Galaxy Clustering for Combined Probes of Cosmology and Sterile Neutrino Constraints

GALAXY CLUSTERING FOR COMBINED PROBES OF COSMOLOGY AND STERILE NEUTRINO CONSTRAINTS

A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering

2018

Jack Elvin-Poole School of Physics and Astronomy Contents

Abstract 8

Declaration 9

Acknowledgements 11

1 Introduction 12 1.1 Cosmology ...... 13 1.1.1 Introduction to cosmology ...... 13 1.1.2 Structure growth ...... 20 1.1.3 Cosmic microwave background ...... 22 1.1.4 The large scale structure of the universe ...... 25 1.1.5 Weak gravitational lensing ...... 32 1.1.6 Other cosmological probes ...... 38 1.1.7 Concordance within the standard model ...... 39 1.1.8 Current and future surveys ...... 40 1.2 Neutrinos ...... 43 1.2.1 Neutrino Oscillations ...... 45 1.2.2 Impact of neutrinos on cosmology ...... 50 1.3 Science motivation ...... 53

2 A Combined View of Sterile-Neutrino Constraints from CMB and Neutrino Oscillation Measurements 55 2.1 Data sets ...... 57 2.1.1 Cosmological Data sets ...... 57

2 2.1.2 Oscillation Data sets ...... 60 2.2 Thermalisation of sterile neutrinos ...... 61 2.3 Results ...... 62 2.4 Conclusions ...... 65

3 Cosmology from combining galaxy clustering and weak lensing in the Dark Energy Survey Year 1 data 66 3.1 Motivation ...... 67 3.2 Galaxy samples ...... 68 3.3 Multi-probe Covariance ...... 70 3.4 Analysis choices and scale cuts ...... 72 3.4.1 Analysis choices ...... 72 3.4.2 Scale cuts ...... 72 3.5 DES Year 1 LSS cosmology results ...... 75 3.6 Conclusions ...... 78

4 Galaxy clustering for combined probes in the Dark Energy Survey Year 1 data 80 4.1 Introduction ...... 80 4.2 Theory ...... 82 4.3 Data ...... 85 4.3.1 Y1 Gold ...... 85 4.3.2 redMaGiC sample ...... 85 4.4 Analysis methods ...... 88 4.4.1 Clustering estimators ...... 88 4.4.2 Covariances ...... 90 4.5 Systematics ...... 91 4.5.1 Survey property (SP) maps ...... 91 4.5.2 Systematic corrections ...... 94 4.6 Results: Galaxy Bias and Stochasticity ...... 101 4.7 Demonstration of Robustness ...... 107 4.7.1 Selection of threshold ...... 107 4.7.2 Estimator bias ...... 108

3 4.7.3 False correlations ...... 110 4.7.4 Impact on covariance ...... 111 4.8 Conclusions ...... 113

5 Neutrino constraints from the Dark Energy Survey Year 1 data 115 5.1 Model ...... 115 5.2 Data ...... 116 5.3 Systematics ...... 117 5.4 Blinding ...... 119 5.5 Results ...... 120 5.5.1 Massless sterile neutrino model ...... 121 5.5.2 Massive sterile neutrino model ...... 124 5.5.3 Comparison to oscillation data ...... 126 5.6 Conclusions ...... 126

6 Conclusions 128

Bibliography 131

Word count 50414

4 List of Tables

1.1 Cosmological parameter constraints from Planck 2015 ...... 16

3.1 Table describing the priors in DES year 1 ...... 73

4.1 Details of the DES lens sample in each redshift bin ...... 87 4.2 Details of the ﬁducial parameters used for covariance and parameter constraints ...... 91 4.3 List of the maps used in the survey property weights ...... 95 4.4 The χ2 for each redshift bin and for all bins combined ...... 105 4.5 The measurements of galaxy bias bi is DES lenses ...... 105

5 List of Figures

1.1 Matter power spectrum ...... 21 1.2 Plank 2015 CMB temperature power spectrum ...... 23 1.3 Plank 2015 CMB polarization power spectrum ...... 24 1.4 Galaxy clustering in the 6dF Galaxy Survey ...... 30 1.5 BAO angular diameter distance measurements ...... 31 1.6 BAO cosmology measurements ...... 32 1.7 Cosmic shear measurements from DES Year 1 ...... 37 1.8 DES Year 1 cosmic shear cosmological constraints ...... 38 1.9 Tension between low-redshift LSS probes and the CMB ...... 41 1.10 Measurements of the Hubble constant ...... 42 1.11 Footprints of a number of galaxy and CMB surveys ...... 43 1.12 Neutrino mass ordering schematic ...... 47 1.13 Neutrino constraints from disappearance experiments ...... 49 1.14 Neutrino constraints from appearance experiments ...... 50

2.1 CMB power spectrum for diﬀerent neutrino scenarios ...... 57 2.2 Conversion between the oscillation and cosmological parameter spaces . 59 2.3 Neutrino parameter constraints in the oscillation and cosmological parameter spaces ...... 63

3.1 DES Year 1 redshift distributions ...... 70 3.2 ΛCDM constraints from clustering and weak lensing in DES Year 1 data 76 3.3 wCDM constraints from clustering and weak lensing in DES Year 1 data 77 3.4 ΛCDM constraints from DES and Planck ...... 78 3.5 wCDM constraints from DES and Planck ...... 79

6 4.1 DES lens sample angular distribution ...... 88 4.2 DES lens sample redshift distribution ...... 89 4.3 Maps of potential sources of systematics ...... 93 4.4 Correlations between DES lens density and survey properties ...... 95 4.5 Correlations between DES lens density and stellar density ...... 96 4.6 Examples of the impact of observational systematics correction on survey property correlations ...... 96 4.7 The signiﬁcance of each systematic correlation for diﬀerent correction thresholds ...... 98 4.8 Impact of weights on all one dimensional correlations ...... 100 4.9 Two-point auto correlation functions for DES lenses ...... 102 4.10 Two-point cross correlation functions for DES lenses ...... 104 4.11 Constraints on the ratio, r, of galaxy bias measured on w(θ) and galaxy- galaxy lensing ...... 106 4.12 Impact of systematics weight threshold on clustering constraints . . . . 108 4.13 Impact of systematics weight threshold on DES Y1 3x2pt constraints . 109 4.14 Impact of correction biases on clustering constraints ...... 111 4.15 The impact of systematic corrections on the clustering covariance . . . 112

5.1 DES Y1 3x2pt analysis on simulated data vectors contaminated with different systematics ...... 120 5.2 DES Y1 3x2pt+Planck+BAO+SN analysis on simulated data vectors contaminated with different systematics ...... 121 5.3 DES Y1 3x2pt constraints with simulated DES data compared to Planck P TT + lowTEB + lensing in the mν ∆Neff model ...... 122 5.4 DES Y1 3x2pt+Planck+BAO+SN constraints with simulated DES data P in the mν ∆Neff model ...... 123 5.5 Constraints from DES simulated data and Planck TT + lowTEB real P P data in the mν ΛCDM parameter space and the mν ∆Neff ΛCDM model ...... 124 5.6 DES Y1 3x2pt+Planck+BAO+SN constraints with simulated DES data

eﬀ in the msterile ∆Neﬀ model...... 125

7 The University of Manchester

Jack Elvin-Poole Doctor of Philosophy Galaxy clustering for combined probes of cosmology and sterile neutrino constraints September 12, 2018

This thesis describes a study into large scale structure cosmology with the dark energy survey and its use in sterile neutrino cosmology. Cosmological probes can be used to probe the nature of sterile neutrinos. Currently the most constraining data sets are from the cosmic microwave background (CMB) such as Planck. Low-redshift galaxy surveys that probe the large scale structure (LSS) can also be used to constrain neutrino properties but are currently limited by systematics. This thesis explores the CMB neutrino constraints, large scale structure systematics, and a combination of the large scale structure measurements and the CMB. I study the existing sterile neutrino constraints from the CMB and their comparison to oscillation experiments. I find that the CMB constraints from Planck are more constraining than many oscillation experiments. Planck is a complimentary probe of sterile neutrinos to the Main Injector Neutrino Oscillation Search (MINOS) experiment which is sensitive to low mass sterile neutrinos. However, the CMB constraints are model dependent and can be weakened by using a more complex model such as one with non-zero lepton asymmetry. I then study the clustering of galaxies in the Dark Energy Survey (DES) year 1 data, investigating the impact and mitigation of observational systematics. A number of correlations with observational properties are found and removed by applying weights to the galaxy sample. This method is tested for bias in parameter inference for both galaxy clustering alone, and in combination with DES weak lensing. I then present a simulated neutrino analysis combining simulated DES Y1 clustering and weak lensing signals with real data from the CMB and other cosmological probes. I use these measurements to constrain both massless and massive sterile neutrino models. I find that DES year 1 does not significantly add to the statistical power of a massless sterile neutrino analysis. However, the real data could tighten constraints by mitigating slight tensions between probes. The DES year 1 data, in combination with the CMB, baryon acoustic oscillations (BAO) and supernovae, are able to obtain constraints in a massive sterile neutrino model without the need for informative priors.

8 Declaration

No portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualiﬁcation of this or any other university or other institute of learning. Parts of this thesis have been published, or submitted for publication, or will be submitted for publication in the following journal articles

S. Bridle, J. Elvin-Poole, et al. A combined view of sterile-neutrino constraints from CMB and neutrino oscillation measurements. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 764, 2017 In this publication I executed the analysis, produced all ﬁgures apart from Fig 1a, and wrote most of the text. I am the corresponding author on this alpha- betical author list paper. Chapter 2 is based on my contribution to this publication.

DES Collaboration et al. Dark Energy Survey Year 1 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing. Phys. Rev. D, 98:043526, 2017 In this preprint I contributed the galaxy clustering data vector, contributed to the development of the analysis pipeline, and produced Figure 7. Much of Chapter 3 is based on this publication and the references therein.

J. Elvin-Poole et al. Dark Energy Survey Year 1 Results: Galaxy clustering for combined probes. Phys. Rev. D, 98:042006, 2018 In this preprint I executed the analysis, produced all ﬁgures, and wrote the text. Chapter 4 is based on this publication.

DES Collaboration et al. Dark Energy Survey Year 1 Results: Constraints on cosmological extensions from Galaxy Clustering and Weak Lensing. in prep, 2017 In this in-prep preprint I contributed the simulated systematics contaminated data vector, development of the neutrino analysis pipeline, executed the neutrino analysis and wrote the text for the neutrino section. The sections of Chapter 5 that relate to massless sterile neutrino models are based on this paper.

9 Copyright Statement

i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the “Copyright”) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes.

ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intellectual property (the “Intellectual Property”) and any reproductions of copyright works in the thesis, for example graphs and tables (“Reproductions”), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and com- mercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any relevant Thesis restriction declarations deposited in the University Library, The Univer- sity Library’s regulations (see http://www.manchester.ac.uk/library/aboutus/regulations) and in The University’s Policy on Presentation of Theses.

10 Acknowledgements

I thank my supervisor, Sarah Bridle, for all her help and advice. I thank Michael Troxel, Niall MacCrann, Joe Zuntz, Richard Rollins, Eleonora Di Valentino, Nicolas Tessore, Simon Samuroﬀ, Juan Pablo Cordero, Justin Evans, Susana Fernandez, Pawel Guzowski, Brían Ó Fearraigh, and Stefan Söldner-Rembold. I thank the DES working group coordinators Martin Crocce, Ashley Ross, Dragan Huterer, Elisabeth Krause, Scott Dodelson and Gary Bernstein. I thank Anthony Holloway and Robert Dickson from the JBCA IT department for their help with computing resources. I also thank Frances Poole, Martyn Elvin, Emily Elvin-Poole, Cerian Lindsay, Jesspreet Thethi, Lyra Thethi and all my friends and family. This PhD was funded by the Science and Technology Facilities Council. It uses data from the Dark Energy Survey.

This thesis was typeset using LATEX. Many ﬁgures were produced with chainconsumer [5].

11 Chapter 1

Introduction

This thesis is structured into the following chapters,

Introduction: The introduction contains a summary of large scale structure • cosmology, neutrino physics relevant to this work.

A Combined View of Sterile-Neutrino Constraints from CMB and • Neutrino Oscillation Measurements: A study into combining CMB sterile neutrino constraints with those from oscillation experiments. Much of this chapter was published in [1].

Cosmology from combining galaxy clustering and weak lensing in the • Dark Energy Survey Year 1 data: An additional introductory chapter sum- marising the Dark Energy Survey combined probes cosmology results from galaxy clustering and weak lensing.

Galaxy clustering for combined probes in the Dark Energy Survey • Year 1 data: A study into galaxy clustering systematics in the DES Y1 data. Much of this chapter was submitted for publication in [3].

Neutrino constraints from the Dark Energy Survey Year 1 data: A • study into sterile neutrino constraints from cosmology using the DES Y1 data and other cosmological probes. Much of this chapter is to be submitted for publication in [4].

Conclusions: Concluding remarks. • 12 1.1. COSMOLOGY 13

This thesis concerns the study of the large scale structure of the universe, and how it can be used to infer cosmological information on dark energy, structure growth and neutrino properties. This chapter will introduce the cosmology and neutrino physics used throughout this thesis. I will brieﬂy cover the Hot Big Bang model of the universe, structure formation, and the parameterisation of the existing concordance model. The theoretical framework and introduction to structure growth is discussed in Sec. 1.1.1 and 1.1.2. A number of major cosmological probes, including the CMB (Cosmic Microwave Background), galaxy clustering, and weak gravitational lensing, are discussed in Sec. 1.1.3, 1.1.4, 1.1.5, and 1.1.6. The methods and motivation behind combining cosmological probes are discussed in Sec. 1.1.7. A summary of signiﬁcant, current and future surveys is presented in Sec. 1.1.8. The background neutrino physics is discussed in Sec. 1.2. The latest results from oscillation experiments is discussed in Sec. 1.2.1 and the impact on cosmology is discussed in 1.2.2.

1.1 Cosmology

This section will introduce the basic concepts of cosmology that are being tested by observations. This section assumes general relativity and that the universe is homogeneous and isotropic on large scales unless stated otherwise. Much of this section is based on the introductory chapters of [6], [7] and [8].

1.1.1 Introduction to cosmology

The expansion of the universe (ﬁrst discovered by [9]) is locally described by Hubble’s law,

v = H0d (1.1) where v is the object’s radial velocity away from the observer, d is the distance between the object and the observer, and H0 is the Hubble parameter at present day, often 1 1 presented as H0 = h 100 kms− Mpc− . When discussing cosmology, it can be useful to remove this expansion and work in co-moving coordinates. In this coordinate system, the distance between objects does not change due to the Hubble expansion. Co-moving distances can be related to real distances by a scale factor, a in the relation d(x, t) = a(t)r, where d is the real distance 14 CHAPTER 1. INTRODUCTION

and r is the co-moving distance. a is often deﬁned so that a0 = a(t = 0) = 1. The dependence of a on time t, shows how the ‘size’ of universe changes with time. The rate of this expansion is given by the Hubble parameter H(t) =a/a ˙ (t). It is observed on large scales, that the universe is homogeneous and isotropic. From this, one can show (e.g. in [7]) that the metric of the isotropic universe must be the FRW (Friedmann-Robertson-Walker) metric which can be written as,

dr2 ds2 = c2dt2 a(t)2[ + r2(dθ2 + sin2 θdφ2)] (1.2) − 1 kr2 − where ds is a small interval of space-time, r, θ and φ are co-moving spacial coordinates, t is time, and k is the curvature of space-time, where k = 0 is a spatially flat universe, and k = +1 and k = 1 are positively (spherical) and negatively (hyperbolic) curved − universes respectively. A photon will travel on a null geodesic (ds = 0) and one R t cdt can therefore define a particle horizon, ∆r = 0 a(t) , which is the finite distance that particles can travel since the big bang (t = 0). One can also use the metric to introduce the concept of redshift. Much like a Doppler shift, the expansion of the universe induces a shift in the wavelength of light. From the metric in Equ. 1.2, one can define this redshift z to be νemit = (1+z) = 1 , νobs a(temit) where a(temit) is the scale factor at the time the light was emitted. For objects at large radial distances, the redshift z can be used as a measure of distance. By applying the general relativity field equations to the FRW metric (Equ. 1.2), one can show that the rate of expansion of the universe can be described by the Friedmann equations,

a˙ 2 8πGρ k Λ H(t)2 = = + (1.3) a(t) 3 − a(t)2 3 a¨ 4πG Λ = − (ρ + 3p) + (1.4) a(t) 3 3 where H(t) is the Hubble parameter, G is the gravitational constant, ρ is the total energy density (including contributions from matter and radiation), p is the total pressure, and Λ is the cosmological constant. Note that the notation c = 1 has been used. From Equ. 1.3, it is clear that in a ﬂat universe, where k = 0, the energy density 1.1. COSMOLOGY 15 can be written as, 3H(t)2 ρ = (1.5) crit 8πG where the cosmological constant term in Equ. 1.3 has been absorbed into the ﬁrst term as a vacuum energy density. ρcrit is known as the critical density and can be used to normalise ρ. This is shown in the density parameter,

ρ(t) Ω(t) = . (1.6) ρcrit(t) If Ω = 1 the universe is spatially ﬂat, if Ω > 1 the universe has positive curvature and if Ω < 1 the universe has negative curvature. In the standard cosmological model, Ω consists of contributions primarily from four diﬀerent components of the universe: baryonic matter, the name given by cosmologists to all the atomic matter in the universe; dark matter, matter with gravitational interactions but no electromagnetic interactions; radiation, all the photons and relativistic matter; and dark energy, the cause of the universe’s accelerated expansion. The contribution of each component to the total energy density can then be written as,

Ω(t) = Ω (t) + Ω (t) + Ω (t) = 1 Ω (1.7) m r Λ − k where Ωm,r,Λ,k correspond to the matter (baryonic and dark), radiation, cosmological constant and curvature energy density respectively. These parameters can be measured by cosmological probes and are used to describe the contents of the universe. In order to solve Equations 1.3 and 1.4, a relationship between the energy density and pressure (an equation of state) for each component is required. This is given by the equation of state, p = wρ. (1.8) w is known as the equation of state parameter and it is equal to 0 for matter (non- relativistic,) and 1/3 for radiation and relativistic matter. Any component with w < 1/3 is known as dark energy as this will induce an accelerating scale factor in Equ. − 1.4. The cosmological constant, Λ has w = 1. The value of w determines how the − density of the component evolves with the scale factor (and therefore also with time) by the equation

3(1+w) ρ a(t)− (1.9) ∝ 16 CHAPTER 1. INTRODUCTION

4 Therefore, radiation density decreases as ρ a− , matter density decreases as ρ r ∝ m ∝ 3 a− , and ρΛ remains constant.

ΛCDM

The standard cosmological model is known as the ΛCDM model. The ΛCDM model consists of a flat universe (Ωk = 0) which started with the big bang, and continues to expand today. In this model, the universe contains cold dark matter (CDM), and dark energy in the form of a cosmological constant (Λ) in addition to baryonic matter and radiation. ΛCDM can be parametrised using 6 parameters. One typical parametrisation is: Ωm0, the present day total matter density; Ωb0, the present day baryonic matter density; H0, the present value of the Hubble parameter; ns, the spectral index of the primordial power spectrum (see Sec. 1.1.2); τ, the optical depth to re-ionisation (see Sec. 1.1.3); and σ8, the present rms amplitude of matter fluctuations 1 in spheres of radius 8h− Mpc. Many modern cosmological probes aim to constrain these parameters within the ΛCDM model, and to find discordance between this model and the data in order to search for new physics. An example of some one dimensional constraints of these parameters from the Planck experiment [10] is shown in Table 1.1. Constraints from Planck 2015 results TT,TE,EE+lowP 68% limits

Ωb 0.02225 0.00016 τ 0.079 ±0.017 n 0.9645± 0.0049 s ± H0 67.27 0.66 Ω 0.6844± 0.0091 Λ ± Ωm 0.3156 0.0091 σ 0.831 ±0.013 8 ± Table 1.1: Cosmological parameter constraints from Planck 2015 release on the main six ΛCDM parameters and ΩΛ. Constraints are from TT, TE, and EE power spectra (see Sec. 1.1.3).

Dark matter

Dark matter is the component of the universe which does not interact electromagnet- ically, but can still be detected due to its gravitational interactions. The ﬁrst evidence for dark matter was found by Fritz Zwicky [11], who found that the velocity dispersion of galaxies in the Coma Cluster implied a higher overall mass 1.1. COSMOLOGY 17 than the cluster’s luminosity suggested. Additional evidence came from measurements of galaxy rotation curves in the 1970’s [12] [13]. The rotation curve, a measure of stellar (or Hi gas) velocity against distance from the galactic centre, of the galaxy M31 was found to be approximately ﬂat for large radii, R. In the absence of dark matter, the radial velocity would be proportional to 1/√R. This can be explained by a halo of dark matter at large R. This observation has since been found for many other galaxies [14]. Modern cosmological probes (see Sec. 1.1.3, 1.1.4, 1.1.5 and 1.1.6) have shown that this dark matter component is mostly cold dark matter (CDM), meaning it consists of heavy particles, traveling at non-relativistic speeds (e.g. see [15]). This cold dark matter has an equation of state parameter w = 0. This rules out low-mass, active neutrinos as the primary cold dark matter particle candidate, although they do have some impact on structure formation as a hot dark matter (HDM) component (see Sec. 1.2.2). Gravitational microlensing has been used to show that the additional mass cannot be baryonic (e.g. massive compact halo objects (MACHOS), [16]), so the CDM must be an undiscovered particle. Candidates for this particle include supersymetric particles [17] or massive sterile neutrinos [18] [19].

Dark energy

Dark energy was first suggested by Einstein as a cosmological constant, Λ, in the GR field equations in order to produce a static universe. After Hubble’s work on recession velocities [9], Λ went out of favour as a theory until the 1990s when [20] and [21] discovered from observations of type 1a supernovae, that the expansion rate of the universe was accelerating (see Sec. 1.1.6). Various theories have been suggested for the cause of this acceleration, the simplest of which is Einstein’s cosmological constant Λ, which can be interpreted as a constant scalar field with an equation of state parameter w = 1, unchanging in time or space. These observations have since been updated − with larger numbers of supernovae [22] (see Sec. 1.1.6). CMB experiments such as Planck [10] and WMAP [23], have measured a very flat universe Ω 0 with a matter density parameter Ω 0.3 and a negligible radiation k ≈ m ≈ component at present time. From Equ. 1.7, this results in Ω 0.7, where dark Λ ≈ 18 CHAPTER 1. INTRODUCTION energy is the dominant component at present time. Some alternative dark energy theories to the ΛCDM model have been suggested. One of the simplest is a model that allows w = 1. Note from Equ. 1.4 that any 6 component with w < 1 will produce an accelerating universe, a¨ > 0. This can − 3 also be extended to allow w to be redshift dependent, w(a) = w0 + wa(1 + a). No cosmological probe has yet found a strong detection of w = 0 or w = 1. a 6 0 6 Another alternative is that the acceleration is not caused by dark energy, but is a result of new gravitational physics. The alternative theories of gravity are known as modified gravity. Many modified gravity theories exist (see [24, 25, 26]). One of the simplest modifications is to consider the two potentials Ψ and Φ in the metric,

h 2i ds2 = a(t) (1 + 2Ψ)dτ 2 a(t)(1 2Φ)dx~ (1.10) − − − where dx~ is the co-moving spacial vector. In GR Φ = Ψ, and so making Φ = Ψ would 6 allow a different metric, and therefore a different Friedman equation. This modification to the potentials is often parameterised as,

k2Ψ = 4πGa2(1 + µ(a, k))ρδ , (1.11) − k2(Ψ + Φ) = 8πGa2(1 + Σ(a, k))ρδ . (1.12) − where µ and Σ are the new parameters to be constrained by observations. Any modified gravity theory must be consistent with the early time DM and radiation dominated universes. Therefore any changes in the Friedmann equation must only apply at late times, [27]. This makes late time cosmological probes such as galaxy clustering (Sec. 1.1.4) and weak gravitational lensing (Sec. 1.1.5), powerful modified gravity and dark energy probes, [28]. Another possible cause of the acceleration would be if the assumption that the universe is homogeneous on large scale were not true. This could cause unexpected changes to the redshift-distance relation, matching the results from supernovae experiments, [27, 29]. At the present time, the majority of cosmological probes are consistent with ΛCDM and alternative acceleration theories are not accepted by mainstream cosmology, [27, 30]. However, some recent large scale structure surveys have found some preference for 1.1. COSMOLOGY 19 models other than ΛCDM. For example [31] and [32] using Baryon Acoustic Oscillation measurements (see Sec. 1.1.4). One issue with the cosmological constant, is that when interpreted as a vacuum energy, cosmological data give a value for the vacuum energy many times lower than the quantum prediction. This is known as the cosmological constant problem [33] and is still present in many modified gravity theories.

Inﬂation

A well established part of the standard cosmological model is inﬂation [34, 35, 36]. The following section will give a brief, qualitative overview of this theory. Inﬂation is a period of rapid exponential expansion in the early universe (t <

32 10− s), mathematically similar to late time dark energy. This theory could solve a number of cosmological problems. One such problem is the horizon problem. It appears that the universe is homogeneous and isotropic on large scales. This means if we observe the distant universe in one direction (e.g. by observing the CMB temperature anisotropies), it looks statistically similar to the universe in the opposite direction. However, the horizon distance at these positions is not large enough for them to be in causal contact. It is then not clear how they can be in equilibrium. Inflation solves this problem by allowing these points to be in causal contact in the very early universe, then move out of each other’s horizon distance during the exponential expansion. Another problem solved by inflation is the flatness problem. The curvature density parameter Ωk has been constrained by CMB experiments to be very close to 0. Planck +0.0040 CMB + external data sets in 2015 find this to be Ωk = 0.0008 0.0039 [10]. However, if − the value of Ωk was close to, but not exactly 0, it’s value would exponentially diverge from 0 over time. The flatness problem then states that the initial Ωk must either be exactly 0, or incredibly close to 0, in order to be consistent with current observations. There is no theoretical reason for either to be true. Inflation solves this problem by allowing our observable universe to be a small part of a much larger, curved universe, so that space-time appears to be locally flat. Inflation would cause gravitational waves in the early universe to imprint B-modes into the polarization of the CMB (see Sec. 1.1.3). If these primordial B-modes were 20 CHAPTER 1. INTRODUCTION to be detected, it would provide evidence for cosmic inflation. In order the detect primordial B-modes, the B-modes from CMB lensing and foreground dust must be subtracted from the signal. A number of CMB experiments are currently in search of this B-mode detection such as BICEP [37], the South Pole Telescope (SPT) [38] and POLARBEAR [39].

1.1.2 Structure growth

Initial perturbations in the matter density field, caused by quantum fluctuations in the early universe, grow due to gravitational collapse. This growth of structure is damp- ened by the expansion of the universe. Studying this structure growth can therefore provide much cosmological information [40, 7]. Density fluctuations δ about the mean matter density ρ¯ are defined as,

ρ(~x) =ρ ¯(1 + δ(~x)) (1.13) where ρ is the matter density at a point in space ~x For density perturbations of δ << 1, the structure formation can be treated with | | linear approximations. Larger density perturbations behave non-linearly and must be studied with N-body simulations. The growth of δ is described by the Poisson equation,

2Φ = 4πGa2ρδ¯ (1.14) ∇ where Φ is the gravitational potential. This can be solved to ﬁnd the linear growth factor D where δ(a) D(a)δ . ∝ 0

Matter power spectrum

The matter power spectrum can be used to describe fluctuations in the matter density field in Fourier space. It is defined as,

P (k) = δ 2 (1.15) h| k| i where δk is the Fourier transform of the real-space density ﬂuctuations. The primordial

ns matter power spectrum is given by a power law, P (k) = Ask [7]. As is the power spectrum normalization and ns is the spectral index, predicted to be very close to 1. 1.1. COSMOLOGY 21

The following is a qualitative description of the evolution of the matter power spectrum. During the radiation dominated era, the baryons and photons are tightly coupled in a plasma. This plasma is influenced by gravity, an attractive force, but also by photon pressure, a repulsive force. This causes the fluid to oscillate in what is known as acoustic oscillations. Therefore, during this period, small scale density perturbations did not grow. However, perturbations larger than the horizon scale do grow, as the gravitational collapse happens before the photon pressure has time to travel across the overdensity. Therefore the large scale power (small k) increases while the small scale power remains constant. This induces a peak in the matter power spectrum (see Figure 1.1). Over time, the distance that can be traveled by the radiation pressure (the Jeans length) increases and the oscillations happen on larger scales. This causes the peak to shift to larger scales. After matter-radiation equality, perturbations on all scales grow at the same rate and the shape of the matter-power spectrum is fixed.

The linear matter power spectrum can be calculated numerically by solving what are known as the Boltzmann equations (e.g. [41]). A number of Boltzmann codes are available to do this, e.g. CAMB [42] or CLASS [43].

Figure 1.1: Linear and non-linear matter power spectra. Generated with the CAMB module in CosmoSIS [44] using the Planck cosmology from Table 1.1. The non-linear power spectrum uses the Takahashi haloﬁt model [45]. The turnover of this matter +0.0041 power power spectrum is consistent with the measured value of k = 0.0160 0.0035 from the WiggleZ Survey [46] − 22 CHAPTER 1. INTRODUCTION

Because of the non-linear growth of structure on small scales, the linear matter power spectrum is not suﬃcient at high k. A non-linear correction must be made by measuring the power spectrum of N-body simulations and calculating a cosmology dependent transfer function to produce the non-linear matter power spectrum from the linear one [47, 45]. An example of a linear and non-linear matter power spectrum can be seen in Figure 1.1. The matter power spectrum can be measured with late-time cosmological probes (Sec. 1.1.4 and 1.1.5) which can in-turn be used to constrain cosmological parameters.

Most notably Ωm, and σ8 which can be considered to be the normalization of the power spectrum. The matter power spectrum can also be used the constrain any hot or warm dark matter content of the universe. This is because the light WDM particles are able the free-stream compared to the CDM, smoothing out ﬂuctuations on scales smaller than the particles horizon distance. This suppresses power in P (k) at large k. This method has been used to constrain the masses of WDM particles including active neutrinos, [48, 49], and sterile neutrinos, [50, 51]. See Sec. 1.2.2 for further details.

1.1.3 Cosmic microwave background

One of the most powerful cosmological probes has been the cosmic microwave background (CMB) measurements. From the CMB, cosmological information about the very early universe can be obtained. In this section, the science behind the CMB observations is discussed and a summary of the cosmological constraints from diﬀerent CMB experiments is presented.

Introduction to the CMB

In the early universe (t < 105 years), baryons and photons are coupled via Thompson scattering in a baryon-photon plasma. This plasma oscillates via the acoustic oscillations discussed in Sec. 1.1.2. As the universe expands, the density decreases such that the baryons and photons become decoupled and the universe becomes transparent to the photons. This process is known as recombination. The radiation from this decoupling can now be seen as a black-body spectrum with a temperature of T0 = 2.725K. The rarefactions and compressions of the oscillations are left imprinted on the CMB 1.1. COSMOLOGY 23 as small temperature anisotropies. These anisotropies can be measured by CMB experiments and a power spectrum of the CMB temperature can be calculated,

D∆T ∆T E X (2l + 1) (~x) (x~ ) = C P (~x x~ ) (1.16) T¯ T¯ 0 4π l l · 0 l=0

where Pl are the Legendre polynomials, T is the temperature ﬁeld, ~x is a position on the temperature map, and Cl is the power spectrum. An example CMB TT power spectrum measured by the Planck experiment [10] can be seen in Figure 1.2.

Figure 1.2: CMB temperature power spectrum measured by Planck, 2015 data release [10].

The peaks of this spectrum correspond to the compressions and rarefactions of the oscillations. The damping of the peaks at l > 1000 is known as Silk damping [52] and is caused by the diﬀusion of photons. The locations and relative amplitudes of these peaks and troughs can be used to constrain cosmological parameters. Two parameters of particular importance to late-time cosmology, that can be constrained by the CMB

TT power spectrum, are Ωm and As. By measuring the amplitude of ﬂuctuations at recombination, one can predict the amplitude of matter density ﬂuctuations today

σ8(z = 0) and compare to late time measurements. Cosmology can also be constrained by the polarization of the CMB. This can be used to create an E-mode polarization power spectrum (EE) and the cross correlation between the temperature and the polarization (TE). These power spectra are shown in Figure 1.3. 24 CHAPTER 1. INTRODUCTION

Figure 1.3: CMB E-mode polarization power spectrum and the temperature - polarization cross power spectrum from the Planck 2015 data release [10].

As photons from the CMB travel to the observer, their paths will be bent by the gravitational field of the large scale structure of the universe. This causes a lensing of the temperature and polarization fields of the CMB. In the polarization field, this can be seen as a B-mode pattern. It is predicted, in the absence of primordial gravitational waves, gravitational lensing and foreground dust, that the CMB polarization would have a purely E-mode pattern. Measurements of B-mode polarization can therefore be used to constrain cosmology, either by removing the lensing signal and searching for the primordial B-modes (Sec. 1.1.1), or by using the CMB lensing signal to probe the large scale structure density field [53] The CMB power spectra can also be used to provide constraints on cosmological neutrinos. The tightest constraints on neutrinos from a single cosmological probes are currently from CMB experiments [54, 10, 55] (see Sec. 1.2.2). 1.1. COSMOLOGY 25

CMB surveys

The ﬁrst CMB experiment to measure the temperature anisotropies was the Cosmic Background Explorer (COBE) [56]. COBE was a satellite telescope that imaged the full sky at an angular resolution of 7◦. It was able to show the CMB frequency spectrum to be a black-body spectrum and measure the slight temperature anisotropies.

The next full sky CMB survey was WMAP [23]. WMAP had an improved angular resolution of 1300 and probed the TT power spectrum up to l = 1000 as well as providing all sky polarization maps. WMAP provided cosmological parameter constraints within the ΛCDM model, and constraining the curvature to 0.0133 < Ω < 0.0084. − k The most recent full sky CMB survey is Planck [57, 10]. Planck has an angular resolution as low as 500 in its high frequency instrument, and measured the TT power spectrum at l > 2000. Planck was able to further constrain cosmological parameters and found no signiﬁcant deviation from the ΛCDM model. The parameter constraints from the Planck TT power spectrum and low-l polarization are shown in Table 1.1.

A number of CMB experiments over smaller regions of the sky have also taken place. Although these are not able to constrain the low l modes of the power spectrum, but they can image at higher resolution and can focus on measuring polarization (e.g. searching for B-modes, [37]) or detecting galaxy clusters with the Sunyaev-Zeldovich´ eﬀect. These experiments can also provide high quality CMB data for cross-correlating with other probes (See Sec. 1.1.7). Some examples of these CMB experiments are BICEP-2, [37], ACT-Pol, [58], POLARBEAR, [39], and SPT, [38].

1.1.4 The large scale structure of the universe

Low-redshﬁt large scale structure (LSS) surveys can constrain late-time cosmology and the dark energy dominated era. Because the dark matter density ﬁeld cannot be directly observed, these surveys must either measure a tracer of the dark matter (e.g. galaxies, Hi gas) or use an indirect method (e.g. Weak lensing, Sec. 1.1.5). LSS surveys can provide measurements of the matter power spectrum Pδδ(k) and can use this to constrain cosmological parameters. 26 CHAPTER 1. INTRODUCTION

Galaxy Clustering

Galaxies can be used as a tracer of the dark matter ﬁeld. It is assumed that regions where there is a high density of matter ρm, there will also be an high-density of galaxies

ρg. The galaxy number overdensity δg can be related to the matter overdensity δm, to linear order, by a bias factor b,

δg = bδm (1.17)

Here, b is the linear galaxy bias. Expanding out to higher powers of δ, one can also include non-linear bias factors. For example,

2 δg = b1δm + b2δm. (1.18)

Other more complex expansions of galaxy bias can be written, e.g. [59, 60]. Assuming linear galaxy bias, the power spectrum of the galaxy ﬁeld is then simply related to the matter power spectrum by,

P (k) b2P (k) (1.19) gg ≈ δδ In order to measure the galaxy distribution, a location on the sky (RA and DEC angular co-ordinates) and a redshift z must be measured for each galaxy in a particular sample. Due to the difficulties in measuring redshifts in large volume surveys, a galaxy sample is often split into broad redshift bins and the 2D projected angular clustering is measured. In the real space this is often calculated as w(θ), the excess probability of finding two galaxies separated by an angle θ compared to a random catalog of points with the same angular selection function. In spherical harmonic space, the clustering is described by a 2D spherical harmonic power spectrum Cgg(l). The 2D galaxy density field can de decomposed into spherical harmonics,

+l X X δg(~n) = almYlm(~n) (1.20) l=0 m= l −

where Ylm(~n) is the spherical harmonic. The covariance of the coeﬃcients alm forms the power spectrum [61], 1.1. COSMOLOGY 27

a a 0 0 = δ 0 δ 0 C (l). (1.21) h lm l m i ll mm gg The theoretical prediction of this power spectrum can be obtained by projecting the galaxy power spectrum Pgg(k) into the 2D spherical harmonic power spectrum,

1 Z C (l) = 4πk2dkP (k)Ψ2(k) (1.22) gg 2π2 gg l where,

Z 2 Ψl (k) = dzn(z)D(z)Jl(kr(z)) (1.23)

where Jl is the Bessel function of order l, and D(z) is the growth factor [61]. Evaluating Equ. 1.22 is computationally expensive and can be simpliﬁed by making use of the limber approximation [62], where the Bessel functions go to delta functions, giving,

Z n(χ) 2 C (l) = dχ P (l/χ) (1.24) gg χ gg where χ is the co-moving radial distance, and n(χ) is the normalized number of galaxies at χ. The limber approximation is commonly used in both weak lensing and galaxy clustering studies and is valid for l 1 [63]. The Cgg(l) power spectrum can then be transformed in to real space by summing over Legendre polynomials,

X 2l + 1 w(θ) = P (cos θ)C (l) (1.25) 4π l gg l=0 1 where Pl is the legendre polynomial of order l . Alternatively, the clustering signal can be calculated by Fourier transforming into real space by a Hankel transform,

Z ldl w(θ) = C (l)J (lθ) (1.26) 2π gg 0 where J is the 0th order Bessel function. Equ. 1.26 assumes l 1. 0 1This makes use of the spherical harmonic addition theorem P (ˆn nˆ) = 4π Pl Y (ˆn)Y ∗ (ˆn) l · 2l+1 m=−1 l,m l,m [61] 28 CHAPTER 1. INTRODUCTION

Using relations 1.22 to 1.26, galaxy clustering can be used to constrain P (k) and the cosmological parameters and model that deﬁne it [61]. However, from Equ. 1.19, one can see that the amplitude of the galaxy clustering signal will be degenerate with galaxy bias b, limiting the cosmological information that can be obtained from the galaxy clustering signal alone.

Redshift estimates

The redshift z can be obtained in two ways. The most precise is spectroscopic redshifts, which are obtained by measuring the wavelength of the emission lines in the observed spectra and calculating their shift from lab values. As this method requires measuring the spectrum at high resolution, it takes a significant amount of time to measure each z. In order to measure z quickly, for use in very large galaxy surveys, photometric redshifts (photo-z) are used. This involves measuring the flux from the galaxy in multiple wavelength bands. The redshift can then determined by comparing the colours of the galaxy (e.g. magnitude in band 1 - magnitude in band 2) with the colours from galaxy spectra templates. Photo-z estimates are much less accurate than the spectroscopic redshifts and more prone to systematic effects, but quicker and allow the volume of the survey to be much larger than spectroscopic surveys. Machine learning methods can also be trained on spectroscopic redshifts to make photo-z estimates. These methods require the spectroscopic sample to be representative of the photometric sample in colour space [64].

Baryonic Acoustic Oscillations

In addition to measuring and modelling the full w(θ) signal, LSS galaxy clustering measurements can also be used to detect Baryonic Acoustic Oscillations (BAO). At recombination, the acoustic oscillations seen in the CMB will be imprinted of the baryon density field. As structures form, this results in an excess number of galaxies separated by distances corresponding to the acoustic peaks. The location of the first acoustic peak in the galaxy density field will appear as a bump in the galaxy clustering

1 signal at 100h− Mpc. This can be seen either in the projected 2D w(θ) or in the 3D ∼ clustering signal ξgg(r). When measured in the galaxy distribution, this can provide 1.1. COSMOLOGY 29 a distance measure or standard ruler. By measuring this distance as a function of of redshift one can probe the late-time expansion history of the universe, and therefore constrain dark energy parameters [65, 66]. The BAO signal is imprinted on the galaxy clustering as a distance along the line of sight of Hrs/c and a distance across the sky of rs/DA(z), were rs is the sound horizon and DA is the angular diameter distance. The location of the BAO peak can then be fitted to the data by varying the location of this peak in the modeling of the the galaxy clustering functions. See Figure 1.4 for an example of this fit applied to data from the 6dF galaxy survey [67]. BAO measurements are often made by constraining the angular position of the BAO peak compared to a fiducial cosmology,

[D (z)/r ] α = A s obs (1.27) [DA(z)/rs]ﬁducial where α is the parameter to be constrained, expected to equal 1. A number of nuisance parameters are often introduced to capture other features of the clustering signal rather than marginalising over the cosmological input parameters [68]. This produces a more model-independent BAO measurement. BAO detections have been carried out by a number of spectroscopic surveys including SDSS with luminous red galaxies [69], SDSS III-BOSS [70], SDSS main galaxy sample [71], BOSS [72], wiggleZ [73, 74], eBOSS quasars [75], 2dF galaxy redshift survey [76], 6dF galaxy survey [67] and the Lyman-α forest [77, 78]. A comparison of some of these BAO measurements is shown in Figure 1.5 Although the increased uncertainty in the z in photometric surveys will dilute the BAO signal, it is still detectable in modern photometric surveys. A photometric BAO detection was made with the Dark Energy Survey year 1 data in [79, 80, 81].

BAO measurements are able to break the degeneracy between H0 and w in CMB constraints (e.g. [67, 82]). See Figure 1.6 for an illustration using WMAP CMB data and BOSS BAO data.

Redshift space distortions

The peculiar velocities of the galaxies will cause spurious eﬀects to their redshift distribution due to Doppler shifts. These are called Redshift Space Distortions (RSD). 30 CHAPTER 1. INTRODUCTION

Figure 1.4: 3D galaxy correlation function as measured in the 6dF Galaxy Survey, from [67]. measurement made at an effective redshift of z = 0.106. The significance of the BAO detection is found by comparing to a clustering signal with no BAO feature (no-baryon fit).

They can be caused by either random velocities of galaxies within each cluster, or by the gravitational collapse of the cluster on large scales. These distortions can either be thought of as an eﬀect that needs to be modeled for galaxy clustering measurements, or can be used as a cosmological probe itself, directly measuring the growth of structure [83, 84].

Measuring the LSS clustering

The 2D angular 2-point correlation function, w(θ), can be measured on real data by using the Landy & Szalay estimator [85],

DD 2DR + RR wˆ(θ) = − , (1.28) RR where DD, RR and DR are the number of pairs of galaxies from a real galaxy sample D and a random catalog R with the same angular selection (mask) as the data. The random catalog R should have a density many times greater than the density of D to reduce noise in the measured signal. The number of RR pairs should then be appropriately normalised. 1.1. COSMOLOGY 31

Figure 1.5: Measurements of the angular diameter distance from various BAO measurements compared to the Planck best ﬁt cosmology prediction in ΛCDM. Figure from [79], data from [79, 67, 71, 72, 74, 75, 77, 78].

Any quantity that impacts the selection of galaxies that is not accounted by the angular mask, can lead to a systematic bias the observed clustering signal. This can come from the observing conditions (such as PSF size or extinction), or from contamination from stars. In both of these cases, the systematic can only lead to an increase in the observed clustering signal. The angular correlation function can be corrected for any systematic dependencies by subtracting the auto and cross correlations between maps of the systematic variations and the galaxy catalog [86] or by weighting each galaxy by the inverse of the measured contamination. This is the subject of Chapter 4 of this thesis.

The errors on the correlation function (which propagate through to the parameter constraints) can be calculated either directly from the data using jackknife errors, by measuring w(θ) from simulated data, or analytically using the halo model, where the large scale structure consists of a number of dark matter halos [87]. See Sec 3.3 for details. 32 CHAPTER 1. INTRODUCTION

Figure 1.6: 2D parameter constraints on w and Ωm from WMAP CMB data + combinations of of BOSS 2pt functions. Figure from [82].

1.1.5 Weak gravitational lensing

Another method for measuring the density field is weak gravitational lensing. The gravitational field of the matter density distribution deflects light from distant sources. This will produce a coherent distortion in the image of the background sources. In the regime of weak lensing, this presents itself as a 1% shear of a source galaxy’s ∼ ellipticity. Measurements of the ellipticity of distant galaxies can therefore be used to probe the matter density distribution between the source and the observer. However, because the effect is small, the signal must the averaged over many galaxies. For this reason, weak lensing studies are often performed on photometric galaxy surveys, calculating 2D weak lensing statistics in tomographic bins (redshift bins).

Because weak lensing does not use a tracer of the the dark matter, but directly probes the dark matter distribution, it does not depend of the unknown galaxy bias. In this section, the methods of weak lensing analyses will be introduced and the results from some recent weak lensing surveys will be reviewed. Much of the theoretical framework is base on the review articles [40, 88]. 1.1. COSMOLOGY 33

Gravitational lensing

As a photon travels through a gravitational potential Φ, (in GR) the ray of light is deﬂected by an angle,

2 Z αˆ = p Φdr (1.29) − c ∇⊥ where r is the the path of the light ray in proper co-ordinates.

Equ. 1.29 can be used to derive the 2D lensing (convergence) power spectrum Cκ from the matter power spectrum,

Z χlim q2(χ) l + 1 2 (1.30) Cκ(l) = dχ 2 Pmm k = , χ 0 χ χ where χ is the co-moving distance from the observer and q(χ) is the lensing eﬃciency,

2 Z χlim 3 H0 χ χ0 χ q(χ) = Ωm dχ0n(χ0) − (1.31) 2 c aχ χ χ0 where n(χ) is the normalised distribution of galaxies in co-moving coordinates (derived from the observed redshift distribution of the galaxies). This distribution determines the redshift range that the weak lensing signal is sensitive to (the lensing kernel of the sample). In a tomographic analysis, q2(χ) can be replaced with q(i)(χ)q(j)(χ) in Equ. 1.30, where i and j label redshift bins, to obtain the cross-bin convergence power spectrum

ij Cκ (l). This can then be converted into a two point correlation function,

Z ij 1 ij ξ (θ) = dllJ0/4(lθ)Cκ (l) (1.32) ± 2π where J0/4 are the 0th and 4th order Bessel functions. Equ. 1.32 is equivalent to Equ. 1.26 for galaxy clustering. When measuring on the data, the two-point shear-shear correlation functions are deﬁned as,

ξ (θ) = γtγt (θ) γ γ (1.33) ± h i ± h × ×i 34 CHAPTER 1. INTRODUCTION

where the average is over all pairs of galaxies in angular bin θ, and γt, are the × tangential and cross-component of the galaxy shear relative to the line separating the galaxies in question. The ξ can therefore be used to constrain the matter power spectrum. Cosmic ± shear is particularly sensitive to Ωm and σ8 through the convergence power spectrum, Equ. 1.30. By splitting the galaxy sample into tomographic bins, cosmic shear can be used to measure the growth of structure and dark energy parameters.

Shear measurement

The first step in measuring weak lensing is to measure the shapes of the source galaxies. This must be done to very high precision as the the expected shear signal is of order 1% the intrinsic ellipticity of the galaxies. The galaxy shapes can be measured by a number of methods. These can either measure the shape of a stacked image [89] or multiple epochs, simultaneously fitting to a number of images of the same galaxy. The measurement itself can be done either by fitting a model galaxy template to the image [90, 91, 92] or by splitting the pixel data into moments [93]. The performance of different shear measurement codes have been presented and compared in measurement challenges such as STEP [94, 95] and the GREAT challenges [96, 97, 98, 99]. A number of systematics can bias shape measurements. These can be described by,

e = (1 + m)g + αe + c (1.34) h i true PSF where e is the mean measured shear, m is the multiplicative error, e is the PSF h i PSF ellipticity, α is the leakage of the PSF shape into the galaxy shape measurement and c is the additional additive error. The biases m, α, and c can be caused by model bias, where the underlying galaxy model does not ﬁt the true galaxy shape [100]. This can be caused by irregularly shaped galaxies or spiral arms. Another is leakage of the PSF ellipticity into the shape measurement (α from Equ. 1.34). Other sources of error include noise bias [101], missing pixels, and contamination from nearby galaxies [102]. These can be measured and calibrated by testing the shape measurement method on realistic image simulations (e.g. [103]) or by using self-calibrating methods [104, 105]. 1.1. COSMOLOGY 35

The residual multiplicative bias m can then be modeled by,

ij i j ij ξ ,obs = (1 + m )(1 + m )ξ ,true. (1.35) ± ±

Weak lensing systematics

In addition to the shear measurement bias, a number of other systematics can effect cosmic-shear measurements. One major systematic is the photo-z error. When the weak lensing source galaxies are split into tomographic bins i, each with a number density distribution n(z), the overlap between the n(z) is determined by the errors on the photo-z estimates. Any biased photo-z estimates are then present in the n(z), and therefore will effect the cosmology constraints [106]. Another source of systematic errors is the intrinsic alignment of the galaxies (IA). Intrinsic alignments are the primary astrophysical systematic in cosmic shear studies. Reviews of this subject include [107, 108]. If pairs of galaxies are aligned in the absence of lensing, this could be interpreted as a cosmological signal. The intrinsic alignment effects the 2-point correlation function in Fourier space by,

GG GI II Cl = Cl + Cl + Cl (1.36)

II GI where Cl is the intrinsic alignment correlation and Cl is the cross-correlation between the galaxy shear and the intrinsic shear. The shape of these power spectra depend on the model used (e.g. [109, 110, 111]). The amplitude and redshift dependance of these spectra will often be parameterised and marginalised over in any cosmology analysis. The intrinsic alignment is expected to be present in the data because the ellipticity of the galaxies will depend on the tidal gravitational field in which they were formed. Any correlations in the gravitational field will therefore induce correlations into the shear field. This effect can be removed by down-weighting galaxies that are physically close [112], from cross-correlating with the galaxy density field [113, 114] or by marginalizing over a parametrized intrinsic alignment model [110]. 36 CHAPTER 1. INTRODUCTION

Another systematic is the eﬀect of baryons on the cosmic shear signal. At very small scales, baryonic cooling increases power in the matter power spectrum. Baryonic pressure can also lead to a suppression of power on intermediate scales [115]. This can cause the cosmological inference to be biased. It can be alleviated by applying cuts to the data on small scales, or by modeling this baryonic feedback in the already non-linear region of the power spectrum [116, 117]. As the statistical power of galaxy surveys increases, these systematic errors will dominate and increasingly sophisticated methods will be needed to mitigate their impact.

Weak lensing detections and constraints

A number of observations in the early 2000s provided measurements of weak gravitational lensing [118, 119]. These early measurements were systematics dominated. Some recent examples of cosmic shear measurements came from, the Deep Lens Survey (DLS) [120, 121], SDSS [122, 123], the CFHTLenS survey [124, 40], the kilo-degree survey (KiDS) [125, 126], and the Dark Energy Survey [127, 128], DES and KiDS being the current state-of-the-art results. Looking at the DES year 1 cosmic shear results as an example, the ξ signal is ± shown in Fig. 1.7 and the cosmological constraints in the Ωm,σ8 plane are shown in 0.5 Fig. 1.8. The ﬁducial DES analysis provides a constraint of S8 = σ8(Ωm/0.3) = 0.782 0.027 in ΛCDM. ±

Galaxy Galaxy lensing

Cosmological information can also be obtained by measuring the stacked weak lensing signal around foreground tracers. When the foreground tracer is another galaxy, this is known as galaxy-galaxy lensing [130]. This can be thought of as either a measurement of the halo structure of the foreground (lens) galaxy or as a cross correlation between the shear ﬁeld and the lens galaxy ﬁeld. In this chapter I will focus on the latter. One expects the background (source) galaxy to be sheared tangentially to the 2D line drawn between the lens position and the source position. The simplest galaxy- galaxy lensing measurement is the average tangential shear signal γt, 1.1. COSMOLOGY 37

2 1,4 1,3 1,2 1,1 4 − 10 / + 0 θξ

2,4 2,3 2,2 4

− 2.5 10 / +

θξ 0.0

3,4 3,3 4,4 5 4 4 − −

10 2.5 10 / / + −

θξ 0.0 0 θξ 5 4,4 3,3 3,4 4 4 − −

10 2.5 10 / / + −

θξ 0 0.0 θξ 2,2 2,3 2,4 4

2.5 − 10 / −

0.0 θξ 1,1 1,2 1,3 1,4 2 4 − 10 / 0 − θξ

101 102 101 102 101 102 101 102 θ (arcmin) θ (arcmin) θ (arcmin) θ (arcmin)

Figure 1.7: 2-point correlation functions ξ+ and ξ measured on the DES Year 1 shear catalog split into 4 tomographic bins. −

P i wiet,j γt(θ) = P (1.37) h i i wi where et,j is the ellipticity of a source galaxy tangential to the lens galaxy direction for source-lens pair i, and wi is a weight associated with this pair. The sum is over all lens-source pairs separated by an angle θ within an angular bin.

The γt signal can be modelled in a similar way to w(θ) (Equ. 1.26) and ξ (Equ. ± 1.32) using the limber approximation,

Z dl l γij(θ) = bi J (lθ)Cij (l) (1.38) t 2π 2 gG where i and j label the lens and source bins, J2 is the 2nd order Bessel function, and

CgG(l) is the shear-position 2D power spectrum given by,

Z qj(χ) l + 1 Cij (l) = dχni(χ) P k = 2 , χ (1.39) gG H(z)χ2 mm χ where ni(χ) is the lens sample distribution, and qj(χ) the the lensing eﬃciency from Equ. 1.31. Galaxy-galaxy lensing has been measured in many recent galaxy samples including, SDSS [131, 132, 133], CFHTLenS [134, 135, 136], KiDs [137, 138] and the Dark Energy 38 CHAPTER 1. INTRODUCTION

1.00 DES Y1 0.95 KiDS-450 Planck 0.90

0.85

0.80 8 σ 0.75

0.70

0.65

0.60

0.55 0.2 0.3 0.4 0.5 ΛCDM 0.90 Ωm

0.85 5 . 0 3) . 0 / 0.80 m (Ω 8 σ ≡

8 0.75 S

0.70

0.65 0.2 0.3 0.4 0.5 Ωm

Figure 1.8: ΛCDM cosmological constraints from DES Y1 cosmic shear [128] and KiDS-450 cosmic shear [126] in the Ωm, σ8 plane, comparing to Planck CMB constraints [10]. The degeneracy between Ωm and σ8 and be seen in Equ. 1.30 and 1.31. Figure from [128]. Both the DES and KiDS results have since been updated in [129].

Survey [139, 140].

1.1.6 Other cosmological probes

A number of other cosmological probes can provide parameter constraints complimentary to galaxy surveys and the CMB. One of the most successful of these is type 1a Supernovae experiments. Already mentioned in Sec. 1.1.1, measurements of the luminosity of type 1a supernovae can be used to measure the expansion rate of the universe. Because type 1a supernovae have a known luminosity curve, they can be used as a standard candle. The luminosity of the supernovae can then be used to calculate the 1.1. COSMOLOGY 39 luminosity distance to the object. By plotting luminosity distance against redshift, the expansion history of the universe (Equ. 1.1) and dark energy parameters can then be constrained. Dark energy constraints from supernovae are limited by systematics including dust extinction from the milky way [141]. Analysis of the joint-light curve analysis (JLA) from SDSS-II and the Supernovae Legacy Survey finds w = 1.018 − ± 0.057 when combining with CMB data [142]. Another cosmological probe is Hi intensity mapping. This is a similar concept to galaxy clustering, but neutral hydrogen is used as a tracer of the dark matter instead of galaxies. Neutral hydrogen radiates with a wavelength of 21cm from transitions in the hyperfine structure [143]. This radiation can be detected by radio telescopes and used to probe the large scale structure of the universe, including BAO. Hi intensity mapping will be able to probe the large scale structure up to higher redshifts than galaxy surveys. The Square Kilometre Array (SKA), expects to be able to map Hi for BAO measurements up to z 3 [144]. ∼ Another cosmological probe is cluster counting [42]. Structure growth, and therefore dark energy, can be probed by measuring the dependence of the number density of clusters with redshift and cluster mass. Cluster counting can either be performed directly from galaxy surveys (e.g. [145]) or from the CMB via the Sunyaev-Zel’dovich (SZ) effect. The SZ effect is caused by CMB photons gaining energy from inverse Compton scattering with electrons in galaxy clusters. This can then be used to count the number of clusters along the line of sight [146].

1.1.7 Concordance within the standard model

Each cosmological probe is sensitive to diﬀerent combinations of parameters. For

α example, weak lensing is sensitive to the combination of Ωm(σ8) where alpha is a constant power, meaning there is a degeneracy between Ωm and σ8. In the same plane, measurements of the CMB anisotropies have a different degeneracy direction. By combining the constraints from these probes, the degeneracy can be broken, and much tighter parameter constraints can be found. Combined constraints can be calculated either by separately analyzing the two data sets and combining the resulting constraints (e.g. via importance sampling), or by doing a joint likelihood analysis. The latter method is required when the two probes 40 CHAPTER 1. INTRODUCTION are correlated. Data sets can also be combined by cross-correlating two data sets. For example, cross correlations of DES-SV data with the CMB data from the SPT have been released in 2015. This included a cross-correlation between cosmic shear with galaxies and CMB lensing from SPT [147] where both lensing signals should be lensed by the same structure. This analysis found the amplitude of cross-correlation A = 0.88 0.30, SPT ± where A = 1 is the ΛCDM prediction. In future surveys, this cross-correlation could be used to constrain the IA signal. Also a cross correlation between DES galaxy clustering w(θ) and CMB lensing from SPT was performed [148] finding the amplitude of cross- correlation A = 0.73 0.16. A number of other cross-correlation analyses have been ± performed for the CMBxLSS and between different LSS probes such as RSD, w(θ), and shear-shear weak lensing [149, 150]. One way to test the validity of the ΛCDM model without introducing alternative models is to compare different cosmological data sets in search of tension. For example, some hints of tension were found between a combination of low-redshift LSS probes such as BAO, lensing and cluster counts and CMB data from Planck and WMAP by [151, 152]. This tension was in found to be most noticeable in the Ωm, σ8 plane. This mild tension was also found by [153] between Planck CMB data and CFHTLenS cosmic shear data. The recent releases of DES and KiDS have lessened this tension, though the exact significance quoted depends on which data sets are considered and which consistency metric is used to measure it. This potential inconsistency can be seen in Figure 1.9. A number of suggestions for ways to resolve these tensions have been proposed. These include systematic effects that have not been accounted for or new physics including sterile neutrinos [154, 153]. Another tension which has received attention is the seemingly inconsistent measurements of the Hubble constant H0 from the CMB and the distance ladder. This potential inconsistancy can be seen in Figure 1.10.

1.1.8 Current and future surveys

A number of cosmological galaxy surveys are ongoing or planned for the near future. Here, some of the major experiments will be discussed, comparing each survey. 1.1. COSMOLOGY 41

Figure 1.9: Left: potential hints at tension between a combination of LSS probes and CMB data from Planck and WMAP, from [151] Right: Potential tension between CFHTLenS cosmic shear and CMB data from Planck. Figure from [153]. These plots show the LSS-CMB constraints in 2D (Ωm,σ8) marginalizing over the other standard ΛCDM parameters and nuisance parameters for each data set. As this is a multi- dimensional space, the direction in which the tension is strongest is not necessarily aligned with any parameter plane. This explains why the LSS data in the left panel overlap with the CMB data despite the consistancy metric suggesting tension.

For weak gravitational lensing, there are three major photo-z surveys ongoing, The Dark Energy Survey (DES) [159], the Kilo Degree Survey (KiDS) [160] and the Hyper Suprime-Cam (HSC) [96]. Of these, DES will have the largest survey area of 5000 square degrees and is expected to detect 300 million galaxies ( 200 of which are ∼ ∼ expected to be used in cosmic shear measurements) by the end of its 5 year observing time. This will provide DES with the best statistical power of these surveys. However, KiDS and HSC both beneﬁt from the high quality of their observations, including low PSF at the VST site for KiDS, and the deep limiting magnitude of the HSC observations. Cosmological parameter constraints have already been released by DES and KiDS. A comparison of the KiDS and DES cosmic shear constraints is shown in Fig 1.8. The DES year 1 analysis is discussed in detail in Chapter 3 and throughout this thesis. Some large spectroscopic surveys are the SDSSIII-BOSS survey [70], WiggleZ [73] and the Two-degree-Field Galaxy Redshift Survey (2dF) [76]. By design, the photo-z surveys partially overlap with these spectroscopic surveys. The DES footprint overlaps with stripe 82 of BOSS and the KiDS survey overlaps with 2dF. These overlaps can be seen in Figure 1.11. Figure 1.11 also shows the considerable overlap between DES and SPT that was 42 CHAPTER 1. INTRODUCTION

Figure 1.10: Measurements of the Hubble constant from a combination of DES LSS + BAO + Big Bang Nucleosynthesis [155], Planck CMB data [10], SPT high-l polarisation [156], SHOES supernovae [157], HOLiCOW strong gravitational lensing [158]. It has been suggested that the Planck and SHOES measurement are inconsistent with each other. Figure from [155]. used for the cross-correlation analyses discussed in Sec. 1.1.7. In the coming years, a number of large galaxy surveys will take place, increasing the number of objects detected for both weak lensing and galaxy clustering by an order of magnitude. One such survey is the Large Synoptic Survey Telescope, a ground based telescope that will observed 10000 square degrees of the sky at an optical depth of ∼ > 2 magnitudes deeper than SDSS [161]. Another is ESA’s Euclid satellite, a space telescope that will observe > 15000 square degrees and is expected to image up to 1 billion galaxies. Both LSST and Euclid intend to measure cosmic shear and galaxy clustering [162]. The Wide Field Infrared Survey Telescope (WFIRST) [163] is a planned infrared space telescope designed for spectroscopy, that will provide weak lensing on 2400 ∼ square degrees of sky. In radio astronomy, the Square Kilometre Array (SKA) will provide large data sets for use in all areas of astrophysics including cosmology. In particular interest to this work, the statistical power of an SKA1 weak lensing analysis is expected to 1.2. NEUTRINOS 43

Figure 1.11: Upper panel: The footprint of current photometric surveys plus the South Pole Telescope CMB experiment. Lower panel: The Footprint of current spectroscopic surveys. Figure from [159]. be competitive with the current generation of surveys (e.g. DES). Cross correlations between the SKA and optical surveys could remove wavelength depended systematics which are expected to be dominant for such large surveys [164, 165].

1.2 Neutrinos

Neutrinos are particles that interact via the weak interaction and gravity but do not interact via electromagnetism. As a result, neutrinos have long been proposed as candidates for dark matter particles. In this section I will outline the basics of neutrino physics, the latest measurements from particle physics experiments, and the impact of neutrinos on cosmology. The standard model contains three ﬂavours of neutrino, ν , ν and ν . Each | ei | µi | τ i is produced through the decay of its associated lepton; electron, muon and τ leptons 44 CHAPTER 1. INTRODUCTION respectively. These are sometimes referred to as the ‘active’ neutrinos. In the standard picture of neutrino mass, the three neutrino ﬂavour states do not each have an associated mass. Rather, they are a superposition of three mass states. This superposition is described by the Pontecorvo Maki Nakagawa Sakata (PMNS) matrix [166, 167, 168],

      νe Ue1 Ue2 Ue3 ν1             (1.40)  νµ  =  Uµ1 Uµ2 Uµ3   ν2        ντ Uτ1 Uτ2 Uτ3 ν3 where ν1, ν2 and ν3 represent the three mass states, corresponding to masses m1, m2 and m3. U is a unitary 3x3 matrix, the elements of which are the coeﬃcients of the superposition. In the simple case of two neutrino states, the PMNS matrix is a simple rotation matrix parameterised by a single mixing angle. In the case of three neutrino states, U can be written as the product of three rotation matrices, with a phase introduced in the second rotation.

      iδCP 1 0 0 c13 0 s13e c12 s12 0             (1.41) U =  0 c23 s23   0 1 0   s12 c12 0       −  iδCP 0 s c s e− 0 c 0 0 1 − 23 23 − 13 13 where cij = cos(θij) and sij = sin(θij). When written as the product of rotations, U can be parameterized my four parameters θ12, θ13, θ23, and δCP . The θ parameters are known as mixing angles, and they describe the the degree of overlap between the states. For example, if θ12 = 45◦, θ13 = 0, and θ23 = 0, would correspond to maximal mixing between the mass states ν and ν distributed equally | 1i | 2i between the ν and ν flavour states, with no mixing in the third state, ν = ν . | ei | µi | 3i | τ i The δCP parameter is a phase describing charge-parity violations. This introduces an asymmetry between neutrino and anti-neutrino oscillations (see Sec. 1.2.1). The oscillation and cosmological data studied in this work have no sensitivity to the CP violation phase [169]. One can also introduce additional flavours of neutrino in the form of sterile neutrinos. These are neutrinos that do not interact via the weak interaction, and therefore do not have associated leptons. Sterile neutrinos have been proposed as a candidates 1.2. NEUTRINOS 45 for cold, warm or hot dark matter. The simplest sterile neutrino extension is to add an additional sterile flavour state ν and mass state ν with mass m , to the PMNS matrix in Equ. 1.40. This | si | 4i 4 introduces three new mixing angles, θ14, θ24, and θ34. It is assumed that all of these new mixing angles are very small such that for most cases ν ν . It is an aim of | si ≈ | 4i recent oscillation experiments to constrain, and potentially detect, these angles.

1.2.1 Neutrino Oscillations

As neutrinos propagate through space, the composition of the neutrinos in the mass basis (P a ν ) will change. Classically, this can be thought of as states with different i | ii masses but equal kinetic energy, propagating with different velocities. As the neutrino’s coefficients in the mass basis change, so to will the coefficients in the flavour basis (P b ν ). When a neutrino, starting in a pure flavour state, begins to propagate α | αi through space, there is a non-zero probability of measuring this neutrino to be in a different flavour state.

For simplicity, consider the case of two neutrino ﬂavours ν and ν , two masses | αi | βi m1 and m2, and a single mixing angle θ. We consider a neutrino that begins in a pure ν state and then propagates with an energy E. One can show (e.g. in [170]) the | αi probability of measuring the neutrino ﬂavour ν at a distance L from the source to | βi be,

∆m2L P (ν ν ) = sin2 (2θ) sin2 (1.42) α → β 4E where ∆m2 = m2 m2 referred to as the mass splitting. Starting with a known | 1 − 2| neutrino source, one can measure the oscillation probability by measuring the ﬂux of neutrinos after distance L. From this probability, constraints on the mass splitting and mixing angle of the neutrinos can be obtained, but not on the absolute value of the neutrino mass. Equ. 1.42 can be extended to three neutrino ﬂavours (see e.g. [169]). However, many oscillation experiments use the two-neutrino approximation as the dependence on the other angles, and CP phase is weak. 46 CHAPTER 1. INTRODUCTION

Active neutrino experiments

The oscillations of active neutrinos have primarily been measured by three methods, solar neutrino detections, atmospheric neutrino detections, and long baseline oscillation experiments.

Solar neutrinos originate in the sun as νe. The number of these neutrino detected by Earth based experiments was much less than expected from the standard solar model and it was found that neutrino oscillations explained this deﬁcit. Solar neutrinos can be used to constrain θ and ∆m2 . These are sometimes referred to θ and ∆m2 . 12 | 21| sol | sol| Results from the Sudbury Neutrino Observatory in combination with other solar neu-

2 +0.027 2 +0.20 5 2 trino experiments ﬁnd tan θsol = 0.427 0.024 and ∆msol = 7.46 0.19 10− eV [171]. − | | − × For reviews on solar neutrinos see [172, 173].

Atmospheric neutrinos are produced when cosmic rays (mostly protons) interact with nuclei in the upper atmosphere ( 15 km), producing π and K mesons. These ∼ mesons are unstable and quickly decay into electrons and muons and their associated neutrinos and anti neutrinos ν , ν , ν¯ , ν¯ . For typical neutrino energies of 1 GeV, µ e µ e ∼ the expected ratio of (νµ +ν ¯µ)/(νe +ν ¯e) is 2. In the presence of neutrino oscillations,

νµ oscillate into ντ reducing the detectable flux and altering the νµ/νe ratio. These measurements can be used to constrain θ and ∆m2 . These are sometimes referred 23 | 32| to θ and ∆m2 . Constraints from the neutrino detector Super-Kamiokande (SK) atm | atm| 2 3 2 [174] find best fit values of θ = 49 and ∆m = 2.6 10− eV assuming normal atm | atm| × heirachy [175]. For reviews on atmospheric neutrinos see [176, 177].

The active neutrino parameters can also be constrained by long baseline oscillation experiments. These use a source of neutrinos in a pure flavour state, and detect the flux of a particular flavour after propagating over a long distance L, directly measuring the probabilities in Equ. 1.42. These experiments require long baselines (high L) to be sensitive to small mass splittings. For reviews on long baseline experiments see [178, 179, 180].

Given the measurements of the active mixing angles and mass splitting from oscillation experiments, one can draw a schematic of the potential neutrino mass hierarchy and mixing, seen in Figure 1.12. Oscillations experiments are only sensitive to the diﬀerence in neutrino masses, not the absolute values. 1.2. NEUTRINOS 47

Figure 1.12: Schematic diagram demonstrating the two potential mass orderings for the active neutrinos. If the lightest mass is m1 (left), this is known as the normal hierarchy P (NH). In this case the minimum sum of the the neutrino masses is mν & 0.06eV. If the lightest mass is m3 (right) this is known as the inverted hierarchy (IH). In this P case the minimum sum of the the neutrino masses is mν & 0.1eV. The colours in each bar represent the different flavour states that make up each mass state. The mixing shown here is for the best fit mixing angles from oscillation experiments [181] and δCP = 0. 48 CHAPTER 1. INTRODUCTION

Sterile neutrino experiments

In addition to the standard neutrino constraints, oscillation experiments can also be used to search for sterile neutrinos. When searching for sterile neutrinos, oscillation experiments with short baselines (low L) are typically used in order to be sensitive to large mass splittings [182]. These sterile neutrino searches broadly come in two categories; appearance experiments and disappearance experiments. Appearance experiments measure the flux of neutrinos in one flavour state, from a source of a different flavour state. In the presence of a sterile neutrino an excess flux in the new flavour will be detected. In disappearance experiments, the flux of the neutrinos is measured in the same flavour as the source. In the presence of sterile neutrinos, the additional oscillation into the sterile flavour state causes a reduction in detected flux. This reduction can be used to constrain sterile neutrino models.

A νµ disappearance experiment is sensitive to the mixing angle θ24 and the mass splitting ∆m24.A νe disappearance experiment is sensitive to the mixing angle θ14 2 and the mass splitting ∆m14 .A ν¯e or νe appearance experiment is sensitive to a combination of these two mixing angles,

2 2 2 sin 2θµe = sin 2θ14 sin θ24 (1.43) where θµe is the angle describing the mixing measured by appearance experiments.

The MINOS experiment [183] is a neutrino oscillation experiment. The νµ disappearance component of the experiment is sensitive to oscillations into the sterile state.

2 The latest results from MINOS provide an exclusion region in the ∆m14,sin θ24 space [184] shown in Figure 1.13.

The Daya Bay experiment [185] is a νe disappearance experiment that has produced 2 an exclusion region in the ∆m14, sin θ14 space [186] shown in Figure 1.13.

The LSND experiment observed an excess of ν¯e appearance in a ν¯µ beam in 2001

[187]. Results from the MiniBooNE experiment have shown an excess of both νe [188] and ν¯e [188, 189, 190, 191]. When analyzed using a model with 1 sterile neutrino, these appearance experiments collectively show a 6.1σ excess signal.

2 −3 Since the mass splitting ∆m12 has already been measured to be 10 eV, one can easily convert ∼ between the mass splitting results in νe and νµ disappearance experiments. 1.2. NEUTRINOS 49

Figure 1.13: The exclusion regions from disappearance experiments. Left: Results from the MINOS ν disappearance experiment probing the ν ν channel. Figure µ µ → sterile from [184]. Right: Results from the Daya Bay νe disappearance experiment probing the ν ν channel. Figure from [186]. e → sterile

A joint analysis of MINOS and Daya Bay data [192] shows the combined result

2 projected onto the sin 2θµe angle so that it can be compared to appearance experi- 2 ments. This result ﬁnds that most of the ∆m14,sin 2θµe space allowed by LSND and MiniBooNE is ruled out by the disappearance experiments (see Fig. 1.14). The reason for sterile oscillation detections in appearance experiments but not in disappearance experiments remains unknown. Constraints on sterile neutrinos have also been obtained by the IceCube experiment [193]. This experiment measures the disappearance of atmospheric muon neutrinos (νµ and ν¯µ) that have travelled through the earth. See Chapter 2.

Absolute neutrino mass from the laboratory

Oscillation experiments are sensitive the diﬀerence in neutrino mass states but not the absolute masses. However, it is possible to obtain limits on the absolute neutrino mass scale from cosmological measurements (see Sec. 1.2.2), or from beta decay experiments.

A beta decay produces an electron e and an anti-electron neutrino ν¯e with a known combined energy. For example in tritium beta decay the electron and neutrino share 18.6 keV of energy. This energy is shared between the electron mass, the eﬀective ∼ 50 CHAPTER 1. INTRODUCTION

Figure 1.14: Sterile neutrino constraints in the θµe appearance space. Left: results from a combination of MINOS and Daya Bay compared to the LSND anomalies. Figure from [192]. Right: Recent results from the MiniBooNE experiment which ﬁnd a 6.1σ excess when combined with LSND. The measurement from appearance experiments has not been found in disappearance experiments. Figure from [191]. neutrino mass3, and the kinetic energy of the the two particles. The energy spectrum of the electrons produced by the beta decay is measured, and the neutrino mass can be determined from the zero-point energy of the spectrum (eﬀective neutrino mass = 18.6 keV - maximum electron energy) [194, 195].

Current experiments are able to constrain the neutrino mass scale to m(νe) < 2 eV [196, 197], signiﬁcantly weaker than cosmological constraints (see Sec. 1.2.2). The KATRIN experiment [198], collecting data since June 2018, will be able to improve this by an order of magnitude, forecasting a constraint of m(νe) < 0.2 eV at 90% conﬁdence.

1.2.2 Impact of neutrinos on cosmology

In addition to oscillation experiments, constraints on neutrino parameters can also be derived from cosmological experiments. In this case, we state that some fraction of dark matter is hot (HDM) or warm dark matter (WDM) rather than cold (CDM), and

3As the neutrino will be in a pure ﬂavour state, this experiment probes the combination p P U 2m2, rather than a speciﬁc mass state. i | ei| i 1.2. NEUTRINOS 51 that this contribution is coming from neutrinos, either active or sterile. The impact of neutrinos on cosmological observables will be described in this section. This section uses the notation of review articles [199, 200, 201]. In the early universe of the standard hot big bang model, neutrinos are coupled the the baryon-photon plasma by weak interactions. As the universe expands, the interaction rate decreases. When the interaction rate falls below the level required to couple the neutrinos to the plasma, they decouple. This temperature is approximately 1MeV. Before decoupling, the momentum distribution of these neutrinos is given by the Fermi-Dirac distribution,

h p µ i 1 f(p, T ) = exp − ν + 1 − (1.44) T where p is the neutrino momentum, T is the temperature of the universe, µν is a chemical potential that exists only in the case of neutrino-anti-neutrino asymmetry. At these early times, the neutrinos are relativistic. Since we know neutrinos are massive particles, as the temperature of the universe drops to below the neutrino mass, they will become non-relativistic. In the case of multiple mass states, each state becomes non-relativistic at a diﬀerent temperature/redshift.

When neutrinos are relativistic (mν << Tν) the neutrino density is given by,

7π2 4 4/3 ρ = T 4 (1.45) ν 120 11 γ where the relation T (t = 0) (4/11)1/3T (t = 0) has been used to relate neutrino ν ≈ γ temperature to the CMB temperate. This can be written as a contribution to the overall radiation density,

h 7 4 4/3 i ρ = ρ + ρ = 1 + N ρ (1.46) r γ ν 8 11 eff γ where Neff is the effective number of neutrino species. The standard model contains three flavors of active neutrinos which contribute Neff = 3. An additional contribution to the neutrino temperature comes from election positron annihilation which happens just after neutrino decoupling. This reheats the neutrinos by a small amount. This heating is folded into the Neff parameter in order to keep the definition of neutrino temperature fixed. This gives the standard model prediction of Neff = 3.046. 52 CHAPTER 1. INTRODUCTION

At late times (mν >> Tν) the neutrino are non-relativistic and contribute to the total matter density as,

ρν = mνnν (1.47) where mν is neutrino mass (assuming one mass state), and nν is the number density of neutrinos. Extending to multiple mass states, this can be written as a contribution to Ω and in terms of the sum of the neutrino masses by,

P m Ω = i i . (1.48) ν 94.1h2eV

Free streaming

Relativistic neutrinos free-stream at approximately the speed of light with a free- streaming length equal to the Hubble radius (c/H). The free-streaming lengths grows with the Hubble radius, until the neutrinos become non-relativistic during matter domination, when the free streaming length starts to reduce. As a result, there is a maximum free-streaming length associated with a neutrino of mass m. The maximum length corresponds to a minimum wave number,

0.5 0.5 m 1 k 0.018Ω hMpc− . (1.49) nr ' m 1eV The effect of this on the matter power spectrum is to smooth out structure on scales smaller than knr. Measurements of the matter power spectrum can be used to measure this suppression of power and therefore constrain the neutrino mass. When we consider multiple mass states, the suppression of structure from each mass begins at a different knr, so theoretically one could measure the full neutrino mass hierarchy from the matter power spectrum alone. However, in practice, large scale structure probes such as galaxy clustering, BAO, and weak gravitational lensing, typically probe structure on smaller scales than knr and at less precision than would be required to distinguish between the different masses, meaning these probes are sensitive to the total neutrino contribution Ων but not sensitive to the mass hierarchy. Neutrinos can also impact the CMB power spectra. As neutrinos behave like radiation at early times, and matter at late times, there are a number of ways they 1.3. SCIENCE MOTIVATION 53 can impact the CMB spectra. Individually, some of these effects can be compensated in a cosmological analysis by simultaneously varying the other cosmological parameters

(e.g. varying Ωm mitigates some of the late time impact on the CMB), but cannot account for all these effects simultaneously. As radiation, the presence of extra radiative degrees of freedom alters the matter- radiation equality redshift zeq. This impacts both the position and amplitude of the CMB power spectra peaks. The neutrinos will also alter the expansion history of the universe, impacting the angular diameter distance to the last scattering surface. This causes a shift in the angular locations of the CMB power spectra peaks. The same structure suppression that impacts the matter power spectrum, also impacts the CMB TT power spectrum by suppressing the low l power (20 < l < 500) through the early Integrated Sachs-Wolf Effect. This model of cosmological neutrinos can be used to constrain the sum of the neutrino mass from CMB, BAO, large scale structure, and H0 measurements. One can also constrain sterile neutrino models by allowing the effective number of neutrinos

Neff to vary from the standard prediction of Neff = 3.046 and allowing for additional mass states. When interpreting Neff as coming exclusively from neutrinos, it describes the total number of fully thermalised neutrino species (neutrinos with the same temperature Tν). If the interaction between the active and sterile neutrinos is very small

(e.g. due to small mixing angles θi4), then a single sterile neutrino will contribute < 1 to Neﬀ . This is the subject of Chapter 2.

1.3 Science motivation

The aim of this thesis is to use large scale structure measurements to obtain constraints on sterile neutrino models. I begin by exploring the existing constraints from the CMB and oscillation experiments, and comparing their results (Chapter 2). By making this comparison, one can test the consistancy between the two types of data, informing both particle physics and cosmology. I then discuss the DES year 1 analysis of weak lensing and galaxy clustering (Chap- ter 3) and the systematics analysis of its galaxy clustering data vector (Chapter 4). 54 CHAPTER 1. INTRODUCTION

This demonstrates a number of robustness tests that are required to produce reliable cosmology constraints from large scale structure experiments. I then combine these two topics with a simulated analysis of the DES year 1 data vector, constraining sterile neutrino models (Chapter 5). This demonstrates the potential improvement large scale structure experiments can provide, to the existing cosmological constraints. Chapter 2

A Combined View of Sterile-Neutrino Constraints from CMB and Neutrino Oscillation Measurements

This chapter is based on the publication [1]. The production of future experiment sensitivities was done in collaboration with the other authors. The search for low-mass sterile neutrinos is motivated by several experimental anomalies that are not consistent with the three-flavour paradigm. Sterile neutrinos would change the oscillation probabilities observed by detecting neutrinos from accel- erators, nuclear reactors, or produced in the atmosphere. On a cosmological scale, they would modify the power spectrum of the Cosmic Microwave Background (CMB) (Fig. 2.1). Both types of measurement put severe constraints on the existence of extra neutrino flavours, but they are evaluated in terms of different parameter sets. The CMB measurements constrain the effective number of additional neutrino species, ∆Neff

(above the Standard Model (SM) prediction of Neff = 3.046), and the effective sterile sterile neutrino mass meff . Oscillation experiments parameterize their constraints in terms 2 of mass-squared differences, ∆mij, between the mass eigenstates, and the mixing angles θαβ between mass and flavour eigenstates. Here, I use the calculation of [202] and show the Planck CMB cosmology constraints in the same parameter space as used for

νµ disappearance measurements.

Several experimental anomalies related to the appearance and disappearance of νe

55 56 CHAPTER 2. STERILE NEUTRINOS FROM CMB AND OSCILLATIONS could be explained by light sterile neutrinos with a mass-squared diﬀerence relative to the active states of ∆m2 1 eV2 [50, 203, 204]. The LSND Collaboration observes an ≈ excess of ν¯e appearance in a ν¯µ beam [187], and MiniBooNE measures an excess of both

νe [188] and ν¯e appearance [189, 190, 191]. Reactor experiments observe a deﬁcit of 6% in the ν¯ ﬂux compared to expectations [205]. Furthermore, Gallium experiments ≈ e 71 71 51 37 observe a smaller ν + Ga Ge + e− event rate than expected from Cr and Ar e → sources [206]. The Daya Bay Reactor experiment has searched for ν¯e disappearance 2 2 2 setting limits on the mixing angle sin θ14 in the low ∆m region 0.0002 < ∆m41 < 2 0.2 eV [207]. These results have been combined with νµ disappearances searches by 2 2 MINOS [208] to obtain stringent constraints on the product sin 2θ14 sin θ24 [192].

For this analysis, I focus on recent νµ disappearance results, where no anomalies have 2 2 been found, and assume that sin θ14 = sin θ34 = 0 in order to be consistent with the assumptions that were used for deriving these limits.

Several studies have combined oscillation and cosmological data to constrain sterile neutrinos. Several [209, 210, 211, 212, 213] use the posterior probability distribution on ∆m2 from short-baseline anomalies as a prior in the cosmological analysis. In this analysis, I convert the full CMB cosmology constraints into the oscillation parameterisation and vise versa, focusing on recent νµ disappearance results. This conversion has also been studied in [214, 215]. This analysis diﬀers in several ways: (i) unlike [214]

sterile I use the 2D combined constraints on ∆Neff and meff in the cosmological analysis, rather than converting 1D constraint values in each parameter individually; (ii) I use the latest CMB data from Planck, updating from the WMAP 5-year data used in [215]; (iii) I solve the full quantum kinetic equations, rather than using the averaged momentum approximation [216] used in [214, 215]; (iv) I also consider the impact of non-zero lepton asymmetry, L, and a different sterile mass mechanism. The lepton asymmetry is defined as L = (n n ¯)N /N , where n and n ¯ are the number den- f − f f γ f f sities of fermions and anti-fermions, respectively, and Nf and Nγ are the numbers of fermions and photons. Fermion f can be e, µ or τ. 2.1. DATA SETS 57

Figure 2.1: The CMB temperature power spectrum for different values of the effective number of neutrino species and effective sterile mass, fixing the cold dark matter energy density. The ΛCDM case uses the SM value Neff = 3.046 and an active neutrino mass P sum mν = 0.06 eV. All other cosmological parameters are set to the best-fit values from the Planck 2015 data shown by error bars [225]. The power spectra are generated using the CAMB module [226] in CosmoSIS [227].

2.1 Data sets

2.1.1 Cosmological Data sets

Observations of the CMB radiation are the most powerful probe of cosmology, giving a snapshot of the universe around 300,000 years after the Big Bang. The angular intensity fluctuations are sourced by temperature fluctuations in the plasma, which in turn depend on the constituents of the universe, including sterile neutrinos. Cosmology results are most sensitive to the sum of all neutrino masses, rather than the relative masses of the active and sterile neutrinos. The Planck satellite currently provides the definitive measurement of the CMB temperature anisotropies [217]. The Planck data P have been used to constrain the sum of the active neutrino masses yielding mν < 0.68 eV from CMB temperature data alone [10]. The information from the CMB can also be combined with that from other cosmological observations for even tighter constraints [218, 219, 220, 151, 221, 222, 223, 224]. Here, I use the Planck temperature power spectrum and low multipole polarisation data alone.

To constrain sterile neutrinos, two parameters are added to the baseline Planck 58 CHAPTER 2. STERILE NEUTRINOS FROM CMB AND OSCILLATIONS

sterile 2 analysis: the effective sterile mass, meff = (94.1 Ωsterileh ) eV, and the effective number of additional neutrino species, ∆N = N 3.046. The cosmological model used eff eff − sterile is ΛCDM+meff +∆Neff . Additional cosmological parameters and their degeneracies with neutrino parameters are not considered here. Figure 2.1 shows the power spectrum of the CMB temperature fluctuations. One

sterile observes that increasing the effective number of neutrino species, while fixing meff = 0, shifts the peak structure to higher multipoles, l, due to a change in the matter- radiation equality redshift, zeq. There is also an increase in the integrated Sachs-Wolfe sterile effect at low l [228]. A non-zero meff further changes zeq adding to the shift of the peak locations [229, 50, 228, 199].

sterile The effective mass, meff , can be related to the mass of the sterile neutrino, msterile = m4, in two ways. The first is to assume a thermal distribution with an arbitrary temperature Ts. The quantity ∆Neff is then a measure of the thermalisation 4 of the sterile neutrinos, ∆Neff = (Ts/Tν) , yielding,

3 sterile Ts thermal 3/4 thermal meﬀ = m4 = (∆Neﬀ ) m4 . (2.1) Tν

The second model assumes the extra eigenstate is distributed proportionally to the active state by a scaling factor, χs, here equal to ∆Neﬀ ,

sterile DW DW meﬀ = χsm4 = ∆Neﬀ m4 . (2.2)

This is known as the Dodelson-Widrow (DW) mechanism [51]. I use the thermal distribution as the ﬁducial interpretation and show that the conclusions are robust to this choice. The Planck analysis assumes the normal mass ordering of the active neutrinos with the minimum masses allowed by oscillation experiments, m = 0 eV, m 0 eV, and 1 2 ≈ m3 = 0.06 eV. Any excess mass is considered to be from a single additional state, which implies that ∆m2 m2. I use these assumptions throughout this analysis. Assuming 41 ≈ 4 inverted mass ordering or allowing m1 > 0 would strengthen the Planck constraints on sterile neutrinos. These assumptions allow us to directly compare to the oscillation data. The Planck 95% Conﬁdence Level (CL) contour is shown in Fig. 2.2 (b, d) for

thermal a prior of m4 < 10 eV [10]. 2.1. DATA SETS 59

sterile Figure 2.2: (a, c) Cosmological parameters ∆Neff and meff calculated in the oscil- 2 2 lation space ∆m , sin 2θ24 using LASAGNA. I use the thermal sterile neutrino mass (Eq. 2.1) and L = 0. Also shown are the constraints from the experiments native to this space, MINOS and IceCube, and the SBN sensitivity. The region to the right of the contours is ruled out at the 95% CL. (b, d) ∆m2, sin2 2θ in the cosmological space, sterile meff , ∆Neff . The region above the blue line is excluded by the Planck temperature thermal and low-l polarization data at 95% CL. A prior of m4 < 10 eV has been applied thermal as in [10]. The hatched area corresponds to m4 > 10 eV where ∆Neff was not calculated. 60 CHAPTER 2. STERILE NEUTRINOS FROM CMB AND OSCILLATIONS

2.1.2 Oscillation Data sets

The MINOS experiment [230] reconstructs interactions from a νµ beam created in an accelerator at Fermilab in a near detector (ND), located about 1 km from the source, and a far detector (FD) at 735 km. A sterile neutrino will reduce the νµ survival probability through its mixing with the active neutrinos. In most analyses, the ND serves as a reference point that defines the un-oscillated beam spectrum. However, for mass differences above ∆m2 1 eV2, oscillations occur rapidly and can already ≈ lead to a depletion of the neutrino flux at the ND. MINOS has therefore performed an innovative analysis exploiting the ratio of the neutrino energy spectra measured in the

FD to those in the ND using both charged-current (CC) νµ and neutral-current (NC) neutrino interactions [231, 208]. Limits on sterile-neutrino parameters are obtained by performing a χ2 ﬁt of the far-over-near ratio for both CC and NC data samples. I use the χ2 surface given in [231], which includes the data published in [208] and incorporates the statistical uncertainties, a full covariance matrix of the experimental

2 systematic uncertainties, and a weak constraint on ∆m32, which the data can then constrain. All other three-flavour oscillation parameters are fixed in the MINOS fit. I assume that all uncertainties follow a Gaussian distribution, and derive confidence levels using Gaussian χ2 p-values. The 95% CL contour derived from the MINOS χ2 distribution is shown in Fig. 2.2 (a, c). The IceCube detector [193] comprises 5160 optical modules instrumenting 1 km3 ∼ of ice at the South Pole. Neutrinos are detected using Cherenkov radiation emitted by charged particles produced in CC interactions. This is used to measure the disappearance of atmospheric muon neutrinos (νµ and ν¯µ) that have traversed the Earth. Sterile neutrinos are expected to modify the energy-dependent zenith-angle distribution of the νµ and ν¯µ through resonant matter-enhanced oscillations caused by the MSW effect [232, 233]. IceCube has searched for sterile neutrinos by studying the 2D distribution of the reconstructed neutrino energy and zenith angle [234, 235]. The IceCube likelihood distribution utilizes both shape and rate information, including systematic and statistical uncertainties. The distributions shown in Fig. 2.2

(a, c) are taken from [235]. The IceCube Collaboration also assumes θ34 = 0 in its analysis. It shows that this assumption leads to a more conservative limit and that non-zero values of θ14 have little eﬀect on the results [234]. 2.2. THERMALISATION OF STERILE NEUTRINOS 61

The Short Baseline Neutrino (SBN) programme [236] at Fermilab will study the LSND [187] and MiniBooNE [188, 189, 190] anomalies. It comprises three liquid-argon time projection chambers at diﬀerent baselines in a νµ beam: the already-running MicroBooNE detector, and the SBND and ICARUS detectors that are due to start data-taking in 2018. The programme will primarily search for νe appearance, but can also study the disappearance of νµ. The SBN sensitivity was obtained using the code GLOBES [237]. The details of this calculation are presented in [1].

Only the νµ disappearance channel is used to make a direct comparison with the MINOS and IceCube measurements in Figs. 2.2 (a, c).

2.2 Thermalisation of sterile neutrinos

sterile To relate the cosmological parameterization (meff , ∆Neff ) to the oscillation param- 2 eterization (∆mij, θαβ), I solve the full quantum kinetic equations that govern the sterile neutrino thermalization [202]. I use LASAGNA [238] to solve these equations in the simplified scenario with one active and one sterile neutrino flavour as described in [239, 240]. This scenario contains a single mixing angle, θ, and the flavour states are

ν = cos θν sin θν , (2.3) a 1 − 2

νs = sin θν1 + cos θν2 , (2.4)

where ν1,2 are the mass eigenstates, and νa,s the active and sterile ﬂavour eigenstates, respectively. The LASAGNA input parameters are the mass splitting, ∆m2, between the two mass states, the mixing angle, θ, the lepton asymmetry, L, and the range in temperature, T , over which to evolve the kinetic equations. LASAGNA produces a grid in the parameter x = p/T , where p is the neutrino momentum, upon which the factor

+ ¯ ¯ Ps = (P0 + P0) + (Pz + Pz) (2.5)

is calculated. Here, P0 and Pz are the ﬁrst and last components of the neutrino Bloch vector, (P0,Px,Py,Pz). The factor P0 corresponds to the number density of the mixed 62 CHAPTER 2. STERILE NEUTRINOS FROM CMB AND OSCILLATIONS

state, and Pz is related to the probability that a neutrino is in the sterile or active state, Prob(ν ) = (1 P )/2, and Prob(ν ) = (1 + P )/2. The factors P¯ and P¯ are s − z a z 0 z + the corresponding anti-neutrino values. Ps is used to calculate

R dx x3f P + 0 s (2.6) ∆Neﬀ = R 3 4 dx x f0

x with the Fermi-Dirac distribution function, f0 = 1/(1 + e ). This is valid if the active states are in thermal equilibrium. More details on LASAGNA are given in [202, 238] and on the quantum kinetic equations in [239, 240]. For the ﬁducial analysis, I run LASAGNA with L = 0 in a temperature range

2 2 1 < T < 40 MeV, calculating ∆Neff on the 2D grid of ∆m , sin 2θ values shown in sterile Fig. 2.2 (a). I convert positions in the cosmology parameter space (meff , ∆Neff ) into 2 2 2 2 the oscillation space (∆m41, sin 2θ24), first by using ∆m41 = m4 and Eq. 2.1 to find 2 2 ∆m41, then interpolating sin 2θ24 from the underlying grid in Fig. 2.2. I assume that the sterile-active mixing is dominated by a single angle θ24.

2.3 Results

The experimental inputs have been derived using different statistical approaches. For the oscillation experiments, a ∆χ2 contour is calculated as the difference of the χ2 of the best-fit hypothesis for the data to the χ2 at each model point. For the Planck data, a multi-dimensional Markov Chain MC is produced allowing cosmological, nuisance

sterile and neutrino parameters (∆Neff , meff ) to be varied. The number density of points in this chain is proportional to the posterior. When the priors on the input parameters are flat, this is also proportional to the likelihood . I draw this likelihood surface in L the (∆N , msterile) plane and take the χ2 to be 2 ln( ). The contour describing the eff eff − L 95% CL corresponds to χ2 χ2 = 5.99 in the 2D input distributions. In the Planck − min case, this leads to a dependence on the prior of these parameters. In this analysis, the

sterile priors are flat in the ranges 0 < ∆Neff < 1, and 0 < meff < 3 eV. As a result of this approach the constraints shown in this section are not Bayesian confidence levels, but rather a frequentist interpretation of the Planck results. Several standard assumptions about the cosmological and neutrino models are made that may impact the conclusions of this work. I assume ΛCDM in the Planck 2.3. RESULTS 63

Figure 2.3: (a) Sterile neutrino exclusion regions at 95% CL from Planck, MINOS, IceCube, and the SBN forecast in the oscillation parameter space. The dashed line is the Planck constraint with m4 calculated using the Dodelson-Widrow mechanism. 2 The dot-dash line is the Planck constraint using a large lepton-asymmetry, L = 10− . (b) The same contours in the cosmological space, where the diﬀerence between the thermal and Dodelson-Widrow scenarios is negligible. 64 CHAPTER 2. STERILE NEUTRINOS FROM CMB AND OSCILLATIONS analysis and a single sterile neutrino species, mixing only by one channel, in the conversion between parameter spaces. I also assume that any ∆Neﬀ is caused only by neutrinos and no other light relic particle. Studying the impact of these assumptions is beyond the scope of this analysis. Fig. 2.3 shows the CMB and oscillation experiment exclusion regions on the same axes in the oscillation and cosmology parameter spaces. The CMB data exclude a similar corner of the parameter space to the oscillation experiments, ruling out large mixing angles and large sterile-neutrino masses within the 3+1 model. This conclusion is unchanged by switching from the thermal mass in Eq. (2.1) to the Dodelson-Widrow mechanism in Eq. (2.2).

2 2.4 The region ∆m41 < 10− , leads to m4 < m3, and the active masses can no longer be treated as a single state. Therefore, the Planck contour below this value is too

2 conservative. Ref. [214] discusses cosmological constraints in this ∆m41 range.

2 sterile The Planck contour at large ∆m41 is dominated by the constraint on meff , and 2 at low ∆m41 by the constraint on ∆Neff , as shown in Figs. 2.2 (a, c). This is also illustrated in Fig. 3 of [214], which converts 1D upper limits on each of these parameters separately, instead of the 2D likelihood surface. Comparing these results to the averaged-momentum approximation results of [214], I find that solving the full quantum kinetic equations results in qualitatively similar constraints. In the fiducial analysis, I convert between parameter spaces using the assumption L = 0. In this case, the Planck data are more constraining than the oscillation ex-

2 1 2 periments for large mass-squared diﬀerences, ∆m41 > 10− eV , and less constraining 2 2 1 2 than MINOS in the range 10− < ∆m41 < 10− eV . When the lepton asymmetry is 2 large, L = 10− , the oscillations between sterile and active neutrinos are suppressed, giving a lower ∆Neﬀ for the same oscillation parameters (see Fig. 4 of [202]). This weakens the Planck constraints in the oscillation space such that they are now less constraining than all of the oscillation experiments considered, as shown by the dot- dash line in Fig. 2.3 (a). This demonstrates the model dependent nature of the CMB constraints in the oscillation space. However, a lepton asymmetry this high would introduce other cosmological inconsistencies, particularly in BBN constraints [241]. These inconsistencies are not studied in this work.

2 The MINOS experiment is particularly sensitive to the region of low ∆m41 because 2.4. CONCLUSIONS 65 of its baseline and neutrino energy range. This is the only region where the oscillation data are more constraining than the cosmology data when assuming L = 0. In the

sterile cosmology space, this corresponds to ruling out a region of large ∆Neﬀ at low meﬀ .

2.4 Conclusions

In conclusion, I compare sterile neutrino constraints from oscillation experiments and cosmological constraints. I use the quantum kinetic equations to convert between the standard oscillation parameterization of neutrinos (the mass-squared difference and mixing angle) and the cosmology parameterization (the effective sterile neutrino mass and the effective number of neutrino species). I show the relationship between each of the parameter combinations. I show the Planck 2015 CMB cosmology constraints in the oscillation parameter

2 space and ﬁnd that they rule out large values of ∆m41 and mixing angle θ. For the ﬁducial case, the region of parameter space ruled out by IceCube data is already excluded by the Planck CMB constraints. I show that much of the MINOS exclusion

2 region is also ruled out by Planck CMB constraints, although for low ∆m41 MINOS is more constraining. The forecast constraints for the SBN experiments are not expected to add to the information already provided by Planck CMB results with these model assumptions. However, their main sensitivity will be through the νe appearance searches not considered here. The MINOS data add the most information to that provided by Planck CMB measurements because it probes the lowest ∆m2. The power of the Planck CMB constraint is robust to the choice of eﬀective mass deﬁnition used in the cosmology model, giving similar results from the thermal and Dodelson-Widrow mechanisms. However, if one allows the lepton asymmetry to be

2 very large (L = 10− ), the Planck exclusion region is signiﬁcantly reduced. I also show the oscillation experiment constraints in the cosmology parameter space, where the same eﬀect is observed. In this parameter space the MINOS constraints rule out a larger fraction of the region allowed by the CMB. Chapter 3

Cosmology from combining galaxy clustering and weak lensing in the Dark Energy Survey Year 1 data

The aim of this chapter is to outline the Dark Energy Survey (DES) analysis combining galaxy clustering and weak lensing measurements from the DES Year 1 data set. Much of the work in the following chapters of this thesis relate to this analysis. In this chapter I will give an overview of the full analysis and how it compares to analyses from other experiments. This will elaborate on the physics discussed in Sec. 1.1.4 and 1.1.5. The full details of this analysis are contained in [2, 3, 128, 140, 242, 243, 244, 103, 245]. This was a collaborative work to which I contributed the production of lens galaxy samples, validation of the clustering covariance, and generated the galaxy clustering data vector. The Dark Energy survey [246] is a photometric galaxy survey that uses the Dark Energy Camera (DECAM) [247] on the Blanco telescope at Cerro Tololo Inter-American Observatory (CTIO). The 5 year survey will cover 5000 square degrees of the sky ∼ in 5 photometric ﬁlters grizY . The data collected by DES are used for cosmological measurements including cosmic shear (Sec 1.1.5), galaxy clustering (Sec 1.1.4), BAO (Sec 1.1.4), cluster counts, cluster lensing, and supernovae, as well as other non-cosmological measurements that will not be discussed here. The Year 1 DES data set [245] contains 137 million objects over a footprint ∼ of 1800 square degrees collected in the ﬁrst year of observations. A number of ∼ 66 3.1. MOTIVATION 67 cosmological analyses have been performed on this data set [2, 128]. This chapter focuses on the analysis in [2] which combines cosmic shear, galaxy-galaxy lensing and galaxy clustering to obtain cosmological constraints in a ΛCDM model, and a wCDM model (varying the dark energy equation of state w).

3.1 Motivation

In the ΛCDM model, the large scale amplitude of a 2D galaxy clustering measurement w(θ) is proportional to the product of the square of the amplitude of the matter power spectrum at the redshift of the galaxy sample and the linear galaxy bias squared b2 (see Equ. 1.19, 1.24 and 1.26). The galaxy clustering measurement therefore provides a constraint on σ8b. This degeneracy means the galaxy clustering amplitude cannot be used alone to constrain σ8 and other cosmological parameters.

A galaxy-galaxy lensing signal γt is also proportional to the square of the amplitude of the matter power spectrum but is only proportional to one factor of b (see Equ 1.38).

2 The galaxy-galaxy lensing measurement therefore provides a constraint on σ8b. If the same galaxy sample used in a clustering measurement is used as the lens sample in a galaxy-galaxy lensing measurement, these two probes can be combined to break this degeneracy and simultaneously measure b and σ8 [248, 133]. In the following chapters, this combination will be referred to as either w(θ) + γt or a 2x2pt analysis. In a tomographic analysis, the galaxy-galaxy lensing signal also provides some constraining power from the ratio of the γt signal from different source bins but the same lens bin. With fixed source and lens redshift distributions n(z), this provides a measurement of the power spectrum. If the cosmological parameters are fixed, these ratios can be used to constrain uncertainty in the source n(z). In the ΛCDM model, the tomographic cosmic shear measurements ξ primarily ± 0.5 constrain the parameter combination σ8(Ωm/0.3) . This can be combined with w(θ)+

γt to provide tighter cosmological parameter constraints in the σ8, Ωm plane and on dark energy extensions such as varying the dark energy equation of state w. By combining these probes, a self-calibration of the n(z) uncertainty can occur (e.g. [249]), as well as adding statistical power. In the following chapters, this combination of ξ , ± w(θ), and γt will be referred to as a 3x2pt analysis. 68 CHAPTER 3. COSMOLOGY FROM CLUSTERING AND LENSING

A 3x2pt analysis has recently been released by the KiDs experiment [138] where the KiDs photometric data have been used to produce a source sample for lensing measurements, and a overlapping sample of spectroscopic galaxies is used for the lenses. The advantage of using spectra for the lenses is that you have negligible lens redshift uncertainty. The disadvantage is that you are limited to only use the regions of your photometric sample that overlap with the spectra for the w(θ) and γt measurements. For the KiDs analysis [138], the spectra cover 150 square degrees of the total 450. Another disadvantage of using spectra is that one has no control over the speciﬁcations of the lens sample. For the KiDs analysis [138], the GAMA sample has a maximum redshift of 0.5, giving no overlap between the source distribution nsource(z) and lens distribution nlens(z) in the higher z bins. Another recent 3x2pt analysis was performed by the Dark Energy Survey [2] using a photometrically selected lens sample. A summary of this analysis is the focus of the rest of this chapter. The remainder of this thesis concerns this DES 3x2pt analysis.

3.2 Galaxy samples

The DES Year 1 analysis uses two different samples of source galaxies with measured ellipticities [243]. One source sample uses the shear measurement code im3shape [250], which fits bulge and disc galaxy profiles to single band images, selecting the ellipticity from the fit with the lowest χ2. In order to correct for shear calibration bias (Sec. 1.1.5), these measurements were tested on a suite of image simulations [103]. The im3shape catalog contains 21.9 million galaxies.

The second shape catalog used in the DES Year 1 analysis uses the shape measurement code Metacalibration [104, 105]. Metacalibration measures ellipticity using a Gaussian fit. By deconvolving the galaxy image with the PSF, applying an artificial shear, reconvolving with the PSF and then remeasuring the ellipticity, metacalibration can test the response of the shape estimator to an artificial shear and self-calibrate the shape catalog. Therefore, image simulations are not required for the metacalibration catalog. The DES Y1 metacalibration catalog contains 34.8 million galaxies.

The source samples are split into 4 tomographic redshift bins, 0.2 < z < 0.43, 3.2. GALAXY SAMPLES 69

0.43 < z < 0.63, 0.63 < z < 0.9, 0.9 < z < 1.3 according to redshift point estimates from the BPZ algorithm [251]. The redshift distributions of the source samples nsource(z) are shown in Figure 3.1. The source samples are used for the ξ measurments ± and as the background sample in the γt measurement.

The lens sample is used for the galaxy clustering measurement and as the foreground sample in the galaxy-galaxy lensing γt. In the DES Year 1 analysis a sample of Luminous Red Galaxies (LRGs) was selected using the redMaGiC algorithm [252], split into ﬁve equally spaced tomographic redshift bins between z = 0.15 and z = 0.9. This sample is described in detail in Chapter 4. The redshift distributions of the lens galaxies are shown in Figure 3.1.

Each ξ , γt, w(θ) measurement is evaluated in 20 angular bins between θ = 2.50 ± and θ = 2500. The ξ+ and ξ are measured for each of the source bin combinations −

(10 total); γt is measured for each source-lens combination (20 total); only the auto- correlation is measured for w(θ) in each lens bin (5 total). This results in a data vector of 900. Only 457 of these data points are used in the ﬁnal analysis due to scale cuts (see Sec. 3.4).

The ﬁducial source redshift distributions are determined by stacking pdfs for each galaxy using the BPZ algorithm [251]. The uncertainty in each redshift distribution is parameterised by a single shift parameter per redshift bin ∆zi,

ni (z) = ni (z ∆zi) (3.1) source BPZ −

i where nBPZ is the sum of the individual probability distributions from the BPZ template ﬁtting for all galaxies in z bin i.

Each ∆zi is marginalized in the MCMC analysis with a Gaussian prior applied. The width and centre of these priors is determined by two methods. One method compares the DES n(z) with the overlapping COSMOS 10-band photometry [253]. The other using the clustering signal between the DES galaxies and an overlapping sample with high precision redshift. These methods are detailed in [242, 254, 255, 256]. 70 CHAPTER 3. COSMOLOGY FROM CLUSTERING AND LENSING

8 7 Lenses 6 redMaGiC 5 4 3 2

Normalized counts 1 0 5 Sources METACALIBRATION 4 IM3SHAPE 3

Normalized counts 1

0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Redshift

Figure 3.1: The normalised redshift distributions of the galaxy samples used in the DES year 1 3x2pt analysis. The top panel shows the lens galaxy distributions, calculated by summing the estimated Gaussian pdf of each galaxy from the redMaGiC algorithm. The bottom panel shows the source galaxy distributions for the two source catalogs. These have the same redshift bin selection but diﬀerent n(z) due to diﬀering quality cuts. The black lines show the n(z) for the full samples, the coloured lines show the n(z) of each individual redshift bin. The galaxies are selected by point estimates of the redshift and the n(z) is calculated by stacking the full pdf of each galaxy. Therefore, the mean of the n(z) is not necessarily in the centre of the selection bin. Figure from [2].

3.3 Multi-probe Covariance

The likelihood L(D Θ) for the DES Y1 data vector D at a given parameter vector Θ | is given by,

1 T 1 L(D Θ) = N exp [(D M) C− (D M)] (3.2) | − 2 − − where M is the model prediction at parameter vector Θ, C is the covariance matrix for the data, and N is a normalisation factor. The covariance C includes the covariance matrix of each two-point function individually, as well as the covariance between functions. C is calculated analytically using the Cosmolike likelihood framework [257]. This ﬁrst calculates the covariance in Fourier space, 3.3. MULTI-PROBE COVARIANCE 71

(α,β) (ν,µ) G (α,β) (ν,µ) Cov(CAB (l),CCD (l)) = Cov (CAB (l),CCD (l)) NG (α,β) (ν,µ) (3.3) +Cov (CAB (l),CCD (l))

SSC (α,β) (ν,µ) +Cov (CAB (l),CCD (l)) where A, B, C, D label the ﬁelds (galaxy density δg or shear δs), α, β, ν, and µ label the redshift bins. CovG is the Gaussian term (including shot noise), CovNG is the non-Gaussian term calculated using the halo model, and CovSSC is the super sample covariance. The Gaussian term is given by,

2π CovG(C(α,β)(l),C(ν,µ)(l)) = (( C(α,ν)(l) + δD δD N α )( C(β,µ)(l) + δD δD N β ) AB CD ldlA AC α,ν AC A BD β,µ BD B (α,µ) D D α (β,ν) D D β +( CAD (l) + δα,µδADNA )( CBC (l) + δβ,νδBC NB )) (3.4)

D α where δ is the delta function and NA is the noise term given by:

α 2 α Nδs = σ /n¯source (3.5)

α α Nδg = 1/n¯lens (3.6)

The Non-Gaussian term CovNG uses the projected trispectrum, calculated using the halo model [87]. Because the region of the universe sky being probed by DES could sit in a large- scale over (or under) density, the super sample covariance term CovSSC is also required. This depends on the variance of the underlying density ﬁeld. CovNG and CovSSC are described in detail in [257]. The Fourier space covariance is then converted into real space by integrating over Bessel functions,

1 Z l 2 Cov(ξα,β (θ ), ξν,µ (θ )) = dl J (lθ )J (lθ )Cov(Cα,β,Cν,µ ) (3.7) AB(i) 1 CD(j)) 2 A 2π i 1 j 2 AB CD where Ji is a Bessel function of order i, and i and j correspond to the relevant Bessel functions for the C(l) ξ transformation in question (e.g. i = 0 for w(θ), i = 2 for → γt(θ) ). 72 CHAPTER 3. COSMOLOGY FROM CLUSTERING AND LENSING

This analytic covariance is then validated on log-normal realisations of the shear and galaxy density ﬁelds with the FLASK simulations [258]. The details of this covariance validation are described in [244].

3.4 Analysis choices and scale cuts

3.4.1 Analysis choices

The analysis was performed varying a total of 26 parameters. The cosmological pa-

2 rameters Ωm, H0, Ωb, As, Ωνh , ns each have ﬂat priors (DES large scale structure measurements are not sensitive to τ). Galaxy bias is modelled with a single linear bias parameter for each lens bin bi. Scales for which this assumption is not valid were removed from the likelihood calculation. The shear calibration bias is marginalised over using a single new parameter for each source bin mi. Details and testing of this model were presented in [128] and [243]. Intrinsic alignments are modelled with the non-linear alignment model [109]. Two parameters were allowed to vary; AIA, which determines the overall amplitude of the

IA signal; and αIA which determines the redshift dependance of the IA signal. Details and testing of this model were presented in [128] and [2]. The uncertainty of each of the nine redshift distributions (ﬁve lens, four source) was captured by a single redshift bias parameter ∆zi. This is applied as a shift in z to each n(z). A Gaussian prior is applied to each ∆zi as discussed in Sec. 3.2. A summary of all the parameters and priors in the DES Y1 analysis is given in Table 3.1. The parameter sampling was performed using the multinest nested sampling algorithm [259] which allows easy computation of the Bayesian evidence.

3.4.2 Scale cuts

Due to uncertainties in the small scale modelling of the correlation functions, data points below a minimum angular scale for each of w(θ), γt, ξ are removed from ± the likelihood calculation. The small scale modelling can be aﬀected by a number of 3.4. ANALYSIS CHOICES AND SCALE CUTS 73

Table 3.1: Table describing the priors on all of the cosmological and nuisance parameters in the DES Y1 3x2pt analysis. Gauss(µ, σ) denotes a Gaussian prior with mean µ and width σ. ﬂat(a, b) denotes a ﬂat prior between a and b. Table from [2].

Parameter Prior Cosmology Ωm flat (0.1, 0.9) 10 9 A flat (5 10− , 5 10− ) s × × 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− ) w flat (× 2, 0.33) Lens Galaxy Bias − − bi(i = 1, 5) flat (0.8, 3.0) Intrinsic Alignment αIA AIA(z) = AIA[(1 + z)/1.62] AIA flat ( 5, 5) α flat (−5, 5) IA − 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) i mim3shape(i = 1, 4) Gauss (0.0, 0.035) 74 CHAPTER 3. COSMOLOGY FROM CLUSTERING AND LENSING potential systematics. The systematics tested in the DES Y1 analysis at the theoretical modelling level are listed below.

Baryonic feedback: The impact of baryons on the matter power spectrum, • either through AGN feedback or Supernovae feedback, can be propagated to the 2pt functions measured by DES. The impact of baryons is measured on hydro- dynamical simulations and compared to Dark-matter only N-body simulations.

Non-linear galaxy bias: On small scales, the assumption of linear galaxy bias • is no longer valid.

Non-local γ : The galaxy-galaxy lensing signal is non-local and contains signif- • t icant contributions from non-linear scales. The small scales of γt are aﬀected by the so called "1-halo term" which depends on the halo mass and the halo mass proﬁle [244, 248].

Intrinsic alignment model: The DES analysis assumes a relatively simple • IA model (NLA), with redshift dependance determined by a single parameter. More complex intrinsic alignment models such has tidal-alignent tidal-torquing models [111] or diﬀerent IA amplitudes for red and blue galaxies are considered as systematics for the DES 3x2pt analysis.

Biased n(z): If the prior on shift in the n(z) was misestimated, this can lead • to a bias in cosmology.

Limber approximation and RSD: The limber approximation (Equ. 1.24) is • assumed for all 2pt correlation function measurements. Redshift space distortions are also ignored in the modelling. These impact a wide range of scales in the clustering 2pt function, not just small scales.

For each of these potential small scale systematics, a simulated data vector is produced, contaminated by the systematic. These simulated vectors simply sample the modelled correlation function at the same angular separations as the data. The details of the modelling of each of these systematics is described in Chapter 5. A full MCMC likelihood analysis is then performed on each of the contaminated data vectors, plus a baseline data vector with no contamination. The scale cuts are selected 3.5. DES YEAR 1 LSS COSMOLOGY RESULTS 75 at a ﬁxed physical scale for each probe, such that the contaminated vectors give an unbiased S8 posterior distribution in ΛCDM and an unbiased w posterior distribution in wCDM, compared to the baseline case. For DES Y1 these scale cuts are at 8Mpc for w(θ) and ξ , and 12Mpc for γt. The initial estimates for the scales cuts were decided ± by the fractional impact of the largest contaminate. For ξ the limiting systematic ± was the impact of baryons. For w(θ) the limiting systematic is non-linear galaxy bias. The details of the scale cuts analysis are presented in full in [244].

3.5 DES Year 1 LSS cosmology results

The ﬁnal posterior distributions of the DES Y1 3x2pt analysis and the comparison with external data sets are shown and summarized in this section. These results were presented in full in [2]. Consistency between data sets and model comparison are assessed using the Bayes evidence ratio. For model comparison, the evidence ratio is ,

P (D M ) R = | 1 (3.8) P (D M ) | 2 where D is the data and M1 and M2 are the two models being compared. For consistency tests between two data sets in the same model, the evidence ratio is,

P (D ,D M) R = 1 2| (3.9) P (D M)P (D M) 1| 2| where D1 and D2 are the two data sets being compared and M is the model. Each evidence was calculated during the nested sampling of the posterior distribution, using the Multinest algorithm. Consistency was assessed on the Jeﬀreys scale where R < 0.1 is strong evidence of inconsistency, 0.1 < R < 0.31 is substantial evidence of inconsistency, 0.32 < R < 10 is substantial evidence of consistency, and R > 10 is strong evidence of consistency. Data sets were combined unless strong evidence for inconsistency was found (R < 0.1). Figure 3.2 shows the cosmic shear constraints from ξ , the combination of galaxy- ± galaxy lensing and clustering γt + w(θ) (2x2pt) and the full 3x2pt combination in 76 CHAPTER 3. COSMOLOGY FROM CLUSTERING AND LENSING

ΛCDM. The evidence ratio for combining ξ and 2x2pt is R = 583, which indicates ± strong evidence for consistency.

0.90

0.85

0.80 8 S 0.75

0.70

1.05 DES Y1 Shear DES Y1 w + γt DES Y1 All 0.90 8 σ

0.75

0.60

0.2 0.3 0.4 0.5 0.70 0.75 0.80 0.85 0.90 Ωm S8

Figure 3.2: ΛCDM constraints from the individual DES probes, cosmic shear and w(θ)+γt and the 3x2pt combination. Contours are shown for 68% and 95% conﬁdence levels. Figure from [2].

Figure 3.3 shows the ξ , γt + w(θ) and 3x2pt constraints for the wCDM model ± (allowing the dark energy equation of state to vary). The evidence ratio for combining these two data sets is R = 1878, which indicates strong evidence for consistency. The left panel of Figure 3.4 shows the DES 3x2pt constraints compared to the Planck 2015 results using the full TT and low-l polarisation power spectra, in ΛCDM model. The evidence ratio for combining these two data sets is R = 6.6, which indicates substantial evidence for consistency. The right panel of of Figure 3.4 shows a comparison between DES and the combination of Planck, Planck lensing, BAO (from the 6dF Galaxy Survey [260], the SDSS Data Release 7 Main Galaxy Sample [261], and BOSS Data Release 12 [262]), and 3.5. DES YEAR 1 LSS COSMOLOGY RESULTS 77

DES Y1 Shear DES Y1 w + γt DES Y1 All

.4 0 − .7 0 − .0 1

w − .3 1 − .6 1 −

.90 0

.84 0

.78

8 0 S .72 0

.66 0

.6 .3 .0 .7 .4 .16 .24 .32 .40 .48 1 1 1 0 0 .66 .72 .78 .84 .90 0 0 0 0 0 0 0 0 0 0 − − − − − Ωm w S8

Figure 3.3: wCDM constraints from the individual DES probes, cosmic shear and w(θ)+γt and the 3x2pt combination. Contours are shown for 68% and 95% confidence levels. Figure from [2]. supernovae constraints (from JLA [142]). Figure 3.5 shows the DES 3x2pt constraints, Planck, and the combination of DES, Planck, BAO and supernovae results in wCDM model. The evidence ratio for combining these two data sets is R = 10.3, which indicates strong evidence for consistency. The evidence ratio values obtained when testing for consistency between DES and Planck are close to the Jeffreys scale boundaries. It is therefore difficult to make any definitive statements about consistency without considering other consistency metrics. The evidence ratio comparing the two models ΛCDM and wCDM for the full combined data set DES+Planck+BAO+JLA is R = 0.7. Therefore the wCDM model is not currently favored by the data. The DES 3x2pt constraint on the dark energy equation of state w is,

+0.21 w = 0.82 0.20. (3.10) − − 78 CHAPTER 3. COSMOLOGY FROM CLUSTERING AND LENSING

0.96 DES Y1 0.90 Planck (No Lensing) DES Y1 DES Y1 + Planck (No Lensing) Planck+BAO+JLA DES Y1+Planck+BAO+JLA 0.90 0.85

8 0.84 8 S 0.80 S

0.78 0.75

0.72

0.70 0.24 0.30 0.36 0.42 0.48 0.24 0.28 0.32 0.36 Ωm Ωm

Figure 3.4: Left: ΛCDM constraints from DES 3x2pt compared to those from Planck temperature and low-l polaristation spectra. Right: ΛCDM constraints from DES 3x2pt compared to the combination of Planck, BAO and SN data. In both of these comparisons, no strong evidence for inconsistency was found from the evidence ratio test. Figure from [2].

Combining DES 3x2pt with Planck+BAO+SN, the w constraint narrows to,

+0.05 w = 1.00 0.04. (3.11) − −

3.6 Conclusions

This chapter presents and summarizes the DES year 1 analysis of weak lensing and galaxy clustering. This analysis was found to be both internally consistent, and consistent with the CMB and other cosmological probes. This analysis focusses on the ΛCDM and wCDM models, ﬁnding no preference for the latter. The DES year 1 analysis required a number of systematic analyses to produce robust results. Chapter 4 will examine the galaxy clustering systematics part of this analysis in more detail. The DES year 1 analysis assumes 3 neutrino masses and marginalises over their sum. Sterile neutrinos are not included in the baseline analysis. I extend the analysis to include sterile neutrinos in Chapter 5. 3.6. CONCLUSIONS 79

Planck No Lensing DES Y1 DES Y1+Planck No Lensing DES Y1+Planck+BAO+JLA

0.80

0.72 h

0.64

0.56

0.6 −

1.0 − w

1.4 −

1.8 −0.90

0.85

8 0.80 S

0.75

0.70 0.24 0.30 0.36 0.42 0.56 0.64 0.72 0.80 1.8 1.4 1.0 0.6 0.70 0.75 0.80 0.85 0.90 − − − − Ωm h w S8

Figure 3.5: wCDM constraints from DES 3x2pt compared to those from Planck, and a combination of DES with all external data sets considered in this work. Figure from [2]. Chapter 4

Galaxy clustering for combined probes in the Dark Energy Survey Year 1 data

This Chapter is based on [3]. The production of the original galaxy catalogs and redshift estimates were done in collaboration with the co-authors.

4.1 Introduction

Galaxies are a biased tracer of the matter density field. In the standard ‘halo model’ paradigm, they form in collapsed over-densities (dark matter halos; [263]), and the mass of the halo they reside in is known to correlate with the luminosity and colour of the galaxy, with more luminous and redder galaxies strongly correlated with higher mass. Therefore, the galaxy ‘bias’ depends strongly on the particular sample being studied. Thus, in cosmological studies the galaxy bias is often treated as a nuisance parameter — one that is degenerate with the amplitude of the clustering of matter. See, e.g., [264] and references therein. The degeneracy can be broken with additional observables. This includes the weak gravitational lensing ‘shear’ field, which is induced by the matter density field. Correlation between the galaxies and the shear field ([130]; often referred to as ‘galaxy- galaxy lensing’) contains one factor of the galaxy bias and two factors of the matter field. The galaxy auto-correlation contains two factors of the galaxy bias and again

80 4.1. INTRODUCTION 81 two factors of the matter field. Thus, the combination of the two measurements can break the degeneracy between the two quantities, and it is a sensitive probe of the late-time matter field (see, e.g., [133, 265]). The auto-correlation of the shear field alone includes no factors of the galaxy bias and can thus be used directly as a probe of the matter field. However, its sensitivity to many systematic uncertainties related to the estimation of the shear field differs from that of the galaxy-galaxy lensing signal. As shown by [266, 267, 268, 249], the impact of such systematic uncertainties can be mitigated by combining the shear auto- correlation measurements with those of galaxy clustering and galaxy-galaxy lensing. Thus there is substantial gain to be obtained from a combined analysis. Such a combined analysis is performed with the Dark Energy Survey (DES1; [246]) Year-1 (Y1) data ([2]; hereafter Y1COSMO). DES is an imaging survey currently amassing data over a 5000 deg2 footprint in five passbands (grizY ). When completed, it will have mapped 300 million galaxies and tens of thousands of galaxy clusters. In this work, I study the clustering of red sequence galaxies selected from DES Y1 data using the redMaGiC [252] algorithm, chosen for its small redshift uncertainty. This is the same sample used to obtain cosmological results in the Y1COSMO combined analysis. In particular, I study the large-scale clustering amplitude and its sensitivity to observational systematics. Following previous studies [269, 270, 271, 272], angular maps are used to track the observing conditions of the Y1 data in order to identify and correct for spurious fluctuations in the galaxy density field. I further determine the effect these corrections have on the covariance matrix of the angular auto-correlation of the galaxies. I present robust measurements of the clustering amplitude of redMaGiC galaxies as a function of redshift and luminosity, thus gaining insight into the physical nature of these galaxies and how they compare to other red galaxy samples. The results of this paper were used for the joint DES cosmological analysis presented in Y1COSMO and Chapter 3. This outline of this chapter is: Sec. 4.2 summarizes the model used to describe the galaxy clustering measurements; Sec. 4.3 presents the DES data used; Sec. 4.4 presents how the clustering statistics are measured and their covariance estimated; Sec. 4.5 summarizes the results of the observational systematic tests. The primary

1http://www.darkenergysurvey.org/ 82 CHAPTER 4. CLUSTERING IN DES results of galaxy bias measurements are presented in Sec. 4.6 and a demonstration of their robustness in Sec. 4.7 before concluding in Sec. 4.8.

In order to avoid conﬁrmation bias, this analysis was performed “blind”: I did not measure parameter constraints or plot the correlation function measured from the data on the same axis as any theoretical prediction or simulated clustering measurement until the sample and all measurements in Y1COSMO were ﬁnalized. The analysis was tested on simulations before running on data.

Unless otherwise noted, I use a ﬁducial ΛCDM cosmology, ﬁxing cosmological pa-

9 rameters at Ω = 0.295, A = 2.260574 10− , Ω = 0.0468, h = 0.6881, n = 0.9676. m s × b s This is consistent with the latest cosmological data from the Planck mission [10] and is used as the ﬁducial cosmology for all the DES Y1 analyses used in Y1COSMO. This cosmology was used to generate Gaussian mocks in Sec. 4.5 for systematics testing.

After un-blinding, the galaxy bias was re-measured, ﬁxing the cosmological param-

9 eters at the mean of the DES Y1COSMO posterior, Ω = 0.276, A = 2.818378 10− , m s × Ωb = 0.0531, h = 0.7506, ns = 0.9939, Ων = 0.00553 (note that these values are shown at a greater precision than can be measured by DES). This cosmology was used for all bias measurements in Sec. 4.6.

4.2 Theory

Throughout this paper, redMaGiC clustering measurements are modelled assuming a local, linear galaxy bias model [273], where the galaxy and matter density fluctuations are related by δ (x) = bδ (x), with density fluctuations defined by δ (n(x) n¯)/n¯. g m ≡ − The validity of this assumption over the scales considered here is provided in [244] and Chapter 3 and shown in simulations in [274].

The galaxy clustering model used in this paper matches that used in Y1COSMO. This model includes 3 neutrinos of degenerate mass.

Multiple galaxy redshift bins i are considered, each characterized by a redMaGiC

i galaxy redshift distribution ng(z), normalized to unity in redshift, and a galaxy bias bi which is assumed to be constant across the redshift bin range. Under the Limber [275] and flat-sky approximation the theoretical prediction for the galaxy correlation 4.2. THEORY 83 function w(θ) in a given bin is, Z Z 2 dl l wi(θ) = bi J (lθ) dχ 2π 0 2 ni (z(χ))dz/dχ l + 1/2 g P , z(χ) , (4.1) × χ2 NL χ where the speed of light has been set to one, χ(z) is the comoving distance to a given redshift (in a flat universe, which is assumed throughout); J0 is the Bessel function of order zero; H(z) is the Hubble expansion rate at redshift z; and PNL(k; z) is the 3D matter power spectrum at redshift z and wavenumber k (which, in this Limber approximation, is set equal to (l + 1/2)/χ). Note that in Eq. 4.1 I have assumed the bias to be constant within each bin, see Fig. 8 in [244] for a test of this assumption. Again, all assumptions and approximations mentioned here have been shown to be inconsequential in [244, 274]. To model cross-correlation between redshift bins, simply change ni (z)2 ni (z)nj (z) and (bi)2 bibj in Eq. 4.1. g → g g → Throughout this paper, I use the CosmoSIS framework [227] to compute correlation functions, and to infer cosmological parameters. The evolution of linear density fluctuations is obtained using the Camb module [226] and then converted to a non- linear matter power spectrum PNL(k) using the updated Halofit recipe [45]. The theory modeling assumes the Limber approximation, and it also neglects redshift space distortions. For the samples and redshift binning used in this paper, those effects start to become relevant at scales of θ & 1 deg [276, 277, 278]. In the DES paper [244] it is explicitly shown that they have negligible impact in derived cosmological parameters for this galaxy sample, given the statistical error bars of DES Y1. Con- cretely, a theory data vector was produced with the exact (non-Limber) formula (Equ. 1.22) including redshift space distortions and was then analyzed with the baseline pipeline assumed here, and also in Y1COSMO. Figure 8 in [244] shows that including or not including these contributions makes negligible impact in parameters such as

Ωm and S8 for a ΛCDM universe or w in a wCDM one. The impact of these effects on the fixed-cosmology bias measurements in Sec. 4.6 was also tested and found to be negligible. Hence in what follows, such terms are ignored for speed and simplicity. However future data analyses may need to account for these effects due to improved statistical uncertainty. DES models (and marginalizes over) photometric redshift bias uncertainties as an 84 CHAPTER 4. CLUSTERING IN DES

i i additive shift ∆z in the redMaGiC redshift distribution nRM(z) for each redshift bin i.

ni(z) = ni (z ∆zi) (4.2) RM −

The priors on the ∆zi nuisance parameters, are measured directly using the angular cross correlation between the DES sample and a spectroscopic sample. These values are shown in Table 4.2, and the method is described in full in [254]. I use the same ∆zi as Y1COSMO for all tests of robustness of the parameter constraints.

I also compare the measurements of bi to the same quantity measured by galaxy- galaxy lensing using the two-point correlation function γt (see Equ. 1.37 and 1.38 for definition). I use the notation bi for this measurement. The details of this mea- × surement are described in [140] (hereafter Y1GGL). In order to take the off-diagonal elements of the covariance matrix between the two probes into account, I produce joint constraints from w(θ) and γt at fixed cosmology (the mean of the Y1COSMO posterior), using different bias parameters for the two probes, and marginalizing over the same nuisance parameters as were used in the fiducial analysis of Y1COSMO (intrinsic alignments, source and lens photo z bias, and shear calibration). To test the − consistency between the two probes the parameter r is used,

i i b r = × . (4.3) bi

If r = 1, this indicates an inconsistency between the two bias measurements and would 6 thus suggest a breakdown of the simple linear bias model. This test informs the choice of ﬁxing r = 1 in the Y1COSMO analysis.

A combination of galaxy clustering and galaxy-galaxy lensing, provides a measurement of galaxy bias and σ8 only if you assume that r = 1. This test provides a measurement of r which informs the choice of ﬁxing r = 1 in the Y1COSMO analysis. In principle, this test could also be performed by including the shear-shear correlation which also measures σ8. 4.3. DATA 85 4.3 Data

4.3.1 Y1 Gold

I use data taken in the ﬁrst year (Y1) of DES observations [279]. Photometry and ‘clean’ galaxy samples were produced with these data as outlined by [245] (hereafter denoted ‘Y1GOLD’). The outputs of this process represent the Y1 ‘Gold’ catalog. Data were obtained over a total footprint of 1800 deg2; this footprint is deﬁned by ∼ a Healpix [280] map at resolution Nside = 4096 (equivalent to 0.74 square arcmin) and includes only pixels with minimum exposure time of at least 90 seconds in each of the g,r,i, and z-bands, a valid calibration solution, as well as additional constraints (see Y1GOLD for details). A series of veto masks, including among others masks for bright stars and the Large Magellanic Cloud, reduce the area by 300 deg2, leaving ∼ 1500 deg2 suitable for galaxy clustering study. I explain further cuts to the angular ∼ mask in Sec. 4.3.2. All data described in this and in subsequent sections are drawn from catalogues and maps generated as part of the DES Y1 Gold sample and are fully described in Y1GOLD.

4.3.2 redMaGiC sample

The galaxy sample used in this work is generated by the redMaGiC algorithm, run on DES Y1 Gold data. The redMaGiC algorithm selects Luminous Red Galaxies (LRGs) in such a way that photometric redshift uncertainties are minimized, as is described in [252]. This method is able to achieve redshift uncertainties σz/(1 + z) < 0.02 over the redshift range of interest. The redMaGiC algorithm produces a redshift prediction zRM and an uncertainty σz which is assumed to be Gaussian. This sample was chosen instead of other DES photometric samples because of its small redshift uncertainty, which is obtained at the expense of number density. The redMaGiC algorithm makes use of an empirical red-sequence template generated by the training of the redMaPPer cluster finder [281, 282]. As described in [282], training of the red-sequence template requires overlapping spectroscopic redshifts, which in this work were obtained from SDSS in the Stripe 82 region [283] and the OzDES spectroscopic survey in the DES deep supernova fields [284]. For the redMaGiC samples in this work, I make use of two separate versions 86 CHAPTER 4. CLUSTERING IN DES of the red-sequence training. The first is based on SExtractor MAG_AUTO quantities from the Y1 coadd catalogs, as applied to redMaPPer in [285]. The second is based on a simultaneous multi-epoch, multi-band, and multi-object fit (MOF) (see Sec. 6.3 of Y1GOLD), as applied to redMaPPer in [286]. In general, due to the careful handling of the point-spread function (PSF) and matched multi-band photometry, the MOF photometry yields lower colour scatter and hence smaller scatter in red-sequence photo-zs. For each version of the catalog, photometric redshifts and uncertainties are primarily derived from the fit to the red-sequence template. In addition, an afterburner step is applied (as described in Sec. 3.4 of [252]) to ensure that redMaGiC photo-zs and errors are consistent with those derived from the associated redMaPPer cluster catalog [252].

As described in [252], the redMaGiC algorithm computes colour-cuts necessary to produce a luminosity-thresholded sample of constant co-moving density. Both the luminosity threshold and desired density are independently conﬁgurable, but in practice higher luminosity thresholds require a lower density for good performance. Note that in [252], the co-moving density was computed with the central redshift of each galaxy (zRM). For this work, the density was computed by sampling from a Gaussian distribution z σ , which creates a more stable distribution near ﬁlter transitions. RM ± z This is the only substantial change to the redMaGiC algorithm since the publication of [252].

I use redMaGiC samples split into five redshift bins of width ∆z = 0.15 from z = 0.15 to z = 0.9. I define the footprint such that the data in each redshift bin will be complete to its redshift limit across the entire footprint. To make this possible, I define samples based on a luminosity threshold. Reference luminosities are computed as a function of L , computed using a [287] model for a single star-formation burst ∗ at z = 3 (see Sec. 3.2 [282]). Naturally, increasing the luminosity threshold provides a higher redshift sample as well as decreasing the comoving number density. Using a different luminosity threshold in each redshift bin allows us to maximize signal to noise while also providing a complete sample in each redshift bin. The details of these bins are given in Table 4.1.

The 5 redshift bins were chosen so that the width of the bins is signiﬁcantly wider than the uncertainty on individual galaxy redshifts, but smaller than the diﬀerence 4.3. DATA 87

2 z range Lmin/L ngal (arcmin− ) Ngal Photometry ∗ 0.15 < z < 0.3 0.5 0.0134 63719 MAGAUTO 0.3 < z < 0.45 0.5 0.0344 163446 MAGAUTO 0.45 < z < 0.6 0.5 0.0511 240727 MAGAUTO 0.6 < z < 0.75 1.0 0.0303 143524 MOF 0.75 < z < 0.9 1.5 0.0089 42275 MOF

Table 4.1: Details of the sample in each redshift bin. Lmin/L describes the minimum ∗ luminosity threshold of the sample, ngal is the number of galaxies per square degree, and Ngal is the total number of galaxies. between the maximum redshifts of the luminosity thresholds used.

In addition to the primary redMaGiC selection, I also apply a cut on the luminosity L/L < 4 as this was shown for DES Science Verification to reduce the stellar ∗ contamination in the sample, although this is mostly superfluous for Y1 Gold. During testing, I find that the observational systematic relationships for the 0.5L sample, ∗ used for z < 0.6, are minimized for the MAG_AUTO sample, with a very minor impact on photo-z performance. For L 1.0, used for z > 0.6, I instead find that the observa- ∗ ≥ tion systematic relationships are minimized for the MOF sample, and that the photo-z performance is also improved. Consequently, I use MAG_AUTO for the z < 0.6 sample and MOF for z > 0.6. See Sec. 4.5 for further discussion.

I build the area mask for the redMaGiC samples based on the depth information produced with the redMaGiC catalogs. This information is provided by the zmax quantity, which describes the highest redshift at which a typical red galaxy of the adopted luminosity threshold (e.g. 0.5L ) can be detected at 10σ in the z-band, at ∗ 5σ in the r and i-bands, and at 3σ in the g-band, as described in Sec. 3.4 of [282].

The quantity zmax varies from point to point in the survey due to observing conditions.

Consequently, I construct a zmax map, speciﬁed on a HealPix map with Nside = 4096. In order to obtain a uniform expected number density of galaxies across the footprint,

I only use regions for which zmax is higher than the upper edge of the redshift bin under consideration. The footprint is deﬁned as the regions where this condition is met in all redshift bins. Thus, I only use pixels that satisfy each of the conditions where the 0.5L sample is complete to z = 0.6, the 1.0L sample is complete to z = 0.75, and ∗ ∗ the 1.5L sample is complete to z = 0.9. I also restrict the analysis to the contiguous ∗ region shown in Figure 4.1. 88 CHAPTER 4. CLUSTERING IN DES

30 0.30 35 ] 0.25 2 n

40 i m c r DEC 0.20

45 a [

g n 50 0.15

0.10 120 100 320 300 RA Figure 4.1: Galaxy distribution of the redMaGiC Y1 sample used in this analysis. The ﬂuctuations represent the raw counts, without any of the corrections derived in this analysis. This analysis was restricted to the contiguous region shown in the ﬁgure. The area is 1321 square degrees.

An additional 1.6% of the footprint is vetoed because it has extreme observing conditions. The selection of these cuts is detailed in Sec. 4.5. After masking and additional cuts, I obtain a total sample of 653,691 objects distributed over an area of 1321 square degrees, as shown in Fig. 4.1. The average redshift uncertainty of the sample is σz/(1 + z) = 0.0166. The redshift distribution of each bin can be seen in Figure 4.2. The number of objects in each bin increases up to z = 0.6 due to the increase in volume, and decreases at higher redshift due to the increased luminosity threshold.

4.4 Analysis methods

4.4.1 Clustering estimators

The correlation functions w(θ) are calculated using the Landy & Szalay estimator [85]

DD 2DR + RR wˆ(θ) = − , (4.4) RR where DD, RR and DR are the number of pairs of galaxies from the galaxy sample D and a random catalog R. This is calculated in 20 logarithmically separated bins 4.4. ANALYSIS METHODS 89

1.8 0.15 < z < 0.3 1.6 0.3 < z < 0.45 0.45 < z < 0.6 1.4 0.6 < z < 0.75 0.75 < z < 0.9 1.2 6 −

0 1.0 1 ×

) 0.8 z (

n 0.6

0.4

0.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 z

Figure 4.2: Redshift distribution of the combined redMaGiC sample in 5 redshift bins. They are calculated by stacking Gaussian PDFs with mean equal to the red- MaGiC redshift prediction and standard deviation equal to the redMaGiC redshift error. Each curve is normalized so that the area of each curve matches the number of galaxies in its redshift bin. in angle θ between 2.5 arcmin and 250 arcmin to match the analysis in Y1COSMO. I use 60 times more randoms than data. The pair-counting was done with the package tree-corr [288]2.

I also calculate w(θ) on Gaussian random ﬁeld realizations which are described in a pixelated map format. For these correlations I use a pixel-based estimator. Using the notation of [272], the correlation between two maps N1 and N2 of mean values

N¯1, N¯2 is estimated as

Npix Npix X X (Ni,1 N¯1)(Nj,2 N¯2) wˆ (θ) = − − Θ , (4.5) 1,2 N¯ N¯ i,j i=1 j=i 1 2

where the sum runs through all pairs of the Npix pixels in the footprint, Ni,1 is the value of the N1 map in pixel i, and Θi,j is unity when the pixels i and j are separated by an angle θ within the bin size ∆θ. I have tested that the diﬀerence between the estimators in Equ. 4.4 and 4.5 is negligible for this analysis.

2available at https://github.com/rmjarvis/TreeCorr 90 CHAPTER 4. CLUSTERING IN DES

4.4.2 Covariances

The fiducial covariance matrix used for the w(θ) measurement is a theoretical halo model covariance, described and tested by [244] and in Chapter 3. The covariance is generated using CosmoLike [257], and is computed by calculating the four-point correlation functions for galaxy clustering in the halo model. Additionally, an empirically determined correction for the survey geometry’s effect on the shot-noise component has been added. The presence of boundaries and holes decrease the effective number of galaxy pairs as a function of pair separation, which in turn raises the error bud- get associated to shot noise over the standard uniform sky assumptions. This same covariance is used for the combined probes analysis and is detailed in Y1COSMO. For the analysis of observational systematics and their correlation with the data, I use a set of 1000 mock surveys (hereafter ‘mocks’) based on Gaussian random field realizations of the projected density field. These are then used to obtain an alternative covariance, which includes all the mask effects as in the real data. The mocks used were produced using the following method. I first calculate, using Camb [226], the gg galaxy clustering power spectrum Ci (`), assuming the fiducial cosmology with fixed galaxy bias bi for each redshift bin i; the galaxy bias values are listed in Table 4.2. The angular power spectrum is then used to produce a full-sky Gaussian random field of δg. I apply a mask to this field corresponding to the Y1 data, as shown in Fig. 4.1.

This is converted into a galaxy number count Ngal as a function of sky position, with the same mean as the observed number count N¯o in each redshift bin, using

Ngal = N¯o(1 + δg). (4.6)

Shot noise is finally added to this field by Poisson sampling the Ngal field. In order to avoid pixels with δ < 1, which cannot be Poisson sampled, I follow g − the method used by [244]: before Poisson sampling, I multiply the density field by

2 a factor α, where α < 1; I then rescale the number density ngal by 1/α in order to preserve the ratio of shot-noise to sample variance; I then rescale the measured w(θ) by 1/α2 to obtain the unbiased w(θ) for each mock. This procedure is summarized by

δ αδ , (4.7) g → g n n /α2 , (4.8) gal → gal w(θ) w(θ)/α2 . (4.9) → 4.5. SYSTEMATICS 91

i i z range bﬁd ∆z 0.15 < z < 0.3 1.45 Gauss (0.008, 0.007) 0.3 < z < 0.45 1.55 Gauss ( 0.005, 0.007) 0.45 < z < 0.6 1.65 Gauss− (0.006, 0.006) 0.6 < z < 0.75 1.8 Gauss (0.00, 0.010) 0.75 < z < 0.9 2.0 Gauss (0.00, 0.010)

Table 4.2: Details of the fiducial parameters used for covariance and parameter con- i straints. Here, bfid is the fiducial linear galaxy bias for bin i applied to the Gaussian mock surveys I use to construct the covariance matrices. The ∆zi prior is a Gaussian prior applied to the additive redshift bias uncertainty. These were selected to match the analysis in ([2]; Y1COSMO).

The mocks are used to estimate statistical errors in galaxy number density as a function of potential systematics. Alternatively, I “contaminate” each of the 1000 mocks with survey properties as discussed in Sec. 4.5 to assess the impact of systematics on the w(θ) covariance. Note that these mocks would not be fully realistic for w(θ) covariance and cosmological inference as they are basically Gaussian realizations. These mocks allow us to quantify significances (i.e., a ∆χ2) to null tests, which are a necessary step in the analysis. Further, given such a large number of realizations I am able to obtain estimates of both the impact of the systematic correction on the resulting statistical uncertainty and any bias imparted by this methodology to well below 1σ significance (e.g., given 1000 mocks, a 0.1σ bias can be detected at 3σ significance).

4.5 Systematics

4.5.1 Survey property (SP) maps

The number density of galaxies selected based on their imaging is likely to fluctuate with the imaging quality due to fluctuations in the noise (e.g., Malmquist bias) and limitations in the reduction pipeline. Such fluctuations can imprint the structure of certain survey properties onto the galaxy field, thereby producing a non-cosmological signal. In order to quantify the extent of these correlations and remove their effect from the two-point function, maps of DES imaging properties were produced using the methods described in Ref. [289]. I consider the possibility that depth, seeing, exposure time, sky brightness and airmass, in each band griz, affect the density of 92 CHAPTER 4. CLUSTERING IN DES galaxies selected. In total, I consider 21 survey property maps. I refer to these as SP maps from here on:

depth: the magnitude limit at which it is expected to be able to detect a galaxy • to 10σ signiﬁcance;

seeing FWHM: the full width at half maximum of the PSF of a point source; •

exposure time: total exposure time in a given band; •

sky brightness: the brightness of the sky, e.g., due to background light or the • Moon phase;

airmass: the amount of atmosphere a source has passed through, normalized to • be 1 when pointing at zenith.

Where relevant, I use the weighted average quantity over all exposures contributing to a given area. These maps can be seen in Fig 4.3. I also consider Galactic extinction and stellar contamination (or obscuration [269]) as potential systematics. The stellar density map was created by selecting moderately bright, high conﬁdence stars. Using the notation of Y1GOLD, this selection is MODEST_CLASS = 2 with 18.0 < i < 20.5, FLAGS_GOLD = 0, and BADMASK 2. I also ≤ include an additional colour cut of 0.0 < g i < 3.5 and g r > 0. These stars − − were binned in pixels with Nside = 512 (equivalent to 47 square arcmin), and the corresponding area for each pixel was computed at higher resolution (Nside = 4096) from the Y1 Gold footprint and pixel coverage fraction, as well as the bad region mask. Together, this yields the number of moderately bright stars per square degree that can be used to cross-correlate galaxies with stellar density. Using MODEST_CLASS to select stars means this map could potentially contain DES galaxies. For this reason, I test for correlations with the astrophysical maps separately to the SP maps. As I ﬁnd no correlation between stellar density and galaxy density, I do not take this contamination into account. For Galactic extinction, I use the standard map from [290]. 4.5. SYSTEMATICS 93

25 1.2 25 23.25 ]

30 c 30 e 23.00 1.1 s c d 35 r 35 n a 22.75 a [ b

40 1.0 d 40

n 22.50 i a b

22.25 m 45 i 45 0.9 l i g

22.00 a M M 50 0.8 H 50

W 21.75 F 21.50 120 100 320 300 120 100 320 300

25 1.40 25 1600 30 1.35 30 d d n

35 n 1400 1.30 35 a a b b 1200 40 1.25 40 i i t s 1.20 1000 h s g i

45 a 45 r b 1.15 m r

800 y i k A 50 1.10 50 S 600 1.05 400 120 100 320 300 120 100 320 300

25 25 4.0 ] s

450 [ 30 30 3.5 d ] 400 n 3.0 2 35 a 35 b n 350 2.5 i i 40 m

40 c

300 r e 2.0 a [ m i

45 250 t 45 1.5 s r e a r t

200 u

1.0 S s

50 o 50 150 p

x 0.5 E 100 0.0 120 100 320 300 120 100 320 300

Figure 4.3: Maps of potential sources of systematics. Shown here for i-band only. Maps in other bands show ﬂuctuations on similar scales. Each SP map is shown at Nside = 1024. The stellar density map is shown at Nside = 512. 94 CHAPTER 4. CLUSTERING IN DES

4.5.2 Systematic corrections

This section describes the method used to identify and correct for observational systematics. I also discuss the uncertainty on this correction and its impact on the w(θ) covariance. The approach is to first identify maps that are correlated with fluctuations in the galaxy density field at a given significance. I then correct for the contamination using weights to be applied to the galaxy catalog. As demonstrated by [291], when testing a large number of maps one expects there to be some amount of covariance between the maps and the true galaxy density field due to chance. Consequently, it is possible to over-correct the galaxy density field using the type of methods employed in this work. To limit this effect, I do not correct for all possible maps, and limit the analysis to those maps that are detected to be correlated with the galaxy density field at high significance (above a given threshold). I test the robustness of the results on the choice of threshold in Sec. 4.7.1 and I test for biases due to over-correction in Sec. 4.7.3. The end result of this procedure is a measurement of w(θ) that is free of systematics above a given significance (in this concrete case, a galaxy density free of two sigma correlations with SP maps, as defined below, and visualized in Fig. 4.3) and that can be directly utilized in combination with weak lensing measurements for cosmological analyses. I identify the most significant SP maps as follows. First, given an SP map of some quantity s, I identify all pixels in some bin s [s , s ]. I then compute the average ∈ min max density of galaxies in these pixels. By scanning across the whole range of possible s- values for the SP map, one can directly observe how the galaxy density field scales with s. Examples of these are shown in Figs. 4.4, 4.5 and 4.6. I first remove regions of the footprint that display either especially significant (> 20%) changes in galaxy density from the mean, or are poorly fit by a monotonic function. These regions are defined from the cuts shown in Fig 4.4. I remove regions of the footprint with i-band FWHM > 4.5 and i-band exposure time > 500s. These cuts remove 1.6% of the Y1 area. After cutting the footprint, I determine which SP maps most significantly correlate with the data by fitting a linear function to each number density relationship. I minimize a χ2 where the model is N A s + B. I determine the significance of model gal ∝ a correlation based on the difference in χ2 between the best-fit linear parameters, and 4.5. SYSTEMATICS 95

z range Maps included in Maps included in 3∆χ2(68) weights 2∆χ2(68) weights 0.15 < z < 0.3 Depth (r) Exptime (i) FHWM (z) FWHM (r) Airmass (z) 0.3 < z < 0.45 Depth (g) Depth (g) 0.45 < z < 0.6 FWHM (z) FWHM (z) Exptime (g) Exptime (g) FWHM (r) FWHM (r) Skybright (z) Skybright (z) Depth (i) 0.6 < z < 0.75 FWHM (gri) PCA-0 FWHM (gri) PCA-0 Skybright (r) Skybright (r) FWHM (z) FWHM (z) Exptime (i) Exptime (z) 0.75 < z < 0.9 Airmass (i) Airmass (i) FWHM (r) FWHM (r) FWHM (g)

Table 4.3: List of the maps used in the SP weights. Each of these has been determined to impart ﬂuctuations in the galaxy sample at > 3∆χ2(68) or > 2∆χ2(68) signiﬁcance. The weights were applied serially for each map in the order shown, starting from the top of the table. ‘FWHM’ refers to the full-width-half-maximum size of the PSF. The photometric band of each SP map is in parentheses.

1.2 ® l

a 1.0 g N

0.8 / l a

g 0.6

N 0.15 < z < 0.3 0.3 < z < 0.45 0.45 < z < 0.6 0.6 < z < 0.75 0.75 < z < 0.9 0.4 3.0 3.5 4.0 4.5 3.0 3.5 4.0 4.5 3.0 3.5 4.0 4.5 3.0 3.5 4.0 4.5 3.0 3.5 4.0 4.5 FWHM i [pixels] FWHM i [pixels] FWHM i [pixels] FWHM i [pixels] FWHM i [pixels]

1.4 ® l

a 1.2 g N

1.0 / l a

g 0.8

N 0.15 < z < 0.3 0.3 < z < 0.45 0.45 < z < 0.6 0.6 < z < 0.75 0.75 < z < 0.9 0.6 200300400500600700 200300400500600700 200300400500600700 200300400500600700 200300400500600700 Exptime i [s] Exptime i [s] Exptime i [s] Exptime i [s] Exptime i [s]

Figure 4.4: Correlations of volume-limited redMaGiC galaxy number density with seeing FWHM and exposure time before any survey property (SP; see text for more details) cuts (illustrated with the red vertical lines) were applied to the mask. In the absence of systematic correlations, the results obtained from these samples are expected to be consistent with no trend (the reference green dashed line). The cuts removed regions with i-band FWHM > 4.5 pixels and i-band exposure time > 500s as these showed correlations that differed significantly from the mean (> 20%) or were not well fit by a monotonic function. No SP weights were used in this figure. 96 CHAPTER 4. CLUSTERING IN DES

1.2 1.2 1.2 1.2 1.2 0.15 < z < 0.3 0.3 < z < 0.45 0.45 < z < 0.6 0.6 < z < 0.75 0.75 < z < 0.9 ® l

a 1.1 1.1 1.1 1.1 1.1 g N

1.0 1.0 1.0 1.0 1.0 / l a

g 0.9 0.9 0.9 0.9 0.9 N 0.8 0.8 0.8 0.8 0.8 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 2 2 2 2 2 nstars(arcmin− ) nstars(arcmin− ) nstars(arcmin− ) nstars(arcmin− ) nstars(arcmin− )

Figure 4.5: Galaxy number density divided by the mean number density across the footprint for each redshift bin, split by the number density of stars. The points with error-bars display the results for the 3∆χ2(68) weighted sample, the cyan curves display the results without these weights. For the weighted sample, the χ2 of the line Ngal/ Ngal = 1 with the data points shown for each bin is 24.9, 16.0, 13.1, 6.6 and h i 2 10.9 with Nd.o.f = 10. The ∆χ between the null signal and a linear best ﬁt is 0.99, 0.95, 0.24, 0.013, and 0.082. This does not meet either of the ∆χ2 thresholds used in this analysis. Therefore, no evidence for stellar contamination or obscuration was found in this sample.

1.3 No weights 1.3 No weights Weights: airmass-i, FWHM-r Weights: airmass-i, FWHM-r 1.2 1.2 ® l

a 1.1 1.1 g N

/ 1.0 1.0 l a g

N 0.9 0.9

0.8 0.8

0.7 0.7 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 FWHM r [pixels] FWHM i [pixels]

Figure 4.6: Galaxy number density in the highest redshift bin, 0.75 < z < 0.9, as a function of two example SP maps, FWHM r-band and FWHM i-band. The black points correspond to the 3∆χ2(68) weighted sample, the cyan line is the unweighted sample. In this redshift bin, the SP maps used in the 3∆χ2(68) weights were Airmass i and FWHM r. The left panel demonstrates the eﬀect of the weights on the FWHM r correlation. The right panel demonstrates that correlations with SP maps that were not included in the weights are still reduced due to correlations among the SP maps. The full set of SP correlations for the maps in Table 4.3 are shown in Fig. 4.8. 4.5. SYSTEMATICS 97 a null test of N / N = 1, gal h gali

∆χ2 = χ2 χ2 . (4.10) null − model

The ∆χ2 is then compared to the same quantity measured on the Gaussian random fields described in Sec. 4.4.2. I then label each potential systematic to be significant at 1σ if the ∆χ2 measured on the data is larger than 68% of the mocks respectively. I denote this threshold as ∆χ2(68) and quote significances as ∆χ2/∆χ2(68); the square- root of this number should roughly correspond to the significance in terms of σ. Some examples of these tests for the observational systematics can be seen in Fig. 4.7. The full set of tests can be seen in Fig. 4.8. I see no significant correlation with stellar density in the sample, as shown in Fig. 4.5. Similarly, I find no correlations with Galactic extinction. Thus, the main tests are against SP maps, which are particular to DES observations. Once the most significant contaminant SP maps have been identified, I define weights to be applied to the galaxy sample in order to remove the dependency, following a method close to that of the latest LSS survey analysis [292, 293, 265, 294]. Note however that I identify systematics using a rigorous χ2 threshold significance criteria, based on a large set of Gaussian realizations, which has not been done for most LSS analyses. For this method, I apply the following steps to each redshift bin separately. The correlation with a systematic s is fitted with a function N / N = F (s). gal h gali sys For depth and airmass, the function used was a linear fit in s. For exposure time and sky brightness, the function was linear in √s, as this is how these quantities enter the depth map. For the seeing correlations, I fit the model

N / N = F (s ) gal h gali sys FWHM s B F (s ) = A 1 erf FWHM − , (4.11) sys FWHM − σ where sFWHM is the seeing full-width half-max value, and A, B and σ are parameters to be ﬁtted. This functional form matches that applied to BOSS [295, 293]; it is thus the expected form when morphological cuts are applied to reject stars (as this is what causes the relationship for BOSS).

Each galaxy i in the sample is then assigned a weight 1/Fsys(si) where si is the 98 CHAPTER 4. CLUSTERING IN DES

0.15 < z < 0.3 Ndof = 8 0.3 < z < 0.45 Ndof = 8 8 7 no weights no weights 7 2 6 2

3∆χ (68) weights ) 3∆χ (68) weights ) 8 8 6 2 5 2 2∆χ (68) weights 6 2∆χ (68) weights 6 ( (

5 2 2 4 χ χ 4

∆ 3 ∆ /

/ 3 2

2 2 χ

χ 2

∆ 1 ∆ 1 0 0

fwhm r fwhmfwhm z fwhm g i fwhm i airmass i fwhm r fwhm g exptime r exptime gexptimeairmass zexptime rairmass i g airmass z fwhm z airmass i skybrightskybright z i skybrightskybright r g exptimeexptime rexptime g z airmass r airmass g MOF depth r MOF depth i exptime i airmass z skybrightskybright r g skybrightskybright i z MOF depth g MOF depth z MOF depthMOF i depth r MOF MOFdepth depth z g MAGAUTO depth r MAGAUTO depth i MAGAUTO depth z MAGAUTOMAGAUTO depth r depth i MAGAUTO depth g MAGAUTO depth z MAGAUTO depth g

0.45 < z < 0.6 Ndof = 8 0.6 < z < 0.75 Ndof = 8 5 8 no weights no weights 2 7 2

) 3∆χ (68) weights 3∆χ (68) weights 4 ) 8 2∆χ2(68) weights 8 6 2∆χ2(68) weights 6 6 ( (

2 5

3 2 χ χ 4 ∆ 2 ∆ /

/ 3 2 2 χ 1 χ 2 ∆ ∆ 1

0 0

fwhm i fwhm z fwhm r fwhm g exptime i airmass r airmass i fwhm r fwhm i fwhm z airmass z exptime z skybrightairmass i g exptime r fwhm g airmass i exptime g skybright gskybright r airmass r airmassairmass g z exptime r exptime g MOF depth z skybright z MOF depth g MOF depthMOF depth r i exptime i skybright i skybright g exptime z skybright z MOF depth i skybrightMOF depth r r MOF MOFdepth depth z g

MAGAUTO depthMAGAUTO z depth g MAGAUTO depth r MAGAUTO depth i MAGAUTO depth i MAGAUTO depth r MAGAUTO depth z MAGAUTO depth g

0.75 < z < 0.9 Ndof = 8 14 13 no weights 12 3∆χ2(68) weights ) 11

8 2 10 2∆χ (68) weights 6

( 9 2 8 χ 7

∆ 6

/ 5 2 4 χ 3

∆ 2 1 0

fwhm i fwhm z fwhmfwhm r g airmass i airmassairmass r g exptime r airmass z exptime i exptime g exptime z skybrightskybright i z skybright skybrightr g MOF depthMOF r depth g MOF depth z MOF depth i

MAGAUTO depthMAGAUTO r depth g MAGAUTOMAGAUTO depth depth i z

Figure 4.7: The significance of each systematic correlation. The significance is calculated by comparing the ∆χ2 measured on the data to the distribution in the mock realizations. I find the 68th percentile ∆χ2 value, labeling it ∆χ2(68), for each map obtained from the mock realizations. I quote the significance for the relationship obtained on the data as ∆χ2/∆χ2(68). Weights are applied for the SP map with the largest significance, with the caveat that I do not correct for both depth and the components of depth (e.g. exposure time, PSF FWHM) in the same band. For example, in the bin 0.15 < z < 0.3, correcting for r-band depth (the most significant contaminant) did not remove all the r-band correlations with ∆χ2/∆χ2(68) > 2, so is not included in the final 2∆χ2/∆χ2(68) weights. This is repeated iteratively until all maps are below a threshold significance, shown here for thresholds of 2∆χ2/∆χ2(68) and 3∆χ2/∆χ2(68). The x axis is shown in order of decreasing significance for the unweighted sample. The labels in bold are the SP maps included in the 2∆χ2/∆χ2(68) weights. In the second redshift bin, 0.3 < z < 0.45, the 3∆χ2/∆χ2(68) and 2∆χ2/∆χ2(68) weights are the same because correcting for only g-band depth removes all correlations with ∆χ2/∆χ2(68) > 2. 4.5. SYSTEMATICS 99 value of the systematic at the galaxy’s location on the sky. This weight is then used when calculating w(θ) and in all further null tests. In this sample, I find evidence of multiple systematics at a significance of ∆χ2/∆ χ2(68) > 3, some of which are correlated with each other. To account for this, we first apply weights for the systematic with the highest ∆χ2/∆χ2(68). Then, using the weighted sample, I remeasure the significance of each remaining potential systematic and repeat the process until there are no systematics with a significance greater than a ∆χ2/∆χ2(68) = 3 threshold. The final weights are the product of the weights from each required systematic. I also produce weights using a threshold of ∆χ2/∆χ2(68) = 2, allowing us to determine if using a greater threshold has any impact on the clustering measurements. I refer to these weights as the 3∆χ2(68) and 2∆χ2(68) weights respectively. The final weights used in this sample are described in Table 4.3. The SP maps are either the depth or properties that contribute to the depth (e.g. holding everything else fixed, a longer exposure time will result in an increased depth). Thus, in bins where multiple SP weights were required, I avoided correcting for both depth and SPs that contribute to the depth in the same band. In these cases, I only apply weights for the SPs that contribute to the depth. Fig. 4.8 shows the correlation between the sample density and the SP maps used in Table 4.3, both with and without weights. Fig. 4.7 summarizes the results of the search for contaminating SPs, for each redshift bin. The blue points show the significance for each map, prior to the application of any weights. The black and red points display the significance after applying the 3∆χ2(68) and 2∆χ2(68) weights respectively. In Sec. 4.7, I will test the results with both choice of weights and whether to expect any bias from over-correction from either choice.

When Fsys(s) is a linear function, the method described above, hereby referred to as the weights method, should be equivalent to the method used in [270, 272]. This has been shown in [265] for the DES science veriﬁcation redMaGiC sample.

The impact of the SP weights on the w(θ) measurement can be seen in Figure 4.9. The dashed line displays the measurement with no weights applied. One can see that in all redshift bins, the application of the SP weights reduces the clustering amplitude and that the eﬀect is greatest on large scales. This is consistent with expectations 100 CHAPTER 4. CLUSTERING IN DES

1.3 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2 1.2 ® ® l l a a

g 1.1 1.1 1.1 1.1 1.1 g N N 1.0 1.0 1.0 1.0 1.0 / / l l a a g 0.9 0.9 0.9 0.9 g 0.9 N N 0.8 0.8 0.8 0.8 0.8 0.15 < z < 0.3 0.15 < z < 0.3 0.15 < z < 0.3 0.15 < z < 0.3 0.3 < z < 0.45 0.7 0.7 0.7 0.7 0.7 100 200 300 400 3.0 3.5 4.0 4.5 3.0 3.5 4.0 4.5 5.0 1.1 1.2 1.3 22.5 23.0 23.5 Exptime i [s] FWHM z [pixels] FWHM r [pixels] airmass z mag lim g-band

1.3 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2 1.2 ® l a

g 1.1 1.1 1.1 1.1 1.1 N

1.0 1.0 1.0 1.0 1.0 / l a

g 0.9 0.9 0.9 0.9 0.9 N 0.8 0.8 0.8 0.8 0.8 0.45 < z < 0.6 0.45 < z < 0.6 0.45 < z < 0.6 0.45 < z < 0.6 0.45 < z < 0.6 0.7 0.7 0.7 0.7 0.7 3.0 3.5 4.0 4.5 100 200 300 400 500 600 3.0 3.5 4.0 4.5 5.0 1500 2000 2500 3000 21.5 22.0 22.5 FWHM z [pixels] Exptime g [s] FWHM r [pixels] skybright z [ADU] mag lim i-band

1.3 1.3 1.3 1.3 1.3

1.2 1.2 1.2 1.2 1.2 ® l a

g 1.1 1.1 1.1 1.1 1.1 N

1.0 1.0 1.0 1.0 1.0 / l a

g 0.9 0.9 0.9 0.9 0.9 N 0.8 0.8 0.8 0.8 0.8 0.6 < z < 0.75 0.6 < z < 0.75 0.6 < z < 0.75 0.6 < z < 0.75 0.6 < z < 0.75 0.7 0.7 0.7 0.7 0.7 7.5 7.0 6.5 6.0 5.5 250 300 350 400 450 3.0 3.5 4.0 4.5 100 200 300 400 100 200 300 400 500 600 FWHM gri PCA 0 skybright r [ADU] FWHM z [pixels] Exptime i [s] Exptime z [s] −

1.3 1.3 1.3

1.2 1.2 1.2 ® l a

g 1.1 1.1 1.1 N

1.0 1.0 1.0 / l a

g 0.9 0.9 0.9 N 0.8 0.8 0.8 0.75 < z < 0.9 0.75 < z < 0.9 0.75 < z < 0.9 0.7 0.7 0.7 1.1 1.2 1.3 1.4 3.0 3.5 4.0 4.5 5.0 3.5 4.0 4.5 5.0 5.5 airmass i FWHM r [pixels] FWHM g [pixels]

Figure 4.8: Galaxy number density as a function of diﬀerent SP maps. Only the correlations with SP maps used in the 2∆χ2(68) weights calculation are shown. The cyan line is the correlation of the sample without weights. The black points show the correlation after correction with the 2∆χ2(68) weights. The error bars were calculated by measuring the same correlation on the Gaussian mock surveys described in Sec. 4.4.2. The signiﬁcance of these correlations are shown in Figure 4.7. 4.6. RESULTS: GALAXY BIAS AND STOCHASTICITY 101

(see, e.g. Ref. [269]).

4.6 Results: Galaxy Bias and Stochasticity

In this section I present measurements of galaxy bias bi and stochastic bias ri. The amplitude of the galaxy clustering signal is determined by the combination of parame-

i 2 i 2 ters (b σ8) . Equivalently the galaxy-galaxy lensing signal γt is sensitive to b (σ8) . In × the Y1COSMO combined probes analysis, cosmic shear provides a measurement of σ8 meaning that galaxy clustering and galaxy galaxy lensing can each provide an independent measurement of galaxy bias (and therefore one could measure r). In this analysis

i I ﬁx σ8 at the mean of the Y1COSMO posterior (σ8 = 0.81) to measure b (σ8/0.81) i from clustering and r from γt. This provides a cosmology dependent measurement of bias from clustering alone, and test of the assumption r = 1 in Y1COSMO. The w(θ) auto-correlation functions of the redMaGiC galaxy sample are shown in Figure 4.9. I show the auto-correlation calculated with and without a correction for

i observational systematics, as described in Sec 4.5. A minimum angular scale θmin has 1 2 been applied to each redshift bin i. These were chosen to be θmin = 430, θmin = 270, 3 4 5 θmin = 200, θmin = 160, and θmin = 140 to match the analysis in Y1COSMO. These minimum angular scales, varying with redshift, correspond to a single minimum co-

1 i i i moving scale R = 8 Mpch− such that θ = R/χ( z ), where z is the mean min h i h i redshift of galaxies in bin i [244]. The scale was chosen so that a signiﬁcant non-linear galaxy bias or baryonic feedback component to the Y1COSMO data vector would not bias the cosmological parameter constraints.

i The angular correlation function has been calculated on scales below θmin, but these were removed in all parameter constraints. Figure 4.10 shows the galaxy clustering signal between redshift bins. For these cross-correlations, I use a covariance matrix calculated from log-normal simulations described in [244]; the square root of the diagonal of this covariance matrix yields the error-bars shown in the figure. These are the same simulations used to validate the Y1COSMO covariance matrix. Overplotted is the cross-correlation prediction both from the best fit bias values from the auto-correlations, and the best fit cosmology and bias from Y1COSMO. The cross-correlation measurements were not used in the 102 CHAPTER 4. CLUSTERING IN DES

2.5 1.5 2.0 1.5 2,2 3,3 ) 1.0

( 1.5 1.0 w 1.0 0.5 0.5 0.5 1,1 0.0 0.0 0.0 101 102 101 102 101 102 [arcmin] [arcmin] [arcmin]

1.5 1.5 4,4 5,5

) 1.0 1.0

( 0.5 0.5 w 0.0 0.0

101 102 101 102 [arcmin] [arcmin]

Figure 4.9: Two-point correlation functions for the fiducial analysis in each of the 5 redshift bins. These panels show the auto-correlation used in Y1COSMO and the galaxy bias measurements presented in this work. A correction for correlations with survey properties is applied according to the methodology in Sec. 4.5. The grey dashed line is the correlation function calculated without the SP weights. The black 2 points use the 2∆χ (68) weights. I show correlations down to θ = 2.50 to highlight the goodness of the fit towards small scales, but data points within grey shaded regions have not been used in bias constraints or the galaxy clustering part of Y1COSMO. 1 This scale cut has been set in co-moving coordinates at 8 Mpc h− . The solid red curve is the best-fit model using only the w(θ) auto-correlations at fixed cosmology, using ∆zi priors from [254]. The solid blue curve is the best-fit model from the full cosmological analysis in Y1COSMO. 4.6. RESULTS: GALAXY BIAS AND STOCHASTICITY 103 combined probes analysis and so the robustness tests were not performed on these measurements. I present these results to demonstrate that there is a clustering signal in adjacent redshift bins (2,1), (3,2), (4,3), and (5,4) and not as a robustness test, hence I have not included a goodness-of-fit for this measurement. The amplitude of this signal is determined by the overlap in the n(z) between redshift bins (see Figure 4.2). These correlations could be used in future analyses to constrain the redshift bias parameters ∆zi.

Fixing all cosmological parameters, including Ωm, at the Y1COSMO values, I measure the linear bias to be b = 1.40 0.07, b = 1.60 0.05, b = 1.60 0.04, 1 ± 2 ± 3 ± b = 1.93 0.04, and b = 1.98 0.07. The χ2 values of the combined ﬁt and the indi- 4 ± 5 ± vidual bins are shown in Table 4.4. We note that the bin with the smallest probability is bin 1.

The combined goodness-of-ﬁt χ2 of the bias measurements is χ2 = 67 and the number of degrees of freedom is ν = 54 10 (the 10 parameters are b , ∆z ). These − i i values provide a probability to exceed of 1.4%. As in Y1COSMO, note that the formal probabilities of a χ2 distribution are not strictly applicable in this case due to the uncertainty on the estimates of the covariance. Further, because the ﬁve ∆zi are nuisance parameters with tight priors, I also consider ν = 49, which yields a probability to exceed of 4.5%. These probabilities are very similar to the values obtained by the full Y1COSMO data vector, of which this is one part.

I also note that the χ2 is sensitive to the inclusion of the shot-noise correction applied to the covariance detailed in Y1COSMO whereas the bi values and uncertainty were insensitive to this change.

For the L/L > 0.5 sample, the bias is nearly constant as a function of redshift, ∗ though there is a decrease at low redshift that has more than 2σ significance (the correlation in the measured bias for bins 1 and 3 is only -0.04, so we can safely ignore it in this discussion). The difference between bin 1 and bin 3 is less significant if we determine the expectation for a passively evolving sample as in [296, 297], which predicts a bias of 1.52 at z = 0.24 given a bias of 1.61 at z = 0.53. The bias increases for the higher luminosity sample, as expected. The results are broadly consistent with previous studies of the bias of red galaxies at low redshift (see, e.g., [298] for a review) and BOSS at intermediate redshifts (see, e.g., [299]). Further study of the details of 104 CHAPTER 4. CLUSTERING IN DES

4 2,1 2 0

4 3,1 3,2 2

) 0 ( w

0 4 4,1 4,2 4,3 0

1 2 0

4 5,1 5,2 5,3 5,4 2 0 101 102 101 102 101 102 101 102 [arcmin]

Figure 4.10: The two-point cross correlations between redshift bins. These measurements are expected to be non-zero, with a significance related to the degree of overlap in the n(z) displayed in Fig. 4.2. The numbers in each panel correspond to the redshift bins used in the cross-correlation, Adjacent bins are shown on the diagonal. The error- bars were calculated using the log-normal mock surveys used for Y1COSMO covariance validation [244]. The solid red curve is the best-fit model from the auto-correlation using only w(θ) at fixed cosmology. The solid blue curve (mostly under the red) is the best-fit model from the full cosmological analysis in Y1COSMO. For many of the cross-correlation panels, these predictions are indistinguishable. These measurements were not used in any of the parameter fits in this work or in Y1COSMO. 4.6. RESULTS: GALAXY BIAS AND STOCHASTICITY 105

2 z range χ Ndata prob 0.15 < z < 0.3 14.8 8 2.2% 0.3 < z < 0.45 6.9 10 55% 0.45 < z < 0.6 17.7 11 3.9% 0.6 < z < 0.75 11.0 12 35.9% 0.75 < z < 0.9 16.5 13 12.2% w(θ) all bins 67.2 54 1.4%

Table 4.4: The χ2, and probability of obtaining a χ2 exceeding this values for each redshift bin and for all bins combined. For the combined χ2, the number of free 2 parameters is 10 (5 bi and 5 ∆zi). The individual z bin χ values are calculated using the best ﬁt to all z bins combined. The covariance between between z bins is suﬃciently small that we can treat these as independent. I have therefore considered each individual bin to have 2 free parameters. It is expected that measuring the bias in each bin separately would have resulted in a smaller χ2.

i i z range b (σ8/0.81) r 0.15 < z < 0.3 1.40 0.072 1.10 0.08 0.3 < z < 0.45 1.60 ± 0.051 0.97 ± 0.06 0.45 < z < 0.6 1.60 ± 0.039 0.91 ± 0.08 0.6 < z < 0.75 1.93 ± 0.045 1.02 ± 0.13 0.75 < z < 0.9 1.98 ± 0.070 0.85 ± 0.28 ± ±

Table 4.5: The measurements of galaxy bias bi and the ratio of bias from clustering and galaxy-galaxy lensing ri for each redshift bin i, calculated with cosmological parameters ﬁxed at the mean of the Y1COSMO posterior, varying only bias and nuisance parameters with lens photo z priors from [254]. − the redMaGiC samples is warranted, especially if one wishes to use w(θ) at scales smaller than those studied in Y1COSMO.

I compare these bias constraints to those measured from the galaxy-galaxy lensing probe of the same redMaGiC sample, presented in Y1GGL. I parameterize the difference between the two measurements with the cross-correlation coefficients ri, which are presented in Figure 4.11. Beyond linear galaxy bias, r can deviate from 1 and acquire scale dependences, and it must be properly modeled to constrain cosmology with combined galaxy clustering and galaxy-galaxy lensing (e.g. [300]). I constrain ri at fixed cosmology using the Y1COSMO covariance, which includes the covariance between the two probes. All the nuisance parameters discussed in Y1COSMO are varied for this constraint. With this choice of scale cuts, no evidence of tension between the two bias measurements is found. This provides further justification for fixing r = 1 in 106 CHAPTER 4. CLUSTERING IN DES

8 7 6 5 4 3 2 1 0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.4 0.6 0.8 1.0 1.2 1.4 1.6 r1 r2 r3 8 2.8 7 Lmin = 0.5L L 1.5L 2.6

∗ ∗ ∗ bias Galaxy 2.4 6 w(θ) 2.2 5 γt 2.0 4 1.8 3 1.6 2 1.4 1 1.2 0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 r4 r5 Redshift

Figure 4.11: Constraints on the ratio, r, of galaxy bias measured on w(θ) and measured from the galaxy-galaxy lensing signal (see [140] denoted Y1GGL in the text) in each redshift bin. The histograms show the posterior distributions of ri from an MCMC fit for each z in i. The bottom-right panel displays the individual measurements for each bin (purple for the w(θ) measurements and orange for those obtained in Y1GGL). All cosmological parameters were fixed at the DES Y1COSMO posterior mean values, and all nuisance parameters were varied as in Y1COSMO. The constraints were calculated using the full Y1COSMO covariance matrix, so the covariance between the two probes has been taken into account. No significant evidence for r = 1 is found. 6 4.7. DEMONSTRATION OF ROBUSTNESS 107 the Y1COSMO analysis.

4.7 Demonstration of Robustness

I apply a number of null tests to the weighted sample to demonstrate its robustness. I do so by obtaining constraints on the galaxy bias and Ωm. These parameters are sensitive to both multiplicative and additive shifts in the amplitude of w(θ) and therefore they should encapsulate any potential systematic bias that could aﬀect the cosmological analysis of Y1COSMO. I thus perform joint ﬁts to the data in each redshift bin to

i obtain constraints on the five b and Ωm. For these fits, I marginalize over an additive redshift bias uncertainty described in Table 4.2. All other cosmological parameters are fixed at the Y1COSMO cosmology and as such, this should not be interpreted as a measurement of Ωm to be used in further analyses. Results are obtained using the analysis pipeline described in [244] and Chapter 3. I describe how w(θ) is altered to perform each test throughout the rest of this section.

4.7.1 Selection of threshold

I test two thresholds used to determine when to apply weights based on a given SP map: 3∆χ2(68) and a more restrictive (i.e., more maps weighted for) 2∆χ2(68). After reaching a certain threshold, I expect that the only effect from adding extra weights would be to bias the measurements (from over-correction) and add greater uncertainty. I test for those effects in the following subsections. Here, in order to demonstrate that the results are insensitive to the choice in threshold, the change in the measured bi and Ωm must be negligible compared to its uncertainty. Figure 4.12 shows the difference between the 3∆χ2(68) and 2∆χ2(68) SP weights. Because the weights correction can only decrease the w(θ) signal, applying a stricter threshold significance is expected to move the contours towards smaller values of bi. Figure 4.12 shows that this impact is very small compared to the overall Y1 uncertainty and we can conclude that the choice between 3∆χ2(68) and 2∆χ2(68) weights will have negligible impact on the Y1COSMO parameter constraints (The final weights used in Y1COSMO are the 2∆χ2(68) weights). Figure 4.12 also shows the impact of not including SP weights on the parameter 108 CHAPTER 4. CLUSTERING IN DES

no weights no weights no weights 300 0. 2∆χ2(68) weights 2∆χ2(68) weights 2∆χ2(68) weights 3∆χ2(68) weights 3∆χ2(68) weights 3∆χ2(68) weights

275 0. m Ω

250 0.

225 0. 2 4 6 8 50 65 80 95 50 65 80 95 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. b1 b2 b3

no weights no weights 300 0. 2∆χ2(68) weights 2∆χ2(68) weights 3∆χ2(68) weights 3∆χ2(68) weights

275 0. m Ω

250 0.

225 0.

80 95 10 25 8 0 2 4 1. 1. 2. 2. 1. 2. 2. 2. b4 b5

Figure 4.12: Parameter constraints showing the impact of the SP weights, varying Ωm, 5 linear bias parameters bi, and 5 nuisance parameters ∆zi. Contours are drawn at 68% and 95% confidence level. These constraints use the same ∆zi priors as Y1COSMO. The blue contour shows the constraints on w(θ) calculated with no SP weights. The gray and red contours use SP weights removing all 2∆χ2(68) and 3∆χ2(68) correlations respectively. In this parameter space, ignoring the correlations with survey properties would have significantly biased the constraints from w(θ). As expected, the best fit when using the 2∆χ2(68) weights is at smaller values of bi than the 3∆χ2(68) weights, although the difference is not significant compared to the size of the contour. constraints. Ignoring the SP correlations would have resulted in significantly biased

i constraints on b and Ωm. In every redshift bin, the shift is greater than 2σ along the major axis of the ellipses. Figure 4.13 shows the impact of the choice of threshold on the full 3x2pt analysis from Y1COSMO.

4.7.2 Estimator bias

I also test for potential bias in w(θ) induced by over-correcting with the weights method and from correlations between the SP maps. This was done using the Gaussian mocks described in Sec. 4.4.2 using the following method. After the galaxy over-density 4.7. DEMONSTRATION OF ROBUSTNESS 109

Figure 4.13: DES Y1 3x2pt full parameter constraints showing the impact of the SP weights, varying the parameters in Table 3.1. Contours are drawn at 68% and 95% conﬁdence level. This demontrates that the SP contamination does impact 3x2pt cosmology, but the arbitrary choice of signiﬁcance threshold did not. The data vectors were evaluated at the centres of the angular bins rather than at the mean of the pair distribution. This has negligible impact of the parameter contours. 110 CHAPTER 4. CLUSTERING IN DES

ﬁeld has been generated in each realization, I insert the systematic correlation using

2 Fsys(s) and the best-fit parameters for each of the systematics in Table 4.3 at 2∆χ (68) significance. This is equivalent to dividing each mock galaxy map by a map of the SP weights. I then produced a galaxy number count as before, also adding shot noise. I fit the parameters of Fsys(s) to each realization and apply weights to the maps using the same method that is applied to the data. I measure w(θ) using the pixel estimator in Eqn. 4.5 on mocks with systematic contamination and correction, wweights, and on mocks with no systematics added, wno sys. The resulting bias in w(θ) if defined to be,

N N ! 1 X X w = w w (4.12) est bias N no sys,i − weights,j i=1 j=1 where N is the total number of realizations. I then add west bias to the measured w(θ) i and measure b and Ωm. This is designed to test for any bias in w(θ) induced by the the estimator when using weights. This result can be seen in Figure 4.14 where it shows negligible impact on the parameter constraints.

4.7.3 False correlations

Given the large number of SP maps being used in the systematics tests, it is possible that chance correlations will appear signiﬁcant and weights will be applied where no contamination has occurred, biasing the measured signal. To test this, I use the same Gaussian mocks as in Sec. 4.7.2 with no added systematic contaminations. I measure the correlation of each mock with each of the 21 SP maps in Sec. 4.5.1, identifying any correlations above a 2∆χ2(68) threshold signiﬁcance.

The false correction bias wfalse bias, is then deﬁned as the average diﬀerence between the w(θ) measured with no corrections, and the w(θ) measured correcting for all correlations above the threshold using the weights method. I then add wfalse bias to the i measured w(θ) and test the impact on b and Ωm constraints. This test is designed to test for any bias in w(θ) induced by falsely correcting for SP maps that were only correlated with the galaxy density by chance.

2 This result is shown in Figure 4.14 where wfalse bias for the 2∆χ (68) SP maps 4.7. DEMONSTRATION OF ROBUSTNESS 111

fiducial fiducial fiducial fiducial + west fiducial + west fiducial + west .30 0 fiducial + wfalse fiducial + wfalse fiducial + wfalse

.28 m 0 Ω

26 0.

24 0. 20 35 50 65 5 6 7 8 5 6 7 8 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. b1 b2 b3

fiducial fiducial fiducial + west fiducial + west .30 0 fiducial + wfalse fiducial + wfalse

.28 m 0 Ω

26 0.

24 0. 8 9 0 1 80 95 10 25 1. 1. 2. 2. 1. 1. 2. 2. b4 b5

Figure 4.14: Parameter constraints showing the impact of the estimator bias, west and false correction bias wfalse. The fiducial data vector and was calculated using the 2 2∆χ (68) weights on the data. The west and wfalse were measured on Gaussian mock surveys using a 2∆χ2(68) threshold significance. No evidence for significant bias in i i the b , Ωm plane is found. These constraints use the same ∆z priors as Y1COSMO.

has been used. This shows a negligible impact on the constraints. The wfalse bias for the 3∆χ2(68) SP maps is not shown as it has an even smaller impact. This demonstrates that selecting a 2∆χ2(68) threshold does not induce a bias in the inferred bias parameters for the set of SP maps used in this analysis.

4.7.4 Impact on covariance

Correcting for multiple systematic correlations can alter the covariance of the w(θ) measurement in various ways. The scatter in the best fit parameters should increase the variance, while the removal of some clustering modes should decrease it. I test the significance of any changes to the amplitude and structure of the covariance matrix using the Gaussian mocks. For this test I use the same mocks as in Sec. 4.7.2 which are ‘contaminated’ with the same systematic correlations found in the data. I fit the Fsys(s) function to each 112 CHAPTER 4. CLUSTERING IN DES

cov: no sys cov: no sys cov: no sys 30 2 2 2 0. cov: 2∆χ (68) sys cov: 2∆χ (68) sys cov: 2∆χ (68) sys

.28

m 0 Ω

26 0.

24 0. 2 3 4 5 6 5 6 7 8 5 6 7 8 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. b1 b2 b3

cov: no sys cov: no sys 30 2 2 0. cov: 2∆χ (68) sys cov: 2∆χ (68) sys

.28

m 0 Ω

26 0.

24 0. 8 9 0 1 8 9 0 1 2 1. 1. 2. 2. 1. 1. 2. 2. 2. b4 b5

Figure 4.15: Parameter constraints showing the impact of the systematics correction on the covariance. Both contours use the ﬁducial theory data vector. The blue contour uses the covariance from mock surveys with no contamination added (labeled "cov: no sys"). The gray contour uses the covariance determined from mock surveys with the 2∆χ2(68) contaminations added (labeled "cov: 2σ sys"). These constraints use the same ∆zi priors as Y1COSMO.

mock and correct using weights. I then measure the correlation function wweights and calculate the covariance matrix of this measurement. I also measure the correlation on mocks with no systematics added, wno sys, and calculate the covariance matrix i from each measurement. I calculate the galaxy bias b and Ωm constraints for each covariance matrix and test if the resulting contours are significantly different. This test determines whether this additional uncertainty needs to be considered in the Y1COSMO analysis by marginalizing over the fitted parameters.

The results of this test are shown in Figure 4.15. I show that for the SP maps selected in this analysis, the impact on the size of the contours is negligible. Therefore, no additional parameters were included in the MCMC analysis to account for the uncertainty in the correction. 4.8. CONCLUSIONS 113 4.8 Conclusions

I have presented the 2-point angular galaxy correlation functions, w(θ), for a sample of luminous red galaxies in DES Y1 data, selected by the redMaGiC algorithm. This yielded a sample with small redshift uncertainty, a wide redshift range, and wide angular area. This sample is split into ﬁve redshift bins and analyzed its clustering. The ﬁndings can be summarized as follows:

I ﬁnd that multiple systematic dependencies between redMaGiC galaxy density • and survey properties must be corrected for in order to obtain unbiased clustering measurements. I correct for these dependencies by adding weights to the galaxies, following [292, 293].

I demonstrate both that the methods used sufficiently remove systematic con- • tamination (no significant differences are found between applying a 2∆χ2(68) and 3∆χ2(68) threshold; see Fig. 4.12) and that any bias resulting from this method removing true clustering modes is insignificant (see Fig. 4.14). I further demonstrate that the weighting method imparts negligible changes to the covariance matrix (see Fig. 4.15).

I ﬁnd the redshift and luminosity dependence of the bias of redMaGiC galaxies • to be broadly consistent with expectations for red galaxies.

I ﬁnd that the large-scale galaxy bias is consistent with that determined by the • Y1GGL galaxy-galaxy lensing measurements. This is consistent with r = 1 at linear scales, in agreement with basic galaxy formation theory, and a key assumption in the Y1COSMO analysis. (See Fig. 4.11.)

These results give an unbiased w(θ) data vector which was provided to the • Y1COSMO analysis, and future DES year 1 combined probes analyses.

The methods presented, both correcting for systematic dependencies and ensuring the robustness of these corrections, can be used as a guide for future analyses. Possible improvements to the work include incorporating image simulations [301] and using mode projection techniques [271].

The galaxy bias results can be extended to study luminosity dependence within redshift bins and to use smaller scale clustering in order to determine the halo oc- cupation distribution (relating the number of galaxies to halo mass) of redMaGiC 114 CHAPTER 4. CLUSTERING IN DES galaxies. Already, the bias measurements can be used to inform simulations (e.g., for the support of DES Y3 analyses) and additional HOD information would be of further beneﬁt. Finally, the results presented here have been optimized for combination with other cosmological probes in Y1COSMO and this work has ensured the galaxy clustering measurements do not bias the Y1COSMO results. The analysis followed a strict blinding procedure and has yielded cosmological constraints when combined with the other 2-point functions. Chapter 5

Neutrino constraints from the Dark Energy Survey Year 1 data

In this chapter I present simulated results of the DES 3x2pt analysis in cosmological models that allow for sterile neutrinos. The sections of this chapter that relate to massless sterile neutrino models are based on the in prep article [4]. Active and sterile neutrino properties can be measured by a number of cosmological probes. In the early universe neutrinos behave as radiation, contributing extra radiative degrees of freedom and some mass eﬀects. At late times they behave as light matter, suppressing structure in the small scales of the matter power spectrum. See Sec. 1.2.2 for a review of the impact of neutrinos on cosmological observables. The aim of this chapter is to present the latest constraints of sterile neutrino cosmology from LSS, CMB, and supernovae experiments, and demonstrate what the DES Year 1 3x2pt analysis can add to this picture. With the current uncertainty in the DES analysis, it is not expected that DES Y1 3x2pt will be able to competitively constrain neutrino properties without the combination with other probes.

5.1 Model

In the baseline DES Y1 3x2pt analysis [2], neutrinos are modeled by fixing the effective number of radiative species to the standard model prediction Neff = 3.046 and varying P P the sum of the neutrino masses mν. It is assumed that mν consists of three

115 116 CHAPTER 5. NEUTRINOS IN DES neutrino states of degenerate mass, which is a valid assumption for the masses cosmological experiments are currently able to probe (> 0.1 eV). The analysis is performed

2 with a flat prior on Ωνh . In this chapter, this will be referred to as the baseline model P or the mν model. P This analysis extends the mν model by allowing the number of radiative degrees of freedom Neff to vary. Any detection of Neff > 3.046 could indicate either a massless sterile neutrino or other light relic particle. This is a phenomenological model designed to pick up extra radiative degrees of freedom without tying the analysis to a particular sterile neutrino model. This will be the primary model used in the analysis and will P be referred to as the massless sterile model or the mν ∆Neff model. This analysis will also contain constraints on a model that assumes a single massive sterile neutrino. In this model I assume the 3 standard neutrinos have the minimal P mass allowed by oscillation experiments, mν = 0.06eV. The sterile neutrino mass eff msterile is then allowed to vary, along with Neff . Using this model allows one to obtain an upper limit on a sterile neutrino mass parameter, but this will be a model dependent statement (assumes only 1 sterile neutrino, minimal standard masses, etc). This model

eﬀ will be referred to as the massive sterile model or the msterile ∆Neﬀ model.

5.2 Data

The focus of this analysis is on the DES Year 1 3x2pt analysis described in Chapters 3 and 4. I also include external data sets to combine with DES in order to show competitive constraints. The external data sets used are,

1. Planck TT+lowTEB: Using the Planck 2015 TT power spectrum for multipoles ` between 30 and 2508, and polarisation spectra EE, BB and TE for ` between 2 and 29 [10].

2. Planck lensing: Measurements of the CMB lensing from Planck 2015 [53]. Using the CMB lensing probes intermediate redshifts (between DES and recombination).

3. BAO: I include angular diameter distance measurements from the Baryon Acous- tic Oscillation (BAO) feature by the 6dF Galaxy Survey [260], the SDSS Data 5.3. SYSTEMATICS 117

Release 7 Main Galaxy Sample [261], and BOSS Data Release 12 [262]

4. SN: Type 1a supernovae measurements are included using the Joint Lightcurve Analysis (JLA) from [142]

These data sets match those used in the DES Y1 3x2pt ΛCDM analysis [2].

5.3 Systematics

A number of small scale systematics can impact large scale structure measurements. A number of robustness tests were performed in [244] which are also presented in Chapter 3. These tested the impact of small scale systematics on the 3x2pt analysis in the ΛCDM and wCDM models. Here I repeat and extend these tests with the P mν ∆Neff model. The systematics tests involve producing a number of simulated DES data vectors, each contaminated with a different potential systematic. These simulated data vectors are simply a theoretical prediction of the two-point functions, interpolated at the mean of the DES Y1 angular bins. A full likelihood analysis is then performed with each of these data vectors and the results are compared to an analysis using a ‘baseline’ data vector with no systematic contamination. This is done for both DES only constraints, and for the combination with all the data sets listed in Sec. 5.2. Because the external data sets use real data, rather than noiseless simulated data, the input cosmological parameters used for the DES simulated vectors are the best fit values of the combination of the external data sets in ΛCDM. The systematics considered in this analysis and details of their modeling are shown below.

1. Baryonic eﬀects: I simulate a DES Y1 3x2pt data vector that includes a contribution from baryonic feedback in the nonlinear power spectrum. This is calculated using the OWLS AGN simulation [302], following the methodology of [128]. The nonlinear power spectrum is scaled by the ratio of the OWLS power spectrum measured on simulations that include AGN and the OWLS dark matter only power spectrum, 118 CHAPTER 5. NEUTRINOS IN DES

P (k, z) P (k, z) DM+baryons P (k, z). (5.1) → PDM ONLY This then propagtes through to the two-point functions.

2. Intrinsic alignments, simple case: I simulate a data vector with IA amplitude

AIA = 0.5 and redshift dependence αIA = 0.5 using the baseline non-linear alignment model of the DES analysis. While these IA parameters are explicitly marginalized over in the analysis, this systematic check is still useful to monitor any potential biases due to degeneracy between the cosmological parameters and

(AIA, αIA).

3. Intrinsic alignments, complex case: I simulated a data vector using the Tidal Alignment Tidal Torquing model (hereafter TATT) from [111]. This model expands the intrinsic alignment field up to second order in the density field. This introduces a tidal torquing term to the IA spectrum that is quadratic in the tidal field. This term is written as,

1 γI = C s + C s s δ s2 (5.2) ij 1 ij 2 ik kj − 3 ij

where sij is the tidal tensor. In the contaminated vector, the TATT amplitudes

were set to C1 = 0, C2 = 10 with no z dependance.

4. Non-linear bias: I simulate a data vector that models the density contrast of galaxies as 1 δ = bi δ + bi [δ2 δ2 ] (5.3) g 1 2 2 − h i

where i refers to the lens redshift bin. The b2 values used for each lens bin were estimated from ﬁts to the DES Buzzard simulations [303, 274] (b(1) = 0.718, 2 − b(2) = 0.648, b(3) = 0.559, b(4) = 0.389, b(5) = 0.094). 2 − 2 − 2 − 2 −

5. Magniﬁcation: I simulate a data vector that includes the contribution from mag-

niﬁcation to γt and w(θ). These are added in Fourier space in the form,

Cij (l) Cij (l) + Cij (l) + Cij (l) + Cij (l) (5.4) gg → gg gm mg mm Cij (l) Cij (l) + Cij (l) (5.5) g → g m 5.4. BLINDING 119

where m denotes lensing magniﬁcation and i and j denote redshift bins. The magniﬁcation C(l)s are calculated using the method of [268],

Cij (l) = 4(αi 1)(αj 1)Cij(l) (5.6) mm − − Cij (l) = 2(αj 1)Cij (l) (5.7) gm − g Cij (l) = 2(αi 1)Cij(l) (5.8) m − (5.9)

where the αi values are the slopes of the luminosity functions in each z bin, approximated using the ﬁt from [304].

6. Limber approximation and RSD: I simulate a data vector that uses the exact (non-Limber) w(θ) calculation and include the contribution from redshift space distortions from [305].

The results of the systematics tests for the DES simulated data only are shown in Fig. 5.1. The results of the systematics tests for the DES simulated data + external data sets are shown in Fig. 5.2.

5.4 Blinding

As with the ΛCDM analysis in [2], this analysis was performed ‘blind’ to prevent conﬁrmation bias. This follows the DES blinding rules for cosmology analyses. Under these rules, the following conditions apply until the collaboration decides to ‘unblind’ the analysis.

1. The real data points must not be plotted on the same axis as a theoretical

P eff prediction with non-zero values of mν ∆Neff or msterile ∆Neff .

2. Systematics tests must be performed on simulated data vectors before running on the real data.

3. When running on the real data, parameter contours must not be plotted on the same axis as a known data set. 120 CHAPTER 5. NEUTRINOS IN DES

Figure 5.1: DES Y1 3x2pt analysis on simulated data vectors contaminated with different systematics. The ‘baseline’ vector has no systematic contamination. The impact is negligible for all systematics apart from the no limber+RSD vector. This had no impact on the new ∆Neff parameter, but does shift the σ8 and Ωm constraints by 0.5σ. Since the primary result for this analysis is the Neff constraint, this is sufficiently∼ small to have passed this null test.

4. Real data contours must have the axis tick labels removed.

The external data sets have already been published and therefore blinding rules only apply when real DES data are being used.

5.5 Results

The results of this analysis are currently still blind. Here I present results using the baseline simulated data vector and the external data sets. This gives an indication of the constraining power of DES for these neutrino models. 5.5. RESULTS 121

Figure 5.2: DES Y1 3x2pt+Planck+BAO+SN analysis on simulated data vectors contaminated with diﬀerent systematics. The ‘baseline’ vector has no systematic contamination. The impact of the systematic is negligible for all those tested.

5.5.1 Massless sterile neutrino model

Fig. 5.3 shows the constraints from simulated DES data only on the massless sterile P neutrino ( mν ∆Neﬀ ) model. It can be seen that DES only does not provide any constraint of either Ων or ∆Neﬀ within the parameter limits used in this analysis

(the samples have filled the prior range). Looking at the ∆Neff , h0 panel, one can see a small degeneracy between these two parameters. It is theoretically possible to obtain a constraint on ∆Neff from DES by applying a Gaussian prior to h0 from local measurements. However, the DES measurements are not currently statistically powerful enough to obtain a constraint within the prior used in this analysis and would therefore not be competitive with other cosmological probes.

Fig. 5.3 also shows the Planck TT + lowTEB. Here it can be seen that Planck temperature and low-l polarization only is not much more constraining than DES only on the neutrino mass (see Ων panel). It is only in combination with other data sets that P these cosmological constraints on Ων or mν are signiﬁcant. However, for ∆Neﬀ , as 122 CHAPTER 5. NEUTRINOS IN DES

Figure 5.3: DES Y1 3x2pt constraints with simulated DES data compared to Planck P TT + lowTEB + lensing in the mν ∆Neﬀ model. Planck is much more constraining on the ∆Neﬀ but the constraint on Ων is weak compared to the combination the external data sets.

expected, Planck is much more constraining due to ∆Neﬀ being a radiative parameter.

Fig. 5.4 shows the constraints from simulated DES data and all the external P data sets considered with the mν ∆Neff neutrino model. The simulated DES data vector does not appear to significantly add to the external data set combination in the neutrino parameters. However, the simulated DES data vector was evaluated at the external data set best fit (in ΛCDM). If the real DES data were to prefer a different

Ωm or σ8 to the external data, this could shift the neutrino parameter posteriors. It is not the statistical power of DES that is interesting in this case, but the potential for the diﬀerent best ﬁts between DES and Planck to be mitigated by the new parameter.

The external data sets alone give constraints of, 5.5. RESULTS 123

Figure 5.4: DES Y1 3x2pt+Planck+BAO+SN constraints with simulated DES data in P the mν ∆Neff model. The simulated DES data does not appear to significantly add to the external data set combination. However, if the real DES data prefers different Ωm, σ8 to the external data set combination, this can potentially shift the neutrino posteriors.

∆Neﬀ < 0.54 95% Planck + BAO + SN X mν < 0.29 95% Planck + BAO + SN (5.10)

Figure 5.5 shows the impact of freeing the ∆Neff parameter on the comparison between Planck and DES in the Ωm, σ8 plane. Since this figure uses simulated DES data, one cannot assess the impact of this parameter on the possible tension between these data sets. However, one can see that freeing ∆Neff extends the Planck contour along the direction of DES degeneracy, making ∆Neff an unlikely parameter to improve consistency after unblinding. 124 CHAPTER 5. NEUTRINOS IN DES

Figure 5.5: Constraints from DES simulated data (left) and Planck TT + lowTEB real P P data (right) in the mν ΛCDM parameter space and the mν ∆Neff ΛCDM model. This shows the impact of freeing the ∆Neff parameter on the Ωm, σ8 plane. Discussions of a potential tension between Planck and large scale structure measurements have been discussed in this plane (see Sec. 1.1.7). It can be seen that ∆Neff has a small impact on the DES constraints in this plane, but a significant impact on Planck. ∆Neff mostly moves the Planck probability tangentially to the DES direction of degeneracy, 0.5 σ8(Ωm/0.3) .

5.5.2 Massive sterile neutrino model

Fig. 5.6 shows the constraints from simulated DES data and external data sets on the

eﬀ massive sterile neutrino msterile ∆Neﬀ model. The external data sets alone give constraints of,

∆Neﬀ < 0.48 95% Planck + BAO + SN

X eﬀ msterile < 0.70 95% Planck + BAO + SN (5.11)

eﬀ It should be noted that the msterile constraint depends somewhat on the upper edge of the prior range due to large number of samples close to the ∆Neﬀ = 0 axis. In this region of the parameter space the sterile neutrino is uncoupled from the active

eﬀ neutrinos (zero thermalisation) and the msterile upper limit rises the to cold dark matter limit (see [10]). When applying the mthermal < 10eV prior as in [10], the constraint reduces to, 5.5. RESULTS 125

Figure 5.6: DES Y1 3x2pt+Planck+BAO+SN constraints with simulated DES data eff eff in the msterile ∆Neff model. Adding the DES data provide an upper limit to msterile, removing the need for an artificial prior on mthermal.

thermal ∆Neﬀ < 0.49 95% Planck + BAO + SN (m < 10eV)

X eﬀ thermal msterile < 0.50 95% Planck + BAO + SN (m < 10eV) (5.12)

eﬀ thermal Both of these use an arbitrary hard prior, one on msterile and the other on m . When adding the DES data, this artiﬁcial limit is no longer required. Adding the

eﬀ simulated DES data (which assumes msterile = 0 eV and ∆Neﬀ = 0), the upper limits on the sterile parameters are,

∆Neﬀ < 0.51 95% DES(simulated) + Planck + BAO + SN

X eﬀ msterile < 0.35 95% DES(simulated) + Planck + BAO + SN (5.13) 126 CHAPTER 5. NEUTRINOS IN DES

5.5.3 Comparison to oscillation data

Recent results from the MiniBooNE ν appearance experiment have found a 6.1σ excess neutrino appearance signal consistent with a sterile neutrino [191]. Converting the full likelihood surface of the ν appearance experiments into the cosmology space is harder to achieve than for the individual ν disappearance results used in Chapter 2. This is due to the two mixing angles in Equ. 1.43. Therefore, in this section, I only consider the best ﬁt values from MiniBooNE and comment on where this lies in the cosmology space. I leave the assessment of consistency between the MiniBooNE results and the cosmological data to future works. The best ﬁt values from MiniBooNE+LSND in the oscillation space are, ∆m2 = 0.040 eV2, sin2 2θ = 0.894 (using the two-neutrino oscillation probability Equ. 1.42) [191]. Assuming minimal active masses, normal heirachy and a thermal neutrino distribution, the MiniBooNE result corresponds to a fully thermalised sterile neutrino1

eff (∆Neff = 1) with msterile = 0.2eV. This appearance oscillation best fit lies outside of eff the cosmological bounds shown in Fig. 5.6 when using the msterile ∆Neff model and the same assumptions used in Chapter 2. This has previously been studied for other data sets (e.g. in [214]). As shown in Chapter 2, more exotic neutrino behaviour such as non-zero lepton asymmetry could potentially reconcile these data sets. The Mini- BooNE best fit also lies in the exclusion region from the ν disappearance experiments, MINOS and Daya Bay (see [192] and Fig. 1.14).

5.6 Conclusions

I have presented a simulated analysis of the Dark Energy Survey Year 1 galaxy clustering and weak gravitational lensing constraining cosmological and neutrino parameters. By using a simulated data vector at the external data set best fit, the gains from DES are minimal in the massless sterile neutrino model. However, with the real data, the different best fits between DES and the external data are expected to increase the impact of the DES data. When considering a massive sterile neutrino model, DES can provide an upper

1This can be shown by using the mean momentum approximation as in [216] 5.6. CONCLUSIONS 127 bound on the neutrino mass when combined with other cosmological experiments without the need for an informative top-hat prior. Due to blinding restrictions, the analysis on real data is not presented in this thesis. This will presented in the publication [4]. The external data presented in this analysis were chosen to match the DES Y1 ΛCDM analysis [2]. There is potential for further constraints by adding the latest Planck 2018 data vector which was not publicly available at the time of writing this thesis and the latest supernovae results from Pantheon [306] or DES itself [307]. Chapter 6

Conclusions

In this thesis I have presented real data results and simulated results for sterile neutrino constraints from the CMB and large scale structure. I have also shown a study of observational systematics for a galaxy clustering sample in the Dark Energy Survey and a measurement of the linear galaxy bias of these galaxies. This demonstrates some of the required systematic and nuisance parameter research required to obtain neutrino constraints from LSS experiments. In Chapter 2, I have presented the sterile neutrino results from the Planck experiment, and have compared them to those from oscillation experiments. This involves

eff transforming from the cosmological parameterisation of sterile neutrinos (∆Neff ,msterile), into the parameter space used by oscillation experiments ( δm 2, θ). I find that the | | region of the cosmological parameter space ruled out by Planck corresponds to a large fraction of the oscillation parameter space currently being probed by the latest oscillation experiments. I find that the MINOS experiment is complementary to the CMB as it can probe small δm 2, whereas the CMB probes high δm 2. These results are model | | | | dependent. For example, the oscillations in the early universe can be suppressed by non-zero lepton asymmetry, weakening Planck’s constraining power in the oscillation space. I note that this lepton asymmetry would introduce other cosmological inconsistencies not studied in this work [241]. The oscillation constraints from IceCube and the predicted SBN sensitivity have very little impact on the cosmological space, as these mostly exclude regions with Neff = 1. The MINOS results do have a small impact on the cosmological constraints due to ruling out some the low mass splitting oscillations.

128 129

In Chapter 3 I have introduced the DES Year 1 3x2pt analysis, combining galaxy clustering and weak lensing. This summarises the analysis used in the two following chapters. In Chapter 4 I have presented original research into the galaxy clustering of DES Year 1 LRGs for use in the DES 3x2pt analysis. I have shown systematic correlations with a number of DES survey properties (SP maps) in the DES redMaGiC sample.

These are corrected by applying weights to each galaxy equal to the inverse of the Ngal vs SP relation at that point. Since a number of correlations are found, a threshold signiﬁcance must be used to determine which SP maps require correction. I demonstrate that the cosmological analysis is robust to the choice of threshold in the DES Y1 sample. I also test for bias in the correction method by repeating the analysis on Gaussian random ﬁeld simulations. I test that the analysis is robust to bias from false corrections, bias from overcorrecting and the impact on the covariance matrix.

I also make measurements of the linear galaxy bias of the DES redMaGiC sample in tomographic bins. I find these to be consistant with those from galaxy-galaxy lensing measurements using the same galaxies. In Chapter 5 I have presented a simulated analysis of the DES Year 1 constraints on sterile neutrino parameters. I have studied both a massless neutrino model, extending the baseline analysis by 1 new parameter ∆Neff ; and a massive sterile neutrino model, fixing the standard masses to their minimum value. I find that DES is not expected add any significant improvement to the existing combination of cosmological data sets studied in this work. Applying a local H0 to the DES Y1 only constraints would not provide competitive ∆Neff constraints. However, due to the noticeable degeneracy between ∆Neff and H0, there is potential for this combination to provide competitive constraints for future LSS data sets. When considering a massive sterile neutrino model, adding DES to other cosmological data sets provides an upper limit on the neutrino mass. This eliminates the need for an informative top-hat prior. This is designed as preparation for the analysis of real DES data which was blind within the collaboration at the time of writing. The real data DES constraints on

∆Neﬀ will be presented in an upcoming article [4]. A number of extensions to the work presented in this thesis can be made. To 130 CHAPTER 6. CONCLUSIONS extend the studies into cosmological and oscillation data presented in Chapter 2, one could extend the neutrino model from 1+1 (1 active and 1 sterile) to a 3+1 model.

This would allow oscillations to occur through channels other than θ24. A 3+1 analysis could provide a comparison between the appearance experiments and the cosmological data. The large scale structure analysis of neutrinos will be vastly improved by upcoming data sets (e.g. future DES data, LSST, Euclid). In order for these data sets to provide improved constarints, further study of small scale modelling and other systematic eﬀects is required. This type of analysis could also be extended by investigating the impact of diﬀerent priors on the neutrino parameters. Overall, this thesis demonstrates both the latest constraints on sterile neutrinos from cosmology, contextualising them with oscillation experiment results, and demonstrates part of the systematics analysis required to make these constraints a reality. Bibliography

[1] S. Bridle, J. Elvin-Poole, et al. A combined view of sterile-neutrino constraints from CMB and neutrino oscillation measurements. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 764, 2017.

[2] DES Collaboration et al. Dark Energy Survey Year 1 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing. Phys. Rev. D, 98:043526, 2017.

[3] J. Elvin-Poole et al. Dark Energy Survey Year 1 Results: Galaxy clustering for combined probes. Phys. Rev. D, 98:042006, 2018.

[4] DES Collaboration et al. Dark Energy Survey Year 1 Results: Constraints on cosmological extensions from Galaxy Clustering and Weak Lensing. in prep, 2017.

[5] S. Hinton. ChainConsumer. JOSS, 1:52, January 2016.

[6] S Dodelson. Modern cosmology. Academic Press, 2003.

[7] J. A. Peacock. Cosmological Physics. Cambridge University Press, 1999.

[8] Andrew R. Liddle and David H. Lyth. Cosmological Inﬂation and Large-Scale Structure. Cambridge University Press, 2000.

[9] E. Hubble. A relation between distance and radial velocity among extra-galactic nebulae. Proceedings of the National Academy of Sciences, 15(3):168–173, mar 1929.

[10] Planck Collaboration et al. Planck 2015 results. XIII. Cosmological parameters. Astronomy & Astrophysics, 594:A13, September 2016.

131 132 BIBLIOGRAPHY

[11] F. Zwicky. Republication of: The redshift of extragalactic nebulae. General Relativity and Gravitation: Rebpublication 2008, 41(1):207–224, nov 1933.

[12] V. C. Rubin and W. K. Jr. Ford. Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions. The Astrophysical Journal, 159:379, feb 1970.

[13] M. S. Roberts and R. N. Whitehurst. The rotation curve and geometry of M 31 at large galactocentric distances. The Astrophysical Journal, 201:327, oct 1975.

[14] Massimo Persic and Paolo Salucci. Rotation Curves of 967 Spiral Galaxies. The Astrophysical Journal Supplement Series, 99:501, aug 1995.

[15] Daniel B. Thomas, Michael Kopp, and Constantinos Skordis. Constraining dark matter properties with Cosmic Microwave Background observations. Astro- phys.J., 830:10, jan 2016.

[16] Kim Griest, Agnieszka M. Cieplak, and Matthew J. Lehner. Experimental Limits On Primordial Black Hole Dark Matter From The First 2 Yr Of Kepler Data. The Astrophysical Journal, 786(2):158, may 2014.

[17] Maxim Khlopov. Cosmological Probes for Supersymmetry. Symmetry, 7(2):815– 842, may 2015.

[18] Xiangdong Shi and George M. Fuller. New Dark Matter Candidate: Nonthermal Sterile Neutrinos. Physical Review Letters, 82(14):2832–2835, apr 1999.

[19] Steen Hannestad. Neutrino physics from precision cosmology. Progress in Par- ticle and Nuclear Physics, 65(2):185–208, 2010.

[20] Adam G. Riess et al. Observational Evidence from Supernovae for an Ac- celerating Universe and a Cosmological Constant. The Astronomical Journal, 116(3):1009–1038, sep 1998.

[21] S. Perlmutter et al. Measurements of Ω and Λ from 42 High-Redshift Supernovae. The Astrophysical Journal, 517(2):565–586, jun 1999. BIBLIOGRAPHY 133

[22] M. Sullivan et al. SNLS3: Constraints On Dark Energy Combining The Su- pernova Legacy Survey Three-year Data With Other Probes. The Astrophysical Journal, 737(2):102, aug 2011.

[23] E. Komatsu et al. SEVEN-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE ( WMAP ) OBSERVATIONS: COSMOLOGICAL INTERPRETA- TION. The Astrophysical Journal Supplement Series, 192(2):18, feb 2011.

[24] Austin Joyce, Bhuvnesh Jain, Justin Khoury, and Mark Trodden. Beyond the cosmological standard model. Physics Reports, 568:1–98, mar 2015.

[25] Bhuvnesh Jain and Justin Khoury. Cosmological tests of gravity. Annals of Physics, 325(7):1479–1516, jul 2010.

[26] B. Jain et al. Novel Probes of Gravity and Dark Energy. ArXiv e-prints, Septem- ber 2013.

[27] Joshua A Frieman, Michael S. Turner, and Dragan Huterer. Dark Energy and the Accelerating Universe. Annual Review of Astronomy and Astrophysics, 46(1):385–432, 2008.

[28] A. Albrecht et al. Report of the Dark Energy Task Force. ArXiv Astrophysics e-prints, September 2006.

[29] A. Peel, M. A. Troxel, and M. Ishak. Eﬀect of inhomogeneities on high precision measurements of cosmological distances. Physical Review D, 90(12):123536, dec 2014.

[30] Planck Collaboration et al. Planck 2015 results. XIV. Dark energy and modiﬁed gravity. Astronomy & Astrophysics, 2016.

[31] Elisa G. M. Ferreira, Jerome Quintin, André A. Costa, E. Abdalla, and Bin Wang. Evidence for interacting dark energy from BOSS. Phys. Rev. D, page 043520, dec 2014.

[32] Timothée Delubac et al. Baryon acoustic oscillations in the Ly α forest of BOSS DR11 quasars. Astronomy & Astrophysics, 574:A59, jan 2015. 134 BIBLIOGRAPHY

[33] Steven Weinberg. The cosmological constant problem. Reviews of Modern Physics, 61(1):1–23, jan 1989.

[34] Robert Brandenberger. Initial Conditions for Inﬂation - A Short Review. Int. J. Mod. Phys., D26(1):1740002, jan 2016.

[35] S. Tsujikawa. Introductory review of cosmic inﬂation. ArXiv High Energy Physics - Phenomenology e-prints, April 2003.

[36] Qing-Guo Huang, Ke Wang, and Sai Wang. Inﬂation model constraints from data released in 2015. Phys. Rev. D, 93(10):103516, dec 2015.

[37] P.A.R. Ade et al. Detection of B -Mode Polarization at Degree Angular Scales by BICEP2. Physical Review Letters, 112(24):241101, jun 2014.

[38] D. Hanson et al. Detection of B -Mode Polarization in the Cosmic Microwave Background with Data from the South Pole Telescope. Physical Review Letters, 111(14):141301, sep 2013.

[39] The Polarbear Collaboration: P. A. R. Ade et al. A Measurement Of The Cosmic Microwave Background B -mode Polarization Power Spectrum At Sub-degree Scales With Polarbear. The Astrophysical Journal, 794(2):171, oct 2014.

[40] Martin Kilbinger. Cosmology with cosmic shear observations: a review. Rept. Prog. Phys, 78:086901, 2014.

[41] Andrew R Liddle and David H Lyth. Cosmological Inﬂation and Large-Scale Structure. Cambridge Univ. Press, Cambridge, 2000.

[42] Anthony Challinor and Antony Lewis. Linear power spectrum of observed source number counts. Physical Review D, 84(4):043516, aug 2011.

[43] J. Lesgourgues. The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview. ArXiv e-prints, April 2011.

[44] J. Zuntz, M. Paterno, E. Jennings, D. Rudd, A. Manzotti, S. Dodelson, S. Bridle, S. Sehrish, and J. Kowalkowski. CosmoSIS: Modular cosmological parameter estimation. Astronomy and Computing, 12:45–59, 2015. BIBLIOGRAPHY 135

[45] R. Takahashi, M. Sato, T. Nishimichi, A. Taruya, and M. Oguri. Revising the Haloﬁt Model for the Nonlinear Matter Power Spectrum. ApJ, 761:152, December 2012.

[46] G. B. Poole et al. The WiggleZ Dark Energy Survey: probing the epoch of radiation domination using large-scale structure. MNRAS, 429(3):1902–1912, jan 2013.

[47] R. E. Smith et al. Stable clustering, the halo model and non-linear cosmological power spectra. MNRAS, 341:1311–1332, June 2003.

[48] Katarina Markovic, Sarah Bridle, Anže Slosar, and Jochen Weller. Constraining warm dark matter with cosmic shear power spectra. Journal of Cosmology and Astroparticle Physics, 2011(01):022–022, 2011.

[49] O Elgaroy and O Lahav. Upper limits on neutrino masses from the 2dFGRS and WMAP: the role of priors. arXiv preprint astro-ph/0303089, 2003.

[50] K.N. Abazajian et al. Light Sterile Neutrinos: A White Paper. FERMILAB- PUB-12-881-PPD, 2012.

[51] Scott Dodelson and Lawrence M. Widrow. Sterile Neutrinos as Dark Matter. Phys. Rev. Lett., 72:17, 1994.

[52] Wayne Hu and Martin White. The Damping Tail of Cosmic Microwave Back- ground Anisotropies. The Astrophysical Journal, 479(2):568–579, apr 1997.

[53] Planck collaboration et al. Planck 2015 results. XV. Gravitational lensing. As- tronomy & Astrophysics, 594, 2015.

[54] Rupert Allison, Paul Caucal, Erminia Calabrese, Joanna Dunkley, and Thibaut Louis. Towards a cosmological neutrino mass detection. Phys. Rev. D, 92:123535, sep 2015.

[55] Matteo Viel, Julien Lesgourgues, Martin G. Haehnelt, Sabino Matarrese, and Antonio Riotto. Constraining warm dark matter candidates including sterile neutrinos and light gravitinos with WMAP and the Lyman-alpha forest. Physical Review D, 71(6), 2005. 136 BIBLIOGRAPHY

[56] N. W. Boggess, J. C. Mather, R. Weiss, C. L. Bennett, E. S. Cheng, E. Dwek, S. Gulkis, M. G. Hauser, M. a. Janssen, and T. Kelsall. The COBE mission - Its design and performance two years after launch. Astrophysical Journal, 397:420–429, 1992.

[57] P. A. R. Ade et al. Planck 2013 results. XVI. Cosmological parameters. Astron- omy and Astrophysics, 571:A16, November 2014.

[58] M. D. Niemack et al. Actpol: a polarization-sensitive receiver for the atacama cosmology telescope. In Wayne S. Holland and Jonas Zmuidzinas, editors, Mil- limeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for As- tronomy V (Conference 7741), pages 77411S–77411S–21, jul 2010.

[59] J. E. McEwen, X. Fang, C. M. Hirata, and J. A. Blazek. FAST-PT: a novel algorithm to calculate convolution integrals in cosmological perturbation theory. JCAP, 9:015, September 2016.

[60] V. Desjacques, D. Jeong, and F. Schmidt. Large-scale galaxy bias. PHYS REP, 733:1–193, February 2018.

[61] Martín Crocce, Anna Cabré, and Enrique Gaztañaga. Modelling the angular correlation function and its full covariance in photometric galaxy surveys. MNRAS, 414(1):329–349, jun 2011.

[62] D. N. Limber. The Analysis of Counts of the Extragalactic Nebulae in Terms of a Fluctuating Density Field. ApJ, 117:134, January 1953.

[63] P. Lemos, A. Challinor, and G. Efstathiou. The eﬀect of Limber and ﬂat-sky approximations on galaxy weak lensing. JCAP, 5:014, May 2017.

[64] Robert C. Nichol. Cosmology with galaxy correlations. General Relativity and Gravitation, 40(2-3):249–267, dec 2007.

[65] Bruce A. Bassett and Renée Hlozek. Baryon Acoustic Oscillations. Dark Energy, Ed. P. Ruiz-Lapuente (2010, ISBN-13: 9780521518888), oct 2009.

[66] Yazhou Hu, Miao Li, Nan Li, and Shuang Wang. Cosmological implications of diﬀerent baryon acoustic oscillation data. Phys. Rev. D, 92:123516, jun 2015. BIBLIOGRAPHY 137

[67] F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell, Q. Parker, W. Saunders, and F. Watson. The 6dF Galaxy Survey: baryon acoustic oscillations and the local Hubble constant. MNRAS, 416:3017–3032, October 2011.

[68] K. C. Chan et al. BAO from angular clustering: optimization and mitigation of theoretical systematics. ArXiv e-prints, January 2018.

[69] Daniel J. Eisenstein et al. Detection of the Baryon Acoustic Peak in the Large- Scale Correlation Function of SDSS Luminous Red Galaxies. The Astrophysical Journal, 633(2):560–574, 2005.

[70] L. Anderson et al. The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: baryon acoustic oscillations in the Data Releases 10 and 11 Galaxy samples. MNRAS, 441(1):24–62, apr 2014.

[71] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Burden, and M. Manera. The clustering of the SDSS DR7 main Galaxy sample - I. A 4 per cent distance measure at z = 0.15. MNRAS, 449:835–847, May 2015.

[72] S. Alam et al. The clustering of galaxies in the completed SDSS-III Baryon Os- cillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample. MNRAS, 470:2617–2652, September 2017.

[73] Chris Blake et al. The WiggleZ Dark Energy Survey: mapping the distance- redshift relation with baryon acoustic oscillations. MNRAS, 418(3):1707–1724, dec 2011.

[74] E. A. Kazin et al. The WiggleZ Dark Energy Survey: improved distance measurements to z = 1 with reconstruction of the baryonic acoustic feature. MNRAS, 441:3524–3542, July 2014.

[75] M. Ata et al. The clustering of the SDSS-IV extended Baryon Oscillation Spec- troscopic Survey DR14 quasar sample: ﬁrst measurement of baryon acoustic oscillations between redshift 0.8 and 2.2. MNRAS, 473:4773–4794, February 2018. 138 BIBLIOGRAPHY

[76] W. J. Percival et al. The 2dF Galaxy Redshift Survey: the power spectrum and the matter content of the Universe. MNRAS, 327(4):1297–1306, nov 2001.

[77] J. E. Bautista et al. Measurement of baryon acoustic oscillation correlations at z = 2.3 with SDSS DR12 Lyα-Forests. AAP, 603:A12, June 2017.

[78] H. du Mas des Bourboux et al. Baryon acoustic oscillations from the complete SDSS-III Lyα-quasar cross-correlation function at z = 2.4. AAP, 608:A130, December 2017.

[79] The Dark Energy Survey Collaboration et al. Dark Energy Survey Year 1 Re- sults: Measurement of the Baryon Acoustic Oscillation scale in the distribution of galaxies to redshift 1. ArXiv e-prints 1712.06209, December 2017.

[80] M. Crocce et al. Dark Energy Survey Year 1 Results: Galaxy Sample for BAO Measurement. ArXiv e-prints, December 2017.

[81] S. Avila et al. Dark Energy Survey Year-1 results: galaxy mock catalogues for BAO. MNRAS, 479:94–110, September 2018.

[82] A. G. Sanchez et al. The clustering of galaxies in the SDSS-III Baryon Oscil- lation Spectroscopic Survey: cosmological constraints from the full shape of the clustering wedges. MNRAS, 433(2):1202–1222, jun 2013.

[83] Yong-Seon Song and Will J Percival. Reconstructing the history of structure formation using redshift distortions. Journal of Cosmology and Astroparticle Physics, 2009(10):004–004, oct 2009.

[84] Héctor Gil-Marín, Will J. Percival, et al. The clustering of galaxies in the SDSS- III Baryon Oscillation Spectroscopic Survey: RSD measurement from the LOS- dependent power spectrum of DR12 BOSS galaxies. MNRAS, 460(4):4210–4219, sep 2015.

[85] S. D. Landy and A. S. Szalay. Bias and variance of angular correlation functions. ApJ, 412:64–71, July 1993. BIBLIOGRAPHY 139

[86] Ashley J. Ross et al. The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: analysis of potential systematics. MNRAS, 424(1):564– 590, jul 2012.

[87] Asantha Cooray and Ravi Sheth. Halo Models of Large Scale Structure. Phys. Rept., 372(February 2008):1–129, 2002.

[88] Henk Hoekstra and Bhuvnesh Jain. Weak Gravitational Lensing and Its Cosmo- logical Applications. Annual Review of Nuclear and Particle Science, 58(1):99– 123, 2008.

[89] Antony Lewis. Galaxy shear estimation from stacked images. MNRAS, 398(1):471–476, sep 2009.

[90] L. Miller et al. Bayesian galaxy shape measurement for weak lensing surveys - III. Application to the Canada-France-Hawaii Telescope Lensing Survey. MNRAS, 429(4):2858–2880, 2013.

[91] J. Zuntz, T. Kacprzak, L. Voigt, M. Hirsch, B. Rowe, and S. Bridle. IM3SHAPE: a maximum likelihood galaxy shear measurement code for cosmic gravitational lensing. MNRAS, 434(2):1604–1618, jul 2013.

[92] E. S. Sheldon. An implementation of Bayesian lensing shear measurement. MN- RAS: Letters, 444(1):L25–L29, jul 2014.

[93] Gary M. Bernstein, Robert Armstrong, Christina Krawiec, and Marisa C. March. An accurate and practical method for inference of weak gravitational lensing from galaxy images. MNRAS, 459(4):4467–4484, aug 2015.

[94] Richard Massey et al. The Shear Testing Programme 2: Factors aﬀecting high- precision weak-lensing analyses. MNRAS, 376(1):13–38, mar 2007.

[95] C. Heymans et al. The Shear Testing Programme - I. Weak lensing analysis of simulated ground-based observations. MNRAS, 368(3):1323–1339, may 2006.

[96] H Miyatake, R Mandelbaum, and B Rowe. The GREAT3 challenge. Journal of Instrumentation, 9(04):C04031–C04031, apr 2014. 140 BIBLIOGRAPHY

[97] Sarah Bridle et al. Results of the GREAT08 Challenge: an image analysis competition for cosmological lensing. MNRAS, 405:2044–2061, July 2010.

[98] T. D. Kitching et al. Image analysis for cosmology: results from the GREAT10 Galaxy Challenge. MNRAS, 423(4):3163–3208, jul 2012.

[99] R. Mandelbaum et al. GREAT3 results - I. Systematic errors in shear estimation and the impact of real galaxy morphology. MNRAS, 450(3):2963–3007, may 2015.

[100] Tomasz Kacprzak, Sarah Bridle, Barnaby Rowe, Lisa Voigt, Joe Zuntz, Michael Hirsch, and Niall MacCrann. Sersic galaxy models in weak lensing shape measurement: model bias, noise bias and their interaction. MNRAS, 441(3):2528– 2538, aug 2013.

[101] T. Kacprzak, J. Zuntz, B. Rowe, S. Bridle, A. Refregier, A. Amara, L. Voigt, and M. Hirsch. Measurement and calibration of noise bias in weak lensing galaxy shape estimation. MNRAS, 427(4):2711–2722, jan 2013.

[102] M. Jarvis et al. The DES Science Veriﬁcation Weak Lensing Shear Catalogs. MNRAS, 460(2):2245–2281, jul 2015.

[103] S. Samuroﬀ et al. Dark Energy Survey Year 1 Results: The Impact of Galaxy Neighbours on Weak Lensing Cosmology with im3shape. submitted to Phys. Rev. D, 2017.

[104] E. Huﬀ and R. Mandelbaum. Metacalibration: Direct Self-Calibration of Biases in Shear Measurement. ArXiv e-prints 1702.02600, February 2017.

[105] E. S. Sheldon and E. M. Huﬀ. Practical Weak-lensing Shear Measurement with Metacalibration. ApJ, 841:24, May 2017.

[106] C. Bonnett et al. Redshift distributions of galaxies in the DES Science Veriﬁ- cation shear catalogue and implications for weak lensing. Physical Review D, 94(4), jul 2015. BIBLIOGRAPHY 141

[107] M.A. A Troxel and Mustapha Ishak. The intrinsic alignment of galaxies and its impact on weak gravitational lensing in an era of precision cosmology. Physics Reports, 558:1–59, 2015.

[108] Benjamin Joachimi, Marcello Cacciato, Thomas D. Kitching, Adrienne Leonard, Rachel Mandelbaum, Björn Malte Schäfer, Cristóbal Sifón, Henk Hoekstra, Alina Kiessling, Donnacha Kirk, and Anais Rassat. Galaxy Alignments: An Overview. Space Science Reviews, 193(1-4):1–65, jul 2015.

[109] S. Bridle and L. King. Dark energy constraints from cosmic shear power spectra: impact of intrinsic alignments on photometric redshift requirements. New Journal of Physics, 9:444, December 2007.

[110] Jonathan Blazek, Zvonimir Vlah, and Uroš Seljak. Tidal alignment of galaxies. Journal of Cosmology and Astroparticle Physics, 2015(08):015–015, aug 2015.

[111] J. Blazek, N. MacCrann, M. A. Troxel, and X. Fang. Beyond linear galaxy alignments. ArXiv e-prints 1708.09247, August 2017.

[112] P. Catelan, M. Kamionkowski, and R. D. Blandford. Intrinsic and extrinsic galaxy alignment. MNRAS, 320(1):L7–L13, 2001.

[113] B. Joachimi and S. L. Bridle. Simultaneous measurement of cosmology and intrinsic alignments using joint cosmic shear and galaxy number density correlations. Astronomy & Astrophysics, 523:A1, nov 2010.

[114] Istvan Laszlo, Rachel Bean, Donnacha Kirk, and Sarah Bridle. Disentangling dark energy and cosmic tests of gravity from weak lensing systematics. MNRAS, 423(2):1750–1765, 2011.

[115] H. Zhan and L. Knox. Eﬀect of Hot Baryons on the Weak-Lensing Shear Power Spectrum. ApJ, 616:L75–L78, December 2004.

[116] Tim Eiﬂer, Elisabeth Krause, Scott Dodelson, Andrew R. Zentner, Andrew P. Hearin, and Nickolay Y. Gnedin. Accounting for baryonic eﬀects in cosmic shear tomography: determining a minimal set of nuisance parameters using PCA. MNRAS, 454(3):2451–2471, oct 2015. 142 BIBLIOGRAPHY

[117] A. J. Mead, J. A. Peacock, C. Heymans, S. Joudaki, and A. F. Heavens. An accurate halo model for ﬁtting non-linear cosmological power spectra and baryonic feedback models. MNRAS, 454(2):1958–1975, oct 2015.

[118] D. J. Bacon, A. R. Refregier, and R. S. Ellis. Detection of weak gravitational lensing by large-scale structure. MNRAS, 318(2):625–640, oct 2000.

[119] L. Van Waerbeke, Y. Mellier, M. Radovich, E. Bertin, M. Dantel-Fort, H. J. McCracken, O. Le Fèvre, S. Foucaud, J.-C. Cuillandre, T. Erben, B. Jain, P. Schneider, F. Bernardeau, and B. Fort. Cosmic shear statistics and cosmology. Astronomy and Astrophysics, 374(3):757–769, aug 2001.

[120] M. J. Jee, J. A. Tyson, M. D. Schneider, D. Wittman, S. Schmidt, and S. Hilbert.

Cosmic Shear Results from the Deep Lens Survey. I. Joint Constraints on ΩM

and σ8 with a Two-dimensional Analysis. ApJ, 765:74, March 2013.

[121] M. J. Jee, J. A. Tyson, S. Hilbert, M. D. Schneider, S. Schmidt, and D. Wittman. Cosmic Shear Results from the Deep Lens Survey. II. Full Cosmological Param- eter Constraints from Tomography. ApJ, 824:77, June 2016.

[122] E. M. Huﬀ. Seeing in the Dark: Weak Lensing from the Sloan Digital Sky Survey. PhD thesis, University of California, Berkeley, 2012.

[123] H. Lin et al. The SDSS Co-add: Cosmic Shear Measurement. ApJ, 761:15, December 2012.

[124] Catherine Heymans et al. CFHTLenS: The Canada-France-Hawaii Telescope Lensing Survey. MNRAS, 427(1):146–166, 2012.

[125] J. T. A. de Jong et al. The Kilo-Degree Survey. The Messenger, 154:44–46, dec 2013.

[126] H. Hildebrandt et al. KiDS-450: cosmological parameter constraints from tomographic weak gravitational lensing. MNRAS, 465:1454–1498, February 2017.

[127] The Dark Energy Survey Collaboration et al. Cosmology from Cosmic Shear with DES Science Veriﬁcation Data. Submitted to MNRAS, 20(July):20, 2015. BIBLIOGRAPHY 143

[128] M. Troxel et al. Dark Energy Survey Year 1 Results: Cosmology from Cosmic shear. Phys. Rev. D, 98:043528, 2017.

[129] M. A. Troxel, E. Krause, et al. Survey geometry and the internal consistency of recent cosmic shear measurements. MNRAS, 479:4998–5004, October 2018.

[130] J. A. Tyson, F. Valdes, J. F. Jarvis, and A. P. Mills, Jr. Galaxy mass distribution from gravitational light deﬂection. ApJ, 281:L59–L62, June 1984.

[131] C. M. Hirata et al. Galaxy-galaxy weak lensing in the Sloan Digital Sky Sur- vey: intrinsic alignments and shear calibration errors. MNRAS, 353:529–549, September 2004.

[132] R. Mandelbaum, U. Seljak, G. Kauﬀmann, C. M. Hirata, and J. Brinkmann. Galaxy halo masses and satellite fractions from galaxy-galaxy lensing in the Sloan Digital Sky Survey: stellar mass, luminosity, morphology and environment dependencies. MNRAS, 368:715–731, May 2006.

[133] R. Mandelbaum et al. Cosmological parameter constraints from galaxy-galaxy lensing and galaxy clustering with the SDSS DR7. MNRAS, 432:1544–1575, June 2013.

[134] B. R. Gillis et al. CFHTLenS: the environmental dependence of galaxy halo masses from weak lensing. MNRAS, 431:1439–1452, May 2013.

[135] M. Velander et al. CFHTLenS: the relation between galaxy dark matter haloes and baryons from weak gravitational lensing. MNRAS, 437:2111–2136, January 2014.

[136] M. J. Hudson et al. CFHTLenS: co-evolution of galaxies and their dark matter haloes. MNRAS, 447:298–314, February 2015.

[137] K. Kuijken et al. Gravitational lensing analysis of the Kilo-Degree Survey. MN- RAS, 454:3500–3532, December 2015.

[138] E. van Uitert et al. KiDS+GAMA: cosmology constraints from a joint analysis of cosmic shear, galaxy-galaxy lensing, and angular clustering. MNRAS, 476:4662– 4689, June 2018. 144 BIBLIOGRAPHY

[139] J. Clampitt et al. Galaxy-galaxy lensing in the Dark Energy Survey Science Veriﬁcation data. MNRAS, 465:4204–4218, March 2017.

[140] J. Prat et al. Dark Energy Survey Year 1 Results: Galaxy-Galaxy Lensing. Phys. Rev. D, 98:042005, 2017.

[141] Ramon Miquel. Cosmology with type-Ia supernovae. Journal of Physics A: Mathematical and Theoretical, 40(25):6743–6755, jun 2007.

[142] M. Betoule et al. Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples. Astron. Astrophys., 568:A22, 2014.

[143] Tzu-Ching Chang, Ue-Li Pen, Kevin Bandura, and Jeﬀrey B. Peterson. Hydro- gen 21-cm Intensity Mapping at redshift 0.8. Nature, 466:463–465, jul 2010.

[144] M. G. Santos et al. Cosmology from a SKA HI intensity mapping survey. Ad- vancing Astrophysics with the Square Kilometre Array (AASKA14), page 19, April 2015.

[145] R. A. Burenin, A. Vikhlinin, A. Hornstrup, H. Ebeling, H. Quintana, and A. Mescheryakov. The 400 Square Degree ROSAT PSPC Galaxy Cluster Sur- vey: Catalog and Statistical Calibration. The Astrophysical Journal Supplement Series, 172(2):561–582, oct 2007.

[146] T. Kitayama. Cosmological and astrophysical implications of the Sunyaev- Zel’dovich eﬀect. Progress of Theoretical and Experimental Physics, 2014(6):6B111, jun 2014.

[147] D. Kirk et al. Cross-correlation of gravitational lensing from DES Science Veri- ﬁcation data with SPT and Planck lensing. MNRAS, 459(1):21–34, dec 2015.

[148] T. Giannantonio et al. CMB lensing tomography with the DES Science Veriﬁ- cation galaxies. MNRAS, 456(3):3213–3244, jul 2015.

[149] A. Amara et al. The COSMOS density ﬁeld: a reconstruction using both weak lensing and galaxy distributions. MNRAS, 424(1):553–563, jul 2012. BIBLIOGRAPHY 145

[150] Anthony R. Pullen, Shadab Alam, Siyu He, and Shirley Ho. Constraining Grav- ity at the Largest Scales through CMB Lensing and Galaxy Velocities. MNRAS, 460(4):4098–4108, nov 2015.

[151] Richard A Battye and Adam Moss. Evidence for massive neutrinos from cosmic microwave background and lensing observations. Phys. Rev. Lett, 112(5):051303, feb 2014.

[152] Richard A. Battye, Tom Charnock, and Adam Moss. Tension between the power spectrum of density perturbations measured on large and small scales. Physical Review D, 91(10):103508, 2015.

[153] N. MacCrann, J. Zuntz, S. Bridle, B. Jain, and M. R. Becker. Cosmic discordance: are Planck CMB and CFHTLenS weak lensing measurements out of tune? MNRAS, 451(3):2877–2888, 2015.

[154] Marco Raveri. Is there concordance within the concordance Λcdm model? Phys- ical Review D, 93(4):043522, oct 2015.

[155] DES Collaboration et al. Dark Energy Survey Year 1 Results: A Precise H0 Measurement from DES Y1, BAO, and D/H Data. ArXiv e-prints, November 2017.

[156] J. W. Henning et al. Measurements of the Temperature and E-mode Polarization of the CMB from 500 Square Degrees of SPTpol Data. ApJ, 852:97, January 2018.

[157] A. G. Riess et al. A 2.4% Determination of the Local Value of the Hubble Constant. ApJ, 826:56, July 2016.

[158] V. Bonvin et al. H0LiCOW - V. New COSMOGRAIL time delays of HE 0435-

1223: H0 to 3.8 per cent precision from strong lensing in a ﬂat ΛCDM model. MNRAS, 465:4914–4930, March 2017.

[159] Dark Energy Survey Collaboration et al. The Dark Energy Survey: more than dark energy - an overview. MNRAS, 460:1270–1299, August 2016. 146 BIBLIOGRAPHY

[160] Konrad Kuijken et al. Gravitational lensing analysis of the Kilo-Degree Survey. MNRAS, 454(4):3500–3532, oct 2015.

[161] LSST Dark Energy Science Collaboration. Large Synoptic Survey Telescope: Dark Energy Science Collaboration. ArXiv e-prints 1211.0310, November 2012.

[162] Luca Amendola et al. Cosmology and Fundamental Physics with the Euclid Satellite. Living Reviews in Relativity, 16:236, jun 2013.

[163] D. Spergel et al. WFIRST-2.4: What Every Astronomer Should Know. ArXiv e-prints 1305.5425, May 2013.

[164] Ian Harrison, Stefano Camera, Joe Zuntz, and Michael L. Brown. SKA Weak Lensing I: Cosmological Forecasts and the Power of Radio-Optical Cross- Correlations. MNRAS, 463(4):3674–3685, jan 2016.

[165] Anna Bonaldi, Ian Harrison, Stefano Camera, and Michael L. Brown. SKA Weak Lensing II: Simulated Performance and Survey Design Considerations. MNRAS: Letters, 463(4):3686–3698, jan 2016.

[166] B. Pontecorvo. Mesonium and Antimesonium. Soviet Journal of Experimental and Theoretical Physics, 6:429, 1958.

[167] B. Pontecorvo. Inverse beta processes and nonconservation of lepton charge. Soviet Journal of Experimental and Theoretical Physics, 7:172–173, 1958. [Zh. Eksp. Teor. Fiz.34,247(1957)].

[168] Ziro Maki, Masami Nakagawa, and Shoichi Sakata. Remarks on the uniﬁed model of elementary particles. Progress of Theoretical Physics, 28(5):870–880, 1962.

[169] H. Nunokawa, S. Parke, and J. W. F. Valle. CP violation and neutrino oscillations. Progress in Particle and Nuclear Physics, 60:338–402, April 2008.

[170] C. Giunti and M. Laveder. Neutrino Mixing. ArXiv High Energy Physics - Phenomenology e-prints hep-ph/0310238, October 2003.

[171] A. Bellerive, J. R. Klein, A. B. McDonald, A. J. Noble, and A. W. P. Poon. The Sudbury Neutrino Observatory. Nuclear Physics B, 908:30–51, July 2016. BIBLIOGRAPHY 147

[172] J. R. Wilson. An Experimental Review of Solar Neutrinos. ArXiv e-prints 1504.04281, April 2015.

[173] W. C. Haxton, R. G. Hamish Robertson, and A. M. Serenelli. Solar Neutrinos: Status and Prospects. ARAA, 51:21–61, August 2013.

[174] Y. Fukuda et al. Evidence for Oscillation of Atmospheric Neutrinos. Physical Review Letters, 81:1562–1567, August 1998.

[175] R. Wendell. Atmospheric results from Super-Kamiokande. In American Insti- tute of Physics Conference Series, volume 1666 of American Institute of Physics Conference Series, page 100001, July 2015.

[176] T. K. Gaisser. Atmospheric Neutrinos. In J. R. Wilkes, editor, Next Genera- tion Nucleon Decay and Neutrino Deterctors NNN06, volume 944 of American Institute of Physics Conference Series, pages 140–142, November 2007.

[177] T. Kajita. Atmospheric neutrinos and discovery of neutrino oscillations. Pro- ceeding of the Japan Academy, Series B, 86:303–321, 2010.

[178] P. Pasquini. Review: Long-baseline oscillation experiments as a tool to probe High Energy Models. ArXiv e-prints 1802.00821, February 2018.

[179] A. Sousa. Long-Baseline Neutrino Oscillation Experiments. ArXiv e-prints 1102.1125, February 2011.

[180] G. J. Feldman, J. Hartnell, and T. Kobayashi. A Review of Long-baseline Neu- trino Oscillation Experiments. ArXiv e-prints 1210.1778, October 2012.

[181] M. Tanabashi et al. The Review of Particle Physics. Phys. Rev. D., 98:030001, 2018.

[182] T. Katori. Short Baseline Neutrino Oscillation Experiments. In Journal of Physics Conference Series, volume 598 of Journal of Physics Conference Series, page 012006, April 2015.

[183] J. Evans and for the MINOS collaboration. The MINOS experiment: results and prospects. ArXiv e-prints, July 2013. 148 BIBLIOGRAPHY

[184] P. Adamson et al. Search for Sterile Neutrinos Mixing with Muon Neutrinos in MINOS. Physical Review Letters, 117(15):151803, October 2016.

[185] F. P. An et al. A side-by-side comparison of Daya Bay antineutrino detectors. Nuclear Instruments and Methods in Physics Research A, 685:78–97, September 2012.

[186] The Daya Bay collaboration et al. Improved Search for a Light Sterile Neu- trino with the Full Conﬁguration of the Daya Bay Experiment. ArXiv e-prints 1607.01174, July 2016.

[187] A. Aguilar-Arevalo et al. Evidence for neutrino oscillations from the observation of anti-neutrino(electron) appearance in a anti-neutrino(muon) beam. Phys. Rev., D64:112007, 2001.

[188] A. A. Aguilar-Arevalo et al. A Search for electron neutrino appearance at the ∆m2 1eV2 scale. Phys. Rev. Lett., 98:231801, 2007. ∼ [189] A. A. Aguilar-Arevalo et al. Event Excess in the MiniBooNE Search for ν¯ ν¯ µ → e Oscillations. Phys. Rev. Lett., 105:181801, 2010.

[190] A. A. Aguilar-Arevalo et al. Improved Search for ν¯ ν¯ Oscillations in the µ → e MiniBooNE Experiment. Phys. Rev. Lett., 110:161801, 2013.

[191] MiniBooNE Collaboration et al. Observation of a Signiﬁcant Excess of Electron- Like Events in the MiniBooNE Short-Baseline Neutrino Experiment. ArXiv e- prints 1805.12028, May 2018.

[192] Daya Bay Collaboration, MINOS Collaboration, et al. Limits on Active to Sterile Neutrino Oscillations from Disappearance Searches in the MINOS, Daya Bay, and Bugey-3 Experiments. Phys. Rev. Lett., page 151801, 2016.

[193] Francis Halzen and Spencer R. Klein. Invited review article: IceCube: An instrument for neutrino astronomy. Review of Scientiﬁc Instruments, 81(8):081101, jul 2010.

[194] E. W. Otten and C. Weinheimer. Neutrino mass limit from tritium β decay. Reports on Progress in Physics, 71(8):086201, August 2008. BIBLIOGRAPHY 149

[195] Ch. Weinheimer. Neutrino mass from beta decay. Nuclear Physics B - Pro- ceedings Supplements, 168:5 – 10, 2007. Proceedings of the Neutrino Oscillation Workshop.

[196] V.M. Lobashev et al. Neutrino mass and anomaly in the tritium beta-spectrum. results of the troitsk ν-mass experiment. Nuclear Physics B - Proceedings Sup- plements, 77(1):327 – 332, 1999.

[197] C. Kraus et al. Final results from phase II of the Mainz neutrino mass searchin tritium β decay. European Physical Journal C, 40:447–468, April 2005.

[198] T. Thümmler. Introduction to direct neutrino mass measurements and katrin. Nuclear Physics B - Proceedings Supplements, 229-232:146 – 151, 2012. Neutrino 2010.

[199] J. Lesgourgues and S. Pastor. Neutrino cosmology and Planck. New Journal of Physics, 16(6):065002, June 2014.

[200] J. Lesgourgues and S. Pastor. Massive neutrinos and cosmology. Physics Reports, 429:307–379, July 2006.

[201] S. Hannestad. Neutrinos in cosmology. Progress in Particle and Nuclear Physics, 57:309–323, July 2006.

[202] Steen Hannestad, Irene Tamborra, and Thomas Tram. Thermalisation of light sterile neutrinos in the early universe. J. Cosmol. Astropart. Phys., 2012(07):025– 025, 2012.

[203] G. H. Collin et al. Sterile Neutrino Fits to Short Baseline Data. Nucl. Phys., B908:354–365, 2016.

[204] G. H. Collin et al. First Constraints on the Complete Neutrino Mixing Matrix with a Sterile Neutrino. Phys. Rev. Lett., 117:221801, jun 2016.

[205] G. Mention et al. The Reactor Antineutrino Anomaly. Phys. Rev., D83:073006, 2011.

[206] Carlo Giunti and Marco Laveder. Statistical Signiﬁcance of the Gallium Anomaly. Phys. Rev., C83:065504, 2011. 150 BIBLIOGRAPHY

[207] F. P. An et al. Improved Search for a Light Sterile Neutrino with the Full Conﬁguration of the Daya Bay Experiment. Phys. Rev. Lett., 117(15):151802, 2016.

[208] P. Adamson et al. A search for sterile neutrinos mixing with muon neutrinos in MINOS. Phys. Rev. Lett., 117:151803, 2016.

[209] M. Archidiacono et al. Testing 3+1 and 3+2 neutrino mass models with cosmology and short baseline experiments. Phys. Rev. D, 86:065028, 2012.

[210] M. Archidiacono et al. Sterile Neutrinos: Cosmology vs Short-BaseLine Exper- iments. Phys. Rev. D, 87:125034, 2013.

[211] S. Gariazzo, C. Giunti, and M. Laveder. Light Sterile Neutrinos in Cosmology and Short-Baseline Oscillation Experiments. J. High Energy Phys., 11:211, sep 2013.

[212] Maria Archidiacono et al. Light sterile neutrinos after BICEP-2. J. Cosmol. Astropart. Phys., 1406:031, 2014.

[213] M. Archidiacono et al. Pseudoscalar - sterile neutrino interactions: reconciling the cosmos with neutrino oscillations. J. Cosmol. Astropart. Phys., 1608(08):067, jun 2016.

[214] A. Mirizzi et al. The strongest bounds on active-sterile neutrino mixing after Planck data. Physics Letters B, 726:8–14, October 2013.

[215] A. Melchiorri et al. Sterile neutrinos in light of recent cosmological and oscillation data: a multi-ﬂavor scheme approach. J. Cosmol. Astropart. Phys., 1:036, January 2009.

[216] A. Mirizzi et al. Light sterile neutrino production in the early universe with dynamical neutrino asymmetries. Phys. Rev. D, 86(5):053009, September 2012.

[217] P. A. R. Ade et al. Planck 2013 results. I. Overview of products and scientiﬁc results. Astronomy and Astrophysics, 571:A1, November 2014.

[218] C. Dvorkin et al. Neutrinos help reconcile Planck measurements with both the early and local Universe. Phys. Rev. D, 90(8):083503, October 2014. BIBLIOGRAPHY 151

[219] N. MacCrann et al. Cosmic discordance: are Planck CMB and CFHTLenS weak lensing measurements out of tune? MNRAS, 451:2877–2888, August 2015.

[220] A. B. Mantz et al. Weighing the giants - IV. Cosmology and neutrino mass. MNRAS, 446:2205–2225, January 2015.

[221] Qing-Guo Huang, Ke Wang, and Sai Wang. Constraints on the neutrino mass and mass hierarchy from cosmological observations. arXiv:1512.05899, 2015.

[222] Antonio J. Cuesta, Viviana Niro, and Licia Verde. Neutrino mass limits: robust information from the power spectrum of galaxy surveys. Phys. Dark Univ., 13:17, 2015.

[223] N. Palanque-Delabrouille et al. Neutrino masses and cosmology with Lyman- alpha forest power spectrum. JCAP, 1511(11):11, 2015.

[224] E. Di Valentino et al. Cosmological limits on neutrino unknowns versus low redshift priors. Phys. Rev. D, 93:083527, 2015.

[225] N. Aghanim et al. Planck 2015 results. XI. CMB power spectra, likelihoods, and robustness of parameters. arxiv:1507.02704, jul 2015.

[226] A. Lewis and S. Bridle. Cosmological parameters from CMB and other data: A Monte Carlo approach. Phys. Rev. D, 66(10):103511, November 2002.

[227] J. Zuntz et al. CosmoSIS: Modular cosmological parameter estimation. Astron- omy and Computing, 12:45–59, September 2015.

[228] Maria Archidiacono et al. Cosmic dark radiation and neutrinos. Adv. High Energy Phys., 2013:191047, jul 2013.

[229] S. Gariazzo et al. Light sterile neutrinos. J. Phys., G43:033001, jul 2016.

[230] D. G. Michael et al. The magnetized steel and scintillator calorimeters of the MINOS experiment. Nucl. Instrum. and Meth. A, 596(2):190–228, may 2008.

[231] Junting Huang. Sterile Neutrino Searches in MINOS/MINOS+ Experiment. PhD Thesis, FERMILAB-THESIS-2015-6, page We use data from Figure 8.3, 2015. 152 BIBLIOGRAPHY

[232] L. Wolfenstein. Neutrino oscillations in matter. Phys. Rev. D, 17:2369, 1978.

[233] S. P. Mikheev and A. Y. Smirnov. Resonant ampliﬁcation of neutrino oscillations in matter and spectroscopy of solar neutrinos. Sov. J. Nucl. Phys., 42:913, 1985.

[234] M. G. Aartsen et al. Searches for Sterile Neutrinos with the IceCube Detector. Phys. Rev. Lett., 117(7):071801, 2016.

[235] Benjamin J.P. Jones. Sterile Neutrinos in Cold Climates. PhD Thesis, FERMILAB-THESIS-2015-17, page We use data from Figure 3.4.10, 2015.

[236] M. Antonello et al. A Proposal for a Three Detector Short-Baseline Neutrino Oscillation Program in the Fermilab Booster Neutrino Beam. arxiv:1503.01520, 2015.

[237] Patrick Huber et al. New features in the simulation of neutrino oscillation experiments with GLoBES 3.0. (General Long Baseline Experiment Simulator). Comp. Phys. Comm., 177(5):432, sep 2007.

[238] Steen Hannestad, Rasmus Sloth Hansen, and Thomas Tram. Can active-sterile neutrino oscillations lead to chaotic behavior of the cosmological lepton asymmetry? J. Cosmol. Astropart. Phys., 2013(04):032–032, apr 2013.

[239] Kari Enqvist, Kimmo Kainulainen, and Mark Thomson. Stringent cosmological bounds on inert neutrino mixing. Nucl. Phys. B, 373(2):498–528, apr 1992.

[240] L. Stodolsky. Treatment of neutrino oscillations in a thermal environment. Phys. Rev. D, 36(8):2273–2277, oct 1987.

[241] N. Saviano, A. Mirizzi, O. Pisanti, P. D. Serpico, G. Mangano, and G. Miele. Multimomentum and multiﬂavor active-sterile neutrino oscillations in the early universe: Role of neutrino asymmetries and eﬀects on nucleosynthesis. Phys. Rev. D, 87(7):073006, April 2013.

[242] B. Hoyle et al. Dark Energy Survey Year 1 Results: Redshift distributions of the weak lensing source galaxies. submitted to Phys. Rev. D, 2017.

[243] J. Zuntz et al. Dark Energy Survey Year 1 Results: Weak Lensing Shape Cata- logues. Submitted to: Mon. Not. Roy. Astron. Soc., 2017. BIBLIOGRAPHY 153

[244] E. Krause et al. Dark Energy Survey Year 1 Results: Multi-Probe Methodology and Simulated Likelihood Analyses. submitted to Phys. Rev. D, 2017.

[245] A. Drlica-Wagner et al. The Dark Energy Survey First-Year Cosmology Data Set. submitted to Astrophys. J. Suppl. Ser., 2017.

[246] The Dark Energy Survey Collaboration. The Dark Energy Survey. ArXiv As- trophysics e-prints astro-ph/0510346, October 2005.

[247] B. Flaugher et al. The Dark Energy Camera. AJ, 150:150, November 2015.

[248] T. Baldauf, R. E. Smith, U. Seljak, and R. Mandelbaum. Algorithm for the direct reconstruction of the dark matter correlation function from weak lensing and galaxy clustering. Physical Review D, 81(6):063531, March 2010.

[249] S. Samuroﬀ et al. Simultaneous constraints on cosmology and photometric redshift bias from weak lensing and galaxy clustering. MNRAS, 465:L20–L24, Febru- ary 2017.

[250] Joe Zuntz, Tomasz Kacprzak, Lisa Voigt, Michael Hirsch, Barnaby Rowe, and Sarah Bridle. IM3SHAPE: A maximum-likelihood galaxy shear measurement code for cosmic gravitational lensing. Mon. Not. Roy. Astron. Soc., 434:1604, 2013.

[251] D. Coe, N. Benítez, S. F. Sánchez, M. Jee, R. Bouwens, and H. Ford. Galaxies in the Hubble Ultra Deep Field. I. Detection, Multiband Photometry, Photometric Redshifts, and Morphology. The Astrophysical Journal, 132:926–959, August 2006.

[252] E. Rozo et al. redMaGiC: selecting luminous red galaxies from the DES Science Veriﬁcation data. MNRAS, 461:1431–1450, September 2016.

[253] C. Laigle et al. The COSMOS2015 Catalog: Exploring the 1 < z < 6 Universe with Half a Million Galaxies. ApJS, 224:24, June 2016.

[254] R. Cawthon et al. Dark energy survey year 1 results: Calibration of redmagic photometric redshifts in des and sdss from clustering. in prep., 2017. 154 BIBLIOGRAPHY

[255] C. Davis, M. Gatti, P. Vielzeuf, R. Cawthon, et al. Dark Energy Survey Year 1 Results: Cross-Correlation Redshifts in the DES - Calibration of the Weak Lensing Source Redshift Distributions. to be submitted to PRD, 2017.

[256] M. Gatti, P. Vielzeuf, C. Davis, et al. Dark Energy Survey Year 1 Results: Cross- Correlation Redshifts in the DES - Methods and Systematics Characterization. to be submitted to PRD, 2017.

[257] Elisabeth Krause and Tim Eiﬂer. CosmoLike - Cosmological Likelihood Analyses for Photometric Galaxy Surveys. MNRAS, 470(2):2100–2112, 2016.

[258] H. S. Xavier, F. B. Abdalla, and B. Joachimi. Improving lognormal models for cosmological ﬁelds. MNRAS, 459:3693–3710, July 2016.

[259] F. Feroz, M. P. Hobson, and M. Bridges. MULTINEST: an eﬃcient and robust Bayesian inference tool for cosmology and particle physics. MNRAS, 398:1601– 1614, October 2009.

[260] Florian Beutler, Chris Blake, Matthew Colless, D. Heath Jones, Lister Staveley- Smith, Lachlan Campbell, Quentin Parker, Will Saunders, and Fred Watson. The 6dF Galaxy Survey: Baryon Acoustic Oscillations and the Local Hubble Constant. Mon. Not. Roy. Astron. Soc., 416:3017–3032, 2011.

[261] Ashley J. Ross, Lado Samushia, Cullan Howlett, Will J. Percival, Angela Burden, and Marc Manera. The clustering of the SDSS DR7 main Galaxy sample -I. A 4 per cent distance measure at z = 0.15. Mon. Not. Roy. Astron. Soc., 449(1):835– 847, 2015.

[262] Shadab Alam et al. The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample. Mon. Not. Roy. Astron. Soc., 470(3):2617–2652, 2017.

[263] S. D. M. White and M. J. Rees. Core condensation in heavy halos - A two-stage theory for galaxy formation and clustering. MNRAS, 183:341–358, May 1978.

[264] V. Desjacques, D. Jeong, and F. Schmidt. Large-Scale Galaxy Bias. ArXiv e-prints 1611.09787, November 2016. BIBLIOGRAPHY 155

[265] J. Kwan et al. Cosmology from large-scale galaxy clustering and galaxy-galaxy lensing with Dark Energy Survey Science Veriﬁcation data. MNRAS, 464:4045– 4062, February 2017.

[266] Wayne Hu and Bhuvnesh Jain. Joint galaxy - lensing observables and the dark energy. Phys. Rev., D70:043009, 2004.

[267] Gary M. Bernstein. Comprehensive Two-Point Analyses of Weak Gravitational Lensing Surveys. Astrophys. J., 695:652–665, 2009.

[268] B. Joachimi and S. L. Bridle. Simultaneous measurement of cosmology and intrinsic alignments using joint cosmic shear and galaxy number density correlations. Astron. Astrophys., 523:A1, 2010.

[269] A. J. Ross et al. Ameliorating systematic uncertainties in the angular clustering of galaxies: a study using the SDSS-III. MNRAS, 417:1350–1373, October 2011.

[270] S. Ho et al. Clustering of Sloan Digital Sky Survey III Photometric Luminous Galaxies: The Measurement, Systematics, and Cosmological Implications. ApJ, 761:14, December 2012.

[271] B. Leistedt, H. V. Peiris, D. J. Mortlock, A. Benoit-Lévy, and A. Pontzen. Es- timating the large-scale angular power spectrum in the presence of systematics: a case study of Sloan Digital Sky Survey quasars. MNRAS, 435:1857–1873, November 2013.

[272] M. Crocce et al. Galaxy clustering, photometric redshifts and diagnosis of systematics in the DES Science Veriﬁcation data. MNRAS, 455:4301–4324, Febru- ary 2016.

[273] J. N. Fry and E. Gaztanaga. Biasing and hierarchical statistics in large-scale structure. ApJ, 413:447–452, August 1993.

[274] N. MacCrann et al. Dark Energy Survey Year 1 Results: Validating cosmological parameter estimation using simulated Dark Energy Surveys. submitted to Phys. Rev. D, 2017. 156 BIBLIOGRAPHY

[275] D. N. Limber. The Analysis of Counts of the Extragalactic Nebulae in Terms of a Fluctuating Density Field. ApJ, 117:134, January 1953.

[276] N. Padmanabhan et al. The clustering of luminous red galaxies in the Sloan Digital Sky Survey imaging data. MNRAS, 378:852–872, July 2007.

[277] K. Nock, W. J. Percival, and A. J. Ross. The eﬀect of redshift-space distortions on projected two-point clustering measurements. MNRAS, 407:520–532, September 2010.

[278] M. Crocce, A. Cabré, and E. Gaztañaga. Modelling the angular correlation function and its full covariance in photometric galaxy surveys. MNRAS, 414:329– 349, June 2011.

[279] H. T. Diehl et al. The Dark Energy Survey and operations: Year 1. In Ob- servatory Operations: Strategies, Processes, and Systems V, volume 9149, page 91490V, August 2014.

[280] K. M. Górski et al. HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere. ApJ, 622:759–771, April 2005.

[281] E. S. Rykoﬀ et al. redMaPPer. I. Algorithm and SDSS DR8 Catalog. ApJ, 785:104, April 2014.

[282] E. S. Rykoﬀ et al. The RedMaPPer Galaxy Cluster Catalog From DES Science Veriﬁcation Data. ApJS, 224:1, May 2016.

[283] S. Alam et al. The Eleventh and Twelfth Data Releases of the Sloan Digital Sky Survey: Final Data from SDSS-III. ApJS, 219:12, July 2015.

[284] F. Yuan et al. OzDES multiﬁbre spectroscopy for the Dark Energy Survey: ﬁrst-year operation and results. MNRAS, 452:3047–3063, September 2015.

[285] B. Soergel et al. Detection of the kinematic Sunyaev-Zel’dovich eﬀect with DES Year 1 and SPT. MNRAS, 461:3172–3193, September 2016.

[286] T. McClintock et al. Dark Energy Survey Year 1 Results: Weak Lensing Mass Calibration of redMaPPer Galaxy Clusters. ArXiv e-prints, April 2018. BIBLIOGRAPHY 157

[287] G. Bruzual and S. Charlot. Stellar population synthesis at the resolution of 2003. MNRAS, 344:1000–1028, October 2003.

[288] M. Jarvis, G. Bernstein, and B. Jain. The skewness of the aperture mass statistic. MNRAS, 352:338–352, July 2004.

[289] B. Leistedt et al. Mapping and Simulating Systematics due to Spatially Varying Observing Conditions in DES Science Veriﬁcation Data. ApJS, 226:24, October 2016.

[290] D. J. Schlegel, D. P. Finkbeiner, and M. Davis. Maps of Dust Infrared Emis- sion for Use in Estimation of Reddening and Cosmic Microwave Background Radiation Foregrounds. ApJ, 500:525–553, June 1998.

[291] F. Elsner, B. Leistedt, and H. V. Peiris. Unbiased methods for removing systematics from galaxy clustering measurements. MNRAS, 456:2095–2104, February 2016.

[292] A. J. Ross et al. The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: analysis of potential systematics. MNRAS, 424:564–590, July 2012.

[293] A. J. Ross et al. The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: observational systematics and baryon acoustic oscillations in the correlation function. MNRAS, 464:1168–1191, January 2017.

[294] Chris Blake et al. The WiggleZ dark energy survey: The selection function and z = 0.6 galaxy power spectrum. MNRAS, 406(2):803–821, mar 2010.

[295] L. Anderson et al. The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: baryon acoustic oscillations in the Data Releases 10 and 11 Galaxy samples. MNRAS, 441:24–62, June 2014.

[296] J. N. Fry. The Evolution of Bias. ApJ, 461:L65, April 1996.

[297] M. Tegmark and P. J. E. Peebles. The Time Evolution of Bias. ApJ, 500:L79– L82, June 1998. 158 BIBLIOGRAPHY

[298] A. J. Ross, R. Tojeiro, and W. J. Percival. Understanding the faint red galaxy population using large-scale clustering measurements from SDSS DR7. MNRAS, 413:2078–2086, May 2011.

[299] S. Saito et al. Connecting massive galaxies to dark matter haloes in BOSS - I. Is galaxy colour a stochastic process in high-mass haloes? MNRAS, 460:1457–1475, August 2016.

[300] Tobias Baldauf, Robert E. Smith, Uros Seljak, and Rachel Mandelbaum. An algorithm for the direct reconstruction of the dark matter correlation function from weak lensing and galaxy clustering. Phys. Rev. D, 81:063531, 3 2010.

[301] E. Suchyta et al. No galaxy left behind: accurate measurements with the faintest objects in the Dark Energy Survey. MNRAS, 457:786–808, jul 2015.

[302] J. Schaye, C. Dalla Vecchia, C. M. Booth, R. P. C. Wiersma, T. Theuns, M. R. Haas, S. Bertone, A. R. Duﬀy, I. G. McCarthy, and F. van de Voort. The physics driving the cosmic star formation history. Mon. Not. R. Astron. Soc., 402:1536–1560, March 2010.

[303] J. DeRose, R. Wechsler, and E. Rykoﬀ. . in prep, 2018.

[304] C. Blake and S. Bridle. Cosmology with photometric redshift surveys. MNRAS, 363:1329–1348, November 2005.

[305] N. Padmanabhan et al. The clustering of luminous red galaxies in the Sloan Digital Sky Survey imaging data. Mon. Not. Roy. Astron. Soc., 378:852–872, July 2007.

[306] D. M. Scolnic et al. The Complete Light-curve Sample of Spectroscopically Conﬁrmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample. ApJ, 859:101, June 2018.

[307] DES Collaboration et al. FIRST COSMOLOGY RESULTS USING TYPE IA SUPERNOVA FROM THE DARK ENERGY SURVEY: RESULTS . in prep, 2018.