci scalar, − 2 gµν is metric tensor, and Λ is the cosmological constant. The energy momentum tensors Tµν reﬂects the matter content of the universe. For a perfect ﬂuid, Tµν can be written as

Tµν = (ρ + p)uµuν + pgµν (1.6) where uµ is the ﬂuid 4-velocity, ρ is the energy density and p is the pressure. By substi-

tuting Equations (1.1) and (1.6) into Equation (1.5), one obtains the ﬁrst dynamical

Friedmann equation from the time-time components of the Einstein equation:

a˙ 2 8πG Λ k H2(t) = ρ + (1.7) ≡ a 3 3 − a2 where an overdot denotes the derivative with respect to time. From the combination of the time-time component and space-space component of the Einstein equation, the second dynamical Friedmann equation (known as the acceleration equation) is given as

a¨ 4πG Λ = (ρ + 3p) + . (1.8) a − 3 3 6 In general relativity, energy conservation law is described by the vanishing of the

µν covariant divergence of the energy momentum tensor, µT = 0, which yields the ∇ continuity equation

a˙ ρ˙ + 3 (ρ + p) = 0 . (1.9) a

Within the context of the cosmic ﬂuid assumed here, one can deﬁne the equation of state w

p w = . (1.10) ρ

The parameter w has diﬀerent constant values for diﬀerent cosmological sources. For example, w = 0 corresponds to cold non-relativistic matter, w = 1/3 corresponds to

radiation. For the cosmological constant as a form of vacuum energy with negative

pressure, w = 1. By solving Equation (1.9) with Equation (1.10), one may obtain −

ρ(a) a(t)−3(1+w) . (1.11) ∝

For a matter dominated era, the energy density of non-relativistic particles will scale

−3 as ρm a , with w = 0. For a radiation dominated era, the energy density of ∝ −4 radiation will scale as ργ a , with w = 1/3. For a cosmological constant, the ∝ equation of state naturally yields a constant energy that does not vary with respect to time. In the context of dynamical dark energy, w is generally allowed to deviate from 1 or vary as a function of time. Further details of dynamical dark energy − models will be introduced in Section 1.5.

To conveniently express the energy budget in the universe today, one can use the density parameter deﬁned as

0 0 ρi Ωi = 0 , (1.12) ρcrit 7 whdre ρcrit is the critical density determined from the ﬁrst Friedmann equation (1.7)

in a spatially ﬂat universe without cosmological constant. The critical density is

deﬁned as

3H2 ρ0 = 0 . (1.13) crit 8πG

The Friedmann equation (1.7) can be rewritten with respect to the density parameter as follow:

2 2 0 −3 0 −4 0 −1 0 −3(1+w) H (a) = H0 [Ωma + Ωγa + Ωka + ΩΛa ] (1.14) where the density parameter for matter is Ω0 Ω0 + Ω0. The subscripts b and c m ≡ b c

indicates baryons and cold dark matter, respectively. ΩΛ is the density parameter

for a cosmological constant (or dark energy). A recent cosmological analysis from

0 0 Planck (Planck Collaboration et al., 2018) found Ωm = 0.315, and Ωk = 0.0007.

These values are well consistent with the spatially ﬂat ΛCDM cosmological model. In

ΛCDM, Ωγ and Ωk are negligible today. Therefore, Equation (1.14) can be reduced

0 0 to Ωm + ΩΛ = 1 with a cosmological constant Λ.

1.3.2 Distances and Geometry

While some observations can probe the expansion history H(a) directly, others

measure H(a) indirectly by measuring the distance-redshift relation or the growth of

structure. Physical distances can be determined by integrating the line element in

Equation (1.2) with dt = dθ = dφ = 0

χ 0 dphys = a(t) dχ = a(t)χ . (1.15) Z0 The radial comoving distance is given as

d χ = phys . (1.16) a(t)

8 For a spatially ﬂat universe, one can derive the comoving distance normalized to be

equal to the physical distance today (a0 1) as ≡ t0 dt0 a0 da0 χ = = . (1.17) a(t0) a02H(a0) Zt Za The angular diameter distance is similarly derived from the FLRW metric (1.2)

in by setting dt = dr = dφ = 0:

δs d (a) = = af (a) . (1.18) A δθ K

The comoving angular diameter distance is given as

d (a) D (a) = A = f (a) . (1.19) A a K where fK (a) is deﬁned in Equation 1.3. Note that DA(a) is also referred as the

comoving distance χ for a spatially ﬂat universe (k = 0).

Finally, the luminosity distance, dL, is widely used to measure the expansion of the universe. For an object with luminosity L at a distance dL from an observer, the

ﬂux measured by the observer is

L F = 2 . (1.20) 4πdL

From photon conservation, the measured ﬂux for a source measured can be found

from the metric and the redshift of the source (Ellis & van Elst, 1999), which leads

to the relation

D (a) d = A = (1 + z)D (z) . (1.21) L a A

By independently measuring dL and z, one can constrain the expansion history H(z)

behind the angular diameter distance DA(z).

9 1.3.3 Density Fluctuations and Structure Growth

On large scale the evolution of matter and metric perturbations follows linear

perturbation theory. In GR, the scalar mode of the metric perturbations are related to

matter ﬂuctuations, thus, this section will focus on a discussion of scalar pertubations.

An excellent pedagogical introduction can be found in Carroll (2004) and Peter (2013).

Scalar perturbations in the Newtonian gauge can be described by two gravitational

potentials Ψ and Φ as

ds2 = (1 + 2Ψ)dt2 + a(t)2(1 2Φ)d~x2 . (1.22) − −

The ﬁrst order perturbed Einstein equations provide two equations relating the two gravitational potentials (Ma & Bertschinger, 1995).

2 2 k Φ = 4πGa ρiδi (1.23) − i X 2 2 k (Ψ Φ) = 12πGa ρi(1 + wi)σi (1.24) − − i X where δ δρ/ρ is the density contrast, ρi and σi are the density and the anisotropic ≡ stress for matter species i.

At late times, we can set w = σ = 0 and the time evolution of the scalar potential is negligible (Φ0 = 0) under the quasi-static approximation. Then the evolution equation of the matter ﬂuctuations δm is given by

δ¨m + 2Hδ˙m 4πGρδm = 0 , ‘ (1.25) − where the dot notation implies the time derivative. This evolution equation has two solutions - decaying and growing modes. Here, we only use the growing modes

(denoted as D+). They are related to the structure growth that we observe today as

10 follow:

D+(t) δ(t) = δ(ti) , (1.26) D+(ti) where ti is some initial time and D+ and δ at the initial time are set by initial conditions.

Another useful quantity for probing structure growth is the growth rate deﬁned as the derivative of the logarithm of the growth factor D:

d ln D f(a) . (1.27) ≡ d ln a

This quantity combined with the amplitude of matter ﬂuctuation σ8(a) is tightly con-

strained by Redshift Space Distortions (RSD) surveys. It therefore plays an important

role in probing dark energy or modiﬁed gravity.

1.3.4 Correlation Function and Power Spectrum

The galaxy correlation function, generally denoted as ξ(r), is a measure of the degree of clustering of galaxies as a function of separation r. In other words, it is

basically the exceess probability of ﬁnding a galaxy in a volume element dV1 and

another one in dV2 separated in space r, which is expressed as

2 dP =n ¯ [1 + ξ(r)]dV1dV2 (1.28)

wheren ¯ is the mean galaxy number density. A more general form of the correlation

function is the ensemble average of the galaxy overdensity as follows:

ξ(r1, r2) δg(r1)δg(r2) (1.29) ≡ h i

where δg(ri) represents the galxy overdensity at a position ri with respect to the mean

density.

11 The power spectrum is the Fourier transform of the correlation function. Since the

power spectrum is easily predicted directly from theories, a widely-used stratedgy in

cosmological analyses is to construct a model in Fourier space ﬁrst and then to convert

the model to the conﬁguration space for a direct comparison with observations. The

conversion between two quantities are given as

P (k) = ξ(r)eik·rd3r (1.30) Z d3k ξ(r) = P (k)e−ik·r , (1.31) (2π)3 Z where k is the wave number in Fourier space. Most theoretical predictions are calculated using the matter power spectrum Pm(k) in Fourier space. The matter power spectrum, Pm(k), is deﬁned by

0 3 3 0 δ(k)δ(k ) = (2π) Pm(k)δ (k k ) (1.32) h i D −

where δ(k) is matter density ﬂuctuation in Fourier space, and δD is the Dirac delta

function. At early times, primordial ﬂuctuations originated from quantum ﬂuctua-

tions have been blown up to macroscopic scales by cosmic inﬂation. These primordial

ﬂuctuations can be described by the power spectrum as (Harrison, 1970; Peebles &

Yu, 1970; Zeldovich, 1972)

ns Pm(k) Ask (1.33) ∝

with ns 1, As is the amplitude of the primordial power spectrum. The matter ≈ power spectrum today has evolved from this primordial power spectrum through a

number of physical processes as follow:

ns 2 2 Pm(k, z) = Ask G (z)T (k) (1.34)

12 where T (k) is the transfer function, which is constant for modes that entered the horizon before the matter-radiation equality (k < 0.01h−1 Mpc) and scales as k−2 at

smaller scales that entered the horizon during radiation domination (Bardeen et al.,

1986; Sugiyama, 1995; Eisenstein & Hu, 1998). The scale of the horizon at the matter-

radiation equality appears as a turn-over in the shape of the matter power spectrum.

Finally, G(z) is the growth factor of structure described in Section 1.3.3.

To compare observations from galaxy surveys with theoretical predictions, one

needs to covert the matter power spectrum into one for galaxies. On large scales,

galaxy clustering and matter clustering can be simply mapped via a linear galaxy

bias b(z) as follow

δg(z, k) = b(z)δm(z, k) (1.35)

where δm is matter density. Additionally, peculiar motion of galaxies adds distortions

that can be accounted for via the factor f(z)µ2 where µ is the consine of angle to

the line of sight. These distortions observed in the redshift space are called redshift

space distortions (RSD). They are very sensitive to the growth of structure. Further

descriptions about RSD as a probe for cosmic acceleration can be found in Section

??. By considering all factors, the galaxy power spectrum can be written as

n 2 2 2 2 Pg(k, µ, z) = Asks G (z)T (k)[b(z) + f(z)µ ] . (1.36)

Deviations from ΛCDM can appear in the transfer function T (k), the growth factor

G(z), and the growth rate f(z). Finally, one needs to include nonlinear eﬀects on small

scales, which are estimated from N-body simulations or ﬁtting formulae (Takahashi

et al., 2012; Mead et al., 2016).

13 1.4 Dark Energy

A cosmological constant Λ is the prime candidate causing the accelerating expan-

sion of the universe. However, its unnaturally small magnitude is inexplicable, as

discussed in Section 4 in Silvestri & Trodden (2009). Cosmologists have been explor-

ing alternative sources that may be responsible for the accelerating expansion (see

Bean et al. (2005) for a discussion of possibilities). The frequently considered alter-

native for cosmologists is that the accleration of the expansion is driven by a new

form of energy such as a scalar ﬁeld φ with potential V (φ). The Lagrangian density

with respetct to such a scalar ﬁeld is given as

1 µν = g ∂µφ∂µφ V (φ) (1.37) L 2 −

with the energy-momentum tensor

1 σρ Tµν = ∂µφ∂νφ gµν g ∂σφ∂ρφ + V (φ) . (1.38) − 2 For the perfect ﬂuid assumption, the energy-momentum tensor leads to these equations for the energy density and pressure

1 ρ = φ˙2 + V (φ) (1.39) 2 1 p = φ˙2 V (φ) , (1.40) 2 − so that the equation of state for the scalar ﬁeld becomes

1 ˙2 2 φ V (φ) wφ = − . (1.41) 1 ˙2 2 φ + V (φ) At late times, when the time evolution of the scalar ﬁeld is negligible (φ˙ 0), the ≈ scalar ﬁeld behaves like a cosmological constant, with wφ 1. In this case, dark ≈ − energy is treated as a form of a new energy component having the negative pressure.

14 There are more general parametrization for a time-varying w that involve two free parameters:

z w(a) = w0 + wa(1 a) = w0 + wa , (1.42) − 1 + z

where w0 is the value of today. With this deﬁnition, the energy density parameter of

dark energy is now expressed as

0 −3(1+w0+wa) −3wa(1−a) Ωde(a) = Ωdea e . (1.43)

The value of w evolves linearly from w0 +wa at high z to w0 at z = 0. Observations

usually provide constraints at some intermediate redshift, so the statistical errors on

w0 and wa are highly correlated. To avoid this problem, a pivot redshift zp is adopted.

Essentially, zp indicates the redshift at which the error on w(z) is the smallest, for

ﬁxed assumptions about the data. The equation of state with a pivot redshift is

w(a) = wp + (ap a)wa, (1.44) − with

Cw0wa zp = , (1.45) −Cw0wa + Cwawa where C is the 2 2 covariance matrix for w0 and wa. The value of the pivot redshift × depends on what data sets are being considered, but in practice it is usually closet to zp 0.4 0.5. The variance at the pivot is given by ∼ − 2 2 Cw0wa σ (wp) = Cw0w0 . (1.46) − Cwawa

There is a broad spectrum of dark energy models including early dark energy and coupled dark energy to dark matter. I do not discuss further details here but refer the interested readers to Sahni et al. (2003) and Corasaniti et al. (2004).

15 1.5 Modiﬁed Gravity

Within the theory of GR, accelerated expansion is only due to a component of the energy density with a negative equation of state, commonly referred to as dark energy.

However, there is a natural alternative to be considered, namely the possibility that the dynamical rules determining how the geometry of spacetime responds to regular matter sources may differ from those given by GR. In other words, we may consider modifying GR in the low curvature regime. General Relativity has been tested in the laboratory, in the solar system and astrophysically, but there are far fewer tests on cosmological scales. It is therefore natural to explore whether the laws of gravity could be modified on the larger scales and whether such modifications could be responsible for the acceleration.

There are two general approaches that have been developed to test departures from GR at cosmological scales. The ﬁrst approach is to choose a speciﬁc class of

MG models such as f(R) or DGP models and derive cosmological perturbations and observables for these models. These are then implemented in cosmological analysis software and compared to the data. For an excellent review of various MG models, see Jain & Khoury (2010). The second approach is to parameterize the deviations in a phenomenological way without detailed knowledge of the speciﬁc alternative theory. Technically, these models modify the growth equations with additional MG parameters that represent the departure from GR. In this paper, we only consider the second, phenomenological approach.

Modiﬁcations to GR are generally encapsulated by adding two MG parameters to the Poisson equations in (1.24) that were derived from the Einstein’s ﬁeld equations.

We will review some of the most ommonly used MG parametrizations.

16 Caldwell et al. (2007), Amendola et al. (2008), and Bean & Tangmatitham (2010)

modiﬁed the Poisson equations with a set of parameters Q(k, a) and R(k, a) as follow:

2 2 k Φ = 4πGa ρiδiQ(k, a) (1.47) − i X 2 2 k (Ψ R(k, a)Φ) = 12πGa ρi(1 + wi)σiQ(k, a) (1.48) − − i X where the index i denotes each matter species. In GR, both parameters take the value of unity. The parameter Q quantifies a modification to the gravitational potential Φ and the parameter R, often referred as ‘the gravitational slip’, represents the difference between the two gravitational potentials. To avoid a strong degeneracy between the two parameters, Equations (1.47) and (1.48) can be combined as follow Amendola et al. (2008):

2 2 2 k (Ψ + Φ) = 8πGa ρiδiΣ(k, a) 12πGa ρi(1 + wi)σiQ(k, a) (1.49) − i − i X X with

Q(k, a)[1 + R(k, a)] Σ(k, a) . (1.50) ≡ 2

The newly introduced parameter Σ(k, a) modiﬁes the Weyl potential ΨW (Ψ+Φ)/2 ≡ which aﬀects the propagation of light.

A similar parametrization often used in the literature is (Σ, η) (Zhao et al., 2009,

2010; Hojjati et al., 2011a; Caldwell et al., 2007; Amendola et al., 2008). In this parametrization, modiﬁcation to GR is implemented to the Newtonian potential Ψ by µ(k, a) and the other potential Φ is constrained indirectly through the gravitational slip η(k, a) as follow

2 2 k Ψ = 4πGa ρiδiµ(k, a) (1.51) − i X Φ = η(k, a) (1.52) Ψ 17 A third parametrization constrains modiﬁcation for non-relativistic particles and

relativistic particles separately (Tsujikawa, 2007; Song & Koyama, 2009; Linder,

2017). The modiﬁed Poisson equations take the form

2 2 k Ψ = 4πGa Gmatter ρiδi (1.53) − i X 2 2 k (Ψ + Φ) = 8πGa Glight ρiδi (1.54) − i X where Gmatter and Glight are the dimensioness parameters that indicate modiﬁcations of General Relativity. Note that the standard GR equations are recovered when

Gmatter = Glight = 1. The gravitational accelerations of non-relativistic particles are determined by Ψ and the paths of photons depend on Φ + Ψ. Therefore, Gmatter is

sensitive to modiﬁcations to the structure growth through the Newtonian potential

Ψ, whereas Glight is sensitive to modiﬁcations to the lensing of light through the Weyl

potential (Ψ + Φ)/2. Note that, the set of the parameters Gmatter and Glight is also referred to as µ and Σ (Daniel et al., 2010; Song et al., 2011; Daniel & Linder, 2013;

Zhao et al., 2012; Simpson et al., 2013).

All three parametrizations described above are interchangeable during the matter dominated era with negligible anisotropic stress σ = 0 as

G 1 µ = QR = matter , η = (1.55) G R Q(1 + R) G Σ = = light , µη = Q. (1.56) 2 G

A more extended discussion of the MG parametrization can be found in Daniel et al.

(2010). Current MG constraints from DES and the comparison with results from

other cosmological surveys will be presented in Section 3.5.

Recently, the Planck Collaboration presented their MG constraints based on their

measurements of the CMB temperature, polarization, lensing data sets combined with

18 several other external data sets such as the BAO and RSD measurements and type Ia supernova measurements (Planck Collaboration et al., 2018). In the following year,

KiDS-450 (Joudaki et al., 2017) and DES (Abbott et al., 2019) reported their own MG constraints using similar parametrizations and external data sets. A brief summary of the DES work and comparisons with results from other surveys will be given in

Section 3.5.

The DMASS galaxy sample introduced in Chapter 2 is specially designed to achieve tight MG constraints from a joint analysis of galaxy clustering and galaxy lensing. To estimate the constraining power of DMASS, I performed a simple test for forecasting MG constraints. The details and result of the forecast will be presented in Section 3.6.

19 1.6 Cosmological Probes of Cosmic Acceleration

Broadly defined, there are two categories of observational probes that used in cosmological analyses. The first category constrains the cosmic geometry and expansion history via standard rulers and standard candles. The second category includes probes that contrain the growth of structure. In this section, I briefly summarize the

ﬁve principal cosmological probes of dark energy and modiﬁed gravity: SNe Ia, CMB,

BAO, RAD and weak gravitational lensing.

1.6.1 Type Ia Supernovae

SNe Ia are one of the most compelling tools for studying cosmic acceleration. They are produced by the thermonuclear explosions of white dwarfs that occur when ac- creted matter from a companion star exceeds the Chandrasekhar mass limit. The peak luminosity of SNe Ia is consistent because of the uniform mass of white dwarfs that explode via the accretion mechanism (Hoyle & Fowler, 1960; Arnett, 1969; Colgate

& McKee, 1969). Due to the stability of the luminosity, SNe Ia has been considered as good standard candles to measure cosmological distances.

The absolute bolometric magnitude of SNe Ia is known to be MB 19.3. To ≈ − estimate the luminosity distance dL, one can use the distance modulus given as

µ = m MB = 5 log dL(z) + 25 , (1.57) − where m is the apparent magnitude. Along with the redshift of an object independently measured from spectroscopy, the measurement of dL(z) helps to constrain the

Hubble constant H0 and density parameters Ωm and ΩΛ via

(1 + z) z dz0 dL(z) = . (1.58) 0 0 3 0 H0 0 Ω (1 + z ) + Ω Z m Λ p20 As mentioned before, the discovery of cosmic acceleration was made by two independent supernovae teams led by Saul Perlmutter (the Supernova Cosmology Project;

Perlmutter et al., 1999) and Brian Schmidt and Adam Riess (the High Z Supernova

Search Team; Riess et al., 1998). They measured the apparent magnitudes of many type Ia supernovae and found that the most distant SNe are dimmer than would be expected in a matter-dominated universe. Assuming a spatially ﬂat universe in

GR, their measurements imply that the universe has entered a phase of accelerated expansion.

This ﬁnding has been strengthened by larger and better supernova samples constructed later over the following decades Knop et al. (2003); Astier et al. (2006);

Wood-Vasey et al. (2007a); Miknaitis et al. (2007a); Kowalski et al. (2008); Scolnic et al. (2018). Many groups have worked on low-redshift (z < 0.1) SNe from surveys such as CfA1-CfA4 (Riess et al., 1999; Jha et al., 2007; Hicken et al., 2009a,b, 2012),

CSP (Contreras et al., 2010; Folatelli et al., 2010; Stritzinger et al., 2011), LOSS

(Ganeshalingam et al., 2013). These low-redshift SNe samples have contributed to our understanding of the explosion mechanism of SNe Ia and helped to mitigate systematic uncertainties. Intermediate-redshift (0.1 < z < 0.7) SNe that measure the strength of cosmic acceleration are observed by ESSENCE (Miknaitis et al., 2007b;

Wood-Vasey et al., 2007b; Narayan et al., 2016), SDSS (Frieman et al., 2008b; Kessler et al., 2009), SNLS (Conley et al., 2011; Sullivan et al., 2011), and PS1 (Rest et al.,

2014; Scolnic et al., 2014). High-z SNe with redshifts (z > 1.0) are from SCP (Suzuki et al., 2012), GOODS (Riess et al., 2004, 2007), and CANDELS/CLASH surveys

(Rodney et al., 2014; Graur et al., 2014; Riess et al., 2018). These high-z SNe ﬁrmly

21 established the cosmic acceleration by ruling out dust as a plausible cause of the dimming of intermediate-redshift SNe.

Recently, Scolnic et al. (2018) constructed the Pantheon sample by assembling a number of SNe Ia samples from the above and other surveys. Figure 1.1 shows the

Hubble diagram for the Pantheon Supernova sample, plotting the distance modulus for each SN over the redshift range 0.01 < z < 2.3. The solid line is the best-ﬁt cosmology. By combining Planck 2015 CMB measurements with their supernova

0 measurement, they showed that their result favors an accelerated universe with Ωm =

0.307 0.012 and w = 1.026 0.041, which is consistent with the current ΛCDM ± − ± model.

In the near future, we expect new data sets from the Large Synoptic Survey Tele- scope (LSST; LSST Science Collaboration et al., 2009) and the Wide Field Infrared

Survey Telescope (WFIRST; Spergel et al., 2015). LSST will collect hundreds of thousands of Type Ia SNe over 20, 000 deg2 which will reduce the statistical errors ∼ substantially. The WFIRST supernova survey will ﬁnd 2700 Type Ia SNe with ∼ spectroscopic redshifts up to z < 1.7 which will yield the best achievable constraints of the cosmic expansion history and dark energy.

22 14

Figure 11. The Hubble diagram for the Pantheon sample. The top panel shows the distance modulus for each SN; the bottom panel shows residuals to the best ﬁt cosmology. Distance modulus values are shown using G10 scatter model.

FigureGiven 1.1:a vector of The binned Hubble distance residuals diagram of the SN for the Pantheon5.1. sample.Calibration The top panel shows sample that may be expressed as µ~ = µ~ µ~ (as the distance modulus for each SN;model the bottomThe ‘Supercal’ panel calibration shows of residuals all the samples to in the this best fit shown in Fig. 11 (bottom)) where µ~ model is a vector of analysis is presented in S15. S15 takes advantage of cosmology. Distance modulus values2 are shown using G10 scatter model. Figure is distances from a cosmological model, then the of the the sub-1% relative calibration of PS1 (Schlafly et al. frommodel Scolnic fit is expressed et al. as (2018). 2012) across 3⇡ steradians of sky to compare photome- 2 T 1 try of tertiary standards from each survey. S15 measures = µ~ C µ.~ (8) · · percent-level discrepancies between the defined calibra- Here we review each step of the analysis of the Pan- tion of each survey by determining the measured bright- theon sample and their associated systematic uncertain- ness di↵erences of stars observed by a single survey and ties. PS1 and comparing this with predicted brightness dif-

23 1.6.2 Cosmic Microwave Background Radiation

The Cosmic Microwave Background (CMB) is the radiation remnant from the

early universe when photons decoupled from the thermal plasma (z 1090). Due to ≈ the time locality of this event, since the event took place well before the cosmic accel-

eration becomes dominant, the CMB anisotropies do not provide direct constraining

power on dark energy or modiﬁed gravity. Nevertheless they play an important role

as a complementary probe by constraining the background geometry and matter and

radiation content of the universe. The CMB angular power spectrum has two major

features: a series of peaks at small scales and the Sachs-Wolfe plateau at large scales.

The following paragraphs will focus on these two features. For information of other

features, see Huterer & Shafer (2018b).

Before decoupling, the baryon-photon ﬂuid underwent dampled oscillations due

to the competing eﬀects of infall gravity and radiation pressure. The series of peaks

at small scales are essentially the imprints of the acoustic oscillations. Although the

spacing and height of the peaks also oﬀer useful information of the matter density,

the most valuable information comes from the ﬁrst and largest peak. The location of

the ﬁrst peak is related to the sound horizon rs that the acoustic wave can travel at

a sound speed cs. The angular size of the sound horizon is given as

rs(z∗) θ∗ = , (1.59) DA(z∗) where DA is the angular diameter distance from the observer to the CMB and z∗ ≈ 1090 is the redshift at recombination. For the ﬁrst peak, the location of the peak

can be obtained as l πθ∗. Since the angular diameter distance is aﬀected by the ≈

24 contents of the universe at z < 1, more dark energy Ωde (or modiﬁed gravity) shifts the peak to smaller scales.

The late-time Integrated Sachs Wolfe (ISW) eﬀect is a secondary anisotropy of the

CMB caused by CMB photons that gain energy due to time-evolving gravitational potentials (Sachs & Wolfe, 1967; Kofman & Starobinskij, 1985). Since dark energy or modified gravity affects the gravitational potentials driving the growth of large scale structure, the signature of them appears on the ISW signals. The ISW effect modifies the CMB power spectrum at very large scales at (l < 20). At this large scale, measuring the ISW signal is challenging because of the large cosmic variance error. One general approach to bypass this difficulty is to cross-correlate the CMB temperature data with large scale structure tracers as done in Boughn & Crittenden

(2004); Fosalba et al. (2003); Nolta et al. (2004). Many following cosmological analyses adopted the same technique to constrain either dark energy or modiﬁed gravity models with ISW. See Ishak (2019) and references therein.

More advanced methods such as using stacking of CMB ﬁelds (Granett et al.,

2008) or the ISW-lensing bispectrum (Planck Collaboration et al., 2016d) have been actively developed along with better observational data. For further details about various eﬀorts for future analyses, we refer readers to read Section 7.8 in Weinberg et al. (2013) and references therein.

1.6.3 Baryonic Acoustic Oscillation

The early universe was hot and dense enough that photons were coupled with electrons and baryons. As the universe expanded, the primordial gas was cooled and trapped photons became free, leaving the baryons in place. Today, this eﬀect is

25 observed as a small excess in number of pairs of galaxies separated by 150 Mpc (the ∼ sound-horizon distance). These features are often referred to as the baryon acoustic oscillations (BAO). For further details of theoretical background, see Section 4.3 in

Weinberg et al. (2013).

The main strength of BAO to constrain dark energy comes from the precise, independent knowledge of the sound horizon. By using the sound-horizon distance rs as a standard ruler, the BAO method measures the angular diameter distance (DA(z)) and the Hubble parameter H(z):

t0 c z∗ c r = s dt = s dz (1.60) s a(t) H(z) Z0 Z∞ where cs is the speed of sound, z∗ is the redshift at recombination. Separations along the line of sight correspond to diﬀerences in redshift, which can be related to the

Hubble parameter as follow:

∆zs = H(z) rs . (1.61)

If the quantity ∆zs can be measured from spectroscopy, one can constrain H(z) accurately at some speciﬁc redshift. Separations transverse to the line of sight correspond to diﬀerences in angle that constrains the angular diameter distance as follow:

rs ∆θs = . (1.62) DA(z)

By measuring the angular diﬀerence ∆θs from surveys, one can infer the angular diameter distance DA(z).

A plethora of BAO measurements have been made and contributed to constraining the background geometry and expansion history. The ﬁrst convincing dection of

BAO was made from the SDSS Data Release 3 (DR3) (Padmanabhan et al., 2012)

26 and and 2dFGRS (Cole et al., 2005) at a median redshift z = 0.15. The 6dFGS survey measured BAO at very low redshift z = 0.106 (Beutler et al., 2011). The

BOSS collaboration performed BAO analyses with large spectroscopic galaxy samples

LOWZ and CMASS and achieved a sub-per cent precision at z = 0.32 and z = 0.57

(Alam et al., 2017b). WiggleZ survey measured BAO at z = 0.6 (Blake et al., 2011b).

Future surveys for BAO require spectroscopic redshift with uncertainty δz < 0.001, and should have suﬃciently large survey volume to suppress sample variance. In

2020s, a number of ambitious surveys scanning vast areas are in development or ready to launch. Among those future or ongoing surveys, PFS (Tamura et al., 2016) and DESI (DESI Collaboration et al., 2016a) are the ones that satisfy both condition. Along with the future CMB and SN measurements, these two surveys will yield signiﬁcant improvements in the precision of BAO.

1.6.4 Redshift Space Distortion

By the gravitational force of large-scale structure, galaxies move with the so-called peculiar velocity. These motions relative to the Hubble recession lead to a Doppler shift in galaxy redshifts. Consequently, the observed spatial distribution of galaxies appears squashed along the line of sight on large scales and stretched out in the radial direction on smaller scales (Kaiser, 1987; Hamilton, 1998). This distortion effect is reffered as ‘redshift space distortion (RSD)’. RSD is very sensitive to the structure growth rate, therefore, sensitive to dark energy/gravity that governs the structure growth at late times. In this section, I briefly review the basic formalism of RSD.

Detailed reviews on the topic can be found in Samushia et al. (2014) and Blake et al.

(2011a) and references therein .

27 The observed position of a galaxy in the redshift space can be connected to the true position in the real space as follows:

s(r) = r + vr(r)ˆr (1.63)

where vr is the peculiar velocity in the radial direction. On large scales, the peculiar velocity is related to the matter density via

βδm + vr = 0 (1.64) ∇ · with β(z) = f(z)/b(z) and f(z) is the structure growth rate deﬁned in Equation 1.27.

Following the Kaiser formula (?Hamilton, 1998), the galaxy density in the redshift space can be mapped to the galaxy density in the real space:

s 2 r δg(k) = (1 + βµ )δg(k) (1.65) where µ is the cosine of the angle with the line of sight. From this relation, the galaxy power spectrum is derived as

s 2 2 r Pg (k, µ, z) = [b(z) + f(z)µ ] Pm(k, z) (1.66)

The nonlinear FoG eﬀect at small scales can be introduced by multiplying the galaxy power spectrum by the gaussian dampling functions below (Percival & White, 2009a):

2 −(kσvµ) FGaussian(k, µ) = e . (1.67)

where σv is the dispersion of the velocity v. Consequenty, the full form of the galaxy power spectrum in the redshift space is

2 s 2 2 r −(kσvµ) Pg (k, µ, z) = [b(z) + f(z)µ ] Pm(k, z) e (1.68)

28 The constraining power of RSD mainly comes from the combination of two parameters f(a)σ8(a) (Song & Percival, 2009; Percival & White, 2009b). This quantity is sensitive not only to dark energy but also to modiﬁcations to GR (Linder, 2008).

RSD showed its ability to constrain modiﬁed gravity in several cosmological analyses.

For example, Okada et al. (2013) utilized RSD to exclude some covariant Galileon

MG models. The current available RSD measurements include 6dFGS (Beutler et al.,

2011), 2dFGRS (Cole et al., 2005), SDSS LRG (Samushia et al., 2012), BOSS LOW and BOSS CMASS (Alam et al., 2017b) and WiggleZ (Samushia, Percival & Rac- canelli, wig).

29 1.6.5 Weak Gravitational Lensing

Weak gravitational lensing is the deﬂection of light from distant objects by fore-

ground matter in the Universe. The eﬀect of weak gravitational lensing is observed

as a subtle distortion of shapes of distant galaxies. Since light from distant objects

must pass by nearby foregound matter distributions, their shapes can be used to infer

the underlying statistical distribution of foreground matter and hence the inﬂucence

of dark energy on the growth of cosmic structure (for a detailed review, see

The distortions of images of background galaxies can be described as

xu 1 κ γ1 γ2 xl = − − − (1.69) yu γ2 1 κ + γ1 yl − − where (xu, yu) is the displacement vector in the source plane (unlensed) and (xl, yl) is the displacement vector in the image plane (lensed). γ1 and γ2 are the real and imaginary components of the total lensing shear γ respectively. The total lensing shear is deﬁned as γ = γ1 + iγ2. The convergence κ describes a change in size of lensed objects, deﬁned for a given point on the sky θ as

χ χW (χ) κ(θ) = dχ δ(χθ) . (1.70) a(χ) Z0 Here χ is the comoving distance, δ is the matter overdensity. The function W (χ) is the geometric weight function describing the lensing eﬃciencty of foreground galaxies deﬁned as

2 ∞ 0 3H Ωm χ χ W (χ) = 0 dχ0n(χ0) − (1.71) 2c2 χ Zχ in terms of the source distribution n(χ0).

The correlation function of distorted galaxy shapes is referred to as ‘cosmic shear’.

To estimate the cosmic shear ﬁeld, we locally average the shape of large number of

30 distant galaxies. The most widely used statistical measure of cosmic shear is the shear angular power spectrum and its Fourier transform, the shear angular correlation

i,j function. The cosmic shear power spectrum Pκ correlating redshift bins i and j takes the form

∞ W (χ)W (χ) P i,j(l) = dχ i j P (k = l/χ, χ) (1.72) κ a2(χ) δ Z0 where l denotes the angular multipole and P (k, χ) is the usual matter power spectrum.

The integral along the line of sight indicates that weak lensing radially projects the density ﬂuctuations between us and the source galaxies. Additional information can be extracted by correlating diﬀerent redshift bins; this is referred to as tomography.

Another eﬀective application of weak lensing is to measure the correlation of the position of the foreground galaxies with the shear from more distant galaxies. This method, which is referred to as “galaxxy-galaxy lensing”, probes the galaxy-shear correlation function across the sky. In the linear regime, the galaxy-galaxy lensing signal

2 is proportional to b Ωmσ . Combining it with galaxy clustering signal which scales ∝ 8 2 2 as b σ enables us to eliminate the galaxy bias and tighten the constraint of Ωmσ8. ∝ 8 The most advanced application of weak lensing is combining three probes: galaxy clustering, galaxy-galaxy lensing and cosmic shear. This novel technique named as the 3x2 pt statistics adopted in the recent analyses of the imaging surveys such as

DES (Dark Energy Survey Collaboration et al., 2016) and KiDS (de Jong et al., 2013).

The statistical error of the shear power spectrum is deﬁned as

2 σ2 ∆P i,j(l) = P i,j(l) + γ , (1.73) κ (2l + 1)f κ n s sky eff where fsky is the fraction of sky covered by the survey, σγ is the standard deviation of the shear, and neff is the effective numer density per steradian of galaxies. Since the

31 −1/2 statistical errors are proportional to fsky , recent surveys covering large sky such as

DES ( 5, 000 deg2) or KIDS ( 1, 500 deg2) are signiﬁcantly improving dark energy ∼ ∼ constraints. Among the Stage IV surveys, LSST is aming to observe an area of

2 2 20, 000 deg with effective source galaxy density of neff = 30 galaxies per arcmin , which would achieve 0.1% accuracy. ∼ Finally, the interpretation of weak lensing measurements faces various observational and theoretical challenges. A variety of systematic errors such as the atmo- spheric blurring of the images, telescope distortions, the uncertainty in the photometric redshift of source galaxies have been actively studied to ameliorate their effects in the future analyses. Astrophysical uncertainties such as intrinsic alignments, predicting clustering in the non-linear regime from N-body simulations, and accounting for non-Gaussian errors on small angular scales are critical topics to be studied. A detailed review for controling systematic errors and theoretical challenges, see Huterer

& Shafer (2018b).

32 Chapter 2: Producing a BOSS-CMASS sample with DES imaging

In this chapter, I will present the full content of the DES-CMASS (DMASS) catalog paper (Lee et al., 2019). The catalog is constructed from the images of

DES taken during its ﬁrst year. The sample mimics the statistical properties of the spectroscopic galaxy sample ‘CMASS’ from the Baryon Oscillation Spectroscopic

Survey (BOSS). A brief summary of the two cosmological surveys and instruments will be presented with the description of data sets. The original abstract of the paper is given below:

We present a sample of galaxies with the Dark Energy Survey (DES) photometry that replicates the properties of the BOSS CMASS sample. The CMASS galaxy sample has been well characterized by the Sloan Digital Sky Survey (SDSS) collaboration and was used to obtain the most powerful redshift-space galaxy clustering measurements to date. A joint analysis of redshift-space distortions (such as those probed by CMASS from SDSS) and a galaxy-galaxy lensing measurement for an equivalent sample from DES can provide powerful cosmological constraints. Unfortunately, the DES and SDSS-BOSS footprints have only minimal overlap, primarily on the celestial equator near the SDSS Stripe 82 region. Using this overlap, we build a robust Bayesian model to select CMASS-like galaxies in the re- mainder of the DES footprint. The newly defined DES-CMASS (DMASS) sample consists of 117,293 effective galaxies covering 1, 244 deg2. Through various validation tests, we show that the DMASS sample selected by this model matches well with the BOSS CMASS sample, specifically in the South Galactic cap (SGC) region that includes Stripe 82. Combin- ing measurements of the angular correlation function and the clustering-z

33 distribution of DMASS, we constrain the diﬀerence in mean galaxy bias and mean redshift between the BOSS CMASS and DMASS samples to be ∆b = 0.010+0.045 and ∆z = 3.46+5.48 10−3 for the SGC portion −0.052 −5.55 × of CMASS, and ∆b = 0.044+0.044 and ∆z = (3.51+4.93) 10−3 for the −0.043 −5.91 × full CMASS sample. These values indicate that the mean bias of galaxies and mean redshift in the DMASS sample is consistent with both CMASS samples within 1σ.

The full list of the original authors can be found in the original paper.

2.1 Introduction

Since the discovery of the accelerating expansion of the Universe two decades ago

(Riess et al., 1998; Perlmutter et al., 1999), observational and theoretical work has led to a concordance cosmological model dominated by 70% dark energy, 25% dark matter, and 5% baryons. Despite the fact that dark energy occupies the majority of the energy density in the universe, little is understood about its physical nature due to the apparent lack of visible properties. Compelling evidence for the presence of dark energy comes from observations of the underlying matter distribution in the

Universe using supernovae, Baryon Acoustic Oscillations (BAO), and measurements of large-scale structure growth (Frieman et al., 2008a; Weinberg et al., 2013; Huterer

& Shafer, 2018a).

To trace out the underlying structure in matter, cosmologists traditionally use galaxies by measuring galaxy clustering as a function of scales. However, using galaxies as tracers results in a biased view of the matter distribution because galaxies form at the peaks of the matter density field where gas reaches high enough density to cool and form stars (Kaiser, 1984). The relation between the spatial distributions of galaxies and the underlying dark matter density field is known as galaxy bias. Galaxy bias varies for different scales and galaxy properties such as luminosity or type, and those

34 quantities are degenerate with each other. In the absence of additional information, galaxy bias is indistinguishable from the overall amplitude of matter ﬂuctuations, which makes galaxy bias a major systematic uncertainty in cosmological analyses

(Seljak et al., 2005).

Fortunately, weak gravitational lensing provides a direct way to measure the matter distribution, avoiding the issue of galaxy bias. Cosmic shear is the subtle shape distortions of background galaxies by the foreground matter distribution. It is thus directly connected to the matter distribution and thereby lets us measure the matter distribution without any galaxy bias (see the review in Weinberg et al. (2013) and references therein). However, cosmic shear is technically challenging to measure due to many sources of systematic errors. Because of the small size of the eﬀect compared to the intrinsic random variation in galaxy orientations and ellipticities, weak lensing measurements require a substantial number of source galaxies to achieve small statistical errors. This results in including source galaxies as faint and small as possible down to the limit where systematic errors need to be controlled (Mandelbaum, 2018).

Galaxy-galaxy lensing has been shown to be a powerful complementary tool to overcome the aforementioned limitations (Baldauf et al., 2010; Yoo & Seljak, 2012;

Choi et al., 2012; van den Bosch et al., 2013; Mandelbaum et al., 2013; Park et al.,

2016; Miyatake et al., 2015; More et al., 2015; Alam et al., 2017a; Singh et al., 2018,

2019; Amon et al., 2018; Jullo et al., 2019). It is the cross-correlation function between foreground galaxies and background shear, which represents a direct measurement of the galaxy-matter correlation function. In combination with accurate galaxy clustering information, lensing observables can fully exert their full constraining power.

35 In galaxy-galaxy lensing, the galaxy bias is tied to the matter clustering in a diﬀer- ent way from galaxy clustering. Combining the two probes breaks the degeneracy between the two constraints.

Some of the sets of galaxies most frequently used as gravitational lenses in cosmological analyses are the BOSS spectroscopic galaxy samples (Reid et al., 2016) from the Baryon Oscillation Spectroscopic Survey (BOSS; (BOSS; Eisenstein et al., 2011).

The large sample size and availability of spectroscopic redshifts for all BOSS galaxies allowed the BOSS collaboration to measure the BAO signature with an uncertainty of only one per cent for the case of the BOSS CMASS sample, which is the most constraining BAO measurement to date (Reid et al., 2016). This led to several follow-up studies that combined the BOSS galaxy clustering results with galaxy-galaxy lensing measurements using the BOSS galaxies as lenses.

Mandelbaum et al. (2013) constrained the amplitude of the matter ﬂuctuations at z < 0.4 using data from the Sloan Digital Sky Survey (SDSS) Data release 7. They utilized two spectroscopic samples BOSS Main and Luminous Red samples as lenses − and combined galaxy-galaxy lensing between those samples and SDSS source galaxies with galaxy clustering from the same samples. Singh et al. (2018) adopted a similar approach. They combined galaxy clustering from BOSS with galaxy-galaxy lensing and galaxy-CMB lensing signals, by utilizing the BOSS LOWZ (0.15 < z < 0.43) and

CMASS (0.43 < z < 0.7) samples as lenses. However, due to the shallow depth of

SDSS imaging, their measurement of galaxy-galaxy lensing was obtained only with

BOSS LOWZ.

Miyatake et al. (2015), More et al. (2015) and Alam et al. (2017a) extended this kind of joint analysis to galaxies at a higher redshift z 0.5 by using BOSS CMASS as ∼

36 lenses with the deeper and better quality imaging data from CFHTLenS (Heymans et al., 2012). Jullo et al. (2019) performed a similar analysis with BOSS CMASS galaxies and two weak lensing data sets - CFHTLenS and CFHT-Stripe 82 (Moraes et al., 2014). Amon et al. (2018) utilized three spectroscopic galaxy samples including

BOSS LOWZ & CMASS with deep imaging data from KiDS (de Jong et al., 2013) to do a joint analysis of galaxy clustering and galaxy-galaxy lensing. However, the lensing measurements of these analyses are limited to the small overlapping area - a few hundreds of deg2. For instance, the overlapping region between BOSS and

CFHTLenS is only 105 deg2 which is about one hundredth of the BOSS area. ∼ The Dark Energy Survey (DES) is a large weak lensing survey that images over

5, 000 deg2 of the southern sky to 24th i-band limiting magnitude in the grizY ﬁlter bands. Precise photometry and the largest survey area among the current genera- tion of experiments makes DES data an excellent source of imaging data for a joint analysis of galaxy clustering and galaxy-galaxy lensing. However, as with previous measurements combining lensing and clustering, the overlapping region between the

DES Year 1 footprint ( 1, 800 deg2) and the BOSS footprint is fairly small, con- ∼ sisting of only 150 deg2 near the celestial equator called Stripe 82 (Dark Energy ∼ Survey Collaboration et al., 2016; Drlica-Wagner et al., 2018). Simply combining

BOSS galaxy clustering with galaxy-galaxy lensing from DES would be limited to the small overlapping area and fail to utilize the full statistical power of DES.

Inspired by the potential power of combining all the available SDSS and DES measurements, we present in this paper an attempt to deﬁne a catalog of DES galaxies from the full footprint of DES, whose properties match with the BOSS CMASS galaxy sample. The resulting DES-CMASS (hereafter DMASS) sample will be the

37 best available for a cosmological analysis combining galaxy-galaxy lensing and galaxy clustering measurements.

We start by using the subset of BOSS CMASS galaxies in Stripe 82 where the

BOSS footprint overlaps with DES. Using galaxies measured by both DES and BOSS we train a galaxy selection model using the DES photometric information. Rather than classifying individual galaxies, the model assigns a membership probability to each galaxy and down-weights galaxies that are less likely to be CMASS. To account for spatial dependence of photometric errors, we use the Extreme Deconvolution algorithm (Bovy et al., 2011a) and obtain underlying color distributions of galaxies from the training sample. The underlying color distributions are convolved with photometric errors of the target region, and thereby the model correctly accounts for the photometric errors in the diﬀerent regions.

This paper is organized as follows. In the following section, we will introduce the BOSS CMASS sample and the DES Y1 GOLD catalog we use for this work and present the selection criteria that were used for the BOSS CMASS sample in detail.

We will address the diﬀerence between the SDSS and DES photometric systems and explain how it will be accounted for in our probabilistic model. Our model construction can be found in Section 2.3. The systematic uncertainties of the DMASS sample will be presented in Section 2.4 and the basic properties of the resulting DMASS catalog and validation tests will be discussed in Section 2.5. We will summarize and conclude in Section 2.6.

The ﬁducial cosmological model used throughout this paper is a ﬂat ΛCDM model with the following parameters: matter density Ωm = 0.307, baryon density Ωb =

38 0.048, amplitude of matter clustering σ8 = 0.8288, spectral index ns = 0.96 and

−1 −1 Hubble constant h H0/100 km s Mpc = 0.677. ≡

39 2.2 Data

2.2.1 BOSS DR12 CMASS Sample

The Baryon Oscillation Spectroscopic Survey (BOSS; Eisenstein et al., 2011;

Bolton et al., 2012; Dawson et al., 2013a) was designed to measure the scale of baryon acoustic oscillations (BAO) in the clustering of matter over a larger volume than the combined eﬀorts of all previous spectroscopic surveys of large-scale structure. BOSS uses the same wide ﬁeld, dedicated telescope as was employed by SDSS I and II (York et al., 2000), the 2.5 m aperture Sloan Foundation Telescope (Gunn et al., 2006) at

Apache Point Observatory in New Mexico. Those surveys imaged over 10, 000 deg2 of high galactic latitude sky in the ugriz bands, using a mosaic CCD camera (Gunn et al., 1998) with a ﬁeld of view spanning 3◦. BOSS consists primarily of two inter- leaved spectroscopic surveys observed simultaneously: a redshift survey of 1.5 million luminous galaxies extending to z = 0.7 and a survey of the Lyman alpha forest toward 150,000 quasars in the redshift range 2.15 < z < 3.5. Description of survey design, target selection, and their implications for cosmological analysis are available in Dawson et al. (2013a) and Reid et al. (2016).

The BOSS DR12 galaxy survey targeted two distinct samples known as LOWZ and CMASS (Reid et al., 2016). The higher redshift sample CMASS covers redshifts

0.43 < z < 0.75 and is designed to select a stellar mass-limited sample of objects of all intrinsic colors, with a color cut that selects almost exclusively on redshift. The

CMASS galaxy sample is selected by the combination of the 7 diﬀerent color and magnitude cuts. Every source satisfying the selection cuts was targeted by the BOSS spectrograph to obtain their redshifts, except for 5.8% of targets in a ﬁber collision

40 group and 1.8% of targets for which the spectroscopic pipeline fails to obtain a robust

redshift (Reid et al., 2016).

The following three cuts simply limit colors or magnitudes to exclude redshift

failures or outliers with problematic photometry:

17.5 < icmod < 19.9 (2.1)

iﬁb2 < 21.5 (2.2)

rmod imod < 2 , (2.3) − where the subscript ‘mod’ denotes model magnitudes, ‘cmod’ denotes cmodel magnitudes, and ‘ﬁb2’ stands for ﬁber magnitude estimated in a 200 aperture diameter assuming 200 seeing. For further details of SDSS magnitudes, we refer readers to the

SDSS survey website2. The following two cuts are applied to reject stars:

ipsf imod > 0.2 + 0.2(20 imod) (2.4) − −

zpsf zmod > 9.125 0.46zmod , (2.5) − − where ‘psf’ stands for magnitudes computed from the point spread function model.

To exclusively select galaxies on redshift, the BOSS target selection utilizes the quantity d⊥ deﬁned as d⊥ = (rmod imod) (gmod rmod)/8.0. This quantity is − − − designed to approximately follow the color locus of the passively evolving LRG model in Maraston et al. (2009) at z > 0.4. Since redshift gradually increases along the color locus, d⊥ is a good indication of redshift for CMASS type galaxies.

The following two cuts use d⊥ to select objects with respect to redshift:

icmod < 19.86 + 1.6(d⊥ 0.8) (2.6) −

d⊥ > 0.55 . (2.7)

2https://www.sdss.org/dr12/algorithms/magnitudes

41 Equation (2.6) selects the brightest objects at each redshift to keep an approximately constant stellar mass limit over the redshift range of CMASS. Equation

(2.7), the so called ‘d⊥’ cut, is the most restrictive cut among all selections described above. This cut isolates intrinsically red galaxies at high redshift. Considering the color/magnitude space occupied by all SDSS objects, this cut slices the densest region of the sample in the gri color plane and determines the sample’s redshift distribution. This is in contrast to the other cuts, which apply mainly to the edges of the color/magnitude distributions. Therefore, our work is mainly focused on character- izing the same cut in the DES photometry. More details about the d⊥ cut can be found in Eisenstein et al. (2001) and Padmanabhan et al. (2007), and our derived d⊥ cut in the DES system will be discussed in Section 2.3.

The colors and magnitudes used in the selection criteria are corrected for Milky

Way extinction by the galactic extinction map (Schlegel et al., 1998).

2.2.2 DES Y1 Gold Catalog

The Dark Energy Survey (DES; The Dark Energy Survey Collaboration, 2005;

Abbott et al., 2018a) is an imaging survey covering 5, 000 deg2 of the southern sky.

This photometric data set has been obtained in ﬁve broadband ﬁlters, grizY , ranging from 400nm to 1, 060nm (Li et al., 2016; Burke et al., 2018), using the Dark ∼ ∼ Energy Camera (DECam; Flaugher et al., 2015) mounted on the Blanco 4m telescope at Cerro Tololo Inter-American Observatory (CTIO) in Chile. The main goal of DES is to improve our understanding of cosmic acceleration and the nature of dark energy using four key probes: weak lensing, large-scale structure, galaxy clusters, and Type

Ia supernovae.

42 The Y1A1 GOLD wide-area object catalog (Drlica-Wagner et al., 2018) we use in this work was published as part of the DES Year 1 public data release (Abbott et al., 2018b). The catalog consists of 137 million objects detected in coadd images ∼ covering two disjoint areas; one overlapping with the South Pole Telescope (SPT;

Carlstrom et al., 2011), and a much smaller area near the celestial equator called

Stripe 82 (Annis et al., 2014).

For this work, we refine the DES Y1 Gold catalog selection by removing imaging artifacts and areas around bright foreground objects such as bright stars and globular clusters. We only keep clean sources with flag bit > 1 in Table 5 in Drlica-Wagner et al. (2018). We also select sources classified as galaxies by the flag MODEST==1.

Furthermore, we remove regions tagged by the DES Y1 BAO study (Crocce et al.,

2019) using veto masks. These additional masks select only the wide area parts of the surveys, namely those overlapping SPT, and remove a patch of 18 deg2 where the airmass computation is highly corrupted. The DES Y1 BAO study additionally removes a few deg2 sized regions where multi-object ﬁtting (MOF; Drlica-Wagner et al., 2018) photometry is unreliable. However, we do not exclude these regions since we do not use MOF measurements. Further details about the Y1 BAO masks can be found in Crocce et al. (2019). The resulting footprint after applying all masks aforementioned occupies 1, 244 deg2 in SPT and 123 deg2 in Stripe 82.

All magnitudes in the DES Y1 GOLD catalog are shifted by stellar locus regres- sion (SLR) which corrects for Galactic dust reddening (Drlica-Wagner et al., 2018).

For consistency with the original CMASS selection, we have removed this SLR correction and instead applied reddening corrections based on the SFD map (Schlegel et al., 1998) as done in SDSS. The correction to the DES magnitude for a band b

43 is Ab = Rb E(B V)SFD with interstellar extinction coeﬃcients for griz bands, × −

Rb = [3.186, 2.140, 1.569, 1.196], computed in Abbott et al. (2018b).

We applied additional magnitude cuts to the DES Y1 Gold catalog to exclude outliers in color space as follows:

17 < GDET < 24 (2.8)

17 < RDET < 24 (2.9)

17 < IDET < 24 (2.10)

0 < GMOD RMOD < 2.5 (2.11) −

0 < RMOD IMOD < 1.5 (2.12) −

IAUTO < 21 . (2.13)

Sources satisfying the magnitude cuts are kept. Subscripts DET and MOD stand for

DES detmodel magnitude and model magnitude respectively, and AUTO stands for

DES auto magnitudes. These three magnitudes are computed by an image-processing software called SExtractor3. We refer interested readers to the documentation of

SExtractor (Bertin & Arnouts, 1996) for further details. Note that all DES quantities are written in upper case to avoid confusion with corresponding SDSS quantities.

These cuts eﬀectively remove galaxies that are not likely to be CMASS galaxies.

Further, these cuts reduce compute time by decreasing the sample size to 10% of the full Y1 GOLD sample, while keeping 99.5% of CMASS galaxies in the overlapping region, Stripe 82.

44 22.5

.6 20.0 1 17.5 15.0 .2 1

12.5 gal r-i 10.0 N .8 0 7.5 5.0 .4 0 SDSS Color 2.5

.6 .2 .8 .4 .0 0 1 1 2 3 g-r

22.5

.6 20.0 1 17.5 15.0 .2 1

12.5 gal r-i 10.0 N .8 0 7.5 5.0 .4 0 DES Color 2.5

.6 .2 .8 .4 .0 0 1 1 2 3 g-r

Figure 2.1: The two-dimensional histograms of CMASS galaxies from Stripe 82 in the g r vs. r i color plane. The left panel shows SDSS, the right panel shows DES colors− of the− same galaxies. The colorbar shows the number of galaxies binned in each histogram bin. The red line is the d⊥ cut. CMASS galaxies look bluer in the DES photometry and the d⊥ cut discards almost half of the CMASS galaxies by crossing the most dense region. The grey contours in the right panel show the full distribution of DES Y1 GOLD galaxies in the color plane. The grey contours show that blindly lowering the d⊥ cut results in accepting more non-CMASS galaxies. 45 2.2.3 Diﬀerences between the SDSS and DES photometry

In the DES imaging pipeline, magnitudes for extended sources are derived from

diﬀerent models of luminosity proﬁles and bands optimized for each source (Drlica-

Wagner et al., 2018). This complicated procedure makes magnitudes in one band

highly correlated with other bands, as well as the shape or size of galaxies and in-

struments for each system and results in magnitudes for the same object being very

diﬀerent in one system from another in a way that is challenging to predict.

Figure 2.1 shows the diﬀerence in the r i vs. g r color space of the two diﬀerent − − imaging systems, using only tagged CMASS galaxies in the overlap region. The DES colors of CMASS galaxies are obtained by cross-matching the DES Y1 GOLD catalog with the CMASS photometric sample in Stripe 82 by position with a 200 tolerance. For the DES data, the detmodel magnitudes (MAG_DETMODEL) are used. The grey contours in the right panel show all sources from the DES Y1 GOLD catalog. The red solid line in both panels is the d⊥ cut given by Equation (2.7). By noting the large fraction of

DES objects below this line, one can clearly see how diﬀerent the two systems are. In the DES data, the d⊥ cut crosses the most dense part of galaxy sample. Notably, this is a dense region for the full gold sample as well. If we were to blindly apply the d⊥ to the DES data, we would remove almost half of the true CMASS sample. Applying a simple transformation that moves the d⊥ cut to lower r i values recovers most of the − CMASS galaxies, but at the cost of introducing many non-CMASS galaxies into the sample. Also noticeable in Figure 2.1 is the larger scatter in the SDSS distribution, especially in g r. − 3https://www.astromatic.net/software/sextractor

46 g r i z 3000 4000 5000 6000 7000 8000 9000 10000 11000 Wavelength (Angstroms)

Figure 2.2: The response functions for the griz SDSS (shaded) and DES (solid lines) filters as a function of wavelength (amstrong) with the spectral energy density (SED) distribution of an elliptical galaxy at z = 0.4 (black solid line). Near the 4000 amstrong break where the g r transition happens, the SDSS r filter (shaded) covers slightly lower wavelength than− the DES r filter (solid lines) does. This implies the same galaxy near z = 0.4 looks redder in the SDSS photometry than in the DES photometry.

47 There are several reasons for the discrepancy in the color space shown in Figure

2.1. One is that despite both surveys using griz ﬁlters, these ﬁlters are not identical.

Figure 2.2 illustrates this fact. The response functions for the ﬁve SDSS (shaded) and

DES (solid line) ﬁlters with the spectral energy density distribution of an elliptical galaxy are shown. The break in the model spectrum at 4, 000, a primary feature

of galaxy spectra, migrates through the g, r and i ﬁlters as the redshift increases

(Eisenstein et al., 2001; Padmanabhan et al., 2007). For elliptical galaxies near z ∼ 0.4, the 4, 000 break is located at wavelengths where the g r transition happens. − Near the 4, 000 break, the SDSS r ﬁlter (shaded) covers slightly lower wavelengths

than the DES r filter (solid lines) does. That implies galaxies near z 0.4 look redder ∼ in SDSS than they do in the DES photometric system. Since the redshift z = 0.4 is also where the d⊥ cut is defined, this discrepancy of the filter transition exacerbates the color mismatch.

A second cause for the discrepancy in color space arises from diﬀerences in the

SDSS and DES imaging pipelines. Magnitudes for extended sources are derived from the flux of a galaxy fitted with a best matched galaxy profile. Widely used galaxy profiles are exponential and de Vaucouleurs profiles (de Vaucouleurs, 1948), which perform better for disc and bulge galaxies, respectively. The SDSS imaging pipeline uses either one of these profiles to model magnitudes depending on the shape of a galaxy and uses a linear combination of two profiles for SDSS cmodel magnitudes. The

DES imaging pipeline uses only the exponential proﬁle consistently for all magnitudes.

The fitting procedure is different as well. For instance, the SDSS pipeline fits galaxies

48 only in the r band to obtain model magnitudes4 but the DES pipeline chooses a diﬀerent band for every source.

The last and most signiﬁcant reason for the mismatch in color distributions is the fact that SDSS has signiﬁcantly larger photometric errors compared to DES. The typical photometric error of SDSS for galaxies as bright as CMASS is 0.20 along the g r axis and 0.08 for the r i axis which is 5 times larger than the typical error of − − ∼ DES. The CMASS selection cuts in Equations (2.1)-(2.7) are simple cuts that do not take into account photometric errors. Ignoring photometric errors does not cause a notable problem for the cuts designed to limit faint magnitudes or to exclude outliers but must be considered thoroughly when it comes to the d⊥ cut. This is due to the location of the d⊥ cut in the densest region of the color space. Many galaxies with true colors outside of the d⊥ cut have scattered into the sample, while a similar amount of galaxies with true colors within the d⊥ cut could have scattered out of the SDSS selection. From this discussion, we infer that the d⊥ cut used to obtain the BOSS

CMASS sample, in terms of true properties, is not a sharp cut shown in Figure 2.1, but should instead be a form of likelihood function that accepts or rejects galaxies in a probabilistic way based on galaxy colors and photometric errors.

Based on the three reasons we listed above, we constructed a model that can handle the color mismatch and probabilistic selection near the d⊥ cut all together.

4 The term ‘model’ magnitudes in this paper indicates ‘modelMag’ magnitudes used in the BOSS selection criteria

49 2.3 Constructing the Model

While BOSS and DES operate in diﬀerent hemispheres, the survey footprints over-

lap in an equatorial area of the sky known as Stripe 82. DES Y1 imaged 123 deg2 of this region, thereby providing a region where data from the two surveys can be matched.

By using the photometric information in the overlapping region, we build an algorithm for probabilistic target selection that uses density estimation in color and magnitude spaces. The general concept of the algorithm is described in Section 2.3.1.

The algorithm is trained in half of the overlapping region and validated in the other half. We discuss the training and validation data sets in Section 2.3.2. The tools and detailed ﬁtting procedures for training are presented in Section 2.3.3. The results of validation and application of the algorithm to the target galaxies can be found in

Sections 2.3.4 and 2.3.5.

2.3.1 Overview of the Algorithm

The probability of being part of the CMASS sample for a source having a property

θ can be written as the combination of the likelihood and the prior according to Bayes’ theorem:

P (θ C) P (C) P (C θ) = | , (2.14) | P (θ) where

P (θ) = P (θ C)P (C) + P (θ N)P (N) . (2.15) | |

The notation C is the class of CMASS, N is the class of non-CMASS galaxies. P (C)

is the prior probability that a selected source is part of the CMASS sample, which

50 can be interpreted as the fraction of CMASS in the total galaxy sample. P (θ C) | is the likelihood of the source under the probability density function (pdf) of the property θ of CMASS. The pdf of the property θ of CMASS can be constructed from a histogram of CMASS as a function of θ. However, since we use noisy quantities such as observed colors and magnitudes, the resulting pdf might be biased by photometric errors that vary by observing conditions. For example, if the training region has a uniquely diﬀerent observing condition from other regions, the pdf model drawn from the training galaxies will not accurately represent CMASS. Therefore, the pdf should take into account measurement errors.

To ensure a uniform selection across the survey, we use the Extreme Deconvolu- tion (XD) technique ﬁrst proposed in Bovy et al. (2011a). The XD algorithm models the observed distribution of data as a mixture of Gaussians, convolved with a multi- variate Gaussian model for the measurement errors on each point. It iterates through expectation and maximization steps to solve for the maximum likelihood estimates of the parameters specifying the underlying mixture model.

The underlying distribution P (θtrue C) derived from XD is an unbiased pdf free | from regional measurement errors in the training set. By convolving the underlying distribution back with the measurement uncertainties of the test sample, we can infer the observed distribution that the same kind of galaxies would have in the test region.

For a given observation θ = θobs, , the observed quantity θobs with a corresponding { } measurement uncertainty , the likelihood of CMASS is written as

P ( θobs, C) = dθtrue p( θobs, θtrue) P (θtrue C) . (2.16) { }| { }| | Z

The ﬁrst factor p( θobs, θtrue) on the right side stands for the distribution function { }| of measurement uncertainty of θ in the presence of known measurement uncertainty

51 . We assume that the measurement uncertainty distribution of bright galaxies such as CMASS is nearly a Gaussian with a RMS width . The same procedure is repeated for non-CMASS galaxies.

Considering all factors, the resulting posterior probability that will be assigned to a target source having a property θobs with a measurement uncertainty is given as

dθtrue p( θobs, θtrue)P (θtrue C)P (C) P (C θobs, ) = { }| | , (2.17) |{ } dθtrue p( θobs, θtrue)P (θtrue) R { }| where R

P (θtrue) = P (θtrue C)P (C) + P (θtrue N)P (N) . (2.18) | | 2.3.2 The Training and Validation Sets

We use the overlapping area between BOSS and DES to train and validate the algorithm. To label DES galaxies as CMASS or non-CMASS galaxies, we cross-match the reﬁned DES Y1 GOLD catalog (described in Section 2.2) to the BOSS CMASS photometric sample5 using a 200 tolerance. The total number of galaxies labelled as

CMASS is 12, 639 over the area of 123 deg2 .

The labelled DES galaxies are split into the training and validation sets. In the overlapping region, the number density of galaxies varies along latitude. Since our probabilistic model assumes that the galaxies are homogeneously distributed in the full sky, we divided the overlapping area into Healpix pixels of resolution Nside = 64 in NEST ordering and took only even values of Healpix pixels as the training sets to populate the training regions uniformly. The total training set contains 6, 325 CMASS

5We do not use the BOSS spectroscopic sample for training. The spectroscopic sample of BOSS CMASS has about 5.8% and 1.8% of missing targets lost by ‘ﬁber collision’ and ‘redshift failure’, respectively (Reid et al., 2016). Since our probabilistic model is color-based, we utilize the BOSS photometric sample for training in order to include the photometry information from those missing galaxies.

52 galaxies and 340, 202 non-CMASS galaxies in 62.5 deg2. The two samples are used

separately to train the algorithm to construct the likelihoods for CMASS and non-

CMASS galaxies. Note that this division is used only to test the algorithm and we

will later switch to the full Stripe 82 region as the training set for the DES SPT

region.

2.3.3 Obtaining True Distributions with the Extreme-Deconvolution Algorithm

The Extreme Deconvolution (XD) algorithm developed by Bovy et al. (2011a) is a

generalized Gaussian-mixture-model approach to density estimation and is designed

to reconstruct the error-deconvolved true distribution function common to all samples,

even when noise is signiﬁcant or there are missing data. Starting from the random

initial guess of Gaussian mixtures, the algorithm iteratively calculates the likelihood

by varying means and widths of Gaussian components until it ﬁnds the best ﬁt of

Gaussian mixtures.

We use the Python version of the XD algorithm in the AstroML6 package (Vander-

Plas et al., 2012). The following four DES properties are ﬁtted by the XD algorithm:

(GDET RDET), (RDET IDET), RMOD, and IMOD. The two DES colors are selected as − − they mirror the SDSS information used for the d⊥ cut. The apparent magnitude IMOD is included to extract information induced by the cut given in Equation (2.1). There is no r band magnitude cut in the CMASS selection criteria, but we include RMOD in order to provide extra information to capture the diﬀerences between the SDSS and

DES ﬁlter bands. Star-galaxy separation was performed on DES photometry with the

6http://www.astroml.org

53 ﬂag MODEST == 1 (Drlica-Wagner et al., 2018), therefore we do not apply any further

cuts to replace cuts (2.4) and (2.5).

The AstroML XD algorithm leaves the initial number of Gaussian mixture com-

ponents as a user’s choice. One of the well-known methods for choosing the correct

number of components is to use the Bayesian Information Criterion (BIC; Schwarz,

1978). We use the Gaussian mixture module in the scikit-learn7 package (Pe- dregosa et al., 2012) to compute the BIC scores for a diﬀerent number of components.

The optimal number of components found by this exercise is 8 components for the

CMASS training set and 26 components for the non-CMASS training set.

The XD algorithm ﬁts the multi-dimensional histogram of the four aforementioned

DES properties with the optimal number of Gaussian mixtures and returns the values of amplitudes, means and widths of the best fit Gaussian mixture model. The resulting best fit model is used as a true distribution P (θtrue C) in Equations (2.17)-(2.18). | Throughout this work, we assume that there is no correlation between different bands.

2.3.4 Application to the Target Galaxies

In this section, we apply our probabilistic model to the test galaxies in order to validate the algorithm. The underlying distributions P (θtrue C) and P (θtrue N) are | | obtained from the XD algorithm as described in the previous section. The Bayesian

priors are given as P (C) = 0.018 and P (N) = 1 P (C). This is based on the fraction − of CMASS galaxies in the training set8.

7https://scikit-learn.org/ 8The fraction of CMASS may vary depending on the observing condition of the selected area, but we take the value in the training sample as a global prior for simplicity, assuming CMASS

54 100

(%) 60 , i

40 CMASS f

0 0 20 40 60 80 100 Model Probability (%)

Figure 2.3: Accuracy of model membership probabilities assigned to the test set. Galaxies in the test set are binned based on their probability assigned by the probabilistic model. The x-axis shows 20 bins of the assigned probability, and the y-axis shows the fraction of true CMASS galaxies in each bin. If the model successfully recovers the observed distribution of CMASS in the test region, the fraction of true CMASS galaxies in each bin should be identical to the assigned model probability. The dashed diagonal line in the ﬁgure stands for this ideal case, and the grey bars are given by Poisson errors.

55 The probability of being part of CMASS for a given property θ is analogous to the probability of ﬁnding a CMASS galaxy in a group of galaxies having the same property

θ. This implies that in a group of galaxies assigned the same model probability, the assigned probability should be identical to a fraction of galaxies labelled as CMASS.

To conﬁrm this argument, we bin test galaxies in 20 bins based on their assigned probability. In Figure 2.3, the x-axis shows the 20 bins of the assigned probability, and the y-axis shows the fraction of true CMASS galaxies in each bin. The grey bars are the fractions of CMASS-labelled galaxies in the test set with Poisson error bars.

The diagonal dashed line represents the ideal case that the probabilistic model would yield if the model successfully recovers the observed distribution of CMASS in the test region. The computed fractions of CMASS show good agreement with the ideal case within error bars.

Once all galaxies in the test set are assigned a model probability, the widely- accepted next step for classification is dividing target sources into two categories with a threshold probability > 50%. A similar probabilistic approach was done in Bovy et al. (2011b) to distinguish quasars from stars. However, we take a different approach since we are not interested in classifying individual galaxies accurately, but instead we focus on matching the statistical properties of groups of galaxies. In order to produce a statistical match, the membership probability we determine must faithfully reflect the probability that an object would be selected into the BOSS CMASS sample based on SDSS imaging.

Figure 2.4 presents the histograms of the d⊥ color of true CMASS in the training set (blue shaded histogram) binned in the diﬀerent ranges of the membership galaxies are homogeneously distributed in the Universe. We will show that this approximation can be justiﬁed through validation tests later in this paper.

56 0 < P (%) < 10 10 < P (%) < 20 20 < P (%) < 30 30 < P (%) < 40 40 < P (%) < 50 1 train DMASS ftrain = 9.3% ftrain = 7.9% ftrain = 7.6% ftrain = 7.1%

ftrain = 14.9% gal n

0 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 d ,DES d ,DES d ,DES d ,DES d ,DES ⊥ ⊥ ⊥ ⊥ ⊥ 50 < P (%) < 60 60 < P (%) < 70 70 < P (%) < 80 80 < P (%) < 90 90 < P (%) < 100 1 train DMASS ftrain = 7.9% ftrain = 9.1% ftrain = 11.3% ftrain = 17.8%

ftrain = 7.0% gal n

0 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 d ,DES d ,DES d ,DES d ,DES d ,DES ⊥ ⊥ ⊥ ⊥ ⊥

Figure 2.4: Histograms of the d⊥ color distributions of CMASS (blue) in the training set and DMASS (solid black line) in 10 membership probability bins. d⊥ on the x-axis consists of only DES quantities (d⊥,DES = (RDET IDET) (GDET RDET)/8.0 where − − − the subscript ‘DET’ denotes DES detmodel magnitude). ftrain in the top-right corner of each panel denotes the fraction of training galaxies binned in each probability bin, deﬁned as ftrain,i = Ntrain,i/Ntrain,total for the ith probability bin.

1 train train train train DMASS DMASS DMASS DMASS gal n

0 18 20 22 16 18 20 0 1 2 0.5 0.0 0.5 1.0 1.5 R I G R − R I MOD MOD DET − DET DET − DET

Figure 2.5: Histograms of color and magnitude distributions of CMASS in the training set (blue) and DMASS in SPT (black solid line). The colors and magnitudes on the x-axis are DES quantities.

57 probability bins. Training galaxies in low probability bins tend to have low d⊥ values because of their proximity to the d⊥ cut. ftrain in the top-right corner of each panel is the fraction of training galaxies binned in each probability bin, deﬁned as ftrain,i = Ntrain,i/Ntrain,total for the ith probability bin. Over the 10 probability bins, training galaxies are distributed uniformly, with a relatively high fraction in the lowest and the highest probability bins. This indicates that galaxies having low membership probabilities contribute to the CMASS sample as signiﬁcantly as galaxies having high membership probability.

From this, we can infer that in order to generate the same noise level that the original CMASS sample intrinsically has, galaxies should be populated based on their membership probability in the same way that ones in the CMASS sample are. In this sense, the model probability suggests a natural way of how we should make use of the assigned probabilities. We can either sample or weight a galaxy by its assigned probability in order to produce a sample that is a statistical match to the

BOSS CMASS sample. Throughout the rest of the work, we use the membership probability as weights.

The black solid lines in Figures 2.4 and 2.5 show the DES Y1 GOLD galaxies in the SPT region weighted by the assigned membership probabilities. Figure 2.4 shows that the weighting scheme successfully reproduces the noisy quantity d⊥ by populating each probability bin with the DES galaxies (black solid) as the CMASS galaxies (blue) are distributed. In Figure 2.5, the resulting DMASS distributions (black solid) of the colors and magnitudes are in good agreement with the distributions of the training sample (blue).

58 2.3.5 Excluding Low Probability Galaxies

In the overlapping region, only 1.8 per cent of DES galaxies are matched with ∼ BOSS CMASS. This implies the majority of the DES galaxies have extremely low

CMASS probabilities. These galaxies are likely to only add noise to the sample and potentially bias our measurements and therefore need to be removed. We carefully test how the low probability portion of the training sample (even HEALPix pixels) aﬀects the number density. We remove all galaxies lower than a given probability threshold and compare the number density of each sample with those of CMASS in the training sample. Including all sources results in the number density of DMASS being about

3% higher than CMASS in the same region, but near a threshold cut P > 1%, the sample yields a similar number density as CMASS. To validate the threshold cut, we construct a model in the same way but by using only the test sample (odd HEALPix pixels). Figure 2.6 shows that the model from the test sample produces low probability galaxies that affects the number density of the sample in a very similar way as the training sample. The similarity of the curves from different samples implies that the model tends to boost the number density of the sample in a predictable way, and this tendency can be remedied by cutting out low probability galaxies below a certain threshold. The same procedure is performed for the full Stripe 82 region and yields the same number density as CMASS where a threshold cut P > 1% (black points in Figure 2.6). Throughout this work, we use a threshold cut P > 1%. This cut excludes 90% of sources in the DMASS catalog, but when considering membership ∼ probabilities as weights, the effective portion of galaxies eliminated is 2.96%.

59 1.030 full 1.025 even hpix odd hpix 1.020

1.015 CMASS 1.010 /N

1.005

DMASS 1.000 N 0.995

0.990

0.00 0.25 0.50 0.75 1.00 1.25 1.50 Probability Cut (%)

Figure 2.6: Number density of DMASS with respect to the probability cut computed from three diﬀrent regions - the training region (even HEALPix pixels; square), the test region (odd HEALPix pixels; circle), and the full region (training+test; black diamond). Galaxies below a given probability cut are excluded. Number densities are divided by the number density of CMASS in the corresponding regions. The extremely similar shape of curves from diﬀerent samples implies that the model tends to boost the number density of the sample in a predictable way and this tendency can be remedied by cutting out low probability galaxies below a certain threshold. The red star at P = 1% on the black diamond curve is our choice of the probability cut.

60 2.4 Systematic Error Characterization

Astrophysical foregrounds, observing conditions, and spatially-varying depth are

potential sources of systematic uncertainty in galaxy survey analyses. They aﬀect

the probability of detecting sources and also their reconstructed properties, and can

thereby result in systematic biases in cosmological analyses (Leistedt et al., 2016;

Crocce et al., 2016).

We follow the procedures described in Elvin-Poole et al. (2018) to identify and

correct for these kind of systematic biases on the DMASS sample in the SPT region9.

To search for potential systematic uncertainties that aﬀect galaxy clustering, we study

the correlations between the galaxy number density and survey properties. If the

galaxy density is independent from a survey property, we do not consider this property

to have an impact on our DMASS sample. We use HEALPix maps (Nside = 4096) of 4 observing conditions (airmass, seeing FWHM, sky brightness, exposure time),

10σ limiting depth in griz bands, and the 2 astrophysical foregrounds of galactic reddening (Nside = 1024) and stellar density (Nside = 512). A detailed description about constructing HEALPix survey property maps can be found in Leistedt et al.

(2016). The construction of stellar density maps is described in Elvin-Poole et al.

(2018). The SFD galactic dust map is available at the LAMBDA website10 (Schlegel

et al., 1998).

We mask HEALPix pixels where the galaxy number density deviates by more than

20 per cent from the mean value (1.0) or changes sharply after some threshold value.

9We do not ﬁnd any systematic biases from CMASS in the training region. Therefore, systematics adderessed in this section were not considered for modeling the probabilistic model. 10https://lambda.gsfc.nasa.gov/product/foreground

61 We mask HEALPix pixels where seeing FWHM in r band > 4.5 pixels, which removes

2% of the total area.

Prior to correcting systematics, we rank survey properties from the most to least

signiﬁcance. The survey properties are ranked in order of the value given by

∆χ2 = χ2 χ2 , (2.19) null − model

2 2 where χmodel is the diﬀerence in χ between the best ﬁt model of the number density

2 2 2 and data points, χ is χ against a null line ngal/ ngal = 1. We minimize χ null h i model

by ﬁtting a linear model Ngal As + B against the calculated number density with ∝ Poisson errors of each data point.

After ranking properties, we correct for them starting from the highest ranked

one using the inverse of the best ﬁt model as a weight. Since survey properties

are correlated with each other, correcting one survey property can introduce new

systematic trends from another survey property. Therefore, the relationship between

the galaxy number density and survey properties is re-calculated after applying a

weight. Then one moves to the next top-ranked survey property and iterates the

procedure.

The weighting scheme we use assumes that the eﬀects of each sytematic are sepa-

rable. However, there is some correlation between systematic maps that may result in

over-correcting the galaxy density ﬁeld for a large number of systematic maps (Elsner

et al., 2016). To avoid this, we calculate the impact of adding a systematic weight

in every iteration to choose the minimum possible number of survey properties to be

corrected.

62 To investigate the impact of including additional systematic corrections, we utilize

the angular correlation function. The angular correlation function wδgδg (θ) is com-

puted with systematic weights using the Landy-Szalay estimator (Landy & Szalay,

1993) as given by

DD(θ) 2 DR(θ) + RR(θ) wδgδg (θ) = − , (2.20) RR(θ)

where DD, DR and RR are the number of galaxy pairs, galaxy-random pairs, random

pairs separated by a distance θ. Systematic weights are applied to individual galaxies as

N N 1 gal gal DD(θ) = w w Θ(θ θ ) (2.21) N (θ) i j i j DD i j − X X Ngal N 1 rand DR(θ) = w w Θ(θ θ ) (2.22) N (θ) i j i j DR i j − X X N N 1 rand rand RR(θ) = w w Θ(θ θ ) (2.23) N (θ) i j i j RR i j − X X where wi denotes systematic weight (wi = 1 for randoms), N(θ) is the total number

of pairs in a given data set in a given angular bin θ, Θ(θi θj) is 1 if a pair lies at − an angular distance θ, otherwise zero. The correlation function is measured in 10 logarithmically spaced angular bins over the range 2.50 < θ < 2500. We adopt the

same scales for cross-correlation functions with other surveys later in the paper. All

two-point calculations are done with the public code TreeCorr11(Jarvis, 2015).

Randoms for DMASS are uniformly generated on the surface of a sphere and

masked by the same masks described in Section 2.2. The number density of randoms

is chosen to be 50 times larger than DMASS, minimizing the impact of any noise from

the ﬁnite number of randoms and matching the relative number of CMASS randoms.

11https://github.com/rmjarvis/TreeCorr

63 To construct a covariance matrix for DMASS, we ﬁrst compute a covariance matrix

for CMASS from the 1000 QPM CMASS mock catalogues used in the BOSS-III

analyses (Alam et al., 2015):

Nmock 1 k k C(ωi, ωj) = (wi w¯i)(wj w¯j) , (2.24) Nmock 1 − − k=0 − X where Nmock is the total number of mocks, wi represents the ith bin of the angular

k correlation function, wi denotes the ith bin of the angular correlation function from the kth mock, andw ¯ is the average value of w over all mocks.

From the resulting CMASS mock covariance matrix, we derive a covariance matrix for DMASS by using the analytic form of the covariance between the angular correlation functions as follows:

2 0 (2l + 1) 0 2 δθ,θ0 C(θ, θ ) = 2 Pl(cos θ)Pl(cos θ )σ (Cl) + (2.25) fsky(4π) npairs l=0 X 2 where σ (Cl) is the variance of the angular power spectrum Cl, fsky is the fraction of the sky, and npairs is the total number of galaxy pairs. Assuming DMASS and

CMASS have the same galaxy bias and redshift distribution, the ﬁrst term can be easily adjusted for DMASS by altering the survey area factor. The second term, the shot noise term, can be directly calculated from the data. We obtain the ﬁrst term of the CMASS covariance matrix by subtracting the inverse of pair counts of the

CMASS galaxies from the mock covariance, and rescaling it by the ratio of the survey areas. The derived form of the covariance matrix for DMASS is

AD δθ,θ0 δθ,θ0 CD = Cmock,C + , (2.26) A − n n C pairs,C pairs,D where A is the survey area, Cmock is the mock covariance, and the subscripts C and

D stand for CMASS and DMASS respectively.

64 1.00

0.75 2 old χ

− 0.50 2 new

χ 0.25

0.00

r z g i g g veto fwhm fwhm fwhm airmassexptime airmass

Figure 2.7: The impact of systematic weights. Starting from the left, the names of the survey properties are listed on the x-axis in the order that they are corrected. The weight for the particular property is applied on top of the other weights applied earlier. The y-axis shows the χ2 measured between the correlation function with the new and old weights. ‘veto’ denotes a veto mask applied to remove regions where fwhm in r band > 4.5.

65 Figure 2.7 shows the impact of including additional systematic corrections through

the value of χ2, computed from the re-scaled covariance matrix and residuals between

the two measurements - before and after correction. The systematic weights are listed

on the x-axis in the order that they are applied to on top of all the previous weights

applied. Survey properties that show notable impacts are FWHM r-band and airmass

z-band. Applying corrections for the rest of the survey properties barely aﬀects the

angular correlation function. Therefore, we apply systematic weights only for the

top two properties. Figure 2.8 shows the galaxy number density vs. survey property

plots before applying the weights (blue) and after (black). We additionally ﬁnd that

correcting the top two systematics removes any trend with stellar number density.

Our interpretation is that any trend with the stellar number density is not from pure

stellar contamination but from strong correlations between the FWHM and airmass

maps12.

In the later sections, we will apply the systematic weights and veto mask computed in this section to the DMASS sample and report results along with the no systematics case.

12The lack of correlation with stellar density is consistent with the results for the DES red- MaGiC galaxies at similar redshifts (Elvin-Poole et al., 2018).

66 1.25 i g n

h 1.00 / g n 0.75 3.0 3.5 4.0 4.5 2.5 3.0 3.5 4.0 3 4 5 6 1.2 1.4 FWHM r [pixels] FWHM i [pixels] FWHM g [pixels] airmass g

1.25 i g n

h 1.00 / g n 0.75 1.0 1.2 1.4 200 400 600 800 0.5 1.0 1.5 airmass z exptime g stellar density

Figure 2.8: Galaxy number density with respect to survey properties having the top six ∆χ2 and stellar density. The solid blue lines are calculated without correction weights. The dashed black lines are calculated with weights. The error bars on the black lines are calculated assuming Poissonian statistics.

67 2.5 Comparison with the BOSS CMASS Sample

In this section, we compare the properties of the DMASS sample to those of the

BOSS CMASS sample. We evaluate the consistency of the overall number density, the amplitude of the auto- and cross-correlation functions, and redshift distribution.

As described in Ross et al. (2011, 2012), the selection functions for BOSS galaxy data in the NGC and the SGC are slightly diﬀerent due to measurable oﬀsets in the

DR8 (Aihara et al., 2011) photometry between the two regions. DMASS tends to mimic SGC CMASS as the training set taken is a sub sample of CMASS in the SGC.

Therefore, we will speciﬁcally compare DMASS with SGC CMASS in addition to comparisons with the full CMASS sample.

2.5.1 Number Density

In this section, we will compare the number density of CMASS in the training data, which is from 123 deg2 of Stripe 82 area, to the mean density in three distinct footprints: 1) BOSS CMASS data in the NGC area; 2) BOSS CMASS data in the

SGC area; and 3) the DMASS data in the SPT area. We divide each of the three regions into many smaller patches that are the size of Stripe 82. This allows us to determine the expected variance between the number density in the training area and the full region.

The three large footprints are divided as follows: each region is split into HEALPix

2 pixels at resolution Nside = 4096 where the size of each pixel is 0.72 arcmin . Then, contiguous sets of 606, 000 pixels are combined to make each patch comparable to ∼ the size of Stripe 82. We adopt a slightly larger size for one patch, 124 deg2, in order to include all of the HEALPix pixels in the SPT region while keeping the size of all

68 CMASS NGC CMASS SGC DMASS 1.15 1.15 1.15 Patch area 124 deg2 Patch area 124 deg2 Patch area 124 deg2 1.10 1.10 1.10

1.05 1.05 1.05 i i i n n n h 1.00 h 1.00 h 1.00 / / / i i i n n n 0.95 0.95 0.95

0.90 train 0.90 0.90 DMASS 0.85 0.85 0.85 0 10 20 30 40 50 0 5 10 15 0 2 4 6 8 patch index i patch index i patch index i

Figure 2.9: Number density and its deviation in the NGC (left), CMASS in the SGC (middle), and DMASS (right). Each region is divided into Stripe 82-size (train region) patches. Red stars represent the number density of BOSS CMASS in the training region and blue stars are the total number density of DMASS. All values are divided by the mean number density of each CMASS sample. The dark-grey-shaded region is 1σ, and the light-grey-shaded region is the 2σ level deviation of the black points in each panel.

patches the same. The same patch size is applied for the SGC and NGC regions and the remaining HEALPix pixels that cannot be a complete patch are discarded. The number of patches used for this calculation is 10 patches for the DES SPT region, 53 patches for the NGC region, and 20 patches for the SGC region.

Figure 2.9 shows the number density deviation for CMASS in the NGC (left),

CMASS in the SGC (middle) and DMASS in the SPT region (right). The black dots are the number density values determined in each of the small patches. All values are divided by the mean number density of each panel. The grey-shaded region is the standard deviation of the distribution of the black dots. This represents an estimate for the 1σ uncertainty in the number density of a Stripe82-sized patch. All three cases show a similar level of deviations. The red star in the ﬁrst and second panels is the number density of the training (BOSS CMASS) galaxies in Stripe 82 and the blue star is the total number density of DMASS. Note that the location of

69 the stars in each panel shows the relative number density in each region. In all panels

the red star is consistently 5 8% away from the total mean value. One can see ∼ that the number density of DMASS is considerably lower than the number density in the training region (red star), but that it is matched to within 1σ of the overall

CMASS number densities. That is, despite the data in the training region having a signiﬁcantly greater number density than the overall CMASS sample, our model obtains the number density of DMASS that is a fairly good match to both CMASS

SGC and CMASS NGC number densities.

2.5.2 Angular Correlation Function

We use the angular correlation function as a test to validate that DMASS matches the CMASS sample. Assuming that the number density and redshift distributions are matched, we should expect consistent amplitudes of the correlation functions if we have indeed matched the samples. We can directly compare the amplitude of the correlation functions of DMASS and CMASS and thus test for consistency without any cosmological assumptions. Three diﬀerent probes were chosen for this comparison: the galaxy angular auto-correlation and the angular cross-correlation with two full sky surveys - The Wide-ﬁeld Infrared Survey Explorer all sky survey (WISE; Wright et al., 2010) and CMB lensing from Planck (Planck Collaboration et al., 2016b).

Auto-angular Correlation Function

We measure the correlation function wδgδg (θ) of CMASS and DMASS galaxies in the same manner as described in Section 2.4. Each galaxy in the CMASS sample is weighted by systematic (systot), close pair (cp), and redshift failure (zp) weights as

70 given by Reid et al. (2016):

wtotal = wsystot (wcp + wzp 1) . (2.27) −

Note that we do not apply these weights in the CMASS training sample because we utilize the BOSS photometric sample for training. The BOSS photometric sample includes all missing galaxies that are dropped from a spectroscopic sample due to ﬁber collisions and redshift failures. wsystot is not considered either as we do not detect any systematic biases from the DES photometry of the CMASS training sample. As done in the previous BOSS analyses (Chuang et al., 2017; Pellejero-Ibanez et al.,

2017), we apply the explicit redshift cut 0.43 < z < 0.75 to the BOSS CMASS sample. This redshift cut is not considered for training because we utilized only matched photometric information in the training sample. In Appendix A, we show that the redshift cut negligibly aﬀects the 3D two point functions of BOSS CMASS, which justiﬁes our choice of the CMASS photometric sample as the training set.

For DMASS, galaxies in the SPT region are weighted by the CMASS membership probabilities and systematic weights calculated in Section 2.4.

The result is shown in Figure 2.10. The blue and black data points are the angular correlations of CMASS in the SGC and full CMASS respectively, and the red data points show the DMASS angular correlations. Error bars are obtained from the aforementioned mock covariance matrices in Section 2.4. We ﬁnd that the angular correlation function of DMASS has a better agreement with CMASS in SGC than full CMASS. The angular correlation function of full CMASS is slightly higher than the other samples on small scales, as expected from the intrinsic diﬀerence between

CMASS in the SGC and the NGC. On large scales, DMASS tends to deviate from

71 18

14 ) θ (

g 12 δ g δ

θw 10 × 8

1000 CMASS (SGC) 6 CMASS (FULL) χ2 / dof = 4.94/10 4 DMASS SGC 2 DMASS sys χFULL / dof = 10.61/10

2 1 0 10− 10 θ (deg) 0.02 ) θ (

ω 0.00 ∆

0.02 1 0 − 10− 10 θ (deg)

Figure 2.10: The top panel shows the angular correlation function calculated with DMASS (red), DMASS corrected by the systematic weights (orange), CMASS SGC (blue), and full CMASS (black). The bottom panel shows residuals between DMASS 2 2 2 and CMASS SGC (blue) or full CMASS (black). χSGC (χFULL) is the χ of the observed diﬀerence of two point functions of DMASS and CMASS SGC (FULL)

72 the two CMASS samples, but adding systematic weights mitigates the diﬀerence by

suppressing the correlation function of DMASS on large scales.

To quantify consistency between CMASS and DMASS, we use a χ2 statistic and its

associated Probability-To-Exceed (PTE) as our primary metric. We take the observed

diﬀerence of binned two point functions ∆d = wC wD (shown in the bottom panel −

in Figure 2.10) and its associated covariance as Ctot = CC + CD. Cross-covariance

between the CMASS and DMASS measurements is not considered since the two sets

of measurements are carried out on diﬀerent areas on the sky. Then we calculate the

χ2 of the diﬀerence deﬁned by

Nbins 2 T −1 χ = ∆di (Ctot )i,j∆dj . (2.28) i,j X and its associated PTE with the degrees of freedom (the number of bins). A probability of (100 PTE)% = 68% (95%) corresponds to 1σ (2σ) diﬀerence. − The χ2/dof obtained between DMASS and CMASS SGC is 4.94/10 (PTE=90%) in the range 2.50 < θ < 2500. For the comparison with the full CMASS sample,

we obtain a χ2/dof of 10.67/10 with PTE=53%. With the systematic weights, we

obtain 2.58/10 (PTE=99%) for CMASS in the SGC and 8.60/10 (PTE=47%) for full

CMASS.

Cross-correlation with WISE Galaxies

The WISE satellite surveys 99.86% of the entire sky at wavelengths of 3.4, 4.6, 12,

and 22 µm (W1 through W4). To have a uniform galaxy dataset, we select sources

to a ﬂux limit of W1 < 15.2 and remove stars with the cuts W 1 W 2 < 0.2 and − W 2 W 3 < 2.9, following Goto et al. (2012). Regions contaminated by scattered − moonlight are excluded by the ‘moonlev’ ﬂag if at least one of the bands has a value

73 25 CMASS (SGC) 20 CMASS (FULL) DMASS DMASS sys ) 15 θ ( WISE

g 10 δ g δ w 5 ×

1000 0

χ2 /dof = 3.745/10 5 SGC − 2 χFULL /dof = 6.888/10

10 1 0 − 10− 10 θ (deg)

1.0 CMASS (SGC) CMASS (FULL) 0.8 DMASS DMASS sys )

θ 0.6 ( CMB κ g

δ 0.4 θw

× 0.2 1000 0.0 2 χSGC / dof = 8.92/7 χ2 / dof = 7.13/7 0.2 FULL − 1 0 10− 10 θ (deg)

Figure 2.11: Cross-correlation measurements of the DMASS (red), DMASS with systematic weights (orange), CMASS SGC (blue) and full CMASS (black) samples with WISE galaxies (left) and CMB convergence map (right). We dropped the ﬁrst three data points of the cross-correlation with CMB lensing (in grey shaded region) from the measurements of χ2 to include only reliable scales where the analytic covariance matrices are valid.

74 higher than 3 (Kovacs et al., 2013). We also remove regions having the extreme level

of galactic extinction, 0.367 E(B V)SFD > 0.05. × − The resulting WISE galaxy sample approximately spans the redshift range from 0

to 0.4 with median redshift z 0.15 (Goto et al., 2012; Kovacs et al., 2013). CMASS ∼ in the SGC is known to have 5.24% of galaxies and CMASS in the NGC has 3.73%

in the low redshift tail z < 0.43. If the probabilistic model eﬀectively reproduces the

d⊥ cut in the DES photometry, the DMASS sample would have a similar fraction of

galaxies in the low redshift tail and this would result in the same cross-correlation

signal.

We adopt the Landy-Szalay estimator for the cross-correlation given as

δ δ DDW DRW DW R + RRW w g gWISE (θ) = − − , (2.29) RRW where DW and RW stand for WISE galaxies and WISE randoms. WISE randoms are uniformly generated on the surface of a sphere within the masked region, with a size

50 times larger than the WISE galaxy sample.

Errors are derived from analytic covariance matrices. We calculate the covariance matrices of the cross-correlation as the sum of the Gaussian covariance and non-

Gaussian covariance, and the super-sample covariance as detailed in Krause & Eiﬂer

(2017). We adopt the measurement of galaxy bias bWISE = 1.06 and the spectroscopic redshift distribution shown in Figure 3 in Goto et al. (2012).

With the same angular binning choice as the auto-correlation function, we measure the cross-correlation function between WISE galaxies and SGC CMASS, full CMASS, and DMASS as shown in Figure 2.11. The cross-correlation of full CMASS shows a slightly lower amplitude than CMASS in the SGC and DMASS on all scales which is expected because CMASS in the NGC has a smaller number density than CMASS

75 in the SGC at the low redshift end. We ﬁnd that the χ2/dof of DMASS computed with respect to CMASS in the SGC is 9.04/10 (PTE=53%), and the one computed with respect to full CMASS is 12.12/10 (PTE=28%). With the systematic weights, the value is 9.70/10 (PTE=47%) with SGC CMASS and 11.42/10 (PTE=33%) with full CMASS. From the results, we do not ﬁnd strong discrepancies between any of the

CMASS samples and DMASS sample. The result also implies that the probabilistic model is successfully reproduces the d⊥ cut in the DES system that excludes low redshift objects.

Cross-correlation with the CMB Lensing Map

The CMB photons released from the time of last scattering (z 1100) are grav- ∼ itationally deﬂected by the foreground mass distribution as they travel through the large-scale structure. The imprint on CMB anisotropies by this deﬂection of photons is called CMB lensing (Planck Collaboration et al., 2016b). The cross-correlation of galaxy positions and CMB lensing has two advantages for this work. First, CMB lensing is extremely homogeneous compared to galaxy catalogs. All information from the

CMB departs from the same redshift z = 1100 (considered as a very thin redshift bin) and travels the same distance until today regardless of the northern or southern part of the sky. Any difference found between the cross-correlation signals between different galaxy samples and the CMB would originate from differences between the galaxy samples themselves. Second, the galaxy bias is tied to the matter-matter correlation function in a different way that might give us complementary information.

This analysis uses the 2015 CMB convergence map provided by the Planck collaboration (Planck Collaboration et al., 2016b). We use the lensing multipole range

0 of 8 < l < 2048 and apply a Gaussian smoothing of θFWHM = 1.71 to the map.

76 The cross-correlation function is calculated in 10 logarithmically spaced bins between

2.50 < θ < 2500 using the estimator (Omori et al., 2018):

wδgκCMB (θ) = Dκ(θ) Rκ(θ) , (2.30) − with

N N 1 gal pix Dκ(θ) = wDwκκ Θ(θ θ ) (2.31) N Dκ i j ,j i j i=1 j=1 − X X N N 1 gal pix Rκ(θ) = wRwκκ Θ(θ θ ) (2.32) N Rκ i j j i j i=1 j=1 − X X where D and R stand for galaxies and randoms respectively, wD and wR are weights

for galaxies and galaxy randoms, N in the denominator is the total number of pairs,

and κj represents the value of convergence at the jth pixel.

The measurements are shown in Figure 2.11. Error bars are from Gaussian covari-

ance matrices computed by cosmoSIS (Zuntz et al., 2015). With the Gaussian covari-

ance matrices and measured cross correlation functions, we estimate the values of χ2 between CMASS and DMASS. We ﬁnd that the χ2/dof value between CMASS SGC

and DMASS is 24.38/10 (PTE < 1%) and the value between full CMASS and DMASS

is 21.56/10 (PTE=2%). The value between CMASS SGC and NGC is 101.48/10

(PTE < 1%), which is even more extreme than the former two cases. This implies

that the large χ2 values between CMASS and DMASS are not from the diﬀerence

between CMASS and DMASS. Since our analytic covariance matrix is Gaussian, we

believe that these large values of χ2 are due to the lack of the non-linear contributions

on small scales. Therefore, we exclude data points on the scales θ < 10 arcmin and

re-calculated χ2. The minimum angular cut is motivated by the measurement of the

angular correlation function in Section 2.5.2. We compare the mock covariance for

77 the auto-correlation function with the analytic calculation and ﬁnd that the analytic calculation underestimates uncertainties by more than 20% at θ < 10 arcmin. We simply utilize this scale to cut out unreliable information, expecting the non-linear contribution to be dominant on a similar scale in this case. The χ2/dof values with the minimum scale cut are improved to 8.92/7 (PTE=26%) between CMASS SGC and DMASS, and 7.13/7 (PTE=42%) between full CMASS and DMASS. These values of χ2/dof are smaller than the χ2/dof between CMASS NGC and CMASS SGC

(shown in Appendix B).

2.5.3 Redshift Distribution

In this section, we evaluate the redshift distribution of the DMASS sample by cross-correlating DMASS galaxies with the DES redMaGiC sample (Rozo et al., 2016;

Elvin-Poole et al., 2018). The concept of this technique called ‘clustering-z’ is to recover redshift distributions of an unknown sample by cross-correlating it with a galaxy sample whose redshift distribution is known and accurate. The technique was

ﬁrst demonstrated in Newman (2008), and has been developed and applied to various cosmological analyses including DES (Rahman et al., 2016; Choi et al., 2016; Johnson et al., 2017; Morrison et al., 2017; Scottez et al., 2018). Gatti et al. (2018) and Davis et al. (2017) calibrated redshift distributions of the DES Y1 source samples by using the DES redMaGiC sample (Elvin-Poole et al., 2018) as a reference sample. Cawthon et al. (2018) calibrated the DES redMaGiC sample by cross-correlating with the BOSS spectroscopic galaxy samples. For further details about the clustering-z method, we refer interested readers to references in Cawthon et al. (2018).

78 We utilize the redMaGiC sample as a reference sample and follow the general procedures described in Davis et al. (2017). The redMaGiC galaxies are red luminous galaxies selected by the redMaPPer algorithm (Rykoﬀ et al., 2014) above three different luminosity threshold cuts (L/L∗ > 0.5, L/L∗ > 1.0, and L/L∗ > 1.5). These galaxies have excellent photometric redshifts with an approximately Gaussian scatter of σz/(1 + z) < 0.02 and cover the entire redshift range of DMASS within the full DES Y1 footprint. This makes them suitable as a reference sample to evaluate the redshift distribution of the DMASS sample. We opt for the higher luminosity redMaGiC sample selected above a luminosity threshold of L > 1.5L∗ because the sample’s redshifts reach up to 0.9.

To obtain the redshift distribution of the unknown sample, we split the reference sample in narrow redshift bins, ∆z = 0.02, and measure cross-correlations between the galaxies in each redshift bin and the unknown sample. The cross-correlation for the ith redshift bin measures the quantity:

Rmax 0 0 wur(zi) = dR dz bu(R, z )br(R, zi) (2.33) ZRmin Z 0 0 nu(z )nr(zi)ξm(R, zi, z ) × where b denotes galaxy bias of the unknown(‘u’) and reference(‘r’) samples, n(z) stands for a normalized redshift distribution, ξm stands for the matter-matter correlation function. zi is the ith redshift bin of the reference sample. R is the comoving distance, R = (1 + z)DA(z)θ. We adopt Rmin = 500 kpc and Rmax = 1500 kpc based on Gatti et al. (2018) and Davis et al. (2017). We also assume that the galaxy biases of both the reference and unknown sample are scale-independent on these scales. Schmidt et al. (2013) demonstrated that implementing a linear bias model in the clustering-z does not signiﬁcantly aﬀect the methodology, even if these scales are

79 non-linear. The same point was made in Gatti et al. (2018) and Cawthon et al. (2018)

as well. For narrow redshift bins nr(zi) = δ(z zi), Equation (2.33) is simpliﬁed as −

wur(zi) = nu(zi)bu(zi)br(zi)wm(zi) (2.34) where wm is the integrated matter-matter correlation function between Rmin and

Rmax. The cross-correlation was measured with the estimator from Davis & Peebles

(1983) as follows:

Rmax N dRW (R)[DuDr(R, z)] wˆ (z) = Rr Rmin 1 (2.35) ur Rmax NDr dRW (R)[DuRr(R, z)] − RRmin

where W (R) is a weighting function,R DuDr and DuRr stand for the number of galaxy-

galaxy and galaxy-random pairs, NRr and NDr stand for the total number of randoms

and galaxies of the reference sample. With the measured cross-correlation, the redshift

distribution of the unknown sample is given as

wˆur(z) nu(z) . (2.36) ∝ bu(z)br(z)w ˆm(z) If the redshift bins are suﬃciently narrow so the biases and matter-matter correlations

can be considered to be constant in each bin, the auto-correlation of the reference and

unknown samples are given as

2 wˆrr(z) = br(z) wˆm(z) , (2.37)

2 wˆuu(z) = bu(z) wˆm(z) . (2.38)

Then Equation (2.36) is re-written as

wˆur(z) nu(z) . (2.39) ∝ wˆuu(z)w ˆrr(z)

For the redshift evolution ofw ˆuu, wep adopt a power law parametrization (Cawthon et al., 2018):

γ wˆuu(z) (1 + z) . (2.40) ∝ p 80 6.0 model (γ = 0) 5.5 measurement

5.0

4.5 CMASS ˆ w

4.0

3.5 χ2/ dof =15.8/13

3.0 0.45 0.50 0.55 0.60 0.65 0.70 z

Figure 2.12: Integrated auto-correlations (Equation 2.35) of the CMASS SGC sample (black points). The grey-dashed line is the redshift evolution model (1+z)γ with γ = 0 (Equation 2.40). The value of χ2 between the model and the measurement is 15.8 for 13 data points, which indicates that the measurement is well consistent with the model.

Since we do not have access to the true redshifts of the DMASS galaxies, we infer

the redshift evolution ofw ˆuu from the auto-correlations of the CMASS galaxies using

their spectroscopic redshﬁts (see Figure 2.12). Based on the nearly constantw ˆ of

CMASS, we adopt γ = 0 for DMASS.

Figure 2.13 shows the result obtained from clustering-z. We ﬁnd an excellent agreement between the clustering-z distribution of DMASS (blue points with error bars) and the spectroscopic redshift of CMASS SGC (solid black curve). The χ2

81 6 χ2/ dof = 46.3/36 DMASS DMASS sys 5 χ2/ dof (sys) = 36.2/36

3 ) z ( n 2

1 − 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 z (redshift)

Figure 2.13: Redshift distribution of DMASS (blue) and DMASS with systematic weights (orange) recovered by the clustering-z method with redMaGiC. The solid black and dashed lines show the spectroscopic redshift distribution of CMASS SGC and full CMASS.

82 obtained when comparing the two is 36.2 for 36 data points. We conclude that the

clustering-z method returns a n(z) for DMASS that is consistent with the BOSS SGC

n(z).

2.5.4 Diﬀerence in Galaxy Bias

In this section, we present the constraint on the mean galaxy bias diﬀerence be-

tween DMASS and CMASS derived from the combination of diﬀerent probes afore-

mentioned. Due to the weaker constraining power of the cross-correlation functions

compared to the auto-correlation function, we utilize only the auto-correlation func-

tion (Section 2.5.2) and the clustering-z distribution (Section 2.5.3) in this section.

To constrain the shifts in galaxy bias and redshift distribution compared to CMASS,

we model the angular correlation function as follows:

wδgδg (θ, b, ∆b, ∆z)

0 0 0 = dz f(z, b, ∆b) dz f(z , b, ∆b) ξm(R, z, z ) (2.41) Z Z with

f(z, b, ∆b) = (b + ∆b) n(z + ∆z) (2.42)

where ξm is the matter angular correlation function, R is the comoving distance deﬁned as R = (1+z)DA(z)θ, n(z) is the normalized redshift distribution, b is galaxy bias. The galaxy bias of CMASS is known to be nearly a constant within the redshift

0.43 < z < 0.7, so we do not consider redshift evolution of the galaxy bias (Salazar-

Albornoz et al., 2017). ∆b and ∆z are shifts in the galaxy bias and the redshift distribution from ﬁducial quantities. For CMASS, ∆b and ∆z are set to zero. Then,

83 the residuals of the angular correlations of CMASS and DMASS is deﬁned as

∆wδgδg (θ, b, ∆b, ∆z)

= wδgδg (θ, b, ∆b, ∆z) wδgδg (θ, b, 0, 0) . (2.43) − We also model the residuals of the redshift distributions to constrain the redshift shift ∆z independently with the clustering-z measurement in the previous section.

The residual model is given as

∆n(z, ∆z) = n(z + ∆z) n(z) (2.44) − where ∆z is the same parameter shown in Equation (2.41). We use the spectroscopic redshift distribution of CMASS as the true distribution.

Using a combination of the residuals of the angular correlation (Section 2.5.2) and clustering-z (Section 2.5.3) measurements, we perform Markov Chain Monte-Carlo likelihood analyses to constrain the parameter set of b, ∆b, ∆z . The likelihood of { } the combined cosmological probe is given by the sum of individual log likelihoods given as

1 2 2 ln (p) = χ δ δ (p) + χ (p) (2.45) L −2 w g g n(z) where p is the set of varied parameters. We estimate χ2 deﬁned in Equation (4.49).

The data vector ∆d is the diﬀerence between the measurement and theoretical prediction given as ∆d = dtrue d. Equations (2.43) and (2.44) are adopted as the true − 2 2 data vector dtrue for χwδgδg and χn(z), respectively. Residuals of the measurements between CMASS and DMASS are used as an input data vector d for a corresponding probe as well. The covariance matrix for the angular correlation probe is given as the sum of the CMASS and DMASS covariance matrices:

Cwδgδg = CDMASS + CCMASS , (2.46)

84 and the covariance matrix for the clustering-z probe Cn(z) is obtained from the clustering-z calculation in Section 2.5.3. To evaluate the likelihood values and matter power spectrum for a given cosmology, we use the DES analysis pipeline in Cosmo-

SIS (Zuntz et al., 2015). Further details of the likelihood framework are described in

Krause et al. (2017).

Figure 2.14 shows the constraints of the bias shift ∆b and redshift shift ∆z. The orange-solid and red-dashed contours show shifts ∆b and ∆z of DMASS when the values of CMASS SGC are fiducial. The black-solid and blue-dashed contours present shifts ∆b and ∆z of DMASS when full CMASS is used as fiducial. Dashed contours of both cases are obtained with the systematic weights of DMASS. The resulting numbers are ∆b = 0.010+0.045 and ∆z = 3.46+5.48 10−3 between CMASS SGC −0.052 −5.55 × and DMASS, and ∆b = 0.044+0.044 and ∆z= (3.51+4.93) 10−3 between full CMASS −0.043 −5.91 × and DMASS. Since adding systematic weights has a negligible effect on numbers as shown in Figure 2.14, we do not report the results separately. As expected, DMASS has a better agreement with CMASS in SGC. The resulting constraints of ∆b show that the mean galaxy bias of DMASS is consistent with both CMASS samples within

1σ. Moreover, ∆b between DMASS and full CMASS is comparable to 2.6% of the intrinsic diﬀerence in CMASS between the SGC and NGC shown in Appendix B.

In this work, we do not consider the redshift bin biases and their uncertainties of the redMaGiC samples, which are known to be ∆z = (0.010, 0.004, 0.004) and − −

σ∆z = (0.011, 0.010, 0.008) for three redshift bins from z = 0.15 to z = 0.6 (Cawthon et al., 2018). Including the redshift uncertainties as priors would widen the ﬁnal contours, but still keep the ﬁnal constraints consistent with CMASS as all the biases are within 1σ. Future analyses using DMASS will likely need to use a similar prior

85 FULL - DMASS FULL - DMASS(sys) SGC - DMASS SGC - DMASS(sys)

02 0.

.01 z 0 ∆ 00 0.

01 0. − 16 08 00 08 16 01 00 01 02 0. 0. 0. 0. 0. 0. 0. 0. 0. − − ∆b − ∆z

Figure 2.14: Constraints on the galaxy bias shift ∆b and redshift distribution shift ∆z from combination of the auto-angular correlation function and clustering-z. The dashed vertical and horizontal lines show the ideal case where DMASS is perfectly matched with CMASS. Orange-solid and red-dashed contours show shifts from the values of CMASS SGC. Black-solid and Blue-dashed contours show shifts from the values of full CMASS. The DMASS systematic weights are added for dashed contours. Adding systematic weights has very little impact on galaxy bias and redshift distributions.

86 on the redshift bias for DMASS as used for DES redMaGiC in DES Y1 (Elvin-Poole et al., 2018).

87 2.6 Conclusion

In this paper, we constructed a catalog of DES galaxies from the full footprint of DES, whose statistical properties match those of the BOSS DR12 CMASS galaxy sample. We developed an algorithm for probabilistic target selection that uses density estimation in color and magnitude spaces. The algorithm was trained and validated by the DES photometry from the overlapping area between the DES and BOSS footprints. From the distribution of the input DES galaxies in the overlapping region, the algorithm predicts an observed distribution that the same kind of galaxies would have in the target region. A membership probability calculated based on the predicted observation was assigned to each source in the DES Y1 GOLD catalog. By weighting galaxies by their assigned probability, the resulting DMASS sample mimics the noise level the original CMASS sample has.

We showed that the resulting DMASS catalog matches well with both the SGC subset of CMASS as well as the full CMASS sample in various aspects: the number density, auto-angular correlation function, cross-angular correlation function with other full sky surveys and redshift distribution. We determined diﬀerences in galaxy bias and shifts in the redshift distribution between DMASS and other CMASS samples by combining the angular correlation function and redshift distribution from the clustering-z method. The resulting constraints of ∆b show that the galaxy bias of the

DMASS sample is consistent with both CMASS samples within 1σ. Furthermore, ∆b between DMASS and full CMASS is comparable to the 2.6% intrinsic diﬀerence of

CMASS between the SGC and NGC regions.

88 The resulting DMASS sample can be used in cosmological analyses in various ways. The most promising application is using DMASS as a lens sample for galaxy- galaxy lensing. The sample can also be used as a reference sample for photometric redshift training. Current photometric redshift algorithms depend on a small subset of spectroscopic galaxies from a minimal overlap area. Having a reference sample in the full DES footprint gives enormous statistical power to compute photometric redshifts. Beyond the sample used in our work, the probabilistic technique used for this work can be easily applied to other image-based and spectroscopic surveys to identify another CMASS-like sample or other speciﬁc types of samples. Future surveys such as LSST (LSST Science Collaboration et al., 2009) can be a great application for this novel approach as the survey footprint of LSST occupies the entire southern sky but has only a small overlapping area with spectroscopic surveys such as eBOSS or DESI (DESI Collaboration et al., 2016b) that view the northern sky. Producing a spectroscopic galaxy sample with the LSST imaging will enable us to utilize almost the entire sky and yield a wealth of information on the accelerated expansion of the

Universe.

89 Chapter 3: Dark Energy Survey Year 1 Results: Constraints on Extended Cosmological Models from Galaxy Clustering and Weak Lensing

In this chapter, I will present a shortened version of the paper “The DES Y1 cosmological analysis for extended models” (hereafter DES Y1 Extension; Abbott et al., 2019), focucing on the aspect of the modiﬁed gravity analysis. The original abstract is given below:

We present constraints on extensions of the minimal cosmological models dominated by dark matter and dark energy, ΛCDM and wCDM, by using a combined analysis of galaxy clustering and weak gravitational lensing from the first-year data of the Dark Energy Survey (DES Y1) in combination with external data. We consider four extensions of the minimal dark energy-dominated scenarios: 1) nonzero curvature Ωk, 2) number of relativistic species Neff different from the standard value of 3.046, 3) time-varying equation-of-state of dark energy described by the parameters w0 and wa (alternatively quoted by the values at the pivot redshift, wp, and wa), and 4) modified gravity described by the parameters µ0 and Σ0 that modify the metric potentials. We also consider external information from Planck cosmic microwave background measurements; baryon acoustic oscillation measurements from SDSS, 6dF, and BOSS; redshift-space distortion measurements from BOSS; and type Ia supernova information from the Pantheon compilation of datasets. Con- straints on curvature and the number of relativistic species are dominated by the external data; when these are combined with DES Y1, +0.0037 we find Ωk = 0.0020−0.0032 at the 68% confidence level, and the upper limit Neff < 3.28 (3.55) at 68% (95%) confidence, assuming a hard prior Neff > 3.0. For the time-varying equation-of-state, we find the pivot value +0.19 +0.93 (wp, wa) = ( 0.91 , 0.57 ) at pivot redshift zp = 0.27 from DES − −0.23 − −1.11 90 +0.04 +0.37 alone, and (wp, wa) = ( 1.01 , 0.28 ) at zp = 0.20 from DES Y1 − −0.04 − −0.48 combined with external data; in either case we find no evidence for the temporal variation of the equation of state. For modified gravity, we find +0.28 the present-day value of the relevant parameters to be Σ0 = 0.43−0.29 +0.08 +0.42 from DES Y1 alone, and (Σ0, µ0) = (0.06 , 0.11 ) from DES Y1 −0.07 − −0.46 combined with external data. These modified-gravity constraints are consistent with predictions from general relativity.

This is a DES Collaboration key paper with an alphabetical author list. I am one of the principal authors of this publication. The full author list can be found in the original paper.

3.1 Introduction

The cosmological constant, Λ in Einstein’s GR, is the simplest form to describe the cosmic acceleration. It is in excellent agreement with current observations as well.

However, there is still a conundrum known as the cosmological constant problem that conflicts with quantum field calculations. Not only that, ithe underlying mechanism that drives the cosmic acceleration is still unclear. There have been various efforts to explain the underlying physics of the cosmic acceleration in the form of dynamical dark energy. Another alternative is that there might be a new theory of gravity that behaves like GR on small scales but mimics cosmic acceleration on cosmological scales.

Both dark energy and modiﬁed gravity models are not relevant to the cosmological constant or make the cosmological constant zero or negligibly small.

In this chapter, I describe how data from DES introduced in the previous chapter can be used to test and to constrain MG models. In 2017, the DES collaboration published its cosmological analyses of the ﬁrst year survey data (Abbott et al., 2018a).

With the combination of the three sets of two point functions referred as the 3x2pt

91 statistics (galaxy clustering, cosmic shear, galaxy-galaxy lensing), they achieved cosmological constraints as tight as the ones recently published by the Planck Collab- oration using full sky measurements of CMB. The DES analysis (hereafter Y1KP) considered only the standard ΛCDM model and a model with the same assumptions but a free, constant value of w of the dark energy equation of state (wCDM). A follow-up paper published by DES last year extended the original Y1KP analysis and provided constraints on four models beyond ΛCDM: time-varying dark energy, nonzero spatial curvature, new relativistic degrees of freedom, and modiﬁcations of the law of gravity (Abbott et al., 2019). I was a principal author of the DES Year 1

Extension paper focusing on modiﬁed gravity. I will discuss my work in the following sections.

This chapter is organized as follows. In the next section, the theoretical background of MG and the parametrization used in this work will be reviewed. The data sets used will be described in section 3.3 followed by a discussion of the maximum likelihood analysis in Section 3.4. The results of the DES Y1 extension paper on mod- iﬁed gravity will be presented in Section 3.5. The DMASS galaxy sample discussed on

Chapter 2 was designed for a promising novel method to constrain models of MG by combining measurements from DES with results from the BOSS spectroscopic survey.

This method including a forecast on expected constraints is presented in Section 3.6 and I conclude the chapter in Section 3.7.

92 3.2 Theory

As reviewed in Section 1.5, the eﬀect of deviations from GR can be described by

two parameters added to Einstein’s ﬁeld equations as shown

2 2 k Ψ = 4πGa Gmatterδρ (3.1) − 2 2 k (Ψ + Φ) = 8πGa Glightδρ . (3.2) −

This parametrization has the advantage that the parameter Glight is directly linked to the gravitational lensing kernel. It is therefore sensitive to the modified behavior of relativistic particles, while Gmatter is more sensitive to modifications to the structure growth driven by non-relativistic particles. Therefore, this parametrization has the advantage of separating the modifications in the relativistic aspect, from the modifications in the non-relativistic aspect. In this work, we adopt time-evolving MG parameters as

Gmatter = 1 + µ(a) (3.3)

Glight = 1 + Σ(a) (3.4) with

ΩΛ(a) ΩΛ(a) µ(a) = µ0 0 , Σ(a) = Σ0 0 . (3.5) ΩΛ ΩΛ

0 where Ω ΩΛ(a = 1) is the dark energy density today. Note that GR is restored Λ ≡ for µ0 = Σ0 = 1.

Modiﬁcations in the gravitational potentials enter in the structure growth through

2 a¨ 3Ω δ00(a) + + δ0(a) m [1 + µ(a)]δ(a) = 0 . (3.6) a a˙ 2 − 2a2

93 The cosmic shear power spectrum correlating redshift bins i and j is deﬁned as

9 H 4 ∞ W (χ)W (χ) l P i,j(l) = Ω2 0 i j P ( , χ)dχ , (3.7) κ 4 m c a2(χ) m f (χ) Z0 K where χ is the radial coordinate distance, fK is the comoving angular distance, and

Pm(l) is matter power spectrum. Modiﬁcations to GR appear in the lensing kernel

Wi as

∞ 0 0 0 fK (χ χ) 0 Wi(χ) = dχ ni(χ ) − [1 + Σ(χ )] , (3.8) f (χ0) Zχ K

where ni is the redshift distribution of source galaxies in the ith redshift bin.

3.3 Data

The primary data used in this analysis is the auto- and cross-correlation functions

of the positions and shapes of the galaxies measured by DES during its ﬁrst year of

ob servations. We will only give a brief summary of data sets in this section. For

further details, we refer the interested readers to Y1KP.

3.3.1 DES data

Catalogs We use two samples of galaxies - lens galaxies and source galaxies - con-

structed from the images taken by the Dark Energy Survey during its ﬁrst year

observations (The Dark Energy Survey Collaboration, 2005; Abbott et al., 2018a).

The images were processed through the DES Data Management (DESDM) system

(Flaugher et al., 2015). As lens galaxies, we adopt luminous red galaxies identiﬁed

by the redMaGiC algorithm (Rykoﬀ et al., 2014; Rozo et al., 2016; Elvin-Poole et al.,

2018). These galaxies tend to be strongly clustered, and have small photometric

redshift uncertainties. For details on the validation tests for primary systematics

94 and redshift uncertainties, see (Prat et al., 2018; Rozo et al., 2016). Source galaxies are selected from the DES metacalibration catalog (Cawthon et al., 2018; Huﬀ

& Mandelbaum, 2017). Redshifts of individual galaxies are estimated by the BPZ algorithm (Coe et al., 2006). Additional systematic uncertainties are quantiﬁed in

MacCrann et al. (2018); Zuntz et al. (2018).

Measurements For the lens and source galaxies, we measure 3 two-point functions referred to as“DES Y1 3x2pt”: the angular auto-correlation of the positions of the lens galaxies, the cross-correlation of the lens positions and the shear of the source galaxies, and the shear correlation of the source galaxies. All measurements are performed in the 20 logarithmic angular bins 2.50 < θ < 2500 using treecorr13 (Jarvis, 2015).

Details of these measurements and systematics are described in (Hoyle et al., 2018;

Troxel et al., 2018; Prat et al., 2018).

3.3.2 External data

We combine DES data with a collection of external data sets: CMB, CMB Lens- ing, BAO RSD, and SNe Ia. The measurements of the cosmic distances by SNe Ia,

BAO and CMB enable us to constrain the geometry of the universe. RSD provides precise information about the structure growth at specific redshifts, which is sensitive to modifications to GR. In this section, we briefly introduce the external data sets selected in this study. For further details, see Y1KP and the original paper of DES

Y1 Extension.

13 https://github.com/rmjarvis/TreeCorr

95 TT CMB & CMB lensing We utilize the temperature spectra Cl (2 < l < 2508),

polarization spectra CTE,CEE,CBB (2 < l < 30), and lensing measurements from the Planck 2015 data (Planck Collaboration et al., 2016a,c). The primary CMB spectra is not sensitive to modiﬁcation to GR, but provides tight cosmological constraints for the background geometry and the early Universe.

BAO+RSD BAO constrains cosmological parameters for cosmic expansion, by providing the measurements of the Hubble parameter H(z) and the comoving distance dA(z), or the combination of the two at a speciﬁc redshift. In this work, we use three sets of the BAO results from the BOSS Data Release 12 (BOSS DR12; Alam et al., 2017b), the 6DF Galaxy survey (6dF; Beutler et al., 2011) and the SDSS Data

Release 7 Main Galaxy Sample (SDSS MGS; Ross et al., 2015). BOSS DR12 provides measurements of H(z) and dA(z) at three redshift points z = 0.38, 0.51, 0.61. 6dF

and SDSS MGS provide the spherically averaged combination of transverse and radial

2 2 1/3 modes DV (z) [cz(1 + z) D (z)/H(z)] at z = 0.106 and z = 0.15 respectively. ≡ A BOSS DR12 also provides a redshift space distortion (BAO) measurement in the form

of f(z)σ8(z) at the aforementioned three redshifts. As described in Section 3.2, the

structure growth rate f(z) is sensitive to the modiﬁed behavior of non-relativistic

particle. It, therefore, mainly constrains the MG parameter µ(a).

SN For further constraints on the expansion history of the universe we also use the

Pantheon SNe Ia sample (Scolnic et al., 2018). The sample consists of 1048 SNe Ia

over the redshift range 0.01 < z < 2.26.

96 3.4 Analysis

Using the 3x2pt results from DES and measurements from the external data sets, we perform a maximum likelihood analysis to constrain the two MG parameters µ0 and Σ0. Thoretical predictions are estimated by CosmoSIS (Zuntz et al., 2015) with a version of MGCamb14 (Hojjati et al., 2011b) modiﬁed to include the Σ, µ parametrization. The modiﬁed pipeline is validated by comparing its results with the results from another pipeline, cosmoLike(?).

We use the emcee(Foreman-Mackey et al., 2013) implementation of the Markov

Chain Monte Carlo method to evaluate the resulting likelihood function. Our analysis

2 spans the parameter set Ωm, Ωb, h, τ, ns,As, Ωνh ,w, µ0, Σ0 , where τ is the optical { } depth, ns is the spectral index, As is the amplitude of the primordial power spectrum,

Ων is the energy density of massive neutrinos. Other cosmological parameters not described here are set to ΛCDM values. In addition to the cosmological parameters, our model for the data includes 20 nuisance parameters - 9 shift biases ∆zi for the source and lens redshift bins, 5 galaxy biases, bi, for the lens galaxies in the 5 redshift bins, the four multiplicative shear biases mi, and two parameters AIA and ηIA to model intrinsic alignments. The priors imposed on those parameters are stated in

Table 3.1. For further descriptions of additional 20 nuisance parameters, we refer the readers to read Section 3.B of the original paper.

Finally, we limit our analysis to the linear scales since the (µ, Σ) parametrization has not been fully established on nonlinear scales. The scales available for our study are determined by estimating the diﬀerence between the nonlinear and linear theory

14http://aliojjati.github.io/MGCAMB/mgcamb.html

97 3 0.9 ,

2 0.8

1 0.7 8 0

0 S 0.6 1 0.5 2 0.4 3 0.3 0.0 0.3 0.6 0.9 1.2 0.24 0.28 0.32 0.36 0 m

Figure 3.1: The left panel shows constraints on the modiﬁed gravity parameters (Σ0, µ0). Blue contour is from DES alone, yellow is from external data alone, and red is the combination of the two. The dashed horizontal and vertical lines show the ΛCDM case. The right panel shows the constraints on the matter density Ωm and S8 for the three cases aforementioned, with the case of ΛCDM (dashed contours).

predictions in the standard ΛCDM model as follow

2 T −1 ∆χ (dnl dlin) C (dnl dlin) (3.9) ≡ − − where dnl and dlin is the linear and nonlinear data vectors and C is the full DES

Y1 error covariance matrix. We exclude data points starting from the smallest scales until ∆χ2 < 1.

3.5 Results

The results of this analysis are constraints on the modifed gravity parameters Σ and µ. DES data alone is mostly sensitive to Σ. Marginalizing over µ we obtain:

+0.28 Σ0 = 0.43−0.29 . (3.10)

98 Table 3.1: Parameters and priors used to describe the measured two-point functions, as adopted from Y1KP. Flat denotes a ﬂat prior in the range given while Gauss(µ, σ) is a Gaussian prior with mean µ and width σ.

Parameter Prior Cosmology Ωm flat (0.1, 0.9) −10 −9 As flat (5 10 , 5 10 ) × × ns flat (0.87, 1.07) Ωb flat (0.03, 0.07) h flat (0.55, 0.91) 2 −4 −2 Ωνh flat(5 10 ,10 ) Lens Galaxy Bias× bi(i = 1, 5) flat (0.8, 3.0) Intrinsic Alignment ηIA AIA(z) = AIA[(1 + z)/1.62] AIA flat ( 5, 5) − ηIA flat ( 5, 5) − Lens photo-z shift (red sequence) 1 ∆zl Gauss (0.008, 0.007) 2 ∆zl Gauss ( 0.005, 0.007) 3 − ∆zl Gauss (0.006, 0.006) 4 ∆zl Gauss (0.000, 0.010) 5 ∆zl Gauss (0.000, 0.010) Source photo-z shift 1 ∆zs Gauss ( 0.001, 0.016) 2 − ∆zs Gauss ( 0.019, 0.013) 3 − ∆zs Gauss (+0.009, 0.011) 4 ∆zs Gauss ( 0.018, 0.022) Shear calibration− i mmetacalibration(i = 1, 4) Gauss (0.012, 0.023)

99 Including external data tightens the Σ result and gives us a measurement of µ as well.

+0.08 Σ0 = 0.06−0.07 (3.11)

+0.42 µ0 = 0.11 . (3.12) − −0.46 Note that the GR values for Σ and µ are 0 and 0, respectively. Figure 3.1 shows the

64 % and 95% conﬁdence contours of the resulting constraint. The left panel shows constraints on the modiﬁed gravity parameters (Σ0, µ0). The blue contour is from

DES alone, yellow is from external data (Ext) alone, and red is the combination of

the two. The dashed horizontal and vertical lines show the ΛCDM case. The right

panel shows the constraints on the matter density Ωm and S8 for the three cases

aforementioned, with the case of ΛCDM (dashed contours). An interesting point

about this result is that the DES+Ext values favors the ΛCDM values, despite both

the DES alone and the Ext-alone values favor higher Σ0. This may arise because

of the highly anti-correlated quantity Σ0S8, where S8 σ8 Ωm/0.3 is a rescaled ≡ p amplitude of mass ﬂuctuation. DES favors a lower value of S8 than the external data

does, as shown in the right panel. Since the DES lensing amplitude is proportional to

the quantity Σ0S8, suppressing one of them causes the other parameter to increase.

Ext strongly favors a larger value of S8. Consequently, the increased value of S8 from

Ext results in moving Σ0 back closer to zero when two cases are combined.

Finally, we compare our result with similar measurements from the KiDS (Joudaki

et al., 2017) and Planck collaborations (Planck Collaboration et al., 2018). The

KiDS analysis adopted similar data sets to ours except for using their data for the

shear measurements. One diﬀerence is that they didn’t include galaxy clustering and

galaxy-galaxy lensing, and have ﬁxed the sum of neutrino masses to mν = 0.06eV.

The MG parametrization that KiDS uses is the (Q, Σ) parametrizationP which can be

100 mapped to our (µ, Σ) parametrization as follow

QKiDS = 1 + 2ΣDES µDES (3.13) −

ΣKiDS = 1 + ΣDES , (3.14) so ΛCDM is restored for (Q, Σ) = (1, 1). Instead of adopting smooth functions for Q

and Σ as done in our work, they use eight piecewise constant values across two bins

in scale and two in redshift. Therefore, we only compare their values of Q2 and Σ2

(corresponding to the low redshift bin and small scale bin) with our constraints. As

shown in the left panel of Figure 3.2, KiDS prefers very positive Q2 which corresponds

to DES’s preference for a positive Σ0 and negative µ0. Their result shows a similar

tendency as DES’s measurement. Due to the diﬀerent parametrization and data sets

used, the comparison between two works is not straightforward but overall two results

seem in broad agreement.

The comparison with Planck 2018 (hereafter P18) is more straightforward since

our choice of parametrization is mostly motivated by their former work in 2015 and

they chose the same SN and RSD data sets to ours. Not only that, their work utilized

our shear measurement. However, their work did not include our measurements of

galaxy clustering and galaxy-galaxy lensing and they have ﬁxed the sum of neutrino

masses. By combining the CMB measurements with the BAO/RSD and weak lensing

measurements, P18 obtained the MG constraints as below

+0.19 µP18 1 = 0.07 (3.15) − − −0.32 +0.63 ηP18 1 = 0.32 . (3.16) − −0.89

101 The constraint of Σ can be derived from the relation Σ = (µ/2)(1 + η) which results in

+0.059 ΣP18 1 = 0.018 . (3.17) − −0.048

The right panel of Figure 3.2 shows the 68% and 95% conﬁdence regions for the two parameters (µP18 1, ηP18 1). Planck’s MG parametrization can be converted to − − ours as

µP18 = 1 + µDES (3.18)

ΣP18 = 1 + ΣDES . (3.19)

Our constraints from the DES+Ext case shows a very good agreement with the central

values of P18. Our errors on Σ0(µ0) are about 30%(80%) weaker than those from P18,

but this diﬀerence is mainly due to our choice of the varying neutrino mass and slightly

diﬀerent data sets.

3.6 Forecast for DMASS

The DMASS galaxy sample introduced in Chapter 2 was speciﬁcally designed to

eﬀectively combine the existing measurements of galaxy clustering with the measure-

ments of weak lensing. In particular, it is well suited for a novel measurement of the

modiﬁed gravity parameters Σ and µ. In this section we forecast the Σ and µ con-

straints that can be obtained with the DMASS sample. The analysis follows the basic

methodology and techniques used for the DES Y1 extension measurement presented

in the previous sections.

102 KiDS extended cosmologies 1273 Planck Collaboration: Cosmological parameters

0.75 Planck TT,TE,EE+lowE+lensing (2015) Planck TT,TE,EE+lowE+lensing 1.0 Planck TT,TE,EE+lowE+lensing +BAO/RSD+WL +BAO+SNe Planck TT,TE,EE+lowE 0.50 +BAO/RSD+WL 0.5 1 1 0.25 0 0 µ 0.0 0.00

Downloaded from https://academic.oup.com/mnras/article-abstract/471/2/1259/3760291 by guest on 17 July 2019 0.5 0.25 1.0 1 0 1 2 3 0.75 0.90 1.05 1.20 1.35 A 0 1 L

Figure 13. Left: Marginalized posterior contours in the σ 8–"m plane (inner 68% CL, outer 95% CL) in a universe with MG for KiDS in green and Planck Fig. 32. Degeneracy between AL and ⌃0 1, computed as a de- in red. For comparison, dashed contours assume fiducial #CDM. Right: Marginalized posterior contours in the Q2–$2 plane for KiDS with fiducial angular scales in green (denoted by ‘FS’), KiDS keeping only the largest angular scales in pink (denoted by ‘LS’), and respectively combined with Planck in grey Planck TT,TE,EE+lowE+BAO/RSD rived parameter in our (µ, ⌘) parameterization. The horizontal 1 1.0 Planck TT,TE,EE+lowE+BAO/RSD(2015) and blue. The indices represent a particular combination of MG bins, such that z < 1andk > 0.05 hMpc− .ThedashedlinesintersectattheGRprediction dashed line includes ⇤CDM (but is also marginalized over one (Q $ 1). Figure 3.2: Left: Marginalized posterior contours inPlanck theTT,TE,EE+lowEQ2 Σ2 (2015)+BAO/RSD(2015)plane measured by = = of the two degrees of freedom in the µ–⌘ space). The vertical − dashed line shows A = 1. The filled contours use the Planck KiDS (Joudaki et al., 2017) with fiducial angular0.5 scales in green (denoted by ‘FS’), L Di Valentino et al. 2016a;Johnsonetal.2016;PlanckCollaboration tomographic bin and angular scale unless the parameter possesses TT,TE,EE+lowE+lensing likelihood, alone and in combination XIV 2016c). To this end, we use ISITGR (Dossett, Ishak & Molden- KiDStime and keeping scale dependence. only In constraining the largest MG, we divide angularQ and scales1 in pink (denoted by ‘LS’), and respec- + / with WL BAO RSD data. The unfilled contours show the con- hauer 2011; Dossett & Ishak 2012), which is an integrated set of $ in two redshift bins and two scale bins each, with transitions 0 Planck + A ↵ tively combined1 with Planck in grey andµ 0.0 blue. The indices represent a particularstraints from TT,TE,EE lowE. Note that L only a ects modified modules in COSMOMC designed to test gravity on cosmic at k 0.05 hMpc− and z 1. Thus, Q1 and $1 correspond to = = −1 CMB lensing of the Planck power spectra by definition, as dis- scales. the lowz, lowk bins, Q2 and $2 correspond to the lowz,highk combination{ } of MG bins, such{ that z} < 1 and k > 0.05h Mpc. The dashed lines We modify gravity in two ways. Given the first-order perturbed bins, Q3 and $3 correspond to the highz, lowk bins, Q4 and $4 cussed in Sect. 6.2. { } 0.5 Einstein equations, the first modification takes the form of an effec- intersectcorrespond to at the thehighz,high GRk predictionbins. This results in (Q eight= MG Σ = 1). Right: Marginalized posterior distribu- tive gravitational constant that enters the Poisson equation: degrees of freedom{ varied in our MCMC} calculations in addition to tionsthe vanilla of and the lensing MG systematics pa rameters parameters (alongµ withand theη CMBfor Planck TT,TE,EE+lowE+lensing data alone 2 2 k φ 4πGa ρi 'i Q(k,a), (6) 1.0 Since ⌃ measures deviations of the lensing potential from the = − degrees of freedom when applicable). We keep the background ex- i and in combination with external data (as indicated1 in the0 legend),1 using2 the late3 timeGR prediction, it is better constrained by WL data than µ and ⌘ ! pansion to be that of #CDM. In calculating the shear correlation where φ is the potential describing spatial perturbations to the metric parameterizationfunctions, we modify our lensing and module neglecting to integrate directly any over scale dependence. The0 dashed1 lines show theseparately. in the conformal Newtonian gauge, ρi is the density of species the power spectrum of the sum of the two metric potentials, which For simplicity we only allow µ and ⌘ to vary with time (as in i, G is Newton’s gravitational constant and Q(k, a)encodesthe standardin GR reduces ΛCDM to the standard model. integration The over the figure matter power is from Planck Collaboration et al. (2018). PDE15). Scale dependence increases the number of degenera- Fig. 31. Top: Marginalized posterior distributions of the MG pa- time and scale-dependent modifications to the Poisson equation spectrum. cies in parameter space and may require, for example, higher- In Fig. 13, we show constraints in the σ –" and Q –$ planes,rameters µ and ⌘ for Planck TT,TE,EE+lowE+lensing data alone (e.g. Jain & Zhang 2008;Bean&Tangmatitham2010;Dossett 8 m 2 2 order statistics in WL observables (Peel et al. 2018) to break et al. 2015, also see Ma & Bertschinger 1995). The rest-frame where the indices represent a particular combination of MG bins,and in combination with external data (as indicated in the leg- 2 1 the degeneracies. We use the late-time DE parameterization of overdensity is given by 'i δi 3Ha(1 wi)θ i/k ,whereδi is such that z < 1andk > 0.05 hMpc− .Sincethereexistsnoadequateend), using the late-time parameterization and neglecting any ≡ + + PDE15 and Casas et al. (2017), where the time evolution of all the fractional overdensity, wi is the equation of state and θ i is the prescription for the matter power spectrum on non-linear scales inscale dependence. The dashed lines show the standard ⇤CDM peculiar velocity divergence. Thus, we can construct an effective acosmologywithbinnedMG(andalsonoscreeningmechanism),model. Bottom: Impact of the BAO/RSD and Planck TT,TE,EE quantities is assumed to be proportional to the relative dark- gravitational constant, Geff(k, a) G Q(k, a), where Q 1 in we consider two distinct cases: one where the fiducial angular scales+ energy density: GR. The second modification to standard= × gravity enters ≡ of KiDS are included (described in Section 2.1), and a second case lowE data, compared to the 2015 results. For the 2018 Planck where effectively only linear scales are included in the analysis.data, the contours shift towards lower values of ⌘0 1, along µ = + ⌦ 2 2 (z) 1 E11 DE(z); (56a) k [ψ R(k,a)φ] 12πGa ρi σi (1 wi )Q(k,a), (7) the maximum degeneracy line (black versus cyan contours) and − = − + For the latter case, we consider the same ‘large-scale’ cuts as in i ! Section 3.1, removing all angular scales except for θ 24.9,shift in the same direction when using the BAO/RSD data (yel- ij ij = { ⌘(z) = 1 + E21⌦DE(z). (56b) where ψ is the potential describing temporal perturbations to the 50.7 arcmin in ξ (θ)andθ 210 arcmin in ξ (θ). low versus black contours). } + = − metric in the conformal Newtonian gauge, and σ i is the anisotropic For consistency with the previous sections, we show the con- This defines the constants E11 and E21. We report results in terms shear stress. Thus, R(k, a) allows the two metric potentials to differ straints in the σ 8–"m plane for KiDS with fiducial choice of angu- of µ µ(z = 0) and ⌘ ⌘(z = 0), which are determined even in the absence of anisotropic stress, and is equivalent to unity lar scales (presenting the results for KiDS with large-scale cut in 0 ⌘ 0 ⌘ in GR. In our MCMC calculations, we substitute R with a parameter Tables 2 and 3). The KiDS and Planck contours completely overlap, from E11 and E21, given the dark-energy density parameter to- that is more directly probed by weak lensing: $ Q(1 R)/2. In both as a result of Planck largely losing its ability to constrain σ 8are able to capture a generic deviation of the perturbation evolu- day. This parameterization is motivated by the assumption that general MG scenarios, the parameters Q and $ can= be functions+ of for a given matter density, but also because the KiDS constraints aretion from ⇤CDM that does not need to correspond to a known the impact of dark energy depends on its density and therefore both scale and time, and affect the growth of structure. extremely weak given the introduction of eight additional degrees ofmodel. This approach is complementary to constraints on action- allows for more deviation of µ and ⌘ from ⇤CDM at late times. We show the impact of the MG parameters on the shear corre- freedom. Thus, the KiDS and Planck S8 constraints agree to withinbased103 models, which are the topic of the next subsection. When The alternative early-time parameterization included in PDE15 lation functions in Fig. 1,findingthatthelensingobservablesare 1σ (for both choices of scale cuts). As shown in Table 3,when⌘ = µ = 1 we recover GR at all times, including when there are led to similar results and is not discussed here for brevity. Our fairly insensitive to changes in the gravitational constant, while $ accounting for the full parameter space, log 0.42 correspond- I = non-zero contribution from photons and neutrinos to the den- effectively boosts or suppresses the observables uniformly across ing to substantial concordance between KiDS and Planck when choice of parameterization, of course, limits the nature of possi- sity perturbation or anisotropic stress. In the parameterization ble deviations from ⇤CDM; however, the choices of Eqs. (56a) adopted here (described further below), the MG contribution to and (56b) allow us to compare our results directly with those of MNRAS 471, 1259–1279 (2017)⌘ is only relevant at late times, when the anisotropic stress from PDE15. relativistic particles is negligible. Figure 31 shows the marginalized constraints on µ0 and ⌘0 In this section we fix the background evolution to that of from di↵erent combinations of data, and also compares with the ⇤CDM (w = 1 at all times), so that any significant deviation of results from PDE15. Marginalized mean values and errors for µ or ⌘ from unity would indicate a deviation from ⇤CDM. We cosmological parameters are presented in Table 7. This table also consider constraints on the derived quantity ⌃, defined as also lists results for d2 1/2, the root-mean-square CMB lens- h i ing deflection angle, and the parameter combination ⌃0S 8 that k2 [ + ] = ⌃(a, k)4⇡Ga2 2⇢ 3(⇢ + P) . (55) is well-constrained by the DES WL data. These quantities allow ⇥ ⇤ 43 As described in Chapter 2, DMASS is the photometric sample constructed from the DES Y1 images that replicates the properties of the large BOSS CMASS spectroscopic sample (Reid et al., 2016). The CMASS galaxy sample was used to obtain the best RSD and BAO measurements to date. DMASS is well suited for a joint analysis of a galaxy clustering measurement (such as BAO and RSD probed by CMASS ) and a galaxy-galaxy lensing measurement for an equivalent sample from DES to achieve powerful cosmological constraints. In the linear regime, the amplitude of galaxy clustering scales as b2σ2, where b is the galaxy bias. The galaxy-galaxy lensing signal ∝ 8 2 is proportional to bΩ8σ . Therefore, combining two probes enables us to eliminate ∝ 8 the unknown galaxy bias and tighten the constraint Ωmσ8.

From the DES side, we only use the cross-correlation between the galaxy position of the DMASS galaxies and the shape of DES source galaxies (galaxy-galaxy lensing) described in Section 3.3. In order to emphasize the constraining power of DMASS, we do not include the DES measurements of galaxy clustering and cosmic shear.

To generate theoretical prediction for galaxy-galaxy lensing, we adopt the spectroscopic redshift distribution of CMASS. This is a fair choice as the redshift distribution of DMASS obtained with the clustering-z technique shows a fairly good agreement with CMASS redshifts as shown in Section 2.5. For this forecast study, we assume galaxy bias b = 2. By cross-correlating one lens bin and ﬁve source bins, we calculate

ﬁve simulated data vectors for galaxy-galaxy clustering over the 20 logarithmic angular bins over 2.50 < θ < 2500. The eﬀects of systematics such as redshift bin biases and multiplicative biases in shear are ignored for simplicity.

As external data sets, we choose the CMB temperature spectra and polarization from Planck 2015 (Planck Collaboration et al., 2016a) and the redshift space distortion

104 (RSD) and BAO measurements from BOSS (Alam et al., 2017b). BOSS provides two

measurements, one measured at low redshift z = 0.32 using their LOWZ sample and

the other one measured at high redshift z = 0.59 using their CMASS sample. We use

only the high redshift measurement from CMASS.

Figure 3.3 shows the forecasted constraints of (Σ, µ) around the ﬁducial ΛCDM values. The blue contour shows our forecast for the DMASS analysis. The resulting projected uncertainties on Σ0 and µ0 are estimated as

σΣ0 = 0.167 (3.20)

σµ0 = 0.396 . (3.21)

The red contour is from Simpson et al. (2013), which is plotted for comparison with

our work. Simpson et al. (2013) used combined structure growth data (CFHTLenS

(Heymans et al., 2013), WiggleZ (Blake et al., 2011b)), redshift space distortions

from 6dFGS (Beutler et al., 2011), BAO data (BOSS (Anderson et al., 2012), SDSS

LRG (Padmanabhan et al., 2012)), and CMB temperature (TT) and polarization

(TE) from WMAP7 (Larson et al., 2011). The resulting errors on constraints are

σΣ = 0.10 and σµ = 0.23. Their constraint of µ0 seems to be 30% stronger 0 0 ∼ than ours. This is primarily because they included RSD from 6dFGS which constrain

cosmology at low redshift very tightly. Σ0 is quite comparable to our results even

though we do not include cosmic shear which is known to be very sensitive to Σ0. As

stated above, direct comparison between two works is not straightforward due to the

diﬀerence in data sets and scales used, but on the whole the constraining power of

DMASS shown here is promising even for this rough forecast.

105 DMASS .2 1 Simpson

6 0. 0

µ 0 0.

6 0. −

2 1. 50 25 00 25 50 − 0. 0. 0. 0. 0. − − Σ0

Figure 3.3: The blue contour is forecasted 68% and 95% conﬁdence contours on (Σ, µ) obtained with DMASS. The red contour shows constraints from Simpson et al. (2013), plotted for comparison with this work.

106 3.7 Conclusion

In this chapter I presented my work on constraining theories of modiﬁed gravity.

In DES Y1 Extension, we found that the DES data alone constrains the modiﬁed

+0.28 gravity giving Σ0 = 0.43−0.29. Combining DES Y1 with the external data sets yields

+0.08 +0.42 Σ0 = 0.06 and µ0 = 0.11 , this result is consistent with the ΛCDM values. −0.07 − −0.46 We also made a forecast for a measurement of the modiﬁed gravity parameters

Σ and µ using the DMASS sample. By replacing the DES 3x2 pt statistics with galaxy-galaxy clustering between DMASS and DES source galaxies, we were able to demonstrate the capability of DMASS to tighten cosmological parameters by can-

celling galaxy bias. We obtained σΣ0 = 0.167 and σµ0 = 0.396 for the DMASS baed analysis and compared our forecast with the result from Simpson et al. (2013). Due to the several diﬀerences in the approach, a direct comparison is not possible, but achieving similar constraining power on Σ as Simpson without even using cosmic shears is a good indication for the strong potential of the DMASS sample for future cosmological analyses.

107 Chapter 4: Complementary Information in Fourier and Conﬁguration Space for Cosmological Observations

In this chapter, I will present the full content of our estimator paper (Lee et al., prep). The original abstract of the paper is given below: We explore the information content of the three-dimensional redshift-space galaxy correlation function and power spectrum. Given an infinite volume with no scale cuts, the information contained in these two statistics is equivalent. We recover this result, to close approximation, when suffi- ciently large and small scales are included (0 h−1Mpc < r < 200 h−1Mpc; 0 < h < 2hMpc−1). Modeling uncertainties typically require scale cuts (e.g., r > 20h1Mpc, k < 0.2hMpc−1). We study the impact of imposing such cuts, both in terms of the individual signal to noise for each statistic and the impact on the correlation between the two statistics. Scale cuts generically decrease the correlation between the two statistics; we illus- trate this by determining the information in the correlation function as a function of wave number k after applying scale cuts. We study both the total amplitude information and the impact on the information about derived parameters such as amplitude of the velocity field, fσ8.

The original author of the paper is S. Lee, H-. J. Seo, A. J. Ross, E. M. Huﬀ,

K. Honscheid and J. A. Blazek.

4.1 Introduction

The clustering of galaxies and other luminous tracers of large-scale structure has emerged as a primary probe of cosmology (c.f. Alam et al. 2017b and references therein). In particular, measurements of the baryon acoustic oscillation (BAO) feature

108 and the anisotropy of clustering due to redshift-space distortions (RSD) are central to current and upcoming experiments (c.f., DESI Collaboration et al. 2016a). These analyses primarily rely on two-point clustering statistics and are typically done in either Fourier space (the power spectrum, PS) or conﬁguration space (the correlation function, CF).

In principle, the information content is the same in Fourier space and configuration space. However, given finite survey sizes and the need to choose a range of scales for any given analysis, the information content can differ. For instance, it is well known that a delta function in one space is oscillatory in the other. Thus, scale cuts will impact the correlation between the two observables, in a manner dependent on the specific information being extracted. In Alam et al. (2017b), analyses of

Baryon Oscillation Spectroscopic Survey (BOSS; Dawson et al., 2013b) data were performed in both configuration- (Ross et al., 2017; Sánchez et al., 2017b; Satpathy et al., 2017; Vargas-Magaña et al., 2018) and Fourier-space (Beutler et al., 2017a,b;

Grieb et al., 2017). Sánchez et al. (2017b) presented a general methodology to combine several Gaussian posterior distributions from these works into a single set of consensus constraints representing their joint information, taking into account the full covariance between the different estimates. These covariances were empirically determined using mock simulations. Measurements of the same quantity were not perfectly correlated due to a combination of differences between Fourier and configuration space, scale cuts, and methodological choices. The obtained constraints were thus more precise and believed to be more robust than those from any one method.

In the work we present here, we study the information content of two-point clustering statistics in Fourier and conﬁguration space analytically. This allows us to

109 predict the level of correlation between configuration- and Fourier-space results, given a chosen range of scales for each. We choose a fiducial redshift of 0.55 which is approximately the mean redshift of BOSS [CMASS]. We use the ‘Gaussian streaming model’ (Peacock & Dodds, 1994) to produce redshift-space model power spectrum and correlation function data vectors and their covariance matrices. Given this specific model, we assess the level of correlation for different scale cuts and determine the gain obtained by combining results from each statistic.

This paper is organized as follows. In Section 4.2, we present general deﬁnitions of power spectrum (PS) and correlation function (CF) and the relation between them.

In Section 4.3, we evaluate their total amplitude information and develop a direct method to compare their relative information in Fourier space. In Section 4.4, we present the predicted cosmological constraints from an RSD analysis when using the

CF, PS, and their optimal combination. We compare our results to those found in

Alam et al. (2017b). We summarize and conclude in Section 4.5.

The ﬁducial cosmological model used in this paper is a ﬂat ΛCDM model with the

−1 following parameters : b = 2, f = 0.74, velocity dispersions σv = 3.5h Mpc/s, matter

−1 −1 density Ωm = 0.31, Hubble constant h H0/(100kms Mpc = 0.676, ﬂuctuation ≡ −4 3 −3 amplitude σ8 = 0.8. We assume a galaxy number density of 3 10 h Mpc and a × total survey volume of 5h−3 Gpc3. With the FKP weight (Feldman et al., 1994) with

−3 3 −3 3 P0 = 10000h Mpc , the nominal eﬀective volume corresponds to 2.81h Gpc .

110 4.2 Power spectrum and Correlation function

In three dimensions, the power spectrum P (k) and the correlation function ξ(r)

are related via

~ P (~k) = d3re−ik·~rξ(~r) (4.1)

Z 3 d k ~ ξ(~r) = eik·~rP (~k) (4.2) (2π)3 Z Since the two are linear transformation of each another, ideally they shall contain the same information. Any real set of data, however, will be contained within a ﬁnite volume. This breaks the complete conjugation due to the missing information beyond the survey volume:

rmax ∞ ~ ~ P (~k) = d3re−ik·~rξ(~r) + d3re−ik·~rξ(~r). (4.3) Z0 Zrmax Such a survey boundary effect manifests itself as a convolution in Fourier space, i.e., a broad window function that couples modes of different scales. While, in principle, all of the information could be included in the analysis, it is more typical to impose some rmax and kmin on the selected or chosen statistics that are significantly smaller scale.

On small scales, diﬃculties in modeling non-linearities typically further necessitate some rmin and kmax cuts to be applied to the analysis.

Therefore one can imagine that, in any realistic situation, the clustering information measured by a Fourier-space based method, e.g., the power spectrum (PS), may not be exactly equivalent to the clustering information measured based on a conﬁguration-space based method, e.g., the correlation function (CF). Comparing the information content of the PS and the CF for realistic scenarios is the very goal of this paper.

111 The Fisher information matrix provides a method for forecasting minimum possi-

ble errors on unbiased estimators of parameters given a set of observables and a model.

We will compare the expected errors from both estimators on the ﬁnal cosmological

parameters, such as the galaxy bias and the structure growth rate.

In order to enable a direct comparison, we project the information from both

statistics to discretized Fourier-space band powers. This way, we are able to iden-

tify the scale/mode over which each estimator contributes in the presence of a limited

survey volume (and therefore to understand the derived errors on cosmological param-

eters). The errors of each band power are then propagated to the ﬁnal cosmological

parameters. We take care to ensure that the projection conserves the full information

content of each statistic.

In this section, we ﬁrst build the broad bandpower model and its covariance matrix

from a Fourier space, Gaussian streaming redshift-space distortion (RSD) model for

the clustering of galaxies observed by a redshift survey (e.g., BOSS). The equivalent

model in conﬁguration space is computed by Fourier transforming it. Even though the

streaming model implies anisotropic clustering beyond the hexadecapole moment of

the galaxy distribution with respect to the line of sight, we will use only the ﬁrst three

even multipole moments, i.e., ` = 0, 2, 4. The information content will be compared by considering both the cumulative signal to noise and the signal to noise in one bin in Fourier space. We present this in Section 4.3.

4.2.1 Modeling estimator data using the streaming model Modeling power spectrum data

The three dimensional distribution of galaxies observed by a redshift survey is distorted by peculiar velocities through their aﬀect on the measured redshift. The

112 velocities are gravitationally induced by inhomogeneous structure and the eﬀect is known as RSD.

On large scales, the Kaiser effect dominates RSD (?). The Kaiser effect refers to the bulk flow of matter/galaxies infalling towards overdense regions coherently. This coherent motion causes an apparent contraction (i.e., amplification) of structure along the line of sight in redshift space. The Kaiser effect is derived as

2 δk δk(b + fµ ) (4.4) → k where µk is the cosine of the angle between the wavevector ~k and the line of sight. b is the bias, f is the structure growth rate that depends on Ωm(z).

On small scales, galaxies within virialized halos have large random motions relative to each other. This random motion leads to an apparent elongation of halos (i.e., damped clustering) in redshift space along the line of sight , which produces ‘fingers of God’ (Jackson 1972). This effect is modeled by Peacock & Dodds (1994) in a simplified form by multiplying an exponential term to damp the power as below:

2 2 2 δk δk exp[ k µ σ /2] , (4.5) → − k where σ is the velocity dispersion along the line of sight. Note that the exponential term goes to 1 for small k.

The Gaussian streaming model (Peacock & Dodds, 1994) is widely used as a simpliﬁed power spectrum model for redshift space distortions, which embraces the

Kaiser eﬀect on large scales and non-linear behaviors on small scales together as expressed below.

2 2 P (k, µk) = (b + fµk) Pm(k) D(k, µk, σ) (4.6)

113 2 2 2 −1 where D(k, µk, σ) = exp[ k µ σ ]. We adopt b = 2, f = 0.74, and σ = 3.5h Mpc − k as ﬁducial values.

The anisotropic matter power spectrum in redshift space can be decomposed into multipole moments using Legendre polynomials, Ll(µ) as below

P (k, µk) = Pl(k)Ll(µk) , (4.7) l=0 X In linear theory (i.e., only with the Kaiser effect) only modes l = 0, 2, and 4 are non zero. Nearly all of the information is contained in these three low multipoles; we therefore include only the first three modes even though our model includes a non-linear term that affects small scales. Integrating over µk gives the spherically averaged monopole moments as

2l + 1 P (k) = dµ P (k, µ )L (µ ) . (4.8) l 2 k k l k Z In order to obtain a signal to noise of the power spectrum as a function of k, we next discretize the power spectrum into band powers and describe each multipole as a sum of the band powers:

P¯l(k) = P¯l,i, (4.9) i=0 X where each band power in k bin i is spherically averaged power spectrum within a thin shell volume Vi as

1 ki+∆k/2 P¯ = dk 4πk2 P (k) (4.10) l,i V l i Zki−∆k/2 with

ki+∆k/2 1 k 2 V = d3k = 4πk2∆k 1 + i . (4.11) i i 12 ∆k Zki−∆k/2 " # 114 Here ki is center value of ith k bin, and ∆k is the size of k bin. The scope of band

powers is limited between ko and kN to simulate a real survey conducted in a limited

volume.

Assuming a Gaussian density ﬁeld, the covariance per mode is (Tegmark, 1997)

1 2 ~ ~0 ~ C (k, k ) = 2 P (k) + δ~ ~ 0 . (4.12) P n¯ k,k One can take the continuum limit as follows (Cohn, 2006)(Cohn, 2006)

3 (2π) ~ ~ 0 δ~k,~k0 δ(k k ) (4.13) → Vs − 1 δ(~k ~k0) = δ(k k0)δ(µ µ0) (4.14) − 2πk2 − −

where Vs is the survey volume. The continuous form of the covariance function of the

streaming model is given by

2 2 0 1 (2π) 0 0 CP (~k, k~ ) = 2 P (k, µ) + δ(k k )δ(µ µ ) . (4.15) n¯ V k2 − − s Since we are working with spherically averaged multipoles, the 3D covariance of the

power spectrum should be changed into a form of isotropic covariance in the same

manner:

0 P 0 2l + 1 2l + 1 C 0 (k, k ) = ll 2 2 0 0 0 dµk dµ CP (~k, k~ )Ll(µk)Ll0 (µ ) (4.16) × k k Z Z 0 0 δ(k k ) = Ill0 (k, k ) − (4.17) 4πk2 with

0 0 2l + 1 2l + 1 I 0 (k, k ) = ll 2 2 (2π)3 1 2 2 dµk P (k, µ) + Ll(µk)Ll0 (µk) . (4.18) × V n¯ s Z 115 In order to obtain the covariance between band powers from the analytic form, this

continuous form turns into a discrete form by taking shell averaging for k bins. Due to

the delta function of k in the covariance, taking shell averaging for k and k0 technically

transforms the delta function to the inverse of a shell volume in the ith k bin as follow:

0 δ(k k ) δij − 2 (4.19) 4πk → Vi

Therefore, the resulting covariance between the ith and jth band powers is given by

P 1 3 1 3 0 P 0 C¯ 0 (k , k ) = d k d k C 0 (k, k ) (4.20) ll i i V V ll i ZVi j ZVj δij = I¯ll0 (ki, kj) (4.21) Vi where I¯ll0 is a shell-averaged quantity of the continuous form in Equation (4.18). Note

0 that the k bin dependence comes in Vi. The shot noise term only appears where l = l :

3 (2π) δij 0 2(2l + 1) 2 δll , (4.22) Vsn¯ Vi

due to the orthogonality properties of Legendre polynomials given as

1 2 dµ l(µ) l0 (µ) = δl,l0 . (4.23) L L 2l + 1 Z−1 Modeling correlation function data

Modeling the correlation function data in discrete conﬁguration space bands, we start by Fourier transforming theoretical power spectrum multipoles:

dk ξ (r) = il k2P (k)j (kr) . (4.24) l 2π2 l l Z This theoretical form of correlation function is discretized into band correlations given

ξ¯l(r) = ξ¯l,i (4.25) i=0 X 116 where the band correlation ξ¯ is spherically averaged within a thin shell volume Vr,i as

1 ri+∆r/2 ξ¯ = dr 4πr2 ξ (r) (4.26) l,i V l r,i Zri−∆r/2

with the volume of ith shell Vr,i

1 ∆r 2 V = 4πr2∆r 1 + . (4.27) r,i i 12 r " i #

Here ri is center value of ith r bin, and ∆r is the size of r bin. In the same manner of the power spectrum, the scope of band correlations is limited between ro and rN that corresponding minimum and maximum pair separations we adopt in real survey analysis.

In order to get covariances between the two correlation multipole band powers, we take the Fourier transform of Equation (4.21):

3 3 0 ξ 0 l+l0 d k d k P 0 0 0 C 0 (r, r ) = i C 0 (k, k )j (kr)j (k r ) ll (2π)3 (2π)3 ll l l Z 3 Z l+l0 d k 0 = i I 0 (k, k)j (kr)j (kr ) . (4.28) (2π)6 ll l l Z To change the continuous form to a discretized form, we calculate shell-averaged

values for each r and r0 values as follow:

¯ξ 1 3 1 3 0 ξ 0 Cll0 (ri, rj) = d r d r Cll0 (r, r ) (4.29) Vr,i Vr0,j ZVr,i ZVr0,j Since only Bessel functions have r dependence, this integration only aﬀects Bessel

functions as follow:

1 j (k, r ) = dr3j (kr) . (4.30) l i V l r,i ZVr,i The shot noise term (1/n¯2) is calculated as

0 2l 2(2l + 1) δ(r r ) i 2 −2 δl,l . (4.31) Vsn¯ 4πr 117 This term includes a Kronecker delta function for l and a delta function for r, result-

ing from the orthogonality properties of Legendre polynomials (Equation 4.23) and

spherical Bessel functions given below:

∞ 2 0 π 0 dk k jl(kr)jl(kr ) = δ(r r ) . (4.32) 2r2 − Z0 Taking shell averages for the ith r bin and jth r0 bin yields the inverse of the shell

volume of the ith r bin as done in Fourier space:

0 δ(r r ) δij −2 , (4.33) 4πr → Vr,i From this notation, the resulting shot noise term in the conﬁguration space is

2l 2(2l + 1) δij 0 i 2 δll . (4.34) Vsn¯ Vr,i Thus, the shot noise term contributes only to the diagonal components of the covariance matrix for the correlation function, and has a dependence on the size of r bin.

Projecting measurements in conﬁguration space onto Fourier space band powers

In this section, the simulated measurement in configuration space from the previous section is projected onto band powers in Fourier space in order to compare different estimators using the same metric. As a first step, we project the covariance matrix of the correlation function for the limited data range between ro and rN (see

Equation (4.28)) into Fourier space.

In order to obtain the band power covariance from the correlation function, the inverse of covariance matrix of correlation function is multiplied by the projection operators as below :

−1 t FP = XC X (4.35)

118 Since we have m band powers for ` = 0, 2, 4, the projection vectors include

X = (X00,0, ,X00,m,X22,0, ,X22,m,X44,0, ,X44,m) (4.36) ··· ··· ···

and the full covariance matrix is given as,

C0,0 C0,2 C0,4 C = C2,0 C2,2 C2,4 , (4.37)   C4,0 C4,2 C4,4   where each block Ci,j represents the m m covariance matrix between two multipoles × l = i and j. The projection operator vector X is a derivative of band correlation ξ¯ in

Equation (4.26) with respect to band power:

¯ ki+∆k/2 2 ∂ξl,n l dk k 0 0 ˆ [Xll ]in = = i δll jl(k, rn) 2 . (4.38) ∂P¯ 0 2π l ,i Zki−∆k/2 Note that the derivative of the correlation function can be simpliﬁed as an integral of

the Bessel function since the shell-averaged bandpower P¯l,i is zero except within the

ith shell. We use ˆjl(k, rn) to denote the spherically averaged Bessel function in n th

r bin. The Kronecker delta δl,l0 in front of integral implies that only diagonal blocks

where l = l0 survive as below :

X0,0 0 0 X = 0 X2,2 0 . (4.39)   0 0 X4,4   The resulting FP¯ from the correlation function has the same shape as the Fisher

matrix from the power spectrum, but is not diagonal. In the following section, we

determine the amount of information that each estimator contains in various cases,

using the Fisher matrices deﬁned in this section.

119 6 1e5 ξ P 5

4 2 )

N 3 / S ( 2

0 0.0 0.2 0.4 0.6 0.8 1.0 1 kmax (h Mpc− )

Figure 4.1: Cumulative SNR of PS (black dashed line) and CF (blue dot points) as a function of the maximum wave number kmax. The blue solid is a smooth ﬁt to the −1 −1 −1 dot points. We assume k0 = 0 h Mpc , rmin = 0.1 h Mpc and rmax = 180 h Mpc, i.e., an almost all scale available for CF. Throughout all relevant calculations, we adopt 0.004 h Mpc−1 for k bin size dk since SNR from correlation function becomes stable where dk is equal and smaller than 0.004h Mpc−1. Convergence tests for other dk bin sizes is in Appendix.

4.3 Signal to Noise of estimators

We obtained two covariance matrices; one for each estimator, both evaluated in

Fourier space. Throughout this section, we assess the amount of information that

each estimator contains by comparing signal to noise from the covariance matrices.

We focus on the signal to noise of the amplitude of the power spectrum rather than the signal to noise on ﬁnal cosmological parameters. Since cosmological parameters

120 6 1e5 ξ (0,200) ξ 5 (0,500) ξ (10, 200) P 4 2 ) 3 N / S ( 2

0 1e3 8 -1 0 10 ξ 1100 (10, 200) 1 × kmax (h Mpc− ) 6 2

) 4 N / S ( 2 ∆

2 10-1 100 1 kmax (h Mpc− )

Figure 4.2: Top: cumulative SNR of CF for different rmin and rmax. Bottom: SNR contribution from individual band power defined as ∆SNR(kn) = SNR(kn) − SNR(kn−1). The black dashed line in the top and bottom panels represent the SNR of PS while the colored solid lines are for CF16. In the top panel, at low k, SNR from correlation function increases slower than power spectrum until it reaches to the maximum value. This can be seen clearer in the bottom panel; the peak of blue line falls at larger k than the peak of black dot-dashed line (for PS) without amplitude change. The colored dashed lines show ∆ SNR when only diagonal components in Fisher matrix are considered. The location of peaks is almost in agreement with the peak of PS which implies that off-diagonal components (more so for a limited range of r) are the main factor that shift information peak to smaller scales.

121 depend on the power spectrum in the same way in both cases, the signal to noise of

the power spectrum amplitude should give an insight into the potential constraining

power for parameters that sensitively depend on the overall amplitude.

The deﬁnition of cumulative signal to noise (hereafter SNR) of P up to kn is n ¯ ¯ SNR(kn) = P (ki) FP¯(ki, kj) P (kj) (4.40) i,j=0 X ¯ where FP¯ is the Fisher matrix of band power and P (k) is a data vector as a function of

k. For the power spectrum, the Fisher matrix of each multipole moment is diagonal,

therefore, SNRP is simply given by n ¯ 2 SNRP¯(kn) = Pl(ki) FP¯ ll(ki, ki) . (4.41) l=0,2,4 i=0 X X The calculation for the correlation function is not as straight-forward. First, the covariance matrix of ξ (in Equation (4.39)) should be cut to scales being considered in conﬁguration space before projecting to band power space. The resulting cumulative

SNR for the correlation function is a function of kn, rmin and rmax as below n ¯ ¯ SNRξ¯(kn, rmin, rmax) = P (ki) FP¯(ki, kj, rmin, rmax) P (kj) (4.42) i,j=0 X Obtaining band power Fisher matrix FP¯ on scale up to kn in the above equation

involves two subsequent matrix inversions (inverting FP from the previous section to

slice the covariance matrix between k0 and kn and reverting it again ). These inver-

sions are not only computationally expensive but contain matrix elements with values

comparable to machine precision. To avoid numerical errors from the inversions, we

adopt an analytic blockwise inversion method, described in Appendix C, that helps

to slice a Fisher matrix without inverting it.

Figure 4.1 compares cumulative signal to noise of the power spectrum (black

dashed line) and the correlation function (blue points) as a function of the maximum

122 wave number kmax. The noise seen in the points is due only to how we obtain the

2 2 cumulative (S/N) at a given kmax. The total (S/N) for the CF at kmax = 2 is

the same independent of dk, so this noise only aﬀects direct comparison for plotting,

not the result of the Fisher matrix analysis that will be presented later. We assume

−1 −1 −1 kmin = 0 h Mpc , rmin = 0 h Mpc and rmax = 180 h Mpc. Throughout all

relevant calculations, we adopt 0.004 h Mpc−1 for the k bin size dk since SNR from the

correlation function becomes stable when dk is smaller than 0.004. Both estimators

converge as kmax approaches unity and beyond, which implies that the correlation

function over 0 h−1Mpc < r < 180 h−1Mpc has almost the same amount of total

information as the power spectrum when including all scales up to k 2 h Mpc−1. ∼

The top panel of Figure 4.2 shows the cumulative SNR of ξ with diﬀerent rrmin

−1 and rmax in comparison to the SNR of P when summed from k = 0 h Mpc to

kmax. The bottom panel shows the SNR contribution from individual band powers.

The SNR for each band power is derived by differencing the cumulative SNR between subsequent kmin in order to account for the off-diagonal correlations, defined as

∆SNR(kn) = SNR(kn) SNR(kn−1) . (4.43) −

The black dot-dashed line in the top and bottom panels is obtained directly from the

power spectrum estimator. The colored solid lines represent SNR for the correlation

function using various rrmin and rrmax values. For a visual clarity, the colored lines are

a smooth ﬁt to the actual cumulative SNR and ∆SNR is derived by diﬀerenciating

the smooth ﬁt. Note that all other calculations in this paper are conducted without

utilizing this smooth ﬁt.

The top panel of Figure 4.2 conﬁrms that the cumulative signal to noise from

the two estimators is almost the same when all k scales included and rmax 0. ≈∼ 123 The convergence happens at a lesser k when the rmax of the correlation function is greater. As rmax increases, we see that the curve representing the cumulative ξ SNR approaches that of power spectrum (at all scales, not just high k), which is consistent with the general assumption that two estimators are same when all the scale range is accessible, e.g., in the case of limitless surveys. The ﬁgure also shows that while the

−1 gain of increase beyond rmax 200h Mpc is not much, the available SNR quickly ∼ decreases as increasing rmin.

This dependence is further inspected in the bottom panel of Figure 4.2; the peak of the green line (i.e., the peak of the information) is at larger k than the black line but with the same ∆SNR peak value. The peak of the blue is at an even larger k value; i.e., the scale from which information in ξ contributes the most shifts to larger k when the range of r is limited whether in terms of rmin or rmax. In order to understand this scale-shift of information, we next take a look at only the diagonal terms.

The colored dashed lines in the bottom panel of Figure 4.2 show the SNR when the only diagonal components in the Fisher matrix are considered. The locations of the peaks are aligned with the peak of power spectrum information. Clearly, the importance of oﬀ-diagonal components (i.e., correlation between Fourier modes) increases as we limit the range of scales used in the conﬁguration space analysis. Due to the increased coupling between Fourier modes for smaller rmax, the information is spread to larger k modes.

Figure 4.3 displays the SNR obtained from ξ as a function of the maximum and minimum scales considered. It conﬁrms that the dependence on rmax is small while the dependence on rmin is particularly strong. Approximately 90 per cent of the

124 2 2 10 10 P P R R N N S S / / ) ) x n i a m m r 1 r ( 10 ( ξ ξ R R N N S S × × 0 0 0 0 1 1

101 100 170 175 180 185 190 195 200 1 rmax, rmin + 170 (h − Mpc)

Figure 4.3: Maximum cumulative SNR from PS and CF as a function of rmax (red line) and rmin (blue line), relative to the maximum cumulative SNR for PS between −1 −1 −1 0.01h Mpc < k < 2 hMpc . When rmax is varied, rmin is 0h Mpc for all points; −1 when rmin is varied, rmax is 200h Mpc for all points. One can see that the dependence −1 on rmax (red curve going from 170 to 200 h Mpc) is small while the dependence on −1 rmin (blue curve going from 0 to 30 = rmax 170h Mpc) is particularly strong. −

125 information is removed when cutting r > 30h−1. As a caveat, we assume the shot

noise level for the BOSS survey.

We next consider CF and PS data over a limited range of scales, typical of those

used in recent large scale structure analyses (see, e.g., Alam et al. 2017b). In Figure

??, we present a square root of the cumulative SNR assuming ξ(r) for 18 < r <

180 h−1Mpc (red line) and P (k) over 0.01 < k < 0.2 h Mpc−1 (blue line). The

blue line stops at k = 0.2h Mpc−1 which is the maximum k value available while the

red line extends to k 2h Mpc−1 since ξ includes information over all wave vectors. ∼ −1 Note that the cumulative SNR of ξ alone with rmin = 18h Mpc is substantially

−1 lower than for the case with rmin 0h Mpc; That is, by missing the information on ∼ r < 18h−1 Mpc, the loss in the available information on the amplitude of clustering is substantial and unrecoverable even at large k. On the other hand, one ﬁnds that

ξ can provide information beyond scale k = 0.2h Mpc−1 used as a cutoﬀ in the PS analysis. When the both estimators are combined, we therefore expect that there will be slight gain on information contributed from k > 0.2h Mpc−1.

The factor of 3 diﬀerence in the √SNR is perhaps surprising, but we will demon-

strate in the next section that this result is consistent with previous empirical and

theoretical results and that most of the gain in signal to noise is removed when one

must account for the eﬀect of shot noise on the power spectrum.

To summarize, we expect that, for the range of scales used for the realistic LSS

analyses, the amplitude constraints such as bias or growth factor would be much

better constrained by the power spectrum estimator than the correlation function

estimator. The gain from combining two estimators will be small but non-zero on such

parameters. On the other hand, the gain could be more substantial for parameters

126 that depend on the relative amplitude as a function of the line-of-sight angle or as

a function of scale rather than the overall amplitude itself. In the next section, we

project the signal to noise of the combined estimator onto cosmological parameters,

b, f, and σv to study the gain of combined estimators for diﬀerent parameters more directly.

The results in this section imply that the correspondence between the conﬁguration space separation r and the Fourier mode k depends on the range of r used in the

correlation function data. The slow convergence of ξ SNR toward P (k) SNR implies,

when analyzing correlation function data with Fourier-space ﬁtting models, the ﬁtting

model will need to encompass large enough k to reach the same level of SNR as P (k).

Modeling high k can be tricky since nonlinearities will become quickly detrimental

on small scales. If a ﬁtting model for the correlation function data is constructed

in conﬁguration space (e.g., Reid & White, 2011), this high k convergence problem

can be avoided; i.e., it appears that a ﬁtting model is better to be constructed in

the same space as the estimator. If the information on a cosmological parameter of

interest damps quickly at some low k, such as the BAO information damping near

k 0.3h Mpc−1, however, we believe that this high k convergence problem will not ∼ apply.

The reverse and positive side of this high k convergence is that the correlation

function over a proper range of separation r (especially when its ﬁtting model is

constructed in conﬁguration space) includes an information over a broad range of k

in the form of the coupling between diﬀerent Fourier modes. That is, we have access

to smaller scales (and larger scales) information with the correlation function data

when compared to power spectrum data over a limited range of k modes.

127 Conversely, one can imagine conducting an equivalent information content com-

parison in conﬁguration space by transforming power spectrum information into the

information in bands of separation r. We will likely ﬁnd that the power spectrum

data has access to smaller and larger separation r than correlation function data over a limited range of r. The two estimators in most realistic surveys (e.g., even with

−1 rmax 500h Mpc) would therefore contain information that are not identical and ∼ even more so if additional scale limits are imposed due to modeling and observa-

tional systematics. This implies that there is a potential gain for information when

combining two estimators, which motivates the next section.

128 4.4 Cosmological Parameter Constraints

In the linear regime, the amplitude of the matter power spectrum in redshift space

is increased by (1 + β2µ2)2 (?) where β = f/b is the growth rate f divided by the

galaxy bias b. White et al. (2009) calculated constraints on bσ8, fσ8 and the peculiar

velocity dispersion σv in Fourier space, using the power spectrum model in redshift

space described in Section 4.2.1. The similar calculation was done in Reid & White

(2011) (hereafter R11) in conﬁguration space. They assess the information contents

their model has by comparing fractional errors on bσ8 and fσ8 with the results in

White et al. (2009) in the quasilinear regime. In this section, we will redo the work in R11 not only in Fourier space but also in conﬁguration space and compare each result with joint constraints to show how much gain we would get by combining the two estimators.

4.4.1 Individual estimator projections

We ﬁrst project the information of individual estimators onto the RSD cosmological parameters to compare the results with Figure 5 in R11. In our Figure 4.4, the blue line shows the marginalized percentage error on f (bottom) and b (top) as a

−1 function of rmin from CF (with ﬁxed rmax = 180h Mpc). The dot-dashed line is the

constraints from PS as a function of kmax which is plotted at rmin = 1.15/kmax. Since

we use the model from White, Song and Percival (2009), we expect our dot-dashed

line be identical to the corresponding line in Figure 5 of R11. We do not ﬁnd any

apparent quantitative discrepancy based on our visual inspection. Our CF constraint

(blue line) seemingly also agrees with the CF constraint from R11, implying that our

129 Figure 4.4: Fractional error on b and f as a function of rmin. This plot is to be compared with figure 5 of R11. The solid line is obtained from three CF multipoles l = 0, 2, 4, being marginalized over σv, using our model based on White et al. (2009). The same calculation has been done in R11 with a configuration-space counterpart to White et al. (2009). Our calculation agrees with R11’s result within 10% by visual inspection. The dot-dashed lines are obtained from three PS multipoles∼ plotted at rmin = 1.15π/kmax. The difference between two fractional errors from different estimators is comparable to what we observe in confidence ellipses in Figure 4.5.

130 Fourier-space based RSD parameters are equivalent to the conﬁguration-space based

RSD parameters in R11.

The results in Figure 4.5 are consistent with R11 and the bias results are consistent

−1 with our findings in Section 4.4. At rmin = 18h Mpc, the uncertainty on the bias is approximately three times greater for the correlation function, compared to the power spectrum. This remains true if we include the effect of random velocities (σv), as shown with green curves. Once we also account for the effect of shot-noise (red curves), the uncertainty obtained from the power spectrum measurements becomes approximately equal to that of the correlation function measurements. Including the effects of shot-noise is the most realistic case and will be our default choice from here-on.

4.4.2 Combining and Comparing Constraints

We consider the parameter space θ = (b, f, σv, n) and project the Fisher matrix of each estimator to the parameter space directly in order to see how each estimator aﬀects the error of each parameter. The Fisher matrix projected to a parameter space

θ is given by

∂Pl(kp) ∂Pl(kq) Fθiθj = [FP¯P¯]pq , (4.44) ∂θi ∂θj p,q,l X where p and q denote the power bands and with the derivatives

∂Pl(kp) 2l + 1 ∂R(kp, µ) = dµ Pm(kp) l(µ) (4.45) ∂θ 2 ∂θ L i Z i

131 2 2 where R(kp, µ) = (b + fµ ) D(kp, µ, σ).

∂R(k , µ) p = 2(b + fµ2)D(k , µ, σ) (4.46) ∂b p ∂R(k , µ) p = 2µ2(b + fµ2)D(k , µ, σ) (4.47) ∂f p

∂R(kp, µ) 2 2 2 2 = ( k µ )(b + fµ ) D(kp, µ, σ) . (4.48) ∂σ − p

The derivatives of ξ is given as Fourier transform of Equation (4.45). We adopt realistic k and r data ranges used in recent cosmological analyses. We assume PS data over 0 < k < 0.25 hMpc−1 as used in Grieb et al (2016) in combination with CF data over 20 < r < 160 h−1 Mpc as used in Sanchez et al (2016). Again, the ﬁducial values

−1 −4 used here are given as b = 2, f = 0.74, σv = 3.5 h Mpc/s, shot noisen ¯ = 3 10 , × −3 3 and the survey volume V0 = 5h Gpc respectively with the nominal eﬀective volume

−3 3 −3 3 of 2.81h Gpc using P0 = 10000h Mpc . All three modes l = 0, 2, 4 are included.

Next, we obtain joint constraints for PS and CF, considering correlations between two estimators. We first generate 1000 of simulated PSs and CFs by shifting each data point randomly within Gaussian errors derived from covariance matrices. Then, we fit each simulated data vector to predictions to find a set of best fit parameters that minimizes χ2 below

Nbins 2 T −1 χ = ∆di (Ci,j )∆dj , (4.49) i,j X where ∆d dﬁd dsim(θ), Nbins is the number of data points, C is covariance matrix ≡ − described earlier. Cross-correlation between a set of parameters θPS and θCF can be obtained as

Nsim i ¯ j ¯ Cov(θPS, θCF) = (θPS θPS)(θCF θCF) . (4.50) i,j − − X

132 where the subscript ‘PS’ and ’CF’ show which estimator the best ﬁt parameter is obtained from, the index i implies the parameter θ is derived from ith simulation, θ¯ is the mean of the θ population. Auto-correlation gives constraint from either PS or CF that is same with the constraint obtained by the aforementioned Fisher projection.

We conﬁrmed that both methods yield the same result. Along with auto- and cross- covariance, the error of the joint constraint of a parameter θ can be obtained via

1 = (C−1) . (4.51) σ2 θ ij θ ij X Figure 4.5 displays 1σ level conﬁdence ellipses for b and f when marginalized over

σv and shot noise for diﬀerent joint combinations. The left panel shows the constraint from the PS using the maximum scale we include, i.e., over 0h Mpc−1 < k < 2h Mpc−1 in comparison to the constraint from the CF over 0h−1 Mpc < r < 180h−1 Mpc.

As shown earlier, with this range of r, the cumulative SNRs of the two estimators are nearly identical and reach the maximum achievable in our parameterization and assumed survey conditions. Since both estimators contain all structure information available, the two error ellipses in the RSD parameter space are indeed nearly the same. The two estimator data are almost perfectly correlated in this case, and the joint constraint would be identical to the individual error ellipse.

For the case with limited k and r ranges, as shown in the right panel of Figure 4.5,

PS data (blue) overall provides a substantially better result for constraining b and f both than CF data (red). This tendency is also shown in S´anchez et al. (2017a).

Conﬁdence ellipses from the combined estimator(green) are located inside whichever smaller, and especially better for constraining bias. This is consistent with the high cumulative SNR of PS in the previous sections (e.g., Figure 4.2).

133 We next compare our predictions with one of the published DR12 BOSS analyses.

Sánchez et al. (2017b) reports a constraint of 0.041 on fσ8(z) for their 0.4 < z < 0.6 bin when marginalizing over the AP effect using the DR12 data17. We obtain an unmarginalized uncertainty by inverting the publicly released Sánchez et al. (2017b)

covariance matrix that includes the AP constraints. We find σfσ8 = 0.033, which is the appropriate quantity to compare with our results, after dividing by σ8(z) and accounting for differences in the effective volume we assume and that of BOSS. Using

−3 3 σ8(z = 0.51) = 0.60, this means that f σf = 0.736 0.055 for Veﬀ = 1.30h Gpc ± ± at this redshift bin (quoted in table 2 of Alam et al. 2017b). In comparison, our

−3 3 −4 3 −3 calculation assumes Vs = 5h Gpc with n = 3 10 h Mpc , which corresponds × −3 3 −3 3 to Veﬀ = 2.81h Gpc assuming an FKP weight with P0 = 10000h Mpc . Rescaling

−3 3 the BOSS constraint to our Veff = 2.81h Gpc , we derive f σf = 0.736 0.037 ± ± while in this paper we predict f σf = 0.74 0.013. The error is discrepant by 65%. ± ± Given that our result agrees with Reid & White (2011) in Figure 4.4 for the same set of parameters, we note that Sánchez et al. (2017b) uses at least three more free nuisance parameters in Fourier space to marginalize over the high k nonlinear effects when modeling the CF data. This would effectively remove the information at large k values that the CF data contains. In other words, we believe that this 65% offset implies a leverage potentially available by improved models for nonlinearity.

We estimated the percentage improvement on constraint by a joint estimator relative to the PS constraint alone. We ﬁnd that the improvement is small for b since the PS data has a substantially more information on b than CF. The improvement is moderate for f, giving 7 8% gain when combining the PS data with − 17Their constraint using the mock data was 0.044 which is very similar to the data constraint.

134 0h Mpc−1 < k < 0.25h Mpc−1 and the CF data with 20h Mpc−1 < r < 160h Mpc−1. ∼ ∼ The gain for σv shows the greatest improvement, about 10%. By the combining the two estimators, the joint estimator acquires leverage on the relative amplitude between the low and high k clustering (from the CF data) with a tight anchor at low k (from the PS data). The constraint on σv is most sensitive to such k dependent

amplitude and therefore can be greatly improved compared to other parameters.

While we adopted the ranges of the BOSS DR12 analyses, our estimation shows

substantially better constraints on f than those obtained by BOSS. This is likely due

to the fact that we use considerably fewer nuisance parameters. However, we ﬁnd

that the relative constraining power between the PS and CF data is quite similar.

For example the CF analysis of S´anchez et al. (2017b) obtained a 13%, 16%, and

25% higher standard deviation on σf recovered from mocks in the redshift ranges

0.2 < z < 0.5, 0.4 < z < 0.6, and 0.5 < z < 0.75 when compared to the PS analysis

of Grieb et al. (2017). We ﬁnd an 18% diﬀerence testing the same scale ranges.

Considering instead the diﬀerences between the Satpathy et al. (2017) and S´anchez

et al. (2017b) CF results, we ﬁnd only at an 17% increase in σf , predominantly

−1 caused by increasing rmin to 25 h Mpc, compared to a diﬀerence of 30% recovered

from mocks. Clearly, modeling choices had a strong impact on the relative uncertainty

for each BOSS CF analysis.

135 p p 78 xi 0. xi 752 com com 0. diﬀ diﬀ 76 0. .744

f 0 f 74 0.

.736 0 72 0. 0 < k < 2 0 < k < 0.25 (Grieb) 728 0. 0 < r < 180 70 25 < r < 160 (Sanchez) 0.

996 998 000 002 004 965 980 995 010 025 1. 1. 2. 2. 2. 1. 1. 1. 2. 2. b b

Figure 4.5: Confidence ellipses (1σ) for b and f for various scale limits used in recent DR12 BOSS analyses. Different color stands for different estimator; PS(blue), CF(red), joint estimator (green). The purple color stands for the joint constraint with zero cross-covariance. The scales included are specified in the left corner of each panel. The left panel shows the projection when all scales are available. As expected, the two estimators give almost the same constraints and there would be no information gain by joint estimator in this case. The right panel shows the projection with realistic scale limits. In these specific choices of ranges, PS (blue) is substantially better for constraining b and f both. When two estimators are combined (green), the gain is small for b while the gain is substantial for f and σv.

136 p 78 0. xi com 76 0. diﬀ f 74 0.

.72 0 0 < k < 0.25 (Grieb) 25 < r < 160 (Sanchez) 70 0.

75 3.

60 3. s

45 3.

30 3.

10 3.

] 05 4 3. − 10

× 00 3. n [ 95 2.

965 980 995 010 025 .70 .72 .74 .76 .78 .30 .45 .60 .75 1. 1. 1. 2. 2. 0 0 0 0 0 3 3 3 3 b f s

Figure 4.6: Confidence ellipses (1σ) for all cosmological parameters for scale limits used in recent DR12 BOSS analyses (25 < r < 160h−1 Mpc and 0 < k < 0.25h Mpc−1). Different color stands for different estimator; PS(blue), CF(red), joint estimator (green). The purple color stands for the joint constraint with zero cross- covariance. PS (blue) is substantially better for constraining b and f both. When two estimators are combined (green), the gain is small for b while the gain is substantial for f and σv.

137 4.5 Discussion and Conclusion

From the results in the previous section, we have found that the combination of a

Fourier- and conﬁguration-space analysis, using realistic scale limits, can somewhat

improve the total cosmological information. The basic reason for this is that scale

cuts have been imposed for each analysis, meaning that the space containing the

information cannot be equal. For instance, if we translate the conﬁguration-space

information to Fourier space, we ﬁnd it contains information that is at higher k than we use in any Fourier space analysis.

Our results further imply that an accurate conﬁguration-space model must contain nuisance parameters that are not directly translatable to Fourier-space and only represent a portion of the k > 0.4h Mpc−1 information (otherwise, its Fourier transform would provide an accurate k-space model to k < 2h Mpc−1).

One interesting ﬁnding of our work is that the PS is considerably more sensitive

to the clustering amplitude (i.e., b) than the CF for the scales used in typical RSD

analyses. This is consistent with the ﬁndings of BOSS data analyses. For example,

Alam et al. (2017b) found greater than 3σ signiﬁcance for a change in the PS am-

plitude between the North and South Galactic cap footprints for the data (traceable

to oﬀsets in the photometry in the two regions, see their appendix). In the CF, no

oﬀset in amplitude was detectable (Ross et al., 2017). Our results help explain why

this information was not accessible to the CF for r > 20h−1Mpc. However, any im-

provement in the ability to constrain the bias disappears once a free shot-noise term

is introduced into the PS analysis.

Based on our work, the practical implications include

138 Fourier and conﬁguration-space analyses that apply realistic scale limits contain • some complementary information for RSD analyses. An optimal analysis will

therefore combine results and obtain improved constraints over either alone.

Predicting the realistic improvement analytically requires more complex mod- • eling than we have presented. Model parameters must be deﬁned in each indi-

vidual space. Empirical approaches to ﬁnding correlation factors, such as those

based on mocks (e.g., S´anchez et al. 2017a), are likely to be required.

The ability to accurately use the same nuisance parameters across spaces is • likely to be of great beneﬁt. Even when employing empirical approaches, in-

cluding such nuisance terms in the correlation matrices is likely to be of beneﬁt,

as it might help break important degeneracies that aﬀect the precision of the

structure growth constraints.

Thus, we recommend the above points be considered in any future RSD analyses.

In fact, they are likely relevant to any analysis (e.g., weak lensing) that uses the amplitude of a field. In the most optimistic case we found that simply taking advantage of the complementary information in Fourier- and configuration-space can double the effective survey volume.

139 Chapter 5: Conclusion

In this thesis, I have briefly reviewed the basics of the modern cosmology and modified gravity as potetial candidates driving cosmic acceleration. In the later part of the thesis, I described my Ph.D research projects, mostly dedicated to constraining modified gravity or develoing tools for that.

Major ﬁndings are:

Modifications to GR is one of the strong candidates of cosmic acceleration that • does not conflict with quantum field calculations and local tests of Einstein’s

GR.

We constructed the DMASS sample that succesfully replicates the BOSS CMASS • spectroscopic sample. The sample is well suited for a joint analysis of the ex-

isting BAO and RSD measurements from BOSS and a galaxy-galaxy lensing

measurement from DES. In this work, we showed the galaxy bias of DMASS is

consistent with CMASS within 1σ.

We tailored our DES Y1 analysis for extended cosmological models, focusing • on modiﬁed gravity. DES Y1 shear measurement signiﬁcantly improves the

140 existing MG constraints. We found the resulting modiﬁed gravity constraints

are consistent with predictions from general relativity.

To test the constraining power of DMASS, we made a forecast on modiﬁed • gravity by following the basic methodology and techniques used for the DES

Y1 analysis for extended models. By adopting the measurement of galaxy-

galaxy lensing around DMASS and dropping the DES 3x2pt statistics, we were

able to demonstrate the capability of DMASS to achieve tighten constraints by

cancelling galaxy bias.

We have explored the information content of the three-dimensional galaxy cor- • relation function and power spectrum analytically. By adopting realistic scale

cuts used in literature, we studied the impact of imposing such cuts, both in

terms of the individual signal to noise for each statistic and the impact on the

correlation between the two statistics. By the combining the two estimators,

we found small improvement for joint constraints.

Looking forward, we are entering the era of precision cosmology. Upcoming galaxy surveys will scan vast areas of sky, generating an enormous volume of data. For example, LSST starting in 2022 will observe the entire southern sky ( 20, 000 deg2) ∼ and measure several billions of galaxy shapes. Increasing statistical power in the data

will enable us to tighten up either dark energy or modiﬁed gravity. However, this

also implies the future surveys will be rather systematic-error dominated. Therefore,

understanding systematics will play a major role in obtaining precise results. Another

eﬀort to make is developing tools for numerical simulations in the frame of dark

energy or modiﬁed gravity, in order to exploit nonlinear regimes. Along with the

141 ongoing eﬀorts in mitigating systematic errors and developing tools, future data from upcoming surveys will certainly bring us answers for the driver of cosmic acceleration.

142 Appendix A: The impact of redshift tails in BOSS CMASS on galaxy bias

The BOSS analyses use the CMASS galaxies only within the redshift range (0.43 < z < 0.75), by applying the spectroscopic redshift cuts on the CMASS targets selected by photometric cuts in Equations (2.1) - (2.7) (Chuang et al., 2017; Pellejero-Ibanez et al., 2017). Through the redshift cuts, nearly 10% of sources are discarded from the photometric targets. As the DMASS algorithm only utilizes the photometric information of galaxies, the resulting DMASS sample includes sources at the low end

(z < 0.43) or high end (z > 0.75), and they cannot be excluded as done in the

BOSS CMASS sample. To combine the BOSS measurements with the weak lensing measurements of DMASS, the eﬀect of the redshift tails on galaxy clustering should be examined. Here, we test the impact of the redshift tails on the galaxy clustering of BOSS CMASS, speciﬁcally on galaxy bias, by computing the correlation function monopole ξ0(r) and quadrupole ξ2(r).

We use the three dimensional, two point correlation function estimator given by

Landy & Szalay (1993):

DD(s, µ) 2DR(s, µ) + RR(s, µ) ξ(s, µ) = − (A.1) RR(s, µ)

143 where s is the separation of a pair of objects and µ is the cosine of the angle between

the directions between the line of sight (LOS) and the line connecting the pair of ob-

jects. DD, DR, and RR represent the normalized galaxy-galaxy, galaxy-random, and

random-random pair counts, for a given separation s and µ. The weights described

in Equation (2.27) are applied.

To derive the monopole and quadrupole, the two-point correlation function ξ(s, µ)

is integrated over a spherical shell with radius s:

N 1 µ ξ (s) = (2l + 1) ξ(s, µ ) P (µ ) (A.2) l N i l i µ i=0 X where Nµ is the number of µ bins, Pl(µ) is the Legendre Polynomial.

Figure A.1 shows the monopole and quadrupole of CMASS with the redshift cuts

(red) and without the redshift cuts (blue). These multipoles are computed in the scale

range 40h−1 Mpc < s < 180h−1 Mpc with the bin size 5h−1 Mpc, the same scales

and bin size adopted in the previous BOSS analyses (Chuang et al., 2017; Pellejero-

Ibanez et al., 2017). Error bars are computed from the MultiDark-PATCHY BOSS

DR12 mock catalogues (Kitaura et al., 2016). The amplitudes of the multipoles are

overall higher on large scales s > 120h−1 Mpc with the redshift tails. This may

indicate systematics associated with sources at high redshift, but their impact should

be negligible as the oﬀset between the two correlation functions is way smaller than

the statistical errors.

From the measured monopole and quadrupole of the correlation function, we con-

strain galaxy bias b and the structure growth rate, f d ln D/d ln a at a median ≡ redshift z = 0.59 for each case and compare the results. In linear theory, the relative

144 100 with z-cut 80 no z-cut

60 ) s

( 40 0 ξ 2 s 20

20 − 0.5

0.0

0.5 ) − s ( 2

sξ 1.0 −

1.5 −

2.0 − 40 60 80 100 120 140 160 180 s (Mpc/h)

Figure A.1: Monopole (top) and quadrupole (bottom) correlation function of the CMASS sample before (red) and after (blue) applying redshift cuts at 40h−1 Mpc < s < 180h−1 Mpc.

145 amplitudes of multipoles depends only on the combination of b and f as follows:

2 1 ξ (s) = b2 + bf + f 2 ξ (s) (A.3) 0 3 5 m 4 4 ξ (s) = bf + f 2 ξ (s) . (A.4) 2 3 7 m As the linear theory is applicable on the scales we are using, we simply adopt the above equations and expect the potential diﬀerence due to the redshift cuts to appear on the constraints of b and f. The matter correlation function ξm(s) at z = 0.59 is estimated by CAMB (Lewis et al., 2000) with the ﬁducial cosmology.

We perform a Markov Chain Monte-Carlo likelihood analysis using emcee (Foreman-

Mackey et al., 2013). The two parameters b and f are varied in the range of b = [0.5, 3] and f = [0.2, 1.0]. The likelihood is taken from χ2 deﬁned as

NX 2 −1 χ = [Xobs,i Xth,i] Cij [Xobs,j Xth,j] (A.5) i,j − − X where NX is the number of points in the data vector, Xth is the vector from the theoretical model, Xobs is the vector from the measurement. The data points from the multipoles are combined to form a vector X as

X = ξ0(s0), ξ0(s1), ..., ξ0(sN ); ξ2(s0), ξ2(s1), ..., ξ2(sN ) (A.6) { } where N is the number of bins in each multipole.

Figure A.2 shows the constraints of b and f at z = 0.59 on the two dimensional plane. The resulting numbers are b = 2.02+0.04 and f = 0.67 0.02 with the redshift −0.07 ± cut, and b = 2.00+0.04 and f = 0.68 0.02 without the redshift cut. From these −0.06 ± results, we do not ﬁnd a big discrepancy between the two cases.

The negligible impact of high redshift galaxies has also been studied by the original

BOSS analyses. Cuesta et al. (2016), Gil-Mar´ın et al. (2016a) and Gil-Mar´ın et al.

146 zcut no zcut

76 0.

72 0.

f 68 0.

64 0.

60 0. 84 92 00 08 16 60 64 68 72 76 1. 1. 2. 2. 2. 0. 0. 0. 0. 0. b f

Figure A.2: Comparison of galaxy bias constraints from the CMASS clustering before (blue) and after (red) applying z-cuts.

147 (2016b) use CMASS within 0.43 < z < 0.7 with the eﬀective redshift z = 0.57. Alam et al. (2015) compared the measurements from these analyses with the ones including higher redshift galaxies up to z < 0.75, with the eﬀective redshift z = 0.59 (Chuang et al., 2017; Pellejero-Ibanez et al., 2017). To compare the measurements directly at the same redshift, Alam et al. (2015) extrapolated the measurements of Chuang et al.

(2017) and Pellejero-Ibanez et al. (2017) at z = 0.57. A summary of their work can be found in Figures 13 and 15 in Alam et al. (2015). All of the BOSS measurements compared in this work show consistency within the 1σ level or better.

148 Appendix B: Diﬀerences between CMASS SGC and NGC

The BOSS CMASS target selection function is applied diﬀerently in the South

Galactic cap (SGC) and North Galactic cap (NGC) due to the color oﬀsets in the

DR8 photometry between two regions. Schlaﬂy et al. (2010) and Schlaﬂy & Finkbeiner

(2011) have estimated the level of color offsets and found that these differences are due to either calibration errors or errors in the galactic extinction corrections (or combination of both). This offset shifts the values of d⊥ (the combination of g r − and r i colors) by 0.0064 magnitudes between the North and South cap, resulting − in a few per cent difference in the number density and the amplitude of the angular correlation function. Ross et al. (2011) and Ross et al. (2012) have shown that the difference in the number density and the angular correlation function can be mitigated by applying the new cut with the offset, d⊥ > 0.5564, in the SGC. However, the final analyses of BOSS-III were completed with the same d⊥ cut in both regions. Therefore, we do not consider the color offset either.

The resulting DMASS is designed to be closer to CMASS in the SGC than NGC since the extreme deconvolution model is trained with the d⊥ color in the SGC.

Therefore, we report the measurements of the angular correlation functions and galaxy biases of CMASS NGC and SGC here in order to show that discrepancy between

DMASS and full CMASS originates from the intrinsic diﬀerence within CMASS.

149 2 2 Table B.1: χ /dof of three probes calculated between two diﬀerent samples. χsys/dof in the third column are calculated with systematic weights of DMASS obtained in Section 2.4. Values in the parentheses are corresponding PTE values. SGC and FULL in bold stand for CMASS in SGC and full CMASS.

2 2 PROBE χ /dof (PTE) χsys/dof (PTE) SGC - DMASS wδgδg 4.94/10 (90%) 2.58/10 (99%) wδgδWISE 9.04/10 (53%) 9.70/10 (47%) wδgκCMB 8.92/7 (26%) 13.25/7 (6%) FULL - DMASS wδgδg 10.61/10 (39%) 8.60/10 (57%) wδgδWISE 12.12/10 (28%) 11.42/10 (33%) wδgκCMB 7.13/7 (42%) 7.68/7 (36%) NGC - SGC wδgδg 14.53/10 (15%) − wδgδWISE 11.76/10 (30%) − wδgκCMB 23.95/7 (0.1%) −

150 SGC - DMASS FULL - DMASS NGC - SGC

16 08 00 08 16 0. 0. 0. 0. 0. − − ∆b

Figure B.1: Diﬀerence in galaxy biases constrained by the angular correlation func- +0.031 tion of CMASS SGC and CMASS NGC (red). ∆bNGC−SGC = 0.056−0.033. The blue- dashed and black-dot dashed histograms display ∆bSGC−DMASS and ∆bFULL−DMASS obtained in Section 2.5, respectively. The redshift bin bias ∆z of DMASS is marginalized for the latter two cases.

151 Table B.1 shows the values of χ2/dof and its corresponding PTE of all three probes computed in Section 2.5. The last column includes χ2/dof between CMASS in NGC and CMASS in SGC. For all three probes, χ2/dof of ‘NGC-SGC’ is either larger than any of the other two cases or comparable to the case of ‘FULL-DMASS’.

Galaxy biases are derived from the model of the angular correlation function given as

δgδg 0 0 0 0 w (θ) = dz b(z)n(z) dz b(z )n(z ) ξm(R, z, z ) (B.1) Z Z where b is galaxy bias, n(z) is normalized spectroscopic redshift distribution, ξm is matter clustering, and R is the comoving distance deﬁned as R = (1 + z)DA(z)θ.

With the covariance matrices calculated from the QPM mock catalogs in Section

2.4, we have estimated bestﬁt values of galaxy bias that minimize χ2 deﬁned in

Equation (4.49). The data vector ∆d in the equation is deﬁned as the residual of the measurement and theoretical prediction given as ∆d = dtrue d. The vector d − corresponds to the measurements of the angular correlation function of CMASS SGC and CMASS NGC computed in Section 2.5, and dtrue is the theoretical data vector from Equation (B.1).

The ﬁnal constraints of galaxy biases are bSGC = 2.035 0.026 and bNGC = 2.088 ± ± 0.017 from CMASS SGC and CMASS NGC, respectively. The derived galaxy bias

+0.031 diﬀerence between NGC and SGC is ∆bNGC−SGC = 0.056−0.033. Figure B.1 shows the constraint of ∆bNGC−SGC (red-solid) plotted with ∆bSGC−DMASS (blue-dashed) and

∆bFULL−DMASS (black-dot dashed) obtained in Section 2.5. The redshift bin bias ∆z of DMASS is marginalized for the latter two cases. The resulting ∆bNGC−SGC implies that the color oﬀset between the SGC and NGC naturally yields 2.6% of the ∼

152 diﬀerence in galaxy bias, and the constraints of ∆b between DMASS and CMASS are safely within this intrinsic diﬀerence.

153 Appendix C: Analytic way of marginalizing high k scales in band power ﬁsher matrix without inverting

In Equation (4.40), to keep information up to kn scales, k bins higher than kn should be cut out in the band power Fisher matrix. This slicing process involves double matrix inversion because marginalizing of information has to be done in the covariance matrix. Following these cuts, the matrix must be reverted to the Fisher information matrix for subsequent calculations. We use an analytic blockwise inversion method to avoid numerical errors from the inversions.

A matrix can be inverted blockwise by using the following analytic inversion formula :

−1 AB (A BD−1C)−1 (A BD−1C)−1BD−1 = (C.1) CD D−1C−(A BD−1C)−1 D−1 +−D−1−C(A BD−1C)−1BD−1 − − − where A, B, C, and D are matrix sub-blocks of arbitrary size. Let’s assume that the inverted matrix in the above equation is the covariance matrix corresponding to the full band power Fisher matrix as follows,

−1 0 0 −1 AB A B C ¯ = [F ¯] = = , (C.2) P P CD C0 D0 and we want to cut out blocks B0, C0, and D0. Then, the ﬁnal form of a Fisher matrix up to a limited k scale is

F = [A0]−1 = A BD−1C (C.3) − 154 where only D block is inverted (Note that D must be square so that it can be inverted).

If D block is a diagonal matrix having zero oﬀ-diagonal terms, the inversion procedure

is perfectly analytic and there should be no numerical error. If the sub-block D is only one number, the above equation is rewritten as

BC F = A (C.4) − d

The resulting Fisher matrix can be used to to marginalize the next smaller k bin information.

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