Integrated Cosmological Probe Combination

DISS. ETH NO. 25195

Integrated cosmological probe combination

A thesis submitted to attain the degree of

DOCTOR OF SCIENCES of ETH ZURICH

(Dr. sc. ETH Zurich)

presented by

Andrina Nicola

Master of Science in Physics

ETH Zürich

born on February 5th, 1988

citizen of Roveredo (GR), Switzerland

accepted on the recommendation of

Prof. Dr. Alexandre Réfrégier

Dr. Adam Amara

Prof. Dr. Eiichiro Komatsu

2018

De sui ipsius et multorum ignorantia. — Francesco Petrarca

Abstract

Our Universe is an exciting laboratory, which we can study by observing many of its properties, called cosmological probes. Recent observational progress has led to the establishment of the ΛCDM cosmological model, which relies on the general theory of relativity, dark matter, dark energy and the existence of primordial perturbations. In spite of this progress, several questions remain unanswered, such as the nature of dark matter and dark energy and the details of the initial conditions. Constraining the key components of the standard model of cosmology is thus one of the main goals of observational cosmology today. In this thesis, we aim to improve our understanding of ΛCDM by combining the information from different cosmological probes, taking into account both the auto- and the cross-correlations. This allows us to go beyond the notion of isolated probes, towards an integrated view of the Universe and cosmology. In a first step, we compare the constraining power of a future spectroscopic galaxy redshift survey when analyzed using the spherical harmonic tomography power spectrum and the spherical Fourier-Bessel power spectrum in a Fisher matrix analysis. We then develop a framework for integrated analysis of cosmological probes in which the probes are combined starting at the map level. In a first implementation, we apply this framework to a combination of CMB temperature anisotropy measurements from the Planck satellite as well as galaxy clustering and weak lensing measured by the Sloan Digital Sky Survey. We then extend the analysis to also include CMB lensing from Planck, weak lensing from Dark Energy Survey Science Verification data as well as background probes. These analyses yield self-consistent and competitive constraints on cosmological parameters and provide a confirmation of ΛCDM through the consistency of different probes. As a complement, we quantify possible tensions between these constraints and other existing results through the relative entropy and find all data sets considered to be consistent in the framework of ΛCDM. Finally, we revisit the relative entropy and propose a novel model selection method, which combines relative entropy and posterior predictive distributions. In a series of toy models and applications to cosmological data, we show that this algorithm gives results consistent with expectations and thus appears promising for model selection in cosmology.

Introduzione - Zusammenfassung - Zämefassig

L’universo é un appassionante laboratorio, che si lascia decifrare studiando le cosiddette osservabili cosmologiche. La cosmologia ha fatto dei forti progressi negli ultimi anni e questi esperimenti hanno stabilito l’attuale modello standard cosmologico, tecnicamente chiamato ΛCDM. Questo modello si basa sulla teoria della relatività generale, la materia oscura, l’energia oscura e l’esistenza di ﬂuttuazioni primordiali. Malgrado questo progresso, molte questioni rimangono aperte: per esempio la natura della materia e dell’energia oscure, e il meccanismo delle ﬂuttuazioni primordiali. Quindi, uno degli scopi principali della cosmologia sperimen- tale é di studiare queste componenti. In questa tesi usiamo l’informazione contenuta in diverse osservabili per migliorare la nostra conoscenza del modello standard della cosmologia. Allontanandoci quindi dallo studio di singole osservabili, arriviamo ad una visione globale dell’universo e della cosmologia. In einem ersten Projekt vergleichen wir Analysemethoden für eine zukünftige spektroskopi- sche Galaxienkartierung. Wir vergleichen deren Informationsgehalt für eine Analyse mittels Kugelfunktionen und sphärischen Fourier-Bessel Funktionen. Wir entwickeln dann eine Methode um verschiedene kosmologische Observablen gemeinsam zu analysieren. In einer ersten Arbeit verwenden wir diese Methode um Daten der kosmischen Hintergrundstrahlung (aufgenommen mit dem Planck Satelliten) mit Karten von Galaxie- positionen und -scherungen der Sloan Digital Sky Survey zu kombinieren. In einer zweiten Arbeit erweitern wir diese Analyse: (i) auf die Scherung der kosmischen Hintergrundstrahlung, gemessen mit Planck, (ii) auf Karten von Galaxiescherungen der Dark Energy Survey und (iii) auf Messungen des homogenen Universums. Diese Studien erlauben uns die Werte kosmolo- gischer Parameter einzuschränken. Zusätzlich liefert die Übereinstimmung der verschiedenen Datensätze eine Bestätigung des kosmologischen Standardmodells. Wir ergänzen diese Arbeiten indem wir unsere Resultate mit Hilfe der Kullback-Leibler (KL) Divergenz statistisch mit anderen Resultaten vergleichen. Diese Studie zeigt, dass alle be- trachteten Datensätze sowohl miteinander als auch mit dem kosmologischen Standardmodell übereinstimmen. In ere letschte Arbet chömmer nomol uf d KL Divergänz zrugg und schlönd en neui Methode vor, wo chan zum Modäll teschte brucht wärde. Die Methode verbindet d KL Divergänz mit de sogenannte posterior predictive distribution. Mir teschte d Methode i mehrere vereifachte Awändige und zeiged, dass sie s Potenzial hett, um ide Zuekunft i de Kosmologie zum Modäll teschte agwändet z wärde.

iii

Zusammenfassung

Das Universum ist ein aufregendes Labor, das wir durch kosmologische Observablen studie- ren können. In den letzten Jahren gab es grosse Fortschritte in der Kosmologie, und diese Beobachtungen haben zur Aufstellung des kosmologischen Standardmodells, ΛCDM, geführt. Dieses basiert auf der allgemeinen Relativitätstheorie, der Existenz der dunklen Materie, der dunklen Energie und anfänglicher Dichteschwankungen. Trotz dieses Fortschritts bleiben viele Fragen bis heute ungeklärt, wie beispielsweise die physikalische Natur der dunklen Mate- rie und Energie. Eine der Hauptmotivationen der heutigen Kosmologie ist es deshalb, diese Hauptbestandteile des kosmologischen Standardmodells besser zu verstehen. In dieser Arbeit kombinieren wir mehrere kosmologische Observablen, was uns eine umfassende Sicht auf das Universums und die Kosmologie erlaubt. In einem ersten Projekt vergleichen wir Analysemethoden für eine zukünftige spektroskopi- sche Galaxienkartierung. Wir vergleichen deren Informationsgehalt für eine Analyse mittels Kugelfunktionen und sphärischen Fourier-Bessel Funktionen. Wir entwickeln dann eine Methode um verschiedene kosmologische Observablen gemeinsam zu analysieren. In einer ersten Arbeit verwenden wir diese Methode um Daten der kosmischen Hintergrundstrahlung (aufgenommen mit dem Planck Satelliten) mit Karten von Galaxie- positionen und -scherungen der Sloan Digital Sky Survey zu kombinieren. In einer zweiten Arbeit erweitern wir diese Analyse: (i) auf die Scherung der kosmischen Hintergrundstrahlung, gemessen mit Planck, (ii) auf Karten von Galaxiescherungen der Dark Energy Survey und (iii) auf Messungen des homogenen Universums. Diese Studien erlauben uns die Werte kosmolo- gischer Parameter einzuschränken. Zusätzlich liefert die Übereinstimmung der verschiedenen Datensätze eine Bestätigung des kosmologischen Standardmodells. Wir ergänzen diese Arbeiten indem wir unsere Resultate mit Hilfe der Kullback-Leibler (KL) Divergenz statistisch mit anderen Resultaten vergleichen. Diese Studie zeigt, dass alle be- trachteten Datensätze sowohl miteinander als auch mit dem kosmologischen Standardmodell übereinstimmen. In einer letzten Arbeit kehren wir zur KL Divergenz zurück und stellen eine neue Methode zur Modellüberprüfung vor. Diese basiert auf der KL Divergenz und der sogenannten posterior predictive distribution. Wir zeigen anhand verschiedener, vereinfachter Modelle, dass diese Methode das Potential hat, um in Zukunft in der Kosmologie angewandt zu werden.

Contents

Abstract i

Introduzione - Zusammenfassung - Zämefassig iii

Zusammenfassungv

1 Introduction 1 1.1 The homogeneous Universe...... 3 1.2 The inhomogeneous Universe...... 7 1.2.1 Linear perturbation theory...... 7 1.2.2 Cosmological structure formation...... 9 1.3 Observational probes...... 12 1.3.1 Statistics of random ﬁelds...... 12 1.3.2 Cosmic Microwave Background...... 14 1.3.3 Galaxy clustering...... 15 1.3.4 Cosmic shear...... 18 1.3.5 Background probes...... 21

2 Three-dimensional spherical analyses of cosmological spectroscopic surveys 23 2.1 Introduction...... 23 2.2 Comparison baseline...... 24 2.3 Three-dimensional spherical power spectra...... 26 2.3.1 The Cartesian power spectrum...... 26 2.3.2 The spherical harmonic tomography power spectrum...... 29 2.3.3 The spherical Fourier Bessel power spectrum...... 31 2.4 Fisher matrices for 3D spherical power spectra...... 34 2.4.1 The Fisher matrix for the spherical harmonic tomography power spectrum 34 2.4.2 The Fisher matrix for the SFB power spectrum...... 35 2.5 Results...... 36 2.5.1 Spherical harmonic tomography power spectrum...... 36 2.5.2 Spherical Fourier Bessel power spectrum...... 40 2.5.3 Comparison between spherical harmonic tomography and SFB power spectrum...... 43

vii Contents

2.5.4 Implementation effects on estimated survey constraining power.... 44 2.6 Conclusions...... 45 2.A The radialization of the SFB power spectrum in the presence of RSDs...... 46 2.B The SFB power spectrum for a generic distance-redshift relation...... 46 2.C Derivation of the Fisher matrix for the SFB power spectrum...... 48 2.C.1 Full Fisher matrix...... 48 2.C.2 Diagonal Fisher matrix...... 50 2.D Comparison of FM computation techniques for the SFB power spectrum: Results 50

3 Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing 53 3.1 Introduction...... 53 3.2 Framework...... 55 3.3 Theoretical predictions...... 56 3.4 Maps...... 57 3.4.1 Cosmic Microwave Background...... 57 3.4.2 Galaxy overdensity...... 60 3.4.3 Weak lensing...... 61 3.5 Spherical harmonic power spectra...... 63 3.5.1 CMB...... 64 3.5.2 Galaxy clustering...... 66 3.5.3 Cosmic shear...... 66 3.5.4 CMB and galaxy overdensity cross-correlation...... 68 3.5.5 CMB and weak lensing shear cross-correlation...... 68 3.5.6 Galaxy overdensity and weak lensing shear cross-correlation...... 69 3.6 Covariance matrix...... 69 3.6.1 Theoretical covariance estimate...... 70 3.6.2 Covariance estimate from Gaussian simulations...... 70 3.7 Cosmological constraints...... 71 3.8 Conclusions...... 77 3.A Theoretical prediction for CMB and weak lensing shear cross-correlation.... 79 3.B Treatment of systematic uncertainties in galaxy clustering data...... 81 3.C PSF correction and construction of weak lensing shear maps...... 81 3.D Transformation of weak lensing shear under rotation...... 85 3.E Choice of PolSpice parameter settings...... 85 3.F Correlated maps of spin-0 and spin-2 ﬁelds...... 86 3.G Validation of spherical harmonic power spectrum measurements...... 89 3.H Spherical harmonic power spectrum robustness tests...... 89 3.H.1 Comparison between spherical harmonic power spectra in equatorial and Galactic coordinates...... 91 3.H.2 Comparison between spherical harmonic power spectra derived from different foreground-reduced CMB maps...... 92 viii Contents

3.H.3 Impact of systematics correction on galaxy clustering power spectrum. 92

4 Integrated cosmological probes: extended analysis 95 4.1 Introduction...... 95 4.2 Framework for integrated probe combination...... 96 4.3 Data...... 96 4.3.1 DES weak lensing...... 97 4.3.2 CMB lensing convergence...... 99 4.3.3 Type Ia supernovae...... 100 4.3.4 Hubble parameter...... 100 4.4 Model predictions...... 100 4.5 Spherical harmonic power spectra...... 103 4.5.1 DES SV cosmic shear...... 104 4.5.2 CMB temperature and DES SV weak lensing shear cross-correlation... 105 4.5.3 CMB lensing convergence and galaxy overdensity cross-correlation... 105 4.5.4 CMB lensing convergence and CMB temperature cross-correlation... 106 4.5.5 CMB lensing convergence and SDSS Stripe 82 weak lensing shear cross- correlation...... 106 4.5.6 CMB lensing convergence and DES SV weak lensing shear cross-correlation106 4.6 Systematics...... 108 4.6.1 CMB temperature anisotropies...... 108 4.6.2 Galaxy overdensity...... 108 4.6.3 Weak lensing...... 108 4.6.4 CMB lensing convergence...... 109 4.6.5 SNe Ia...... 110 4.7 Covariance matrix...... 110 4.8 Parameter inference...... 111 4.9 Cosmological constraint results...... 114 4.10 Conclusions...... 115 4.A Impact of intrinsic alignments...... 123 4.B Impact of baryonic processes on the dark matter power spectrum...... 125 4.C Correlated spin-0 and spin-2 ﬁelds...... 127 4.D Validation of spherical harmonic power spectrum measurement...... 131 4.E Spherical harmonic power spectrum robustness tests...... 131 4.E.1 Comparison between spherical harmonic power spectra in equatorial and Galactic coordinates...... 131 4.E.2 Comparison between spherical harmonic power spectra measured from different foreground-reduced CMB temperature maps...... 134 4.F Impact of unresolved foregrounds on CMB temperature anisotropies...... 134

ix Contents

5 Integrated cosmological probes: concordance quantiﬁed 139 5.1 Introduction...... 139 5.2 Assessing concordance...... 140 5.3 Data...... 143 5.4 Results...... 144 5.5 Conclusions...... 147 5.A Gaussianity tests...... 152

6 Consistency tests in cosmology using relative entropy 155 6.1 Introduction...... 155 6.2 Data and parameter space...... 156 6.3 Data set consistency: fixed model and different data...... 157 6.3.1 Sequential experiments...... 157 6.3.2 Toy model...... 162 6.4 Model rejection: fixed data and different models...... 164 6.4.1 Implementation...... 166 6.4.2 Toy model...... 167 6.4.3 Application to cosmological data...... 169 6.5 Conclusions...... 174 6.A Assessing consistency for arbitrary updates...... 175 6.B Gaussianity tests...... 175 6.C Alternative method for computing the relative entropy...... 176 6.D Toy model implementation details...... 176 6.D.1 Implementation choices...... 176 6.D.2 Choice of fiducial polynomial coefficients...... 177 6.E SNe Ia and CMB posterior means...... 177

7 Conclusions 181

Bibliography 185

List of symbols 207

List of abbreviations 209

Acknowledgements 213

x 1 Introduction

I say, there is no darkness but ignorance. — William Shakespeare, Twelfth night (IV.II)

The last three decades have seen cosmology transit from a mostly theoretical to an experimental science. Recent observations have yielded precise measurements of the Large-Scale Structure (LSS) of the Universe and the Cosmic Microwave Background (CMB) radiation, leading to the establishment of the ΛCDM cosmological model.

ΛCDM is based on General Relativity (GR) as the theory of gravity and the assumption of isotropy and homogeneity of the Universe on large spatial scales (the cosmological principle). Early observations led to the discovery that the Universe is expanding and cooling down in the process, as described by the Big Bang model of the Universe. The evolution of the Universe in time is determined by its constituents: ordinary matter, as included in the standard model of particle physics; a cold dark matter component that interacts gravitationally, but has as yet not been found to interact through any other means; and a dark energy component, which is responsible for the late-time accelerated expansion of the Universe. The inhomogeneities and anisotropies detected in the CMB temperature and the galaxy distribution ﬁnally, are assumed to have been seeded by a period of exponential expansion shortly after the Big Bang, called inﬂation.

This simple model is described by a set of six parameters, θ {h, Ω , Ω , n , σ , τ }, where = m b s 8 reion h is the dimensionless Hubble parameter, Ωm is the fractional matter density today, Ωb is the fractional baryon density today, ns denotes the scalar spectral index of initial perturbations, σ8 1 is the r.m.s. of matter ﬂuctuations in spheres of comoving radius 8h− Mpc and τreion denotes the optical depth to reionization.

The ΛCDM cosmological model currently provides a good ﬁt to observations on large scales.

1 Chapter 1. Introduction

Despite its observational success, this model poses theoretical challenges. Some of the most pressing ones are that no physical theory of cosmology can explain the Universe at times earlier than the Planck time and that current theories of inﬂation face conceptual problems. Furthermore, despite being required by the standard cosmological model, as yet neither dark matter nor dark energy have been detected experimentally and we are far from an understanding of the latter, given the discrepancy between expectation and measurement of the value of the cosmological constant.

Several current and upcoming experiments have set out to answer these fundamental questions, making the current period very exciting for cosmology. These experiments include the Dark Energy Survey (DES), the Dark Energy Spectroscopic Instrument (DESI), the Large Synop- tic Survey Telescope (LSST), Euclid, the Wide Field Infrared Survey Telescope (WFIRST) as well as Planck, the Simons Observatory and CMB Stage 4, which will cover large, overlapping parts of the sky and provide measurements at an unprecedented precision. The tightest constraints on our cosmological model will come from combining the data from all of these surveys, as these cosmological probes are complementary and correlated. It is therefore crucial to develop analysis techniques that allow us to combine the information from different cosmological probes, taking both the auto- and the cross-correlations into account. These methods will be useful for improving our fundamental understanding of cosmology and for understanding possible tensions between data sets.

In the main part of this thesis, we present a framework for integrated probe combination and describe its application to several different data sets.

This thesis is organized as follows. In Chapter1 we give a brief overview of cosmology, starting with a description of the homogeneous and the inhomogeneous Universe. We then review cosmological observations, with a particular focus on CMB observations, galaxy clustering, weak gravitational lensing and background probes. In Chapter2 we present the comparison of the constraining power of a future spectroscopic galaxy redshift survey when analyzed using the spherical harmonic tomography power spectrum and the spherical Fourier-Bessel power spectrum in a Fisher matrix analysis. In Chapters3 and4 we present an integrated framework for cosmological probe combination in which the probes are combined into a common framework starting at the map level. We describe the application of this framework to a combination of CMB temperature anisotropies, CMB lensing, galaxy clustering, weak lensing,

Supernovae Ia data and H0 measurements. In Chapter5 we investigate possible tensions between the derived constraints and other existing results within the framework of ΛCDM. Finally, in Chapter6 we revisit the relative entropy as a consistency measure and propose a novel framework for model selection based on relative entropy and posterior predictive distributions. We conclude in Chapter7.

2 1.1. The homogeneous Universe

1.1 The homogeneous Universe

On large spatial scales, the Universe appears isotropic and homogeneous, implying that perturbations around the average properties of the Universe are small. This notion of statistical isotropy and homogeneity is formalized in the cosmological principle. The discovery of the linear relation between galaxy distance and recession velocity by Edwin Hubble in 1929 [134], showed that the Universe is expanding and is therefore evolving in time. Thus the Universe can be described as homogeneous and isotropic in space, but not in time. The theoretical framework for describing the Universe as a whole is the general theory of relativity developed by Albert Einstein in 1915 [77]. In this introduction we will give a short summary of physical cosmology, mainly following Refs. [71, 27, 242] and we will give additional references where needed.

The cosmological principle within GR determines the three-dimensional spatial geometry of the Universe. There exist only three spacetime geometries that can be foliated into isotropic and homogeneous spatial slices and these are described by the Friedmann-Lemaître-

Robertson-Walker (FLRW) metric gµν. If we choose spherical coordinates for the spatial part of the metric, the line element is given by [47]

ds2 g dxµdxν c2dt 2 a2(t)£dχ2 S2 (χ)dΩ2¤, (1.1) = µν = − + + K where c is the speed of light, a(t) is the scale factor accounting for the expansion of the Universe, χ is the comoving distance, K denotes the spatial curvature, µ,ν {0,...,3} and ∈ we assume the Einstein summation convention. The three different spatial geometries are parametrized by the quantity S and these correspond to ﬂat (K 0), positively (K 1) or K = = negatively (K 1) curved space. We have = −  1 c p χ  H sinh( K c/H ), K 1,  pK 0 0 = − SK (χ) χ, K 0, (1.2) = =  χ  1 c sin(p K ), K 1. p K H0 c/H0 | | | | =

The motion of particles in an FLRW spacetime is fully determined by the metric through the geodesic equation, which for a particle trajectory xµ(λ) is given by [71,27]

2 µ α β d x µ dx dx Γ , (1.3) dλ2 = − αβ dλ dλ

µ where λ parametrizes the particle path. The Christoffel symbols Γαβ are deﬁned as

µν · ¸ µ g ∂gαν ∂gβν ∂gαβ Γ . (1.4) αβ = 2 ∂xβ + ∂xα − ∂xν

We can study the Universe by measuring the electromagnetic radiation and gravitational

3 Chapter 1. Introduction waves emitted by astronomical sources. An important consequence of the expansion of the

Universe is that radiation emitted by a distant source at time temit at a wavelength λemit will be observed at time tobs at a modiﬁed wavelength λobs. Using the FLRW metric we ﬁnd [71,27]

λ a(t ) obs obs . (1.5) λemit = a(temit)

We can deﬁne the redshift z as

λ λemit a(t ) z obs − obs 1. (1.6) = λemit = a(temit) −

Setting t to today, i.e. t t , and letting a(t ) 1, a(t ) a gives obs obs = 0 obs = emit = 1 1 z . (1.7) + = a

Therefore, we can determine the value of the scale factor at emission by measuring the observed wavelength shift in known atomic transition lines.

Defining and measuring distances to astronomical objects in an expanding Universe is non- trivial. In general there exist two types of distances: comoving distances, which remain fixed through the expansion of the Universe and physical distances, which get stretched with time. The comoving coordinate distance is defined as [71, 27]

cdt Z t0 dt dχ χ c , (1.8) = a(t) =⇒ = te a(t) and it only depends on the spatial curvature through a(t) if we choose the form of the FLRW metric as in Eq. 1.1. Since photons travel on null-geodesics (ds2 0), this is equivalent to the = coordinate distance traveled by a radially-moving photon from its emission at time te to today. For t 0, Eq. 1.8 is the largest distance traveled by photons from the Big bang to today, which e = corresponds to today’s comoving horizon, i.e.

Z t0 dt χH c . (1.9) = 0 a(t)

dt The conformal time η is defined as dη /a(t) and is closely related to the comoving distance as = cdη dχ. (1.10) = If we set c 1, the conformal time and the comoving distance both have units of time or length. = The conformal time can thus be used as an alternative time coordinate, where the conformal time from t 0 to a given time t is equivalent to the comoving particle horizon. 0 = The comoving distance defined in Eq. 1.8 is not observable. If we know the intrinsic physical length of a source d or its absolute luminosity L, we can however define two indirect distance measures. The angular diameter distance is defined as the distance that relates observed

4 1.1. The homogeneous Universe angular size ∆θ to known physical extent d, i.e. [71,27]

d SK (χ(z)) dA(z) . (1.11) = ∆θ = (1 z) + The luminosity distance of a distant emitter is deﬁned as the distance relating observed ﬂux F to intrinsic luminosity L of the source, i.e. [71, 27]

s L dL(z) SK (χ(z))(1 z). (1.12) = 4πF = +

The dynamics of the Universe is determined by GR. The Einstein ﬁeld equations (EFE) of GR relate the curvature of four-dimensional spacetime to the matter content of a given physical system and are given by [47]

8πG Gµν Λgµν Tµν, (1.13) + = c4 where G denotes the gravitational constant. Gµν is a measure for spacetime curvature and is called Einstein tensor; Tµν is the stress-energy tensor, which is a measure for the Universe’s energy density content, and Λ is the cosmological constant. For an ideal ﬂuid with density ρ, pressure p and four-velocity U µ, it is given by ³ p ´ Tµν ρ UµUν pgµν (1.14) = c2 + +

The evolution equation for the scale factor a(t) can be obtained by requiring the FLRW metric to be a solution to the EFE for an ideal ﬂuid with energy density ρ and pressure p. This gives us two evolution equations for a(t), called the Friedmann equations [71,27, 242,47]

µ a˙ ¶2 8πG K c2 H 2(t) ρ , (1.15) = a = 3 − a2 a¨ 4πG µ 3p ¶ ρ , (1.16) a = − 3 + c2 where denotes derivative with respect to time t and we have defined the Hubble parameter · a˙ H(t) /a. = Equations 1.15 and 1.16 show that the dynamics of the Universe’s expansion depends on its constituents. Cosmological observations to date suggest that the Universe’s energy density is distributed between several different species, which can be modeled as perfect fluids. Due to energy-momentum conservation, the energy density of these species evolves differently as the Universe expands. We can introduce an equation of state parameter w, which relates energy density and pressure of an ideal fluid, i.e. [47]

p wρc2. (1.17) =

5 Chapter 1. Introduction

The conservation of energy-momentum then implies that [47]

3(1 w) ρ a− + . (1.18) ∝ The most important component for the early-time evolution of the Universe are relativistic species such as photons and massless neutrinos. These have an equation of state parameter 1 4 w and therefore their energy density falls as ρ a− as the Universe expands, making = 3 r ∝ them dynamically important at early times. At intermediate times in the Universe’s history, the energy density is dominated by non-relativistic species, such as baryons. Starting as early as the 1930s, observations showed evidence for an additional matter component in the Universe, which was subsequently called dark matter [286]. As yet, dark matter has only been detected through its gravitational interactions. It is thought to be collisionless and to interact at most very weakly with ordinary matter (e.g. [59, 55, 73, 185, 230]). Both ordinary and dark matter have an equation of state parameter w 0 and thus only dilute 3 = as ρ a− . The discovery of the accelerated expansion of the Universe in the late 1990s m ∝ [225, 205] has shown that the observations can only be explained by invoking an additional component. This component is responsible for the late-time acceleration of the Universe and 1 is called dark energy. From Equations 1.16 and 1.17 we see that a¨ 0 requires w /3, i.e. a > < so-called dark energy component with negative pressure. The case w 1 is special, as it = − gives constant energy density, i.e. ρ const. (c.f. Eq. 1.18). The energy-momentum tensor for = such a component follows from Eq. 1.14 as T pg , which has the same form as both the µν = µν energy-momentum tensor of the vacuum and the cosmological constant Λ. Therefore, the cosmological constant can be interpreted as due to the energy density of the vacuum [47].

The energy density scale factor scalings derived above allow us to rewrite the Friedmann equation. Let us deﬁne the critical density of the Universe as [47]

3H 2 ρcrit . (1.19) = 8πG

The ratio between the total density of the Universe and the critical density determines the spatial geometry of the Universe; we have [47]:

ρ ρ : K 0 (1.20) > crit > ρ ρ : K 0 (1.21) = crit = ρ ρ : K 0. (1.22) < crit <

We can ﬁnally deﬁne the density parameter Ωi for each species including the curvature as [47]

ρi Ωi . (1.23) = ρcrit

6 1.2. The inhomogeneous Universe

Using these deﬁnitions we can ﬁnally rewrite Eq 1.15 as [47]

2 H 4 3 2 Ω a− Ω a− Ω Ω a− , (1.24) 2 = r + m + Λ + K H0 where H0 denotes the present day Hubble parameter. Observations to date show that the Universe is spatially very close to ﬂat [212], in the following discussion we will therefore set the spatial curvature to zero and assume a ﬂat Universe.

1.2 The inhomogeneous Universe

1.2.1 Linear perturbation theory

The observed Universe is not homogeneous. At small scales we see structures formed through the action of gravity such as galaxy clusters, galaxies, stars and planets. In order to understand how these formed, we need to study the generation and evolution of perturbations to these deviations from isotropy and homogeneity. As the Universe is smooth on large scales, we can treat these deviations as perturbations to an otherwise smooth FLRW background g¯µν, i.e. [279, 27, 242]

g g¯ h , (1.25) µν = µν + µν where h is a symmetric four-tensor and h g¯ . Of the ten independent components µν | µν| ¿ µν of hµν, only six constitute physical degrees of freedom, as the rest is eliminated by gauge freedom.

The perturbations hµν can be decomposed in a scalar, vector and tensor part. This decomposition is helpful, as the equations for scalar, vector and tensor perturbations decouple in linear theory. Focusing only on the scalar perturbations, which are the relevant ones for our work, and adopting the Newtonian gauge we can write the perturbed line element as [71]

µ 2Ψ(x,t)¶ µ 2Φ(x,t)¶ ds2 c2 1 dt 2 a2(t) 1 £dχ2 χ2dΩ2¤, (1.26) = − + c2 + + c2 + where the functions Ψ(x,t) and Φ(x,t) are the only two scalar metric perturbations, called curvature and Newtonian potential respectively. Furthermore, the quantity dΩ dθ2 sin2 θdφ2 = + denotes the angular part of the metric. Analogously, we can deﬁne small perturbations around the mean properties of the Universe’s components; for the particle density for example, we deﬁne:

ρi (x,t) ρ¯i (x,t) δi (x,t) − . (1.27) = ρ¯i (x,t)

The metric perturbations are coupled to the perturbations in the particle distributions through gravity. In order to obtain evolution equations for these quantities we need to split the problem

7 Chapter 1. Introduction into two different parts: First, we need to consider the effect of metric perturbations on the distribution of matter and radiation and second, the effect of perturbations to the energy density of the Universe on the metric perturbations. The first part amounts to deriving the Boltzmann equations for the particle distributions, while for the second part we need to derive the EFE for the perturbed FLRW metric. This gives rise to a set of coupled differential equations, called Einstein-Boltzmann equations. In deriving these equations, we can make several simplifications: Firstly, we can work at linear order and drop all higher order terms, as we assume all perturbations to be small. Secondly, we can transform from real to Fourier space in order to remove derivative terms and thus significantly simplify the equations. Working in Fourier space has the additional advantage that different Fourier modes evolve independently in the linear regime.

We adopt the following Fourier convention in our derivation:

Z d3k f (x) eikx f (k), (1.28) = (2π)3 where f is an arbitrary function on R3. With this convention, the Boltzmann equations for radiation, dark matter and baryons ﬁnally become [71]

· ¸ Φ˙ ikµΨ µvb 1 Θ˙ ikcµΘ τ˙ Θ0 Θ P2(µ)Π , (1.29) + = −c2 − c − − + c − 2 Π Θ Θ Θ , (1.30) = 2 + P2 + P0 · 1 ¸ Θ˙ P ikµΘP τ˙ ΘP (1 P2(µ))Π , (1.31) + = − − + 2 − 3Φ˙ δ˙ ikv , (1.32) + = − c2 a˙ v˙ v ikΨ, (1.33) + a = − 3Φ˙ δ˙ ikv , (1.34) b + b = − c2 a˙ τ˙ v˙ ikΨ [v 3icΘ1], (1.35) b + a = − + R b + Φ˙ ikµΨ Θ˙ ν ikcµΘν , (1.36) + = −c2 − c where we have expressed the perturbation variables in terms of conformal time and thus · denotes derivative w.r.t. η. Furthermore, k is the wave vector, µ is the cosine of the angle between the photon momentum and the wave vector, P` is the Legendre polynomial of order `, Θ denotes the perturbations to the photon temperature (assuming an Einstein-Boltzmann distribution at zeroth order), Θi are the multipole moments of the temperature perturbation field, τ is the optical depth, ΘP denotes the perturbations to the radiation polarization field and ΘPi its multipole moments, δ is the dark matter overdensity field, v its velocity, δb is the

3ρb,0 baryon overdensity ﬁeld, v its velocity, R is the baryon-to-photon ratio, R /4ργ,0, and ﬁnally b = Θν denotes the perturbations to the neutrino temperature.

8 1.2. The inhomogeneous Universe

Eq. 1.29 governs the evolution of the perturbations to the photon temperature and includes effects such as Compton scattering between photons and baryons. Eq. 1.31 describes the generation of photon polarization, mainly from the quadrupole moment of the temperature field. Equations 1.32 to 1.35 describe the evolution of dark and baryonic matter perturbations and the associated velocity fields. The dark matter velocity is only affected by gravitational forces, while the baryon velocity field is additionally modified by Compton scattering with photons. Finally Eq. 1.36 describes the evolution of neutrino temperature perturbations, assuming a vanishing neutrino mass.

This set of equations needs to be closed by the Einstein equations describing the effect of perturbations in the Universe’s energy density on the metric perturbations. The linearized EFE give us [71] µ ¶ 2 a˙ a˙ 2 £ ¤ k Φ 3 Φ˙ Ψ 4πGa ρmδm 4ρrΘr,0 , (1.37) + a − a = + k2 (Φ Ψ) 32πGa2ρ Θ , (1.38) + = − r r,2 where the subscript m includes all matter components and r all radiation components. From Eq. 1.38 we see that within GR and in absence of quadrupole moments in the radiation distribution (the so-called anisotropic stress), we have Φ Ψ. This is a very good assumption = − during matter domination [33] and we therefore set Φ Ψ for the remainder of this thesis. = − In general, the set of coupled Einstein-Boltzmann equations needs to be solved numerically and several codes exist and are publicly available (e.g. [243, 167, 36, 221]). To get more intuition about the solutions of these equations, we will brieﬂy discuss the solutions of these equations qualitatively for matter and radiation perturbations.

1.2.2 Cosmological structure formation

The physics of cosmological structure formation can be pictured as an interplay between gravity, the expansion of the Universe and thermal motion/pressure of the particles. As matter is essentially pressureless, while radiation has high pressure, perturbations to these two components evolve differently, as discussed below.

Matter perturbations

The evolution of matter perturbations is mainly determined by the properties of dark matter, as it makes up most of the matter in the Universe. In general, the evolution of a perturbation at scale k can be divided into different phases: At early times, all perturbations will lie outside the horizon. As long as the scale k of a perturbation is larger than the horizon, no causal physics can affect the mode. At intermediate times, two things happen: (i) As the size of the horizon grows, it will reach the size of the perturbation and the perturbation will cross the horizon. When this happens the mode can be affected by causal physics. (ii) The Universe

9 Chapter 1. Introduction transitions from radiation to matter domination. Since gravitational potential perturbations evolve differently during radiation and matter domination, the properties of a given mode depend on horizon-crossing taking place during radiation or matter domination.

On super-horizon scales, matter perturbations do not evolve and stay constant, except for a small change in amplitude during the transition from radiation to matter domination. As the size of the horizon grows, small-scale perturbations enter the horizon, followed by large- scale perturbations. Small-scale perturbations tend to cross the horizon during radiation domination, while perturbation on large-scales only enter during matter domination. The causal growth of matter perturbations during radiation domination is not very efﬁcient as the modes only grow logarithmically with time. This is due to the oscillation of gravitational potentials during radiation domination, as we will see below. When the Universe enters matter domination, the gravitational potentials stay constant. On scales signiﬁcantly smaller than the horizon, perturbations to the matter distribution therefore start growing linearly with scale factor a(t), independently of their scale k. This suggests that the amplitude of matter perturbations at scale k and late times a, δ(k,a), can be written as

δ(k,a) k2Φ (k)T (k)D (a). (1.39) ∝ p 1

The quantity Φp denotes the primordial potential perturbation amplitude, which we will discuss below. The transfer function T (k) accounts for all scale-dependent processes during horizon crossing and radiation domination. Finally the linear growth factor D1(a) accounts for scale-independent growth during matter domination.

Even though baryons make up only a small fraction of the total matter in the Universe, they have a characteristic impact on the matter distribution, which is due to electromagnetic interactions. We will discuss these Baryonic Acoustic Oscillations (BAO) in Chap. 1.3.3.

Radiation perturbations

The evolution of photon perturbations can be divided into two different phases: the evolution before recombination at z 1100 and after recombination. Before recombination, photons ? ≈ and baryons are tightly coupled through Compton scattering and can be treated as a single photon-baryon fluid [132, 71]. During recombination, electrons and protons combine to form neutral hydrogen and the fraction of free electrons falls significantly. Photons thus decouple from baryons and essentially start free-streaming through the Universe. Even though decoupling and last scattering take place during recombination, which lasts a finite time, here we will assume that recombination occurs instantaneously and thus use last scattering and recombination interchangeably.

In order to better understand the physics of radiation anisotropies, it is instructive to study the perturbations in conﬁguration rather than Fourier space [26]. After horizon-crossing overdense regions will start to gravitationally collapse. As the photon density increases, so

10 1.2. The inhomogeneous Universe does its pressure and as soon as the pressure dominates, it will send out a spherical acoustic sound wave in the photon-baryon fluid. This sound wave will travel until recombination. At recombination, photons and baryons decouple and the photons will thus start free-streaming through the Universe, leaving the baryons behind. This process imprints the distance traveled by the photon-baryon fluid until recombination rs(η?) as a characteristic scale on the distribution of radiation perturbations. In configuration space, this gives rise to enhanced correlations between photon properties separated by a distance rs(η?). In Fourier space on the other hand, this leads to oscillations in the radiation distribution. Denoting the sound speed of the fluid as cs(η), we have [71]

Z η? rs(η?) cs(η0)dη0, (1.40) = 0 where the subscript ? denotes evaluation at recombination.

There are several additions to the simpliﬁed picture given above. Firstly, even during recombination, photons and baryons are not perfectly coupled, which means that photons travel a ﬁnite distance between different scattering events. The total distance traveled by photons in a Hubble time, called diffusion length, is determined by their mean free path and the average number of scattering events. Any perturbation on scales smaller than the photon diffusion length will therefore be damped. Secondly the photon distribution is affected by several late-time phenomena after recombination, such as rescattering with free electrons after reionization, temporal changes in gravitational potentials due to dark energy, and gravitational lensing.

Initial conditions

The study of cosmological structure formation immediately raises the question about the origin of primordial fluctuations. The theory of inflation, which was developed in the early 1980s [104, 172, 16], can both explain the flatness and the curvature problem of cosmology as well as provide a mechanism for the generation of primordial perturbations. However, this theory suffers from several, currently unresolved conceptual problems, such as eternal inflation (multiverse) or the trans-Planckian problem (see e.g. [278, 40]).

The theory of inflation postulates that the Universe underwent an early phase of exponential expansion driven by the potential energy of a scalar field with negative equation of state parameter. Such an inflationary phase brings the Universe to spatial flatness and causes the observable Universe to arise from a causally connected region. Furthermore the exponential expansion will stretch the ubiquitous quantum mechanical fluctuations in the scalar field to observable scales, thus providing a natural mechanism for the generation of primordial perturbations and the initial conditions for structure formation. Due to the quantum mechanical nature of the initial conditions, we cannot theoretically predict the value of the perturbation field but only its statistical properties. The primordial fluctuations generated by single-field inflationary models are Gaussian to a very good approximation [179, 10] and can therefore be

11 Chapter 1. Introduction fully described by their two-point function or power spectrum1. Focusing on such models, the predicted primordial potential power spectrum can be written as [71]

2 µ ¶ns 1 µ ¶2 50π ck − 2 Ωm PΦ (k) δ , (1.41) p = 9k3 H H D (a 1) 0 1 = where δH denotes the scalar power spectrum amplitude. The primordial power spectrum thus forms the seeds of all cosmological structures we see today.

1.3 Observational probes

In cosmology, our laboratory is the Universe, which we can observe without any control on the experimental conditions. We can make inference about the Universe by observing its properties, the cosmological probes, and comparing them to theoretical predictions. Since we can only make theoretical predictions for statistical properties, most cosmological analyses focus on the two-point function or power spectrum of the observables. If the observable is a Gaussian random ﬁeld, these statistics capture all of the information. If the observable has non-Gaussian features, it can still be analyzed in terms of the two-point function but a signiﬁcant amount of information will only be accessible through higher-order statistics.

There is a fundamental measurement uncertainty associated with cosmological observations that is due to the fact that we can only observe our particular realization of the Universe and therefore cannot repeat an experiment. This uncertainty is called the cosmic variance [71].

1.3.1 Statistics of random ﬁelds

As a consequence of the cosmological principle (i.e. the symmetry of the background cosmology), the Gaussian primordial fluctuations are statistically isotropic and homogeneous. Linear evolution of perturbation amplitudes preserves these properties, whereas non-linear evolution creates significant non-Gaussianity and can break the isotropy and homogeneity of the Universe. In the work presented in Chapters2-5 we mostly focus on Gaussian fields that can be fully described in terms of two-point correlators. In this section, we thus review the statistical description of random fields in terms of two-point functions, mostly following Ref. [51].

Let f (x) be a random field on R3. The three-dimensional correlation function (two-point function) is defined as the ensemble average of the product of the value of the field at two different points x and x0 x r, i.e. [51] = −

ξ(x,x0) f (x)f (x0) , (1.42) = 〈 〉 where denotes the average over all realizations. The quantity ξ is thus a measure for the 〈···〉

1These two quantities will be discussed in more detail in Sec. 1.3.1.

12 1.3. Observational probes correlation between the ﬁeld values at different positions. If f is statistically isotropic and homogeneous, i.e. its statistical properties are invariant under translations and rotations, then the correlation function only depends on the norm of the separation of the two points i.e. [51]

ξ(x,x0) ξ( x x0 ) ξ(r ) f (x)f (x0) , (1.43) = | − | = = 〈 〉 where r r . Since cosmological perturbation theory generally predicts Fourier-space quanti- = | | ties, it is instructive to consider the analogous relations in Fourier space. Using Eq. 1.28, the three-dimensional power spectrum of f is deﬁned as [51]

P(k,k0) f (k)f ∗(k0) , (1.44) = 〈 〉 where the superscript denotes complex conjugation. For statistically isotropic and homoge- ∗ neous ﬁelds, this relation simpliﬁes and we obtain [51]

f (k)f ∗(k0) P(k)δ(k k0), (1.45) 〈 〉 = − i.e. different Fourier modes are uncorrelated and the power spectrum only depends on the modulus of the wave vector k.

In order to take into account the spherical sky geometry, we can perform analogous steps for fields f (xˆ) defined on the two-sphere S2. The two-point correlation function of f is defined as [51]

ξ(θ,θ0) f (θ)f (θ0) , (1.46) = 〈 〉 2 where θ,θ0 S . As before, the correlation function of an isotropic ﬁeld only depends on the ∈ angular separation of the points θ,θ0, i.e. we have [51]

ξ(∆θ) f (θ)f (θ0) , (1.47) = 〈 〉 where ∆θ θ θ0 . Any function deﬁned on the sphere can be decomposed into spherical = | − | harmonics as

X X f (θ) f`mY`m(θ), (1.48) = ` m ` | |≤ where Y (θ) denotes the spherical harmonic of degree ` and order m, m `. The quantities `m | | ≤ f`m are the spherical harmonic coefficients of f . For a statistically isotropic field, different harmonic modes are uncorrelated and the spherical harmonic power spectrum (or angular power spectrum) C` of f is then defined by [51]

f`m f ∗ C`δ`` δmm . (1.49) 〈 `0m0 〉 = 0 0 This is the Fourier-space equivalent of Eq. 1.47 for random ﬁelds deﬁned on the sphere.

13 Chapter 1. Introduction

1.3.2 Cosmic Microwave Background

The Cosmic Microwave Background is the most constraining cosmological probe to date and it consists of the photons that last scattered with electrons before recombination and have been free-streaming through the Universe since. This relic radiation has been discovered by Penzias and Wilson in 1965 [203] and found to be nearly isotropic. In 1992, the COBE DMR satellite measured its blackbody spectrum to high precision and detected ﬂuctuations in the ∆T 5 CMB temperature at the level of /T 10− [252]. More recently, the WMAP [30, 124] and later ∼ the Planck [210, 215, 212] satellite produced high-resolution maps of the distribution of CMB temperature and polarization anisotropies on the sky and measured its statistical properties.

The CMB is an inherently two-dimensional quantity and can thus be analyzed in terms of the spherical harmonic power spectrum C`. Theoretically, the CMB power spectrum is well described within linear theory, as the amplitude of radiation perturbations is small at all times. These perturbations thus evolve linearly from the initial ﬂuctuations, preserve Gaussianity and can thus be characterized through the angular power spectrum, which is mostly sensitive to early Universe physics. We can separate the direction- and initial condition dependence of the CMB temperature anisotropies T and relate their spherical harmonic power spectrum to the primordial power spectrum generated during inﬂation and the evolution of matter perturbations as [71]

Z ¯ ¯2 2 ¯∆T`(k)¯ C TT dk k2P lin(k)¯ ¯ , (1.50) ` = π δδ ¯ δ(k) ¯ where ∆T` denotes the transfer function of the temperature anisotropies and δ is the dark matter overdensity. The CMB temperature anisotropy power spectrum is mainly sensitive to the spatial curvature density of the Universe, Ω , the physical baryon density ω Ω h2, the K b = b physical matter density ω Ω h2 and the scalar spectral index n . m = m s The observed CMB temperature anisotropies receive contributions not only from early universe (i.e. pre-recombination) physics but also from late-time phenomena. These secondary CMB anisotropies largely fall into two classes, gravitational and scattering phenomena [132]. Scattering secondaries are due to the fact that galaxy formation results in free electrons that rescatter with CMB photons at late times. An example is the Sunyaev-Zel’dovich effect [260, 261], which is due to scattering of CMB photons with free electrons with large bulk or thermal velocities. An example for a gravitational effect is the integrated Sachs-Wolfe (ISW) effect. During matter-domination, the amplitude of gravitational potential perturbations stays constant while it starts to decay as soon as dark energy becomes signiﬁcant for the expansion. The ISW is due to the blue- or redshifting of CMB photons as they travel through these decaying large-scale gravitational potential perturbations and thus provides a direct evidence for accelerated expansion.

The most prominent gravitational effect on the CMB however, is the lensing of CMB photons by the LSS. Analogously to galaxy weak lensing, the CMB photons get deﬂected by the cumulative

14 1.3. Observational probes lensing effect of the LSS. Therefore the observed CMB temperature at position θˆ can be obtained from the unlensed temperature as [206, 166]

T (θˆ) T (θˆ φ(θˆ)) T (θˆ) i φ(θˆ) T (θˆ) O(φ2), (1.51) obs = unl + ∇ = unl + ∇ ∇i obs + where denotes the two-dimensional gradient on the sphere. The quantity φ(θˆ), which gives ∇ rise the deﬂection, is the CMB lensing potential and is given by [206, 166]

Z 2 dz χ? χ(z) φ(θˆ) − Ψ(χ(z)θˆ,z), (1.52) = −c H(z) χ?χ(z) where χ? denotes the comoving distance to the last scattering surface. The CMB lensing convergence is deﬁned as κ(θˆ) 1 2φ(θˆ)[35]. As we know that the statistical properties of = − 2 ∇ the unlensed CMB are those of a Gaussian random ﬁeld, this means that [206]

(i) if we average over realizations of the CMB temperature for ﬁxed φ(θˆ), CMB lensing generates anisotropies in the CMB temperature covariance matrix,

(ii) if we average over the CMB temperature anisotropies and φ(θˆ), CMB lensing leads to a non-vanishing four-point function in the observed CMB data.

These two signatures allow us to reconstruct the CMB lensing potential from CMB temperature anisotropy maps and obtain information from the lensing convergence. This is in contrast to galaxy weak lensing. Since we do not have any information about the intrinsic distribution of galaxies and their initial positions, we cannot reconstruct the deﬂection ﬁeld but need to focus on cosmic shear, which correlates the shapes of nearby galaxies [166].

The CMB lensing potential can be reconstructed from the CMB temperature and polarization ﬁeld using quadratic estimators (e.g. [195]). It is sensitive to the integrated matter distribution from the last scattering surface to today and the CMB lensing kernel peaks at a redshift z 2. ∼ In order to obtain cosmological information from the CMB lensing potential, we can measure its angular power spectrum or the cross-correlation between the CMB lensing potential and any other tracer of the LSS.

CMB lensing together with secondary CMB temperature anisotropies in general, make the CMB a powerful probe of not only the high- but also the low redshift Universe. Finally, the recent progress in CMB polarization measurements opens up a new window for CMB science.

1.3.3 Galaxy clustering

While the primary CMB anisotropies provide a picture of the matter distribution during the epoch of last scattering (z 1100), a wealth of complementary cosmological information is ? ∼ contained in the three-dimensional matter distribution in the Universe at lower redshift. This information can be extracted by measuring the statistical properties of the matter overdensity

15 Chapter 1. Introduction

ﬁeld δm, e.g. through the correlation function ξmm(r ) in real space or the power spectrum Pmm(k,z) in Fourier space.

Observations have shown that dark matter (DM) makes up approximately 80% of the Universe’s matter content, whereas baryons only constitute the remaining 20% (e.g. [124, 212]). However, as DM does not or interacts very weakly, we need to infer the matter distribution in the Universe using the distribution of the luminous matter, such as galaxies or diffuse gas. In this introduction, we focus on galaxies as a tracer of the underlying DM distribution.

Galaxies form at the density maxima of the matter ﬁeld and are therefore biased tracers of the overall matter distribution [24]. Furthermore, as galaxies are discrete tracers of an underlying continuous density ﬁeld, the measured galaxy density receives a contribution due to shot noise. We can thus write the observed galaxy density as [201]

δ (k,z) b(k,z)δ(k,z) n(k,z), (1.53) g = + where b(k,z) is called galaxy bias parameter and parametrizes the relation between galaxy position and the underlying dark matter ﬁeld. The quantity n(k,z) accounts for shot noise.

The galaxy overdensity field is three-dimensional and optimal extraction of cosmological information from galaxy clustering thus requires an analysis in 3D. Measuring galaxy redshifts to high precision however is challenging and requires spectroscopic galaxy redshift surveys, which infer galaxy redshifts by measuring the electromagnetic spectrum of each galaxy and identifying known atomic transition lines. These measurements are time-consuming and spectroscopic surveys thus typically contain significantly less galaxies than photometric surveys, which infer galaxy redshifts by measuring galaxy fluxes in a given number of photometric passbands. Since photometrically determined galaxy redshifts have large uncertainties, galaxies in these surveys are typically analyzed by splitting the sample in redshift and measuring the two-dimensional clustering in each of these bins. Spectroscopic galaxy redshift surveys on the other hand are typically analyzed using the power spectrum or the three-dimensional correlation function.

We can compute the three-dimensional galaxy clustering power spectrum from Eq. 1.53. Noting that δ(k,z) and n(k,z) are uncorrelated, we obtain the power spectrum at a given redshift z as

2 δ (k)δ∗(k0) P (k,z)δ(k k0) b (k,z)P(k,z)δ(k k0) P (k,z)δ(k k0), (1.54) 〈 g g 〉 = gg − = − + SN − where PSN(k,z) denotes the shot noise power spectrum. The galaxy power spectrum and the correlation function can be interpreted as the probability of detecting two galaxies separated by a distance r [201]. In contrast to the CMB, the matter distribution, including galaxies and clusters, shows large perturbations to the underlying FLRW background and therefore, its statistical properties cannot generally be modeled using linear theory. As large scales have not had the time to collapse yet, these non-linear effects are however mostly conﬁned to small

16 1.3. Observational probes scales. Non-linear evolution has generally two effects: first the non-linear power spectrum deviates from the linear one and second a significant amount of cosmological information is contained in non-Gaussian statistics of the matter-density field, such as the bispectrum. Several approaches to modeling the non-linear matter power spectrum have been developed in the literature. These range from analytic perturbation theory approaches (for a review see [31]) to fitting functions calibrated through numerical simulations (e.g. [251, 267, 186, 187]).

Returning to Eq. 1.54, the shot noise power spectrum for Poisson samples of the underlying density ﬁeld is given by

1 PSN(k,z) , (1.55) = n¯ where n¯ is the mean galaxy number density of the sample. In practice however we need to take into account modiﬁcations to the shot noise power spectrum due to survey masks or non-Poissonian nature of the tracers (see e.g. Chapter3,[199]).

In our discussion so far we have ignored the baryons since they are subdominant with respect to the dark matter. However, the sound waves in the photon-baryon ﬂuid prior to recombination leave characteristic imprints on the matter distribution. Let us recall that after recombination, the matter distribution consists of a superposition of spherical dark matter overdensities surrounded by rings of baryons at a distance of rs(η?). As structures grow, baryons will fall into the DM potential wells and equally the DM will fall into the potentials generated by the baryon distribution. This results in an enhanced probability of ﬁnding two galaxies separated by a distance rs(η?), which manifests itself as a peak in the galaxy correlation function and corresponding oscillations in the power spectrum, called baryonic acoustic oscillations (BAO).

As cosmological constraints from measurements of the shape of Pgg(k,z) are generally weaker than those from the CMB, measurements of the 3D power spectrum are mainly used to extract speciﬁc features in the power spectrum, such as the BAOs or redshift space distortions (RSD). The BAO scale, which is imprinted on the galaxy distribution, can be used as a standard ruler to infer the angular diameter distance dA(z) (Eq. 1.11) to an effective redshift. RSDs arise due to peculiar galaxy velocities. When estimating the distance of galaxies from their redshift we make the assumption that it is solely due to the Hubble ﬂow. However, galaxies tend to fall into overdense regions and these peculiar velocities give rise to an anisotropy in the galaxy clustering signal along the line-of-sight. This allows for the measurement of the linear growth rate f (z)σ8(z), which is given by:

dlogD1(a) f (z)σ8(z) a (1 z) 1 σ8(z). (1.56) = dloga | = + −

In photometric data, these features are damped due to galaxy redshift uncertainties and thus the 2D power spectrum is usually used to derive constraints from the power spectrum shape.

17 Chapter 1. Introduction

1.3.4 Cosmic shear

Clustering studies treat galaxies as point-particles and measure the cosmological information contained in the statistical properties of their relative positions. Galaxies however are not point-like but rather extended objects with a variety of shapes and forms. As photons emitted by distant galaxies travel towards us, their paths get deﬂected by the gravitational effect of the intervening matter, causing galaxy shapes to appear distorted. This effect is called gravitational lensing. If the distortions are large, i.e. when photons are deﬂected by a large overdensity such as a galaxy cluster or a massive galaxy, we speak of strong gravitational lensing. Weak gravitational lensing on the other hand, describes the tiny but coherent distortions caused by the cumulative lensing effect of the LSS between the distant galaxy and the observer. These two regimes of gravitational lensing carry complementary information: strong lensing predominantly probes the matter distribution in the lensing galaxy and therefore the small- scale distribution of matter. Weak lensing or cosmic shear is sensitive to the integrated distribution of gravitationally interacting matter and it thus probes the large-scale distribution of both luminous and dark matter in the Universe. In the following, we will focus on weak gravitational lensing only, as it is one of the cosmological probes used in the work presented later in this thesis.

Weak lensing formalism

In order to derive the light deﬂections caused by the LSS, let us look at the situation of a source galaxy located at the true angular position θS as seen from the observer but observed under the angle θ due to the deﬂection of light (see Fig. 1.1). Since gravitational lensing conserves surface brightness we have [71]

I (θ) I (θ ), (1.57) obs = true S where Iobs(θ) is the observed speciﬁc intensity distribution and Itrue(θS) denotes the surface brightness at the source plane. To derive the relation between the observed θ and true angular positions θS, we need to solve the geodesic equation for photons (Eq. 1.3) in the presence of inhomogeneities along the photon path. This gives us [71]

Z χ µ ¶ i i 2 ∂Φ(x(χ0)) χ0 θS θ 2 dχ0 1 , (1.58) = + c 0 ∂xi − χ where i 1,2 and Φ denotes the gravitational potential perturbation. In the weak-field regime, = we can linearize this equation and define the transformation matrix Ai j , which defines a linear mapping between the angular positions on the source plane and the observed positions, as

i Ã ! ∂θS 1 κ γ1 γ2 Ai j − − − , (1.59) = ∂θ j = γ 1 κ γ − 2 − + 1

18 1.3. Observational probes

Iobs(✓)

Itrue(✓S)

✓ ✓ O S

Figure 1.1: Schematic illustration of weak gravitational lensing.

where κ denotes the scalar convergence and γ (γ ,γ ) denotes the spin-two shear. The = 1 2 convergence affects both components of the source’s angular positions equally and it thus describes isotropic magniﬁcation or demagniﬁcation of the source image. The shear on the other hand describes anisotropic stretching of the galaxy shape. As we have no information on intrinsic galaxy sizes, weak gravitational lensing studies focus on measuring cosmic shear γ, which is given by

A11 A22 γ1 − , (1.60) = − 2 γ A . (1.61) 2 = − 12

We can relate the shear to the gravitational potential perturbations along the line of sight by combining Equations 1.58 and 1.59: From Eq. 1.58 we see that the deflection is sensitive to the gravitational potential integrated along the perturbed photon path x(χ0). We can simplify this by making use of the fact that perturbations in the weak-field regime are small and set the perturbed photon path equal to the unperturbed one, i.e. x(χ0) χ0θ [155]. Using this = so-called Born approximation, we finally obtain [71]

Ã ! Z χ 2 µ ¶ κ γ1 γ2 2 ∂ Φ(x(χ0)) χ0 Ai j δi j − − − 2 dχ0 χ0 1 . (1.62) − = γ κ γ = c 0 ∂xi ∂x j − χ − 2 − + 1 Measuring the statistical properties of the cosmic shear ﬁeld thus allows us to infer the statistical properties of the gravitational potential perturbations.

The shear is a spin-2 ﬁeld, i.e. it is invariant under rotations of 180◦ and can be decomposed into spin-2 spherical harmonics. In order to compute its spin-0 angular power spectrum, we

19 Chapter 1. Introduction need to express the information contained in γ in terms of scalar variables. Any spin-2 field on S2 can be decomposed into two scalar fields, the curl-free E-mode and the B-mode field. Since cosmic shear is sourced by a single scalar field in the Born approximation, the B-mode vanishes and we have [25, 155]:

C EE C γγ C κκ, (1.63) ` ≡ ` = ` C BB 0. (1.64) ` = Using the Poisson equation (late-time limit of Eq. 1.37) to relate the potential to the matter overdensity we obtain the cosmic shear E-mode power spectrum for a source at redshift z as ∗ [71]

Ã !2 2 Z 2 µ ¶2 µ 1 ¶ γγ 3 ΩmH0 c χ χ χ nl ` /2 C dz ∗ − P k + ,z , (1.65) ` = 2 c2 H(z) a2 χ δδ = χ(z) ∗ where we have used the Limber approximation to simplify the integral [169, 147, 148]. In weak gravitational lensing surveys, we usually measure the power spectrum by averaging over a large number of source galaxies with redshift distribution n(z), deﬁned as Z dz n(z) 1. (1.66) =

For such a galaxy population, Eq. 1.65 becomes [71]

γ ¡ ¢ γ ¡ ¢ µ 1 ¶ Z c W χ(z) W χ(z) ` /2 C γγ dz P nl k + ,z , (1.67) ` = H(z) χ2(z) δδ = χ(z) where we have deﬁned the weak lensing shear window function as

2 Z χh 3 ΩmH χ(z) χ(z0) χ(z) γ ¡ ¢ 0 − W χ(z) 2 dz0n(z0) . (1.68) = 2 c a χ(z) χ(z0)

We can see from Eq. 1.67 that the cosmic shear power spectrum is indeed sensitive to the matter power spectrum and it thus too becomes increasingly non-linear at small scales. In addition, numerical simulations have shown that the dark matter power spectrum on small scales is signiﬁcantly affected by baryonic processes such as gas cooling and feedback from Active Galactic Nuclei (AGN) and supernovae (e.g. [275]). Both these effects need to be taken into account when modeling the cosmic shear power spectrum.

Galaxy ellipticities

Following the discussion above, we expect weak gravitational lensing to cause the shapes of galaxies to appear distorted and the shear can thus be inferred from the observed galaxy ellipticities i.e. shapes. This is challenging since galaxies show a variety of intrinsic ellipticities 1 with an r.m.s. of the order of ²2 /2 0.3 whereas the cosmic shear distortions to these 〈 〉 ∼

20 1.3. Observational probes intrinsic shapes are on the level of a few percent, i.e. γ 0.03 [155]. Therefore, cosmic shear ∼ cannot be detected on single galaxies and weak lensing measurements need to be performed on large galaxy samples in order to overcome the statistical noise. The intrinsic shapes of galaxies have additionally been found to exhibit signiﬁcant correlations, which are called intrinsic alignments [120, 180, 128, 80, 196, 140, 247] and are due to the tidal gravitational ﬁeld surrounding the galaxy. Weak gravitational lensing assumes all correlations between galaxy shapes to be due to gravitational lensing and therefore intrinsic alignments need to be taken into account in cosmic shear analyses in order to not bias cosmological parameter constraints (see e.g. [159]).

Several different ellipticity estimators have been developed in the literature (e.g. [149, 188, 285, 245, 246]) and they mainly fall into two categories: moment-based or model-based methods. Moment-based methods infer the galaxy ellipticity from the moments of the galaxy light proﬁle, while model-based methods infer ellipticities by ﬁtting a model galaxy to the image. All these methods need to operate in the low signal-to-noise regime and most of them are affected by biases, which need to be calibrated using simulations or estimated theoretically.

1.3.5 Background probes

All the cosmological probes discussed so far measure the cosmological information contained in the inhomogeneities of the Universe. In addition to these, there exist several cosmological probes that measure the homogeneous expansion history of the Universe and thus derive cosmological constraints. Examples include BAOs, which have been discussed in Chap. 1.3.3, Type Ia Supernovae and direct Hubble constant measurements. In the following, we focus on the latter two as these are the most important ones for this work.

Type Ia Supernovae

Type Ia Supernovae are thought to be the result of thermonuclear explosions triggered by mass accretion onto white dwarf stars. Since the electron-degeneracy pressure inside white dwarfs cannot support masses larger than the Chandrasekhar mass [53], any mass accretion leading to the star exceeding this limit will cause it to gravitationally collapse [97]. Although the exact processes leading to supernova explosions are still under debate, SNe Ia have similar light proﬁles and peak magnitudes, and can thus be used as standardizable candles. By measuring the distance moduli µ for SNe Ia at redshifts zSNe we can thus infer the luminosity distance dL(z), i.e. µ ¶ dL(zSNe) µ(zSNe) mSNe M 5log , (1.69) = − = 10 10[pc] where mSNe is the observed SNe Ia magnitude and M is the absolute magnitude. This relation follows directly from Eq. 1.12 upon taking the logarithm. In recent years the absolute magnitudes of Type Ia Supernovae have been found to exhibit correlations with their color, the

21 Chapter 1. Introduction duration of the explosion and host galaxy properties, which need to be taken into account in cosmological analyses.

Hubble constant measurements

The Hubble constant H0 measures the local expansion rate in the Universe and it is the only cosmological parameter that can be measured both assuming a cosmological model and without any assumptions on cosmology. The idea behind these measurements is that the intercept of the logarithmic magnitude-redshift relation of SNe Ia is a measure for H0 if the distance modulus of the Supernovae is known [137]. Since SNe with recession velocities dominated by the Hubble ﬂow typically lie at distances of a few tens of Mpc, their distances cannot be measured directly. Instead, direct Hubble constant measurements rely on distance ladder measurements. The latest measurements have employed a distance ladder build on local parallax measurements, the maser-based distance to the NGC 4258 galaxy and Cepheids to calibrate SNe Ia distances and measure H0 [228]. These analyses result in measurements complementary to the CMB and are derived without the assumption of a speciﬁc cosmological model.

22 2 Three-dimensional spherical analyses of cosmological spectroscopic surveys

We are all in the gutter, but some of us are looking at the stars. — Oscar Wilde, Lady Windermere’s Fan

This chapter appeared in a similar form as a part of Nicola, Réfrégier, Amara & Paranjape, 2014 [190].

2.1 Introduction

Constraining the nature and properties of dark energy and dark matter are amongst the most intriguing tasks of current physics. Spectroscopic galaxy redshift surveys offer a way to probe the matter distribution at low redshift, which is strongly affected by the properties of the dark sector. Upcoming spectroscopic clustering surveys like DESI [165], HETDEX [123] and PFS [266], are therefore amongst the most promising tools to achieve these tasks. As discussed in Chapter1, galaxy surveys are inherently three-dimensional, making their analysis more complex as opposed to the CMB, which can be analyzed through two-dimensional maps on the sky. Depending on the galaxy survey geometry, different analysis methods have thus been proposed. For surveys with limited angular sky coverage, the sky can be approximated as ﬂat. Therefore, the clustering of galaxies can be analyzed in three-dimensional Cartesian coordinates by means of the spatial correlation function ξ(r ) or through its Fourier counterpart, the Cartesian power spectrum P(k,r ) (for details on these quantities, see Chapter1 and references therein).

In recent years, galaxy redshift surveys have become both wider and deeper, leading us to investigate analysis methods other than P(k,r ), which do not rely on the ﬂat-sky approximation and which facilitate combination of galaxy clustering data with other cosmological probes. A statistic which naturally incorporates the curvature of the sky is the spherical harmonic tomog-

23 Chapter 2. Three-dimensional spherical analyses of cosmological spectroscopic surveys

i j raphy (SHT) power spectrum C` , the spherical harmonic transform of the angular correlation function at redshifts zi (for theoretical studies see e.g. [38, 52, 70, 21] and for application to data see e.g. [111, 129]). The three-dimensional information can partly be retrieved from this tomographic analysis by performing the spherical harmonics decomposition at a number of different redshifts. Tomographic analyses of the matter overdensity ﬁeld require subdivision of data into bins, since a ﬁnite redshift resolution is needed to compute angular correlations in practice.

Another common way to analyze the three-dimensional matter overdensity ﬁeld in spherical geometry, which has been applied to galaxy redshift surveys (e.g. [115, 219]), weak lensing (e.g. [112,48]) and the integrated Sachs-Wolfe effect [244], is to measure its spherical Fourier transform. The result is the three-dimensional spherical Fourier Bessel (SFB) power spectrum C`(k,k0) where the angular dependence is encoded in the multipole ` and the radial dependence in the wave vector k. This statistic allows us to retrieve the clustering information without having to adopt the ﬂat sky approximation or the need for redshift binning.

Recently, the spherical harmonic tomography power spectrum has been compared to P(k,r ), showing that the two methods yield consistent results [21] and both these methods have been employed to investigate the complementarity of weak lensing and galaxy redshift surveys (see e.g. [156, 67]).

With the aforementioned galaxy redshift surveys under development, it becomes increasingly important to further test and compare the applicability of these statistics to survey requirements. In this chapter, we compare the two spherical-sky statistics, i.e. the SHT power spectrum and the SFB power spectrum using a Fisher analysis. We study the sensitivity of these statistics to the detail of their implementation, placing particular emphasis on the advantages and disadvantages of each method, some of which we illustrate with simpliﬁed toy models.

This chapter is organized as follows. In Section 2.2 we summarize our comparison baseline model. In Sections 2.3 and 2.4 we review three-dimensional spherical analyses of the matter overdensity ﬁeld as well as Fisher matrix forecasting techniques and present applications to the SFB power spectrum. In Section 2.5 we present a comparison of the spherical harmonic tomography and the SFB power spectrum. We conclude in Section 2.6. Derivations and discussion of employed toy models are deferred to the Appendix.

2.2 Comparison baseline

In this work, we consider a wCDM cosmological model in the framework of general relativity speciﬁed by the set of 7 cosmological parameters θ (h,Ω ,Ω ,w ,w ,n ,σ ), where we ﬁx = m Λ 0 a s 8 the baryon density Ω 0.045. This model allows for a dynamical evolution of dark energy as b = well as curvature and is characterized by 7 parameters: the mean fractional matter density

Ωm, the fractional density of dark energy ΩΛ, the Hubble constant H0 100h km/s/Mpc, the 1 = r.m.s. of matter ﬂuctuations σ8 in spheres of comoving radius 8h− Mpc, the scalar spectral

24 2.2. Comparison baseline

106 4.0 ×

3.2

2.4 ) z ( W 1.6

0.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 z

Figure 2.1: Survey window function with mean redshift z¯ 1.2 and constant galaxy vol- 3 ¡ 1 ¢ 3 = ume density n0 10− h− Mpc − . This corresponds to a galaxy surface density of nA 2 = = 0.49 arcmin− .

index ns and two parameters w0 and wa that characterize the equation of state of dark energy ([173, 56])

w(a) w (1 a)w . (2.1) = 0 + − a We choose fiducial values θ (0.7,0.3,0.69, 0.95,0,1.0,0.8), which are consistent with the fid = − recent results by WMAP 9 [124]. In all calculations we fix w 0. a = We define the baseline survey in terms of total surveyed volume V and number of detected galaxies Ngal by expressing the galaxy number density as [282]

Ngal n (r ) n0 φ(r ) φ(r ), (2.2) = = V where r is the comoving distance and φ(r ) deﬁnes the radial survey selection function with normalization R d3r φ(r ) V as in [219, 282]. The normalization condition deﬁnes the window = function in redshift W (z)[282]. For incomplete sky-coverage, parametrized by the fraction of sky covered in the survey fsky, it becomes

Z Z c Z V dr r 2φ(r ) dz r 2φ(r ) dz W (z) . (2.3) = H(z) = = 4πfsky

For all surveys considered, we assume fractional sky coverage f 0.125, which represents a sky = lower limit to the sky fraction covered by future surveys, and a radial selection function given

25 Chapter 2. Three-dimensional spherical analyses of cosmological spectroscopic surveys by

2 ³ r ´ φ(r ) e− r0 , (2.4) = 3 3 which yields a survey volume of V π 2 r . This choice is motivated by the fact that it allows for = 0 the analytical computation of SFB power spectra in the absence of evolution, which is useful to test the full results. In order to match window functions characteristic of upcoming galaxy 1 redshift surveys, we set r r (z 1) 2354 h− Mpc, as computed in our ﬁducial cosmological 0 = = = model.

Fig. 2.1 shows the angular galaxy density as a function of redshift for a volume density of 3 ¡ 1 ¢ 3 n 10− h− Mpc − ; it further illustrates the binning scheme adopted in the tomographic 0 = analysis. Each bin conﬁguration is chosen by requiring the same number of galaxies in each redshift bin. Our baseline choices are summarized in Table 2.1. For comparison and future reference we show recent choices made in the literature in Table 2.2. This table highlights the breadth of possible choices to make when analyzing a galaxy redshift survey. The most important ones include:

1. Statistic used: spherical harmonic tomography power spectrum, spherical Fourier Bessel power spectrum or Cartesian power spectrum

2. Physical effects included: redshift space distortions, relativistic corrections (see e.g. [38, 282])

3. Implementation scheme: examples include simplifying assumptions like the Limber approximation, Fisher matrix computation method or number of cosmological parameters considered

From Table 2.2 we see that the choices made differ considerably. One of the aims of this chapter is therefore to investigate how much parameter forecasts and constraints are inﬂuenced by some of these choices.

2.3 Three-dimensional spherical power spectra

2.3.1 The Cartesian power spectrum

A common approach to analyze the observed matter overdensity ﬁeld δ(x,r ) is to expand it into its Cartesian Fourier components δ(k,r ), where we use the comoving distance r as a measure of time t. The real-space overdensity ﬁeld is related to its Fourier counterpart through

Z 1 3 ik x δ(x) d k δ(k,r )e · . (2.5) = (2π)3

26 2.3. Three-dimensional spherical power spectra

Table 2.1: Baseline speciﬁcation

Sky coverage: fsky 0.125 = 1 Selection function: φ(r ) as in Eq. 2.4 with r0 2354 h− Mpc 3 ¡ 1= ¢ 3 Survey Galaxy volume density: n0 10− h− Mpc − = 2 Galaxy surface density: n 0.49 arcmin− A = Angular scales covered: ` [2,50] ∈ Cosmological parameters: θ (h,Ω ,Ω ,w ,n ,σ )a = m Λ 0 s 8 Fiducial values: θ (0.7,0.3,0.69, 0.95,1.0,0.8) ﬁd = − Prior: none Model Galaxy bias: b (k,r ) 1 = Redshift space distortions: yes Relativistic corrections: no SFB SHT Implementation k range: 1 Number of bins: 7 k (0.0007,0.2) hMpc− ∈ Redshift range: 0.05 z 3.0 Fisher matrix: diagonal ≤ ≤ a We further ﬁx Ω 0.045 and wa 0. b = =

The Fourier space correlation function is the Cartesian matter power spectrum P (k,r ) deﬁned by

3 δ(k,r )δ(k0,r ) (2π) δ (k k0)P (k,r ). (2.6) 〈 〉 = D − We only focus on the linear matter power spectrum which factorizes into a time and scale dependent part as

P (k,r ) D2(r )P (k), (2.7) = 0 where D(r ) is the linear growth factor and P (k) is the power spectrum at redshift z 0. We 0 = assume a transfer function as summarized in [256] and neglect both the effects of baryon oscillations (BAOs) and neutrinos in our baseline conﬁgurations. In Section 2.5 we investigate the impact of BAOs on our results.

The measurement of this statistic from galaxy redshift surveys bears one complication: the observables in these surveys are the galaxy redshifts z and their angular positions on the sky ¡θ,φ¢. Therefore, in order to compute any three-dimensional power spectrum from data, the redshift needs to be related to a wave number k through the assumption of a radial distance. This transformation depends on the choice of a ﬁducial cosmological model. Any three- dimensional analysis of the matter overdensity ﬁeld therefore requires the assumption of a cosmological model [223], prior to testing it.

Another consequence of the fact that radial galaxy distances are only accessible through their redshift, is that the distance estimates will be affected by peculiar galaxy velocities. The

27 Chapter 2. Three-dimensional spherical analyses of cosmological spectroscopic surveys

Table 2.2: Compilation of different implementations used in the literature.

Rel. Limber a Paper Statistic RSD Bias Np Prior corr. approx. Gaztanaga et i j C ,P(k) b 8 Planck + al., 2012 [90] ` SN-II de Putter et al., i j C ,P(k) b 9 Planck Fisher 2013 [67] ` analysis Cai et al., 2012 i j C ,P(k) b 6 CMB p. s.c [44] ` Kirk et al., 2015 i j C 7 ﬂat [156] ` Font-Ribera et i j C ,P(k) b 8 Planck al., 2013 [84] ` Di Dio et al., i j C 5 none 2014 [70] ` This work i j C` ,C`(k,k0) 6 none Bonvin et al., i j C -- 2011 [38] ` General Challinor et al., i j C -- analysis 2011 [52] ` Rassat et al., C (k,k ) -- 2012 [219] ` 0 Yoo et al., 2013 C (k,k ) -- [282] ` 0 Pratten et al., C (k,k ) -- 2013 [218] ` 0

a Number of parameters used in analysis. i j b Transverse modes with C . RSDs taken into account for the radial modes in each redshift bin using P(k). ` c Assume primordial CMB power spectrum known.

28 2.3. Three-dimensional spherical power spectra comoving galaxy distances s inferred from their redshifts are given by [146] v n s r · , (2.8) = + a H(a) where v is the galaxy velocity due to the linear collapse of overdensities, H(a) is the Hubble parameter and n denotes the line of sight direction. These redshift space distortions (RSDs) lead to an enhancement of the Cartesian matter power spectrum given by [146]

2 P (k,r ) P (k,r )¡1 βµ2 ¢ , (2.9) s = + k

f d logD(a) where β /b. The quantity f /d loga denotes the linear growth rate, b is the galaxy bias, = = discussed below, and µk is the cosine of the angle between the line of sight and the wave vector k. Measuring the power spectrum in redshift space therefore allows us to also estimate the growth rate f of matter perturbations.

Irrespective of the analysis method, galaxy redshift surveys pose an additional complication. Since galaxies are only expected to form inside the peaks of the overdensity field [24] and galaxy formation is not completely understood yet, the galaxy overdensity field δg(k,r ) is expected to constitute a biased tracer of the underlying dark matter distribution δ(k,r ), i.e. δ (k,r ) b(k,r ) δ(k,r ). In this chapter we assume that galaxies perfectly trace dark matter, g = which amounts to setting the bias parameter b(k,r ) 1. Since in this work we focus on = clustering on large scales, where the scale-dependence of galaxy bias is negligible (e.g. [182]), we believe that this simplified assumption is appropriate because the statistics we compare will all equally suffer from the problem of bias. An investigation of the effects of scale-dependent bias on our results would be interesting for future work.

2.3.2 The spherical harmonic tomography power spectrum

The need for assuming a cosmological model, which arises in three-dimensional analyses of galaxy clustering, can be circumvented with a tomographic analysis. This amounts to discretizing the redshift and analyzing the angular dependence of galaxy clustering through i j the spherical harmonic tomography power spectrum C` at a number of different redshifts in order to partly recover the three-dimensional information. In practice, a galaxy catalogue is analyzed by subdividing the galaxies into redshift bins and computing both the auto- and cross-power spectra for all the bins.

Assuming the overdensity ﬁeld to be statistically isotropic and homogeneous, the spherical harmonic tomography power spectrum including RSDs between redshift bins i and j, with radial selection functions φi (r ) and φj (r ) respectively, is given by [198] Z i j 2 2 ³ i i,r ´³ j j,r ´ C dkk P0 (k) W (k) βW (k) W (k) βW (k) , (2.10) ` = π ` + ` ` + ` where the auto power spectra are obtained for i j and the cross power spectra for i j . The = 6=

29 Chapter 2. Three-dimensional spherical analyses of cosmological spectroscopic surveys

5 10−

5 10− z¯ = 0.30 6 10−

z¯ = 0.66

7 10− z¯ = 0.88

10− | l z¯ = 1.10 l 10 8 C

C − |

z¯ = 1.40 9 10− z¯ = 1.80 7 10− 10 10− z¯ = 2.50

11 10− 101 102 101 102 l l

i j Figure 2.2: Spherical harmonic tomography power spectra C` for the 7 redshift bins and survey speciﬁed in Sec. 2.2, where z¯ denotes the mean redshift of each bin. The left panel shows the auto-power spectra, while the absolute value of the cross-power spectra are shown on the right hand side.

R selection functions are normalized i.e. dr φi (r ) 1. Their unnormalized counterparts, the = i redshift distributions for each bin, are shown in Fig. 2.1. W` is the real-space window function i,r whereas W` accounts for the corrections due to RSDs; both window functions are deﬁned in terms of the spherical Bessel functions j` as [198] Z W (k) dr D(r )φ (r )j (kr ), (2.11) ` = i `

Z " ¡ 2 ¢ r 2` 2` 1 W (k) dr D(r )φi (r ) + − j` (kr ) ` = (2` 3)(2` 1) + − `(` 1) (` 1)(` 2) ¸ − j` 2 (kr ) + + j` 2 (kr ) . (2.12) − (2` 1)(2` 1) − − (2` 1)(2` 3) + − + + + i j Fig. 2.2 shows both the auto and the cross (neighboring redshift bins) SHT power spectra C` as a function of angular scale ` for the baseline conﬁguration deﬁned in Sec. 2.2.

The computation of the spherical harmonic tomography power spectrum through Eq. 2.10 can be computationally expensive and it is therefore common to resort to the small angle and wide selection function approximation. At large `, the spherical harmonic tomography power spectrum can be approximated through Limber’s approximation as [168]

Z Ã 1 ! i j φi (r )φj (r ) ` 2 C dr D(r )P0 k + . (2.13) ` ' r 2 = r

30 2.3. Three-dimensional spherical power spectra

2.3.3 The spherical Fourier Bessel power spectrum

An alternative way for analyzing the galaxy overdensity field is through the spherical Fourier Bessel transform (e.g. [115, 219, 218, 282]). The galaxy overdensity field δ(r) can be expanded into its translationally invariant parts i.e. the eigenfunctions of the Laplacian in spherical coordinates. In flat-space these are given by products of spherical Bessel functions and spherical harmonics Y`m(θ,φ) which leads to the expansion r 2 Z X δ(r) dk δ`m(k)k j` (kr )Y`m(θ,φ). (2.14) = π `,m

The coefﬁcients δ`m(k) are given by

r Z Z 1 2 ` 2 3 ¡ ¢ ¡ ¢ δ`m(k) ki dr r d k0 δ k0,r j` (kr ) j` k0r Y ∗ (θk ,φk ). (2.15) = 2π2 π `m 0 0

The SFB power spectrum is defined as the variance of these coefficients as given below, where the last equality holds if the overdensity field δ(r) is statistically isotropic and homogeneous (SIH) [219]

δ`m(k)δ∗ (k0) C`(k,k0)δ`` δmm C`(k)δD(k k0)δ`` δmm . (2.16) 〈 `0m0 〉 = 0 0 = − 0 0 denotes an ensemble average. Under the SIH assumption we further obtain [48] 〈〉 C (k) P (k). (2.17) ` = A similar identity can be obtained in the presence of RSDs in the ﬂat-sky limit and for high radial wavevectors k. In this case we approximately obtain

2 C (k) P (k)¡1 β¢ . (2.18) ` ' + A derivation of this radialization in the presence of RSDs can be found in Appendix 2.A. In cosmology, the SIH condition will usually be violated for two reasons [219] (i) the observed fields are generally confined to finite regions of space defined by the survey selection function and (ii) the field δ(r) and the power spectrum P (k,r ) evolve.

In order to compute the observed SFB power spectrum, constraints on radial survey geometry can be imposed through a radial selection function φ(r ), as deﬁned in Sec. 2.2[219]

δ (r) φ(r )δ(r). (2.19) obs = Accounting for time-evolution as well as the effects of RSDs, the observed SFB power spectrum of the overdensity ﬁeld δ becomes [219]

µ ¶2 Z 2 2 ¡ r ¢¡ r ¢ C`(k,k0) dk00k00 P0(k00) W`(k,k00) W (k,k00) W`(k0,k00) W (k0,k00) . (2.20) = π + ` + `

31 Chapter 2. Three-dimensional spherical analyses of cosmological spectroscopic surveys

r W`(k,k00) is the real-space window function whereas W` (k,k00) accounts for the corrections due to RSDs; they are deﬁned as ([219, 218]) Z 2 ¡ ¢ W (k,k00) dr r D(r )φ(r )k j (kr ) j k00r , (2.21) ` = ` `

Z 2 · 2 r 2 k ` ¡ ¢ W` (k,k00) dr r β D(r )φ(r ) 2 j` 1 (kr ) j` 1 k00r = k00 (2` 1) − − + 2 ¸ `(` 1) © ¡ ¢ ¡ ¢ª (` 1) ¡ ¢ + j` 1 (kr ) j` 1 k00r j` 1 (kr ) j` 1 k00r + j` 1 (kr ) j` 1 k00r − (2` 1)2 − + + + − + (2` 1)2 + + + + (2.22) Z µ ¶ 2 k dφ(r ) ` ¡ ¢ (` 1) ¡ ¢ dr r β D(r ) j` (kr ) j` 1 k00r + j` (kr ) j` 1 k00r . + k dr (2` 1) − − (2` 1) + 00 + + Eq. 2.22 allows for a time-dependence of the overdensity ﬁeld δ(k,r ), since the survey selection functions tend to be broad in redshift as opposed to the redshift bins in Section 2.3.2.

r 1 Fig. 2.3 shows the SFB power spectrum C 0 (k,k) C (k,k)( 0/2p2π)− , both as a function of angu- ` = ` lar scale ` and wave vector k for the selection function deﬁned in Eq. 2.4. The normalization follows [282] and facilitates comparison of the SFB with the Cartesian power spectrum P (k,r ).

Just as for the spherical harmonic tomography power spectrum it is useful to obtain approximations to Eq. 2.20. In [282] it is shown that in the limit of large angular multipoles `, the SFB power spectrum, neglecting RSDs, can be approximated by [282]

Ã 1 ! Ã 1 ! 2 ` 2 ` 2 ¡ ¢ C (k,k0) φ + D + P0(k)δD k k0 . (2.23) ` ' k k −

In the cases considered, there always remains a signiﬁcant difference between Eq. 2.20 and Eq. 2.23, which is the SFB analogue of Limber’s approximation. Nonetheless it proves useful to test the accuracy of the full equation.

The SFB power spectrum for a generic distance-redshift relation

The spherical Fourier Bessel coefﬁcients are functions of the wave vector k, which in turn depends on the measure of separation in real space. It is customary to choose k conjugate to the comoving separation r , making the assumption of a cosmological model inevitable when computing the SFB transform. As an alternative, in this chapter we additionally compute the SFB power spectrum for two distance-redshift relations which can directly be computed from observable quantities.

In order to derive an expression for the SFB power spectrum, we assume a generic distance- redshift relation deﬁned as r˜(z) where r˜ is an arbitrary monotonic function of the redshift z.

32 2.3. Three-dimensional spherical power spectra

Cl0(k = k = 0.0048)

Cl0(k = k = 0.0078)

Cl0(k = k = 0.1700)

] 4

3 10

3 10 Mpc) Mpc) 1 1 − − h h ) [( ) [( k, k k, k

( 3 ( 0 3 l 0 10 l 10 C C

Cl0(k, k, l = 2)

Cl0(k, k, l = 10)

Cl0(k, k, l = 50) 2 2 10 3 2 1 10 10− 10− 10− 101 1 k [hMpc− ] l

Figure 2.3: The SFB auto power spectrum C`0 (k,k) as a function of wave vector k and angular scale `.

With this choice of distance-redshift relation the SFB power spectrum reduces to

µ ¶2 Z 2 2 ¡ r ¢¡ r ¢ C`(ν,ν0) dk00k00 P0(k00) W`(ν,k00) W (ν,k00) W`(ν0,k00) W (ν0,k00) , (2.24) = π + ` + ` where ν denotes the wave vector conjugate to the new separation measure and φ0(s) is the selection function in this coordinate system. W`(ν,k00) is the real-space window function r whereas W` (ν,k00) accounts for the corrections due to RSDs; they are given by Z 2 ¡ ¢ W (ν,k00) dr r D(r )φ0(r˜)νj (νr˜) j k00r , (2.25) ` = ` `

Z ν2 dr˜ H(z) · `2 r 2 ˜ ˜ ¡ ¢ W` (ν,k00) dr r β D(r )φ0(r ) 2 j` 1 (νr ) j` 1 k00r = k00 dz c (2` 1) − − + 2 ¸ `(` 1) © ¡ ¢ ¡ ¢ª (` 1) ¡ ¢ + j` 1 (νr˜) j` 1 k00r j` 1 (νr˜) j` 1 k00r + j` 1 (νr˜) j` 1 k00r − (2` 1)2 − + + + − + (2` 1)2 + + + + (2.26) Z µ ¶ 2 ν dφ0 dr˜ H(z) ` ¡ ¢ (` 1) ¡ ¢ dr r β D(r ) j` (νr˜) j` 1 k00r + j` (νr˜) j` 1 k00r . + k dr˜ dz c (2` 1) − − (2` 1) + 00 + +

Since the selection function transforms as a scalar quantity, φ0 (r˜) is related to the selection function in terms of the comoving distance through φ0 (r˜(z)) φ(r (z)). For a derivation of = these identities, the reader is referred to Appendix 2.B.

As an illustration of the impact of the choice of distance-redshift relation, we consider two

33 Chapter 2. Three-dimensional spherical analyses of cosmological spectroscopic surveys alternatives to the comoving distance

c r˜(z) z, (2.27) = H0 c r˜(z) ln(1 z). = H0 +

The ﬁrst is the linear approximation to the comoving distance valid for low redshifts;, while the second is a logarithmic approximation to r chosen to reproduce both its behavior at low and intermediate redshift. The resulting SFB power spectra are shown in Fig. 2.6. The normalization again follows [282].

2.4 Fisher matrices for 3D spherical power spectra

The Fisher matrix (FM) allows us to forecast the constraints on cosmological parameters obtainable with future surveys under the approximation of Gaussianity (for an overview of Fisher forecasting see e.g. [270, 113] on which this summary is based). This method can be applied to survey optimization or, as done in this chapter, it can be used to assess the constraining power of different data analysis methods. The FM allows for the propagation of uncertainties in the measurement to uncertainties on the model parameters, which here are the parameters of the wCDM cosmological model. Bayes’ theorem allows us to relate the posterior probability distribution p (θ x) around the maximum likelihood (ML) estimator | to the data likelihood L (x;θ). The inverse covariance matrix of the posterior distribution is called the Fisher matrix and given by

∂2 logL Fαβ . (2.28) = 〈− ∂θα∂θβ 〉

When several parameters are simultaneously estimated from the data, the marginalized uncer- q tainty on each parameter θ is bounded by ∆θ F 1 [152]. The ﬁxed uncertainty, obtained α α ≥ αα− when keeping all parameters except one ﬁxed, is smaller or equal to the former and given by 1 ∆θ /pFαα [152]. α ≥

2.4.1 The Fisher matrix for the spherical harmonic tomography power spectrum

The FM for a tomographic survey employing N redshift bins can be derived from Eq. 2.28 assuming a Gaussian likelihood for the spherical harmonics coefﬁcients. The result is [133]

2 1 h i X ( ` )∆` ˜ 1 ˜ 1 Fαβ fsky + Tr D`αC`− D`βC`− , (2.29) = ` 2

34 2.4. Fisher matrices for 3D spherical power spectra where the sum is over bands of width ∆` in the power spectrum and we set ∆` 1. The data = covariance is an N N matrix given by × x x x x [C ]i j C i j N i j , (2.30) ` = ` + ` where the xi ,x j denote the respective bins. The ﬁrst term in Eq. 2.30 represents the innate xi x j 1 cosmic variance, while the second term is due to shot noise and given by N /nA, where ` = nA is the galaxy surface density of the survey.

The matrix D`α contains the dependence of the observables on the parameters θα and has elements given by

x x ∂C i j i j ` [D`α] . (2.31) = ∂θα

The simple scaling with fsky accounts for the fact that angular modes become coupled for incomplete sky coverage. This reduces the number of independent modes at a given angular scale ` and therefore increases the uncertainties due to cosmic variance [238].

2.4.2 The Fisher matrix for the SFB power spectrum

The computation of the FM for the SFB power spectrum from the Gaussian likelihood for the SFB coefficients δ`m(k) is challenging due to the correlations between different k modes, which are due to time-evolution of the overdensity field and finite survey effects. The complication arising from the non-diagonal data covariance matrix can be dealt with in two different ways: (i) by choosing a finite grid in k space and computing the FM on this discrete grid or (ii) by approximating the full FM by assuming a diagonal data covariance matrix. Drawing from previous work ([268, 116, 112, 282]) we can find expressions for the FM in both cases.

The FM for a measurement of the SFB power spectrum for n discrete wave vectors ki can be written as [112]

2 1 · Cˆ Cˆ ¸ X ( ` )∆` ˆ 1 ∂ ` ˆ 1 ∂ ` Fαβ fsky + Tr C`− C`− , (2.32) = ` 2 ∂θα ∂θβ where the sum is over bands of width ∆` in the power spectrum and we set ∆` 1 and the = scaling with fsky accounts for incomplete sky coverage. Cˆ ` is the non-diagonal covariance matrix for given angular multipole `

C˜ (k ,k ) C˜ (k ,k ) C˜ (k ,k ) ` 1 1 ` 1 2 ··· ` 1 n C˜ (k2,k1) C˜ (k2,k2) C˜ (k2,kn) ˆ  ` ` ··· `  C`  . . . , (2.33) =  . . .. .   . . . .  C˜ (k ,k ) C˜ (k ,k ) C˜ (k ,k ) ` n 1 ` n 2 ··· ` n n

35 Chapter 2. Three-dimensional spherical analyses of cosmological spectroscopic surveys and C˜ (k ,k ) C (k ,k ) N (k ,k ). The ﬁrst term is again the cosmic variance and the ` i j = ` i j + ` i j second is the shot noise in a survey with galaxy volume density n¯(r ) given by [282]

µ ¶Z 2ki k j 2 ¡ ¢ 1 N`(ki ,k j ) dr r φ(r )j` (ki r ) j` k j r . (2.34) = π n¯(r )

If we assume a broad window function, such that mode coupling can be neglected [82], we can approximate C (k,k0) 0 for k k0. This allows us to obtain a simpliﬁcation of Eq. 2.32 ` = 6= given by

k Zmax X (2` 1)∆` Ldk 1 ∂C`(k,k) ∂C`(k,k) + Fαβ fsky 2 , (2.35) = ` 2 2π (C`(k,k) N`(k,k)) ∂θα ∂θβ kmin + where the sum is over bands of width ∆` in the power spectrum, L denotes the maximal length scale probed in the survey and kmin, kmax denote the wave vector limits of the survey. For our calculations we set L to the characteristic survey depth i.e. L r 1 and ∆` 1. = 0 = For a detailed derivation of Equations 2.32 and 2.35 the reader is referred to Appendix 2.C. We note that we do not include any optimal weighting of the data [115], a subject which will be interesting for future work.

2.5 Results

As a means for assessing the applicability of both the spherical harmonic tomography and the spherical Fourier Bessel power spectrum to upcoming galaxy redshift surveys, we compare their forecasted performance in a Fisher matrix analysis. From the numerous possible combinations discussed in Section 2.2, we have chosen to place our emphasis on two topics: We ﬁrst focus on each statistic separately and address the main complication associated with it; then we compare the constraining power of both statistics for the baseline survey (Section 2.2).

2.5.1 Spherical harmonic tomography power spectrum

The constraints on cosmological parameters obtained when analyzing the baseline survey (2.2) using the SHT power spectrum and only taking into account auto-power spectra 2 are highlighted in Table 2.3. For the baseline conﬁguration we obtain uncertainties of the order of the parameter value, which is due to the fact that we only consider large-scale information from angular multipoles ` [2,50] 3. Increasing the maximal angular scale probed to ∈

1The maximal length scale probed L is not a well-defined quantity, but parameter constraints seem stable 1 against changing specification, since setting L V 3 changes results by at most 10%. = survey 2We find that including the cross-correlations does not affect results significantly and we thus neglect them in order to match the specifications used for the SFB power spectrum more closely. 3We believe that this reduced range does not affect our results because we are mainly concerned with comparing two different statistics.

36 2.5. Results

Table 2.3: Parameter constraints obtained for different implementations of the SHT and SFB power spectrum. Results for the baseline conﬁguration are marked in bold. Note that in the case of the SFB the full cov. results neglect contributions due to shot noise, whereas all other constraints assume shot noise as speciﬁed in Sec. 2.2.

a Statistic Implementation Radial res. σh σΩm σΩΛ σw0 σns σσ8 Limber ` 50 7 bins 5.5 1.8 1.2 2.8 0.79 0.22 max = no ` 50 7 bins 2.2 0.32 0.50 1.2 1.4 0.15 RSD max = SHT RSD ` 200 7 bins 0.082 0.040 0.10 0.23 0.10 0.030 max = ` 50 7 bins 0.42 0.17 0.48 1.0 0.33 0.15 max = ` 50 max = 7 bins 0.38 0.12 0.45 0.97 0.28 0.11 RSD w/ BAOs ` 50 10 bins 0.078 0.098 0.26 0.59 0.30 0.084 max = ` 50 20 bins 0.15 0.073 0.19 0.38 0.20 0.074 max = `max 50 30 bins 0.12 0.058 0.17 0.34 0.18 0.060 = 1 Log. 0.2 h Mpc− 0.30 0.048 0.32 0.46 0.31 0.67 1 full cov. Lin. 0.2 h Mpc− 0.22 0.047 0.29 0.40 0.25 0.49 1 Com. 0.2 h Mpc− 0.38 0.043 0.55 1.3 0.38 0.86 1 Log. 0.2 h Mpc− 0.33 0.059 0.52 0.71 0.30 0.74 1 SFB Linear 0.2 h Mpc− 0.53 0.086 0.82 1.0 0.39 1.2 Com. 0.2 h Mpc 1 0.37 0.091 0.72 2.7 0.32 0.81 diag cov. − Com. w/ 0.2 h Mpc 1 0.11 0.028 0.31 1.2 0.15 0.26 BAOs − 1 Com. 0.15 h Mpc− 0.48 0.097 0.79 2.8 0.51 1.1 1 Com. 0.1 h Mpc− 0.55 0.10 0.92 2.9 0.62 1.2 a For the SFB power spectrum this corresponds to kmax.

37 Chapter 2. Three-dimensional spherical analyses of cosmological spectroscopic surveys

` 200, which corresponds to the non-linear cutoff for the lowest redshift bin, consider- max = ably improves parameter constraints. The restriction to large-scale angular perturbations is due to the calculation of the SFB power spectrum, which becomes slow for smaller scales. Nonetheless, the matter density of the universe and the power spectrum amplitude are already sensibly constrained whereas the dark energy sector is poorly constrained due to the significant degeneracies present. Before comparing these results to the constraints obtained for the SFB power spectrum, we ﬁrst discuss the main complication associated with tomographic analyses.

The SHT power spectrum necessitates tomographic analyses of galaxy catalogues, which amounts to splitting the data into redshift bins. This additional freedom raises the question of how to optimally perform this subdivision. For a ﬁxed baseline survey and therefore data, we expect to see small differences between binning schemes. As we will show below, on the contrary, we can identify instabilities when implementing different bin conﬁgurations for a given survey, when we estimate their respective constraining power in a Fisher analysis using Eq. 2.29. Not only is this behavior unexpected but it also implies that these instabilities need to be kept in mind when e.g. comparing the forecasted performance of future surveys.

As an example for studying the effects of redshift binning on parameter constraints, we investigate conﬁgurations that differ in the amount of redshift bin overlap. In general there are two causes for bin overlap in galaxy redshift surveys: (i) in spectroscopic surveys, bins can purposely be tailored to have overlap while (ii) in photometric surveys, redshift bins will overlap due to inaccurate redshift measurements. We focus on spectroscopic surveys and therefore only consider case (i). A nonzero overlap between redshift bins will cause them to be correlated, if we assume that they are both located in the same part of the sky. Investigating the impact of bin overlap/correlation on the constraining power of galaxy redshift surveys therefore not only highlights instabilities with data binning but it also addresses the core of the same sky-different sky issue (see e.g. [156, 67]), which is the question of how much correlations between data sets can affect parameter constraints.

We investigate the effects of bin configuration on constraining power using a series of highly simplified toy models, which are based on the Limber approximation and ignore shot noise contributions. For a detailed description of these, the reader is referred to [189]. As shown in [189], we find that increasing the amount of overlap between bins, while keeping their mean redshifts fixed and taking into account correlations, can result in an improvement of cosmological parameter constraints by as much as a factor of 2. This behavior is only found when constraining parameters that exhibit a high level of redshift degeneracy between each other i.e. parameters which can only be distinguished with information at separate redshifts; an example from cosmology is the redshift degeneracy between parameters which control the growth of structure as a function of time and the overall clustering amplitude. On the other hand, we find that constraints on non-redshift degenerate parameters as well as fixed errors are insensitive to changes in bin overlap. It is important to point out that these conclusions do not apply to bin overlap caused by redshift uncertainties (case (ii)). Redshift errors cannot be

38 2.5. Results modeled solely as a redshift bin broadening, since this approach does not take into account the uncertainty introduced in the redshift distribution. If redshift uncertainties are implemented as in [17], we ﬁnd that increased bin overlap due to larger redshift uncertainties worsens parameter constraints as intuitively expected.

These results suggest that the main effect of overlap between redshift bins on spectroscopic surveys is to break redshift degeneracies between parameters. This seems counterintuitive but as we show in Ref. [189], the dependence of parameter constraints on correlation is a generic feature of such data sets. This suggests that the observed sensitivity of parameter constraints on binning scheme is due to the fact that the amount of correlation between redshift bins, which has an effect on parameter constraints, is scheme dependent.

The results presented so far have been based on tomographic analyses consisting of only two redshift bins; as the number of redshift bins is increased, the effect of bin overlap becomes negligible. The more available cosmological information is recovered from a survey, the less sensitive parameter constraints become to binning schemes. In order to obtain stable parameter constraints from a tomographic analysis of galaxy redshift surveys it is therefore essential to ensure that the available information is well recovered by the survey. We will review the limitations imposed on a tomographic analysis returning back to our baseline survey.

1.2 σh

σΩm 1.0 σΩΛ

σw0

0.8 σns

σσ8

σ 0.6

0.4

0.2

7 10 20 30 NBin

Figure 2.4: Uncertainties (1 σ) on cosmological parameters obtained with SHT power spectrum as a function of the number of redshift bins NBin.

The parameter that mainly controls the constraining power of a tomographic survey is its radial resolution; in practice this is the number of redshift bins. Starting from the baseline survey, we increase the number of redshift bins from N 7 to N 30 as shown in Table Bin = Bin = 2.3 and Fig. 2.4. This reduces uncertainties by almost a factor of three, which is in agreement 1 with the rough /pNBin scaling of parameter constraints with bin number when shot noise is not

39 Chapter 2. Three-dimensional spherical analyses of cosmological spectroscopic surveys yet dominant [70]. Therefore the maximal cosmological information retrievable analyzing a survey using the SHT power spectrum is limited by redshift accuracy and shot noise.

As seen from Table 2.3 we reanalyze the baseline survey neglecting redshift space distortions; once performing the full calculation and once assuming the Limber approximation. The results indicate that both changes deteriorate parameter constraints, showing that including RSDs in tomographic analyses increases the amount of cosmological information. Furthermore, the results obtained with the Limber approximation and the exact calculation deviate signiﬁcantly (differences of up to a factor of 5), which suggests that approximations in power spectrum calculations should be used carefully when computing Fisher matrices.

2.5.2 Spherical Fourier Bessel power spectrum

The constraints on cosmological parameters obtained when analyzing the baseline survey (2.2) using the spherical Fourier Bessel power spectrum are emphasized in Table 2.3. In agreement with the results for the SHT power spectrum, we obtain constraints of the same order of magnitude as the parameters themselves, an effect which we again attribute to the small multipole range considered. The best-constrained parameter is the matter density Ωm, whereas the SFB analysis mostly yields larger uncertainties on the remaining cosmological parameters than its tomographic counterpart. As in the previous section, we address particular issues associated with this analysis before turning to the comparison between the two methods.

Comparison of Fisher matrix computation techniques

The computation of Fisher matrices for the SFB power spectrum through Eq. 2.32 4 is time- consuming and numerically challenging because it requires inverting the covariance matrix deﬁned in Eq. 2.33. This step is tricky for two reasons: Firstly, as seen from Fig. 2.3, the SFB

` power spectrum falls off sharply for large scales with k /r0 for given angular scale ` and ¿ 1 survey depth r . This is because in a cone of angular extent θ /`, large radial modes cannot 0 ∼ be measured when the survey depth is ﬁnite. The rapid decrease in large scale power results in a considerable dynamic range in the covariance matrix eigenvalues, making it almost singular. Secondly, neighboring wave vectors k are strongly correlated, which further complicates the inversion of the covariance matrices.

In order to estimate the SFB Fisher matrix through Eq. 2.32 it is therefore inevitable to restrict

` calculations to separated wave vectors with k /r0 to overcome numerical instabilities. In ≥ practice we cut off all large-scale information for each angular scale ` as soon as it causes the covariance matrix condition number, which is a measure for accuracy loss in matrix inversion, to exceed κ 102. This is possible but it seems desirable to investigate alternatives to this crit = 4There is one subtlety involved with Fisher matrix calculations for the SFB power spectrum: as can be seen from Eq. 2.34, the shot noise contribution to the power spectrum is essentially a galaxy number count and therefore cosmology dependent. Since we ignore information from non-linear scales, we won’t include this information when estimating the constraining power of a survey.

40 2.5. Results

“brute-force” approach.

Even though neighboring wave vectors are strongly correlated, the correlations tend to rapidly decrease as we move away from the diagonal. This suggests resorting to the approximation ¡ ¢ C k,k0 0 for k k0 in order to obtain useful approximations to Eq. 2.32. The most straight- ` ' 6= forward implementation of these ideas is given in Eq. 2.35. Despite being an approximation to the full Fisher matrix given in Eq. 2.32, we ﬁnd that Eq. 2.35 yields no-shot noise as well as shot noise constraints which are mostly accurate to better than a factor of 2 for the baseline survey (for detailed results, see Table 2.4 in Appendix 2.D). These results agree with those obtained for the Cartesian matter power spectrum [268] and encourage the use of Eq. 2.35 for fast calculations which allow errors of up to a factor of a few.

Making use of this simplifying approximation, we investigate the impact of the non-linearity wave-vector cut on parameter constraints. As can be seen from Table 2.3 and Fig. 2.5 decreasing the maximal wave vector kmax by a factor of two increases parameter uncertainties by almost the same amount. The increase is larger than theoretically expected, since decreasing the cutoff scale by a factor of two will halve the number of available modes and should thus lead to an increase in uncertainty by a factor p2. We believe that this is due to the fact that increasing the cutoff scale additionally helps breaking parameter degeneracies, since we approximately observe the theoretical scaling for the ﬁxed parameter constraints.

The choice of distance-redshift relation

As discussed in Sec. 2.3.3, the need for assuming a ﬁducial cosmological model can be miti- gated by analyzing surveys using distance-redshift relations that can directly be computed from observable quantities. Using the expressions derived in Sec. 2.3.3 we can investigate the impact of the distance-redshift relation on the obtained power spectra as well as survey constraining power.

We focus on two simple alternatives to the comoving distance as given in Eq. 2.27. Both approximations are fairly accurate at low redshift; at very high redshifts on the other hand, both approximations break down because they considerably overestimate the comoving separation. The SFB power spectra obtained with these two distance-redshift relations are shown in Fig. 2.6. The choice of a different ﬁducial distance causes a shift in the observed SFB power spectra, because the window functions are offset from those in comoving distance.

These simple distance-redshift relations are viable alternatives to the comoving distance only if analyzing a survey in terms of them does not signiﬁcantly reduce its constraining power. To test their performance, we compare their constraints on cosmological parameters for a survey as deﬁned in 2.2 in two different ways: Since the diagonal Fisher matrix is an acceptable approximation to the full calculation, we will employ it to compare the forecasted parameter constraints obtained with all three distance measures taking shot noise into account. As a mere illustration, we additionally compare the constraints obtained from the full Fisher matrix,

41 Chapter 2. Three-dimensional spherical analyses of cosmological spectroscopic surveys

1.6 σh

σΩm

σΩΛ

σns 1.2 σσ8

σ 0.8

0.4

0.0 0.08 0.12 0.16 0.20 1 kmax [hMpc− ]

Figure 2.5: Uncertainties (1 σ) on cosmological parameters obtained using the SFB power spectrum as a function of maximal wave number kmax. Constraints on w0 are not shown for clarity.

neglecting any shot noise contributions.

Both these results are shown in Table 2.3. The constraints are similar, irrespective of the distance-redshift relation chosen. The only parameter exhibiting a signiﬁcant dependence on the way distance is related to redshift is the dark energy equation of state parameter w0. An analogous behavior is seen in the no shot-noise constraints obtained with the full Fisher matrix: we obtain different constraints especially for those parameters, which the comoving distance depends on, whereas the remaining constraints are largely insensitive to the distance- redshift relation of choice. Choosing a distance-redshift relation other than the comoving distance therefore appears to cause the SFB power spectrum to exhibit a stronger dependence on these former parameters because the volume element and the distance in Eq. 2.25 and 2.26 do not change in the same way.

As we include contributions due to shot noise, this potential constraining power is considerably reduced because alternative distance-redshift relations tend to overestimate the comoving separations for large redshift. The shot noise is therefore enhanced, which largely removes the gain from increased sensitivity.

The above considerations illustrate that constraints obtained from an SFB analysis of a galaxy redshift survey seem to be mostly stable against changes in the assumed distance-redshift relation. This suggests that it could be possible to analyze galaxy clustering using distance- redshift relations which only depend on observable quantities, without too large a loss in constraining power.

42 2.5. Results

2.5.3 Comparison between spherical harmonic tomography and SFB power spectrum

After focusing on each of the two statistics separately we can compare their baseline constraints shown in Table 2.3. Unexpectedly, the survey constraining power is signiﬁcantly affected by the choice of analysis method: When baryon acoustic oscillations are neglected, the SFB power spectrum yields weaker constraints, particularly on those cosmological parameters that need redshift leverage in order to be distinguished (i.e. growth and amplitude parameters). This is already evident for our baseline but the effect is enhanced if we consider larger bin numbers in the SHT analysis.

The weakness of constraints on growth as well as amplitude of matter fluctuations seems to be a generic feature of SFB analyses: When the underlying field is statistically isotropic and homogeneous (SIH), the SFB coefficients are given by Eq. 2.36 i.e. they correspond to an angular average of the Cartesian Fourier coefficient. In practice, the SIH condition is not met due to time evolution of the overdensity field and finite survey effects and the SFB coefficients are related to their Cartesian counterparts through Eq. 2.15. Any violation of the SIH condition will therefore introduce a coupling between the considered radial and angular scales k, ` and the redshift at which δ(k) will mostly be measured. Around r0, the decrease in the selection function amplitude breaks the SIH condition which causes modes to add incoherently and leads to cancellations. The contributions to the SFB power spectrum will therefore be preferentially weighted toward lower redshift, which results in smaller redshift leverage and weaker combined constraints.

These results therefore suggest that, even though both analysis methods are equivalent for infinite survey extent and recovery of all available modes, because the information content of the overdensity field does not depend on the basis set in which it is analyzed, they appear not to be equivalent for finite surveys. Analyses of galaxy redshift surveys through the SFB power spectrum are more affected by finite survey effects, which means that some high redshift information will be downweighted. The SHT power spectrum analysis, on the other hand, allows us to probe more efficiently the complete high redshift range of the survey.

We test the impact of baryon oscillations on our results by reanalyzing our baseline configurations using the transfer function as specified in [78]. From Table 2.3 we see that adding BAOs improves the SFB constraints, while leaving the SHT power spectrum constraints mostly unchanged. This suggests that a significant fraction of the information lost due to finite survey effects in SFB analyses can be compensated by the fact that its three-dimensional nature allows us to recover the information contained in the BAOs, while an SHT analysis tends to dilute these features 5. The inclusion of BAOs results in comparable constraints for both methods. This suggests that SFB is better suited for capturing information beyond the smooth

5The change of transfer function from [256] to [78] without BAOs (the so-called “no-wiggles” approximation obtained with fractional baryon density Ω 0 but ignoring the oscillatory contribution to the power spectrum) b 6= does not signiﬁcantly affect parameter constraints, which suggests that the improvement in constraining power can be mainly attributed to the presence of BAOs.

43 Chapter 2. Three-dimensional spherical analyses of cosmological spectroscopic surveys shape of the power spectrum.

2.5.4 Implementation effects on estimated survey constraining power

The constraining power of a particular survey, as estimated from Fisher matrix calculations, is clearly determined by survey specifications and included physics. Nevertheless, details in the particular implementation can also affect parameter constraints and we can investigate the magnitude of this effect using our simplified toy model (see Ref. [189]). Our findings suggest that the magnitude can potentially equal that of changing survey specifications, when one of the following two conditions is fulfilled: (i) As illustrated in Ref. [189], the choice of binning scheme and thus implementation ceases to be relevant as more information is retrieved from a particular survey and parameter constraints become tighter. This therefore suggests that the choice of prior can significantly affect the stability of parameter constraints: applying a tight prior reduces the susceptibility of parameter constraints to implementation. (ii) A second essential choice is the set of constrained parameters: as highlighted by the toy model, parameters which are redshift degenerate with each other are particularly affected by changes in implementation. On the other hand, constraints on non-degenerate parameters are expected to be more stable. It is thus important to be aware of these instabilities whenever constraining a set of degenerate and loosely constrained parameters.

These susceptibilities can further be ampliﬁed due to numerical instabilities in Fisher matrix calculations. Fisher matrices can have large condition numbers κ(F ) i.e. be close to numerically singular, if one or more parameters are not well constrained by the data. Therefore the error introduced by the Fisher matrix inversion can be of order 100%, if the accuracy in the 1 Fisher matrix elements is smaller than κ(F )− [274].

These findings further suggest that care has to be taken when comparing Fisher matrix results. Whenever two different results need to be compared it is essential to make sure that not only the survey specifications are similar but also that priors and set of constrained parameters agree with each other. This is relevant in light of the recent discussion regarding benefits of performing spectroscopic and photometric galaxy surveys in the same part of the sky (see e.g. [156, 67]): a reliable comparison between the results obtained by different groups seems difficult due to the differing choices of priors and constrained parameters. Exactly matching the baseline survey is especially important in this case since it investigates the impact of cross-correlations on parameter constraints, which, as indicated by the toy model, only has an effect when parameters are loosely constrained and degenerate with each other. This suggests that changes in implementation have the potential to even affect qualitative results in this particular case.

44 2.6. Conclusions

2.6 Conclusions

Using a Fisher analysis, we have investigated three-dimensional analyses of galaxy redshift surveys on a spherical sky. In the course of our analysis, it has become evident that Fisher matrix results need to be carefully analyzed and compared. Especially when Fisher matrix methods are used to forecast constraints on large parameter sets, exhibiting degeneracies among one another, the obtained constraints are susceptible to details in implementation.

In particular we have compared the SHT and the SFB power spectrum, two statistics that are designed for the analysis of galaxy redshift surveys in a spherical geometry. By comparing their forecasted constraints on cosmological parameters, we have shown the applicability of approximations, such as the Limber approximation, and the numerical issues associated with computing these statistics. We have also studied the sensitivity of these statistics to the detail of their implementation. Our analysis is based on several simplifying assumptions; in particular, we restrict ourselves to linear, unbiased galaxy clustering and only focus on large scale power spectrum modes. For future work it would be interesting to include a treatment of these effects.

Using toy models, we ﬁnd that constraints obtained from a tomographic analysis of galaxy redshift surveys can be susceptible to implementation effects like redshift bin overlap, if only a limited amount of the total available information is retrieved. This suggests that in order to be stable against changes in implementation, it is important to retrieve as much information as possible from the tomographic analyses, e.g. by using a large number of redshift bins.

The computation of the SFB power spectrum from data relies on the assumption of a distance- redshift relation, usually the cosmology-dependent comoving distance. Analyses of galaxy redshift surveys by means of the SFB power spectrum therefore require the assumption of a cosmological model, prior to testing it. Comparing the SFB parameter constraints obtained using alternative distance-redshift relations, we ﬁnd them to be largely stable against changes in the assumed distance. This suggests that future surveys could in principle be analyzed using distance-redshift relations only relying on observable quantities without too large a degradation in parameter constraints.

For the baseline survey configuration we considered, we find that the SHT power spectrum yields somewhat tighter constraints than the SFB power spectrum. When we add baryon oscillations, on the other hand, the two methods yield comparable constraints. We attribute the former to the fact that the SFB power spectrum is less sensitive to modes at high redshift near the survey boundary, while the SHT power spectrum can be tailored to probe these modes. In the presence of BAOs, this effect can be compensated by the fact that the three- dimensional nature of the SFB transform allows us to resolve the baryonic oscillations, which tend to be diluted in SHT analyses. This fact would make SHT analyses advantageous for future spectroscopic galaxy redshift surveys mainly focusing on the power spectrum shape, while the SFB power spectrum may be well suited for specific applications like BAOs.

45 Chapter 2. Three-dimensional spherical analyses of cosmological spectroscopic surveys

2.A The radialization of the SFB power spectrum in the presence of RSDs

The overdensity field in the absence of RSDs at a constant time r is isotropic and homogeneous and can be directly related to the Cartesian matter power spectrum as in Eq. 2.17[ 48]. The isotropy is broken in the presence of RSDs but an approximate relation between these two quantities still holds. The SFB coefficients of the overdensity field are related to their Cartesian Fourier counterpart through Z 1 ` δ`m(k,r ) ki dΩk δ(k,r )Y`∗m(θk ,φk ). (2.36) = p8π3

The contribution to the Cartesian Fourier coefﬁcient due to RSDs is given by

δ (k,r ) βµ2 δ(k,r ), (2.37) RSD = k ¡ ¢ 2 4 with a power spectrum δ (k,r )δ∗ k0,r β µ P(k,r ) δ (k0 k). The quantity µ is 〈 RSD RSD 〉 = k D − k the cosine of the angle between the line of sight direction and the wave vector direction. In the flat-sky limit we can assume that the line of sight direction is constant. For a fixed angular multipole `, the SFB power spectrum will obtain contributions from increasingly radial wave vectors for larger k. In the flat-sky limit and large radial wavevectors k we can thus approximate µ 1. Inserting this into Eq. 2.36 using the identities k ' 1 Z 3 (k0 k) x δD(k0 k) d xe − · , (2.38) − = (2π)3 and [9]

ik r X ` ˆ e · 4π i j`(kr )Y`∗m(k)Y`m (nˆ), (2.39) = `,m gives the contribution to the SFB power spectrum due to RSDs

2 δ`m(k)δ∗ (k0) β P(k,r ) δD(k0 k)δ`` δmm . (2.40) 〈 `0m0 〉 = − 0 0 In the ﬂat-sky and large wave vector limit even the RSD contribution approximately radializes in absence of a selection function and time-dependence of the overdensity ﬁeld. This behavior is perceivable in Fig. 4 of [282], illustrating that the curvature of the sky is negligible for small scale perturbations.

2.B The SFB power spectrum for a generic distance-redshift relation

The need for assuming a cosmological model before testing it can be avoided choosing a distance-redshift relation which does not depend on cosmology. In order to derive an expres-

46 2.B. The SFB power spectrum for a generic distance-redshift relation sion for the SFB power spectrum, we assume a generic distance-redshift relation deﬁned as r˜(z) where r˜ is an arbitrary monotonic function of the redshift z. The measured redshift will be affected by peculiar galaxy velocities v along the line of sight n [200] i.e. v n z ztrue · , (2.41) obs ' + ac where c is the speed of light. The distance s inferred from the galaxy redshifts therefore becomes

v n dr˜ s r˜(z ) r˜(ztrue) · . (2.42) = obs ' + ac dz

The overdensity ﬁeld can be decomposed in the SFB basis set with coefﬁcients given by

r Z 2 3 δ`m(ν) d r˜φ0(r˜)δ(r˜)νj` (νr˜)Y ∗ (θ,φ), (2.43) = π `m where ν denotes the wave vector conjugate to r˜ and φ0(r˜) is the selection function in the new coordinate system. Since the overdensity ﬁeld is independent of the distance measure we have d3r˜ δ(r˜) d3r δ(r) where r is the comoving distance. Eq. 2.43 therefore reduces to = r Z 2 3 δ`m(ν) d rφ0(r˜)δ(r)νj` (νr˜)Y ∗ (θ,φ). (2.44) = π `m

Following [116], the functions of r˜ can be expanded as

µ ¶ dφ0 v n dr˜ φ0(r˜) φ0(r˜) · , (2.45) ' + dr˜ ac dz dj (νr ) µv n dr˜ ¶ j (νr˜) j (νr˜) ` · , ` ' ` + dr˜ ac dz which can be inserted into Eq. 2.44 to yield to ﬁrst order

r ½Z Z 2 1 3 3 ik0 r δ`m(ν) d r d k’φ0(r˜)δ(k0)e · νj` (νr˜)Y ∗ (θ,φ) = π (2π)3 `m Z Z ¾ 3 3 v(k0) n ik0 r dr˜ d d r d k’ · e · ν [φ0(r˜)j (νr˜)]Y ∗ (θ,φ) . (2.46) + ac dz dr˜ ` `m

The linear continuity equation allows us to relate the Fourier transform of the galaxy velocity ﬁeld to the overdensity through [71]

aH(z)δ(k) v(k) iβ k. (2.47) = k2

Thus the SFB power spectrum for the distance-redshift relation r˜(z) reduces to

µ ¶2 Z 2 2 r 2 C`(ν,ν0) dk00k00 P0(k00) W`(ν,k00) W (ν,k00) δ`` δmm . (2.48) = π | + ` | 0 0

47 Chapter 2. Three-dimensional spherical analyses of cosmological spectroscopic surveys

r W`(ν,k00) is the real-space window function whereas W` (ν,k00) accounts for the corrections due to RSDs; they are given by Z 2 ¡ ¢ W (ν,k00) dr r D(r )φ0(r˜)νj (νr˜) j k00r , (2.49) ` = ` `

Z ν2 dr˜ H(z) · `2 r 2 ˜ ˜ ¡ ¢ W` (ν,k00) dr r β D(r )φ0(r ) 2 j` 1 (νr ) j` 1 k00r = k00 dz c (2` 1) − − + 2 ¸ `(` 1) © ¡ ¢ ¡ ¢ª (` 1) ¡ ¢ + j` 1 (νr˜) j` 1 k00r j` 1 (νr˜) j` 1 k00r + j` 1 (νr˜) j` 1 k00r − (2` 1)2 − + + + − + (2` 1)2 + + + + (2.50) Z µ ¶ 2 ν dφ0 dr˜ H(z) ` ¡ ¢ (` 1) ¡ ¢ dr r β D(r ) j` (νr˜) j` 1 k00r + j` (νr˜) j` 1 k00r . + k dr˜ dz c (2` 1) − − (2` 1) + 00 + +

Since the selection function transforms as a scalar, φ0 (r˜) is related to the selection function in comoving distance through φ0 (r˜(z)) φ(r (z)). = Any measurement of the SFB power spectrum will be affected by shot noise. The number of galaxies Ngal observed in a given survey is independent of the distance-redshift relation of choice i.e. Z Z 3 3 N d r˜ φ0 (r˜)n0 (r˜) d r φ(r )n (r ), (2.51) gal = = where n (r ) is the galaxy volume density in comoving coordinates and n0 (r˜) is the volume density in terms of r˜. Together with the identity φ0 (r˜(z)) φ(r (z)), this implies = ¯ 3 ¯ ¯d r¯ n0 (r˜) ¯ ¯n (r ). (2.52) = ¯d3r˜ ¯

For a generic distance-redshift relation the shot-noise therefore reduces to

Z Z ¯ ¯ ¡ ¢ 2νν0 2 1 ¡ ¢ 2νν0 2 ¯dr˜ ¯ 1 ¡ ¢ N` ν,ν0 dr r φ0 (r˜) j` (νr˜) j` ν0r˜ dr r˜ ¯ ¯φ(r ) j` (νr˜) j` ν0r˜ . = π n0 (r˜) = π ¯dr ¯ n (r ) (2.53)

2.C Derivation of the Fisher matrix for the SFB power spectrum

2.C.1 Full Fisher matrix

The FM obtained from a data likelihood with covariance matrix C and mean µ is given by [270]

· ¸ T 1 1 ∂C 1 ∂C ∂µ 1 ∂µ Fαβ Tr C− C− C− . (2.54) = 2 ∂θα ∂θβ + 〈 ∂θα 〉 〈∂θβ 〉

48 2.C. Derivation of the Fisher matrix for the SFB power spectrum

] 4 ] 4

3 10 3 10 Mpc) Mpc) 1 1 − − h h ) [( ) [( k, k k, k

( 3 ( 3 0 0 l 10 l 10 C C

Cl0(k, k, l = 2) Cl0(k, k, l = 2)

Cl0(k, k, l = 10) Cl0(k, k, l = 10)

Cl0(k, k, l = 50) Cl0(k, k, l = 50) 2 2 10 3 2 1 10 3 2 1 10− 10− 10− 10− 10− 10− 1 1 k [hMpc− ] k [hMpc− ]

(a) Logarithmic distance-redshift relation (b) Linear distance-redshift relation

Figure 2.6: The SFB auto power spectrum C`0 (k,k) for two different distance redshift relation as a function of angular scale `.

Assuming a measurement of the SFB coefﬁcients of the matter overdensity ﬁeld δ`m(k) for a discrete set of radial wave vectors k denoted (k ,k , ,k ), the data covariance matrix is 1 2 ··· n given by

Cˆ 0 0 0 0  `1 ···  ......   ......   . . . .   ˆ   0 C`1 0 0  Cˆ  ··· ··· , (2.55) =  0 0 0 Cˆ 0   `2 ···   ......   ......   . . . . .  0 0 0 0 Cˆ ··· `max where the sub-covariance matrices are deﬁned as

C˜ (k ,k ) C˜ (k ,k ) C˜ (k ,k ) ì 1 1 ì 1 2 ··· ì 1 n C˜ (k2,k1) C˜ (k2,k2) C˜ (k2,kn) ˆ  ì ì ··· ì  Cì  . . . . (2.56) =  . . .. .   . . . .  C˜ (k ,k ) C˜ (k ,k ) C˜ (k ,k ) ì n 1 ì n 2 ··· ì n n Since Cˆ is blockdiagonal and µ δ (k ) 0, Eq. 2.54 yields = 〈 `m i 〉 = 2 1 · Cˆ Cˆ ¸ X ( ` )∆` ˆ 1 ∂ ` ˆ 1 ∂ ` Fαβ fsky + Tr C`− C`− , (2.57) = ` 2 ∂θα ∂θβ where the sum is over bands of width ∆` in the power spectrum.

49 Chapter 2. Three-dimensional spherical analyses of cosmological spectroscopic surveys

2.C.2 Diagonal Fisher matrix

The computation of the SFB FM through Eq. 2.32 is numerically challenging and it is therefore desirable to investigate possible approximations. Although the amount of cross-correlation

C`(k,k0) between neighboring k vectors can be considerable, it tends rapidly to zero for separated wave vectors. If we assume a broad window function, such that mode coupling can 6 be neglected [82], we can approximate C (k,k0) 0 for k k0 . Assuming a measurement of a ` = 6= set of discrete SFB modes δ (k ) up to ` ` which satisfy δ (k ) 0 and deﬁning `m i ≤ max 〈 `m i 〉 =

δ (k ) δ , (2.58) `m i = `m,i δ (k )δ (k ) C (k ,k ) N (k ,k ) ∆2 , 〈 `m i `m i 〉 = ` i i + ` i i = `,i the data likelihood can be written as

δ2 1 P `m,i ,m,i 1 2 ` ∆2 − `,i L (x;θ) n e . (2.59) = (`max 1)(`max 3) 2 Q 2` 1 (2π) − + `,i ∆`,i+

Applying Eq. 2.28, the FM becomes

X (2` 1)∆` 1 ∂C`(ki ,ki ) ∂C`(ki ,ki ) Fαβ fsky + , (2.60) = 2 (C (k ,k ) N (k ,k ))2 ∂θ ∂θ `,i ` i i + ` i i α β where the sum is over bands of width ∆` in the power spectrum and wave vectors ki . To proceed, we assume that the maximal length scale probed by the survey is given by L. Therefore 2π the minimal measurable mode is k /L which also deﬁnes the k-space resolution. The min = 2π maximal measurable mode is determined by the smallest distance ∆L and given by k /∆L. max = Turning the Riemann sum in Eq 2.60 into a continuous integral yields

k Zmax X (2` 1)∆` Ldk 1 ∂C`(k,k) ∂C`(k,k) + Fαβ fsky 2 . (2.61) = ` 2 2π (C`(k,k) N`(k,k)) ∂θα ∂θβ kmin +

2.D Comparison of FM computation techniques for the SFB power spectrum: Results

The parameter constraints obtained for the SFB power spectrum using both Fisher matrix (FM) computation methods and applying the same wave vector cuts, both neglecting and including shot noise contributions, are shown in Table 2.4. Apart from the constraints on w0 the results agree reasonably well, as mentioned in 2.5.2.

6This approximation is equally justiﬁed when we assume that the SFB power spectrum is computed for radial wave vector bins which are broader than the correlation scale due to ﬁnite survey effects.

50 2.D. Comparison of FM computation techniques for the SFB power spectrum: Results

Table 2.4: Comparison of parameter constraints for the SFB power spectrum obtained using discrete (Eq. 2.32) and continuous (Eq. 2.35) Fisher matrix

Statistic Implementation Shot noise σh σΩm σΩΛ σw0 σns σσ8 Comoving no 0.38 0.043 0.55 1.3 0.38 0.86 full cov. Comoving yes 0.48 0.056 0.68 1.5 0.46 1.1 SFB Comoving no 0.30 0.071 0.57 2.2 0.27 0.66 diag cov. Comoving yes 0.39 0.10 0.77 3.0 0.33 0.85

3 Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing

Zwei Dinge erfüllen das Gemüt mit immer neuer und zunehmender Bewunderung und Ehrfurcht, je öfter und anhaltender sich das Nachdenken damit beschäftigt: Der bestirnte Himmel über mir, und das moralische Gesetz in mir. — Immanuel Kant, Kritik der praktischen Vernunft, Beschluss

This chapter appeared in a similar form as Nicola, Réfrégier & Amara, 2016 [191].

3.1 Introduction

The past two decades have seen immense progress in observational cosmology that has lead to the establishment of the ΛCDM model for cosmology (for details, see Chapter1 and references therein). This development is mainly based on the combination of different cosmological probes such as the CMB temperature anisotropies, galaxy clustering, weak gravitational lensing, supernovae and galaxy clusters. Until now, these probes have been, for the most part, measured and analyzed separately using different techniques and combined at late stages of the analysis, i.e. when deriving constraints on cosmological parameters. However, this approach is not ideal for current and future surveys such as the Dark Energy Survey (DES1), the Dark Energy Spectroscopic Instrument (DESI2), the Large Synoptic Survey Telescope (LSST3), Euclid4 and the Wide Field Infrared Survey Telescope (WFIRST5) for several reasons. First,

1http://www.darkenergysurvey.org. 2http://desi.lbl.gov. 3http://www.lsst.org. 4http://sci.esa.int/euclid/. 5http://wfirst.gsfc.nasa.gov.

53 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing these surveys will cover large, overlapping regions of the observable universe and are therefore not statistically independent. In addition, the analysis of these surveys requires tight control of systematic effects, which might be identified by a direct cross-correlation of the probes statistics. Moreover, each probe provides a measurement of the cosmic structures through a different physical field, such as density, velocity, gravitational potentials, and temperature. A promising way to test for new physics, such as modified gravity, is to look directly for deviations from the expected relationships of the statistics of the different fields. The integrated treatment of the probes from the early stages of the analysis will thus provide the cross-checks and the redundancy needed not only to achieve high-precision but also to challenge the different sectors of the cosmological model.

Several earlier studies have considered joint analyses of various cosmological probes. Mandel- baum et al. [181], Cacciato et al. [43] and Kwan et al. [162] for example derived cosmological constraints from a joint analysis of galaxy-galaxy lensing and galaxy clustering while Liu et al. [175] used the cross-correlation between the galaxy shear field and the overdensity field together with the cross-correlation of the galaxy overdensity with CMB lensing to constrain multiplicative bias in the weak lensing shear measurement in CFHTLenS. Recently, Singh et al. [248] performed a joint analysis of CMB lensing as well as galaxy clustering and weak lensing. Furthermore, Eifler et al. [76] and Krause et al. [159] have theoretically investigated joint analyses for photometric galaxy surveys by modeling the full non-Gaussian covariance matrix between cosmic shear, galaxy-galaxy lensing, galaxy clustering, photometric baryon acoustic oscillations (BAO), galaxy cluster number counts and galaxy cluster weak lensing.

Extending beyond this, we present and implement an integrated approach to probe combination. In this first implementation we combine data from CMB temperature anisotropies, galaxy overdensities and weak lensing. We use data from Planck 2015 [209] for the CMB, for galaxy clustering we use photometric data from the 8th data release of the Sloan Digital Sky Survey (SDSS DR 8) [13] and the weak lensing shear data comes from SDSS Stripe 82 [20]. We combine these probes into a common framework at the map level by creating projected 2-dimensional maps of CMB temperature, galaxy overdensity and the weak lensing shear field. In order to jointly analyze this set of maps we consider the spherical harmonic power spectra of the probes including their cross-correlations. This leads to a spherical harmonic power spectrum matrix that combines CMB temperature anisotropies, galaxy clustering, cosmic shear, galaxy-galaxy lensing and the ISW [232] effect with galaxy and weak lensing shear tracers. We combine this power spectrum matrix together with the full Gaussian covariance matrix and derive constraints on the parameters of the ΛCDM cosmological model, marginalizing over a constant linear galaxy bias and a parameter accounting for possible multiplicative bias in the weak lensing shear measurement. In this first implementation, we use some conservative and simplifying assumptions. For instance we include a limited range of angular scales for the different probes to reduce our sensitivity to systematics, nuisance parameters and nonlinear corrections. With this, we work under the assumption of Gaussian covariance matrices and with a reduced set of nuisance parameters.

54 3.2. Framework

Data

Maps

M1 M2 MM3 3

Power Spectra ij C`

Cov[Cij,Ci0j0 ] P(M|D) Theory ` `0

Ωm h

σ8 ΩΛ

Figure 3.1: Synopsis of the framework for integrated probe combination employed in this work.

This chapter is organized as follows. In Section 3.2 we describe the framework for integrated probe combination employed in this work. The theoretical modeling of the cosmological observables is summarized in Section 3.3. Section 3.4 describes the data analysis for each probe, especially the map-making procedure. The computation of the spherical harmonic auto- and cross-power spectra is discussed in Section 3.5 and the estimation of the covariance matrix is detailed in Section 3.6. In Section 3.7 we present the cosmological constraints derived from the joint analysis and we conclude in Section 3.8. More detailed descriptions of data analysis as well as robustness tests are deferred to the Appendix.

3.2 Framework

The framework for integrated probe combination employed in this work is illustrated in Fig. 3.1. In a first step we collect data for different cosmological probes as taken by either separate surveys or by the same survey. For our first implementation described below we use cosmological data from the CMB temperature anisotropies, the galaxy overdensity field and

55 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing the weak lensing shear field. After data collection, we perform probe specific data analysis which involves data selection and systematics removal. We then homogenize the data format by creating projected 2-dimensional maps for all probes considered. The common data format allows us to combine the cosmological probes into a common framework at the map level. We compute both the spherical harmonic auto- and cross-power spectra of this set of maps i j and combine them into the spherical harmonic power spectrum matrix C` . This matrix captures the cosmological information contained in the two-point statistics of the maps. In a last step we compute the power spectrum covariance matrix and combine it with theoretical predictions to derive constraints on cosmological parameters from a joint fit to the measured spherical harmonic power spectra. The details of the implementation for CMB temperature anisotropies, galaxy overdensities and weak lensing are described below.

3.3 Theoretical predictions

The statistical properties of both galaxy overdensity δg and weak lensing shear γ, as well as their cross-correlation can be measured from their spherical harmonic power spectra. These generally take the form of weighted integrals of the nonlinear matter power spectrum nl Pδδ(k,z) multiplied with spherical Bessel functions j`(kχ(z)). Their computation is time- consuming and we therefore resort to the the Limber approximation [169, 147, 148] to speed up calculations. This is a valid approximation for small angular scales, typically ` O(10), > and broad redshift bins [200]. For simplicity, we further focus on ﬂat cosmological models, i j i.e. Ω 0, for the theoretical predictions. The spherical harmonic power spectrum C at k = ` multipole ` between cosmological probes i, j {δ ,γ} can then be expressed as: ∈ g

Z i ¡ ¢ j ¡ ¢ µ 1 ¶ i j c W χ(z) W χ(z) ` /2 C dz P nl k + ,z , (3.1) ` = H(z) χ2(z) δδ = χ(z) where c is the speed of light, χ(z) the comoving distance, H(z) the Hubble parameter and i 0 ¡ ¢ W χ(z) denotes the window function for probe i 0.

For galaxy clustering the window function is given by

H(z) W δg ¡χ(z)¢ b(z)n(z), (3.2) = c where b(z) denotes a linear galaxy bias and n(z) is the normalized redshift selection function of the survey i.e. R dz n(z) 1. We focus on scale-independent galaxy bias since we restrict = the analysis to large scales, which are well-described by linear theory.

The window function for weak lensing shear is

2 Z χh 3 ΩmH χ(z) χ(z0) χ(z) γ ¡ ¢ 0 − W χ(z) 2 dz0n(z0) , (3.3) = 2 c a χ(z) χ(z0) where Ωm denotes the matter density parameter today, H0 is the present-day Hubble parame-

56 3.4. Maps

ter, χh is the comoving distance to the horizon and a denotes the scale factor.

Similarly to the spherical harmonic power spectra of galaxy clustering and weak lensing the spherical harmonic power spectrum of CMB temperature anisotropies T can be related to the primordial matter power spectrum generated during inﬂation as [71]

Z ¯ ¯2 2 ¯∆T`(k)¯ C TT dk k2P lin(k)¯ ¯ , (3.4) ` = π δδ ¯ δ(k) ¯ where ∆T` denotes the transfer function of the temperature anisotropies and δ is the matter overdensity.

The CMB temperature anisotropies are correlated to tracers of the large-scale structure (LSS) such as galaxy overdensity and weak lensing shear primarily through the integrated Sachs- Wolfe effect [232]. On large enough scales where linear theory holds, the spherical harmonic power spectra between these probes can be computed from expressions similar to those above. In the Limber approximation [169, 147, 148], the spherical harmonic power spectrum between CMB temperature anisotropies and a tracer i of the LSS becomes: [62]

2 µ 1 ¶ ΩmH TCMB 1 Z d ` /2 C iT 3 0 dz [D(z)(1 z)]D(z)W i ¡χ(z)¢P lin k + ,0 , (3.5) ` 2 1 2 δδ = c (` /2) dz + = χ(z) + where T denotes the mean temperature of the CMB today, i {δ ,γ} and W i ¡χ(z)¢ repre- CMB ∈ g sents the window functions deﬁned in Equations 3.2 and 3.3. We have further split the linear lin matter power spectrum Pδδ (k,z) into its time-dependent part parametrized by the growth lin factor D(z) and the scale-dependent part Pδδ (k,0). For a derivation of Eq. 3.5 for the galaxy γT overdensity ﬁeld as tracer of the LSS see e.g. Padmanabhan et al. [197]. The derivation for C` is similar and is detailed in Appendix 3.A.

To compute the auto-power spectrum of the CMB temperature anisotropies we use the publicly available Boltzmann code CLASS6 [164]. For the other power spectra we use PYCOSMO [221]. We calculate the linear matter power spectrum from the transfer function derived by Eisenstein and Hu [78]. To compute the nonlinear matter power spectrum we use the HALOFIT ﬁtting function [251] with the revisions of Takahashi et al. [267].

3.4 Maps

3.4.1 Cosmic Microwave Background

We use the foreground-reduced CMB anisotropy maps provided by the Planck collaboration [210] in their 2015 data release. We choose these over the uncleaned single-frequency maps because they allow to perform the foreground correction on the maps rather than the power spectrum level. This is important when considering probe combination. The Planck

6http : //class-code.net.

57 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing

Table 3.1: Summary of used data.

CMB Survey: Planck 2015 [210] temperature Fiducial foreground-reduced map: Commander anisotropies Sky coverage: f 0.776 sky = Survey: SDSS DR8 [13] Sky coverage: f 0.27 galaxy sky = Galaxy sample: CMASS1-4 overdensity Number of galaxies: N 854063 gal = Photometric redshift range 0.45 z 0.65 ≤ phot < Survey: SDSS Stripe 82 co-add [20] Sky coverage: f 0.0069 sky = weak lensing Number of galaxies: N 3322915 gal = Photometric redshift range: 0.1 . zphot . 1.1 r.m.s. ellipticity per component: σ 0.43 e ∼

foreground-reduced CMB anisotropy maps have been derived using four different algorithms: Commander, NILC, SEVEM and SMICA. The maps are given in HEALPix7 [99] format and are provided in Galactic coordinates at two different resolutions of NSIDE 1024 and NSIDE = 2048. These correspond to pixel areas of 11.8 and 2.95 arcmin2 respectively. Different data = configurations are available [210]; we use both the half-mission half-sum (HMHS) maps, which contain both signal and noise, and the half-mission half-difference maps (HMHD), which contain only noise and potential residual systematic uncertainties. All four maps yield consistent estimates of both the spherical harmonic power spectrum of the CMB temperature anisotropies as well as the spherical harmonic cross-power spectrum between CMB temperature anisotropies and tracers of the LSS [210, 215, 213]. Since the Planck collaboration found the Commander approach to be the preferred solution for studying the CMB anisotropies at large and intermediate angular scales, we also choose it for our analysis. Each of the four foreground reduction methods also provides a confidence mask inside which the CMB solution is trusted. Following the Planck collaboration [210], we adopt the union of the confidence masks for Commander, SEVEM and SMICA. This is referred to as the UT78 mask and covers 77.6% of the sky at a resolution of NSIDE 2048. To downgrade the mask to NSIDE 1024, we follow = = the description outlined in Planck Collaboration et al. [210]. The HMHS CMB anisotropy map derived using Commander is shown in the top panel of Fig. 3.2 for resolution NSIDE 1024 and = the corresponding HMHD map is shown in Fig. 3.18 in the Appendix.

7http : //healpix.sourceforge.net.

58 3.4. Maps

31.0 CMB temperature 4 10 ⇥ 4.5 60

120 60 0 60 120

b 33.5 0

60 4.5 36.0 55.5 53.0 50.5 T [K] l

31.0 Galaxy density 10 60

120 60 0 60 120

b 33.5 0

60 1 36.0 55.5 53.0 50.5 g [1] l

31.0 Weak lensing 0.1000 60

120 60 0 60 120

b 33.5 0

60 0.0003 36.0 ˆ [1] 55.5 53.0 50.5 | | l

Figure 3.2: Summary of the three maps in Galactic coordinates used in this analysis. The all-sky maps are in Mollweide projection while the zoom-in versions are in Gnomonic projection. The HMHS map of CMB temperature anisotropies as derived using Commander is shown in the top panel. It is masked using the UT78 mask. The middle panel shows the systematics-corrected (see text) galaxy overdensity map for CMASS1-4 galaxies. Grey areas have been masked either because they lie outside the survey footprint or are potentially contaminated by systematics. The lower panel shows the map of the SDSS Stripe 82 shear modulus γˆ . Grey areas have been | | masked because they are either unobserved or do not contain galaxies for shear measurement. The zoom-in ﬁgures (left) are enlarged versions of the 5 5 deg2 region centered on (l,b) × (53°, 33.5°) shown in the maps. The zoom-in for the galaxy shear map is overlaid with a = − whisker plot of the galaxy shears. All three maps have resolution NSIDE 1024. =

59 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing

3.4.2 Galaxy overdensity

The SDSS [283, 79, 102, 103] obtained wide-ﬁeld images of 14555 deg2 of the sky in 5 photometric passbands (u,g,r,i,z [89, 250, 72]) up to a limiting r-band magnitude of r 22.5. The ' photometric data is complemented with spectroscopic data from the Baryonic Oscillations Spectroscopic Survey (BOSS) [79, 65, 249]. BOSS was conducted as part of SDSS III [79] and obtained spectra of approximately 1.5 million luminous galaxies distributed over 10000 deg2 of the sky. The SDSS photometric redshifts for DR8 [13] are estimated using a local regression model trained on a spectroscopic training set consisting of 850000 SDSS DR8 spectra and spectroscopic data from other surveys8. The algorithm is outlined in Beck et al. [28].

In our analysis, which is described in the following, we largely follow Ho et al. [129]. We select objects classified as galaxies from the PHOTOPRIMARY table in the Catalog Archive Server (CAS9). To obtain a homogeneous galaxy sample we further select CMASS galaxies using the color-magnitude cuts used for BOSS target selection [79] and outlined in Ho et al. [129]. This selection isolates luminous, high-redshift galaxies that are approximately stellar mass limited [280, 229]. We further restrict the sample to CMASS galaxies with SDSS photometric redshifts between 0.45 z 0.65, i.e. we consider the photometric redshift slices CMASS1-4. This ≤ < selection yields a total of N 1096455 galaxies. gal = To compute the galaxy overdensity field, we need to characterize the full area observed by the survey and mask regions heavily affected by foregrounds or potential systematics. The area imaged by the SDSS is divided into units called fields. Several such fields have been observed multiple times in the SDSS imaging runs. The survey footprint is the union of the best observed (primary) fields at each position and is described in terms of MANGLE [106, 107, 263] spherical polygons. Each of these polygons is matched to the SDSS field fully covering it10. In order to select the survey area least affected by foregrounds and potential systematics we follow Ho et al. [129] and Ross et al. [229] and restrict the analysis to polygons covered by fields with score11 0.6, full width at half maximum (FWHM) of the point spread function (PSF) ≥ PSF-FWHM 2.0 arcsec in the r-band and Galactic extinction E(B V) 0.08 as determined < − ≤ from the extinction maps from Schlegel et al. [235].

To facilitate a joint analysis between the LSS probes and the CMB, which is given as a map in Galactic coordinates, we transform both the galaxy positions as well as the survey mask from equatorial (RA, DEC) to Galactic (l, b) coordinates. We construct the continuous galaxy overdensity field by pixelizing the galaxy overdensities δ δn/n¯ onto a HEALPix pixelization g = of the sphere with resolution NSIDE 1024. We mask the galaxy overdensity map with a = HEALPix version of the SDSS survey mask, which is obtained by random sampling of the MANGLE mask. To account for the effect of bright stars, we use the Tycho astrometric catalog [131] and define magnitude-dependent stellar masks as defined in Padmanabhan et al. [198].

8More details can be found on http://www.sdss3.org/dr8/algorithms/photo-z.php. 9The SDSS Catalog Archive Server can be accessed through http://skyserver.sdss.org/CasJobs/SubmitJob.aspx. 10This information is found in the ﬁles window_unified.fits and window_flist.fits. 11http://www.sdss3.org/dr10/algorithms/resolve.php.

60 3.4. Maps

2.0 W γ(χ(z)) γ 1.5 n (z) nδg (z) ) z

( 1.0 n

0.5

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 z

Figure 3.3: Redshift distribution for the LSS probes. The ﬁgure shows the redshift selection function of SDSS CMASS1-4 galaxies, the redshift selection function for the SDSS Stripe 82 galaxies as well as the weak lensing shear window function deﬁned in Eq. 3.3. The redshift selection function for CMASS1-4 galaxies as well as the weak lensing shear window function have been rescaled relative to the Stripe 82 redshift selection function.

We remove galaxies inside the bright star masks and correct for the area covered by the bright stars by removing the area covered by the star masks from the pixel area A pix,corr = A A when computing the galaxy overdensity. The ﬁnal map covers a fraction pix,uncorr − stars f 0.27 of the sky and contains N 854063 galaxies. sky ≈ gal = Even after masking and removal of high contamination regions, there are still systematics left in the galaxy overdensity map. The correction for residual systematic uncertainties in the maps follows Ross et al. [229] and Hernández-Monteagudo et al. [119] and is described in Appendix 3.B. The ﬁnal map is shown in the middle panel of Fig. 3.2.

As well as the maps we need an estimate for the redshift distribution of the galaxies in our sample. To this end we follow Ho et al. [129] and match photometrically detected galaxies to galaxies observed spectroscopically in SDSS DR9 [12]. We then estimate the redshift distribution of the photometric galaxies from the spectroscopic redshift distribution of the matching galaxies. The selected CMASS1-4 galaxies have spectroscopic redshifts 0.4 . z . 0.7 as can be seen from the redshift distribution shown in Fig. 3.3.

3.4.3 Weak lensing

We take weak lensing data from the SDSS Stripe 82 co-add [20], which comprises 275 deg2 of co-added SDSS imaging data with a limiting r-band magnitude r 23.5 and r-band median ≈ seeing of 1.1 arcsec. The shapes of objects detected in the SDSS were measured from the

61 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing adaptive moments [32] by the PHOTO pipeline [176] and are available on the CAS12. Photo- metric redshifts for all detected galaxies were computed using a neural network approach as described in Reis et al. [224] and are available as a DR7 value added catalog13.

In the following analysis we closely follow the work by Lin et al. [170]. We select objects identified as galaxies in the co-add data (i.e. run 106 or run 206) from the CAS and = = we restrict the sample to galaxies with extinction corrected i-band magnitudes in the range 18 i 24. Further we select only objects that pass the clean photometry cuts as defined < < by the SDSS14 and do not have flags indicating problems with the measurement of adaptive moments as well as negative errors on those. The former cuts especially exclude galaxies containing saturated pixels. We use shapes measured in the i-band since it has the smallest seeing (1.05 arcsec) [20, 170] and further consider only galaxies with observed sizes at least 50% larger than the PSF. This requirement is quantified by requiring the resolution factor R 1 mRrCcPSF/mRrCc [32] to satisfy R 0.33, where mRrCc and mRrCcPSF denote the sum = − > of the second order moments in the CCD column and row direction for both the object and the PSF.

For the above galaxy sample we compute PSF-corrected galaxy ellipticities using the linear PSF correction algorithm as described in Hirata and Seljak [125]. For weak lensing shear measurement we follow Lin et al. [170] and restrict the sample to galaxies with PSF-corrected ellipticity components e ,e satisfying e 1.4 as well as e 1.4 and photometric redshift 1 2 | 1| < | 2| < uncertainties σ 0.15. This sample has an r.m.s. ellipticity per component of σ 0.43. We z < e ∼ then turn the PSF-corrected ellipticities for this sample into shear estimates. The details of the analysis are described in Appendix 3.C.

After computing weak lensing shear estimates from the ellipticities we apply a rotation to both the galaxy positions and shears from equatorial to Galactic coordinates15 to allow for combination with the CMB. We pixelize both weak lensing shear components onto separate HEALPix pixelizations of the sphere choosing a resolution of NSIDE 1024 as for the galaxy = overdensity map. At this resolution the mean number of galaxies per pixel is about 38, which 2 corresponds to n 3.2 arcmin− . We apply a mask to both maps, which accounts for both gal ' unobserved and empty pixels. The ﬁnal maps are constructed using N 3322915 galaxies gal = and cover a sky fraction f 0.0069. The map of the shear modulus γˆ is shown in the sky ≈ | | bottom panel of Fig. 3.2 together with a zoom-in region with overlaid whisker plot illustrating the magnitude and direction of the weak lensing shear.

We follow Lin et al. [170] and estimate the redshift distribution of the galaxies from their photometric redshift distribution. The redshift distribution is shown in Fig. 3.3 together with the window function for weak lensing shear deﬁned in Eq. 3.3. We see that the selected galaxies have photometric redshifts z . 1.0.

12See footnote9. 13http : //classic.sdss.org/dr7/products/value_added/, http : //das.sdss.org/va/coadd_galaxies/. 14http://www.sdss.org/dr12/tutorials/flags/. 15The exact rotation of the shears is described in Appendix 3.D.

62 3.5. Spherical harmonic power spectra

3.5 Spherical harmonic power spectra

We calculate the spherical harmonic power spectra of the maps presented in the previous section using the publicly available code PolSpice16 [265, 57]. The PolSpice code is designed to combine both real and Fourier space in order to correct spherical harmonic power spectra measured on a cut-sky from the effect of the mask. The algorithm can be summarized as follows: starting from a masked HEALPix map, PolSpice ﬁrst computes the so-called pseudo power spectrum, which is then Fourier transformed to the real space correlation function. In order to correct for the effects of the mask, the latter is divided by the mask correlation function. In a last step, the demasked correlation function is Fourier transformed back to the spherical harmonic power spectrum. This approach ensures that PolSpice can exploit the advantages of real space while still performing the computationally expensive calculations in Fourier space.

Demasking can only be performed on angular scales on which information is available, which translates to a maximal angular scale θmax for which a demasked correlation function can be computed. This maximal scale leads to ringing when transforming back from real to Fourier space, which can be reduced by apodizing the correlation function prior to inversion. Both these steps lead to biases in the power spectrum recovered by PolSpice. The kernels relating the average PolSpice estimates to the true power spectra can be computed theoretically for a given maximal angular scale and apodization prescription and need to be corrected for when comparing theoretical predictions to observed power spectra.

An additional difﬁculty arises in the computation of spherical harmonic power spectra of spin- 2 ﬁelds. Finite sky coverage tends to cause mixing between E-and B-modes. The polarization version of PolSpice is designed to remove E- to B-mode leakage in the mean [57]. Details on our earlier application of PolSpice to LSS data are described in Appendix A of Becker et al. [29].

In order to calculate both the auto- and cross-power spectra for all probes, we need to estimate the maximal angular scale θmax. This is not a well-deﬁned quantity but we can separately estimate it for each probe from the real space correlation function of its mask. The real space correlation function of the survey mask will fall off signiﬁcantly or vanish for scales larger than

θmax. We therefore estimate θmax as the scale around which the mask correlation function signiﬁcantly decreases in amplitude. Appendix 3.E illustrates this analysis for the example of the SDSS Stripe 82 weak lensing shear mask. In order to reduce Fourier ringing we apodize the correlation function using a Gaussian window function; following Chon et al. [57] we choose

θ the FWHM of the Gaussian window as θ max/2. Survey masks with complicated angular FWHM = dependence might not exhibit a clear fall-off, which complicates the choice of θmax. We therefore validate our choices of θmax and θFWHM with the Gaussian simulations as described in Appendix 3.F and 3.G. We ﬁnd our choices to allow the recovery of the input power spectra for all the probes and settings.

16http://www2.iap.fr/users/hivon/software/PolSpice/.

63 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing

Table 3.2: Spherical harmonic power spectrum parameters and angular multipole ranges.

Power spectrum θmax [deg] θFWHM [deg] `-range ∆` TT C` 40 20 [10, 610] 30 δg δg C` 80 40 [30, 210] 30 γγ C` 10 5 [70, 610] 60 δg T C` 40 20 [30, 210] 30 γT C` 10 5 [70, 610] 60 γδg C` 10 5 [30, 210] 60

All spherical harmonic power spectra are corrected for the effect of the HEALPix pixel window function and the power spectra involving the CMB map are further corrected for the Planck effective beam window function, which complements the CMB maps.

We now separately describe the measurement of all the six spherical harmonic power spectra. To compute the power spectra, we use the maps and masks described in Section 3.4 at resolution NSIDE 1024, except for the CMB temperature power spectrum. For the latter = we use the maps at resolution NSIDE 2048, but we do not expect this to make a signiﬁcant = difference. The PolSpice parameter settings used to compute the power spectra are summarized in Tab. 3.2. This table further gives the angular multipole range as well as binning scheme employed for the cosmological analysis. For all probes considered, the uncertainties are derived from the Gaussian simulations described in Section 3.6.2 and Appendix 3.F.

3.5.1 CMB

We use the half-mission half-sum (HMHS) map to estimate the CMB signal power spectrum and the half-mission half-difference (HMHD) map to estimate the noise in the power spectrum of the HMHS map.

The minimal angular multipole used in the cosmological analysis is chosen such as to minimize demasking effects and the cut at ` 610 ensures that we are not biased by residual = foregrounds in the maps as discussed in Section 3.7. The resulting power spectrum is shown in the top panel of Fig. 3.4. In Appendix 3.H we compare the CMB auto-power spectrum computed from the different foreground-reduced maps. As illustrated in Fig. 3.16 in Appendix 3.H we ﬁnd that the measured CMB auto-power spectrum is unaffected by the choice of foreground-reduced map.

64 3.5. Spherical harmonic power spectra

6000 ] 2 TT K 5000 µ )[

π 4000 (2 / 3000 TT ` C 2000 + 1)

` 1000 ( ` 0 0 100 200 300 400 500 600 7 10− 4 1.0 × 10− 0.8 δgT δgδg 0.6 g [K]

0.4 δ g T δ ` g δ ` 0.2 C `C 0.0

0.2 5 − 10− 0.4 − 50 100 150 200 3 101 102 9 ×5 10− 10− 2.5 × 1.5 × 2.0 γT γδg γγ 1.0 ) 5 1.5 π 10− (2 /

g 0.5 [K] γγ 1.0 ` γδ ` T C γ ` 0.5 `C 0.0 6 `C 10− + 1)

0.0 ` ( 0.5 ` 0.5 − − 7 1.0 1.0 10− − 100 200 300 400 500 600 − 50 100 150 200 102 6 102 × ` ` `

Figure 3.4: Spherical harmonic power spectra for all probes used in the cosmological analysis. The top left panel shows the power spectrum of CMB anisotropies computed from the Commander CMB temperature map at resolution of NSIDE 2048. The middle left panel shows = the cross-power spectrum between CMB temperature anisotropies and galaxy overdensity computed from the systematics-reduced SDSS CMASS1-4 map and the Commander map at resolution NSIDE 1024. The middle right panel shows the spherical harmonic power spectrum = of the galaxy overdensity computed from the systematics-reduced SDSS CMASS1-4 map at NSIDE 1024. The bottom left panel shows the spherical harmonic power spectrum between = CMB temperature anisotropies and weak lensing shear measured from the Commander CMB map and the SDSS Stripe 82 weak lensing maps at resolution NSIDE 1024. The bottom- = middle panel shows the spherical harmonic power spectrum between galaxy overdensity and galaxy weak lensing shear computed from the systematics-reduced SDSS CMASS1-4 map and the SDSS Stripe 82 galaxy weak lensing shear map at resolution NSIDE 1024. The bottom = right panel shows the spherical harmonic power spectrum of cosmic shear E-modes computed from the SDSS Stripe 82 weak lensing shear maps. The angular multipole ranges and binning schemes for all power spectra are summarized in Table 3.2. All power spectra are derived from the maps in Galactic coordinates. The solid lines show the theoretical predictions for the best-ﬁt cosmological model determined from the joint analysis which is summarized in Tab. 3.3. The theoretical predictions have been convolved with the PolSpice kernels as described in Section 3.5. The error bars are derived from the Gaussian simulations described in Section 3.6.2 and Appendix 3.F.

65 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing

3.5.2 Galaxy clustering

The galaxy overdensity maps described in Section 3.4 are estimated from discrete galaxy tracers. Therefore, their spherical harmonic power spectrum receives contributions from the galaxy clustering signal and Poisson shot noise. To estimate the noise power spectrum, we resort to simulations. We generate noise maps by randomizing the positions of all the galaxies in the sample inside the mask. Since this procedure removes all correlations between galaxy positions, the power spectra of these maps will give an estimate of the level of Poisson shot noise present in the data. In order to obtain a robust noise power spectrum, we generate 100 noise maps and estimate the noise power spectrum from the mean of these power spectra.

The spherical harmonic galaxy clustering power spectrum contains significant contributions from nonlinear structure formation at small angular scales. The effects of nonlinear galaxy bias are difficult to model and we therefore restrict our analysis to angular scales for which nonlinear corrections are small. We can estimate the significance of nonlinear effects by comparing the spherical harmonic galaxy clustering power spectrum computed using the nonlinear matter power spectrum as well as the linear matter power spectrum. Since galaxies are more clustered than dark matter this is likely to underestimate the effect. We find that the difference between the two reaches 5% of the power spectrum uncertainties and thus becomes mildly significant at around ` 250. This difference is smaller than the difference max ∼ derived in Ho et al. [129] and de Putter et al. [66] which is likely due to the fact that we consider a single redshift bin and do not split the data into low and high redshifts. In order not to bias our results we choose ` 210 which is comparable to the limit used in Ho et al. [129] and max = de Putter et al. [66]. To determine the minimal angular multipole we follow Ho et al. [129], who determined that the Limber approximation becomes accurate for scales larger than ` 30. = The middle right panel in Fig. 3.4 shows the spherical harmonic galaxy clustering power spectrum computed from the systematics-corrected map in Galactic coordinates. In Appendix 3.H, we compare the spherical harmonic power spectrum derived from the systematics- corrected maps in Galactic and equatorial coordinates. We find small differences at large angular scales, but the effect on the bandpowers considered in this analysis is negligible, as can be seen from Appendix 3.H (Fig. 3.14). To test the procedure for removing systematic uncertainties, we compare the spherical harmonic power spectra before and after correcting the maps for residual systematics. We find that the removal of systematics marginally reduces the clustering amplitude on large scales, which is expected since Galactic foregrounds exhibit significant large scale clustering. Small angular scales on the other hand, are mostly unaffected by the corrections applied. These results are shown in Appendix 3.H (Fig. 3.17).

3.5.3 Cosmic shear

The power spectrum computed from the weak lensing shear maps described in Section 3.4.3 contains contributions from both the cosmic shear signal and the shape noise of the galaxies, which is due to intrinsic galaxy ellipticities. In order to estimate the shape noise power

66 3.5. Spherical harmonic power spectra spectrum we follow the same methodology as for galaxy clustering and resort to simulations. We generate noise-only maps by rotating the shears of all the galaxies in our sample by a random angle. This procedure removes spatial correlations between galaxy shapes. Since the weak lensing shear signal is at least an order of magnitude smaller than the intrinsic galaxy ellipticities, the power spectrum of the randomized map gives an estimate of the shape noise power spectrum. As for galaxy clustering, we compute 100 noise maps and estimate the shape noise power spectrum from the mean of these 100 noise power spectra.

For the cosmological analysis we choose broader multipole bins than for the CMB temperature anisotropies and galaxy clustering since the small sky fraction covered by SDSS Stripe 82 causes the cosmic shear power spectrum to be correlated across a signiﬁcantly larger multipole range. The low and high ` limits are chosen to minimize demasking uncertainties and the impact of nonlinearities in the cosmic shear power spectrum.

The spherical harmonic power spectrum of the weak lensing shear E-mode is displayed in the bottom right panel of Fig. 3.4 and the B-mode power spectrum is shown in the Appendix (Fig. 3.19). We see that the E-mode power spectrum is intrinsically low as compared to the best-ﬁt theory power spectrum. These results are similar to those derived by Lin et al. [170], 0.7 who found a low value of Ωm σ8 for Stripe 82 cosmic shear. As can be seen, we do not detect a signiﬁcant B-mode signal.

When comparing the weak lensing shear E-mode power spectra computed from the maps in Galactic and equatorial coordinates, we ﬁnd discrepancies. These are mainly caused by the correction for additive bias in the weak lensing shears. As described in Appendix 3.C, the PSF-corrected galaxy shears are affected by an additive bias. Following Lin et al. [170], we correct for this bias by subtracting the mean shear of each CCD camera column from the galaxy shears. This correction is performed in equatorial coordinates and ensures that the mean shear vanishes in this coordinate system. When the galaxy positions and shears are rotated from equatorial to Galactic coordinates, this ceases to be true. Therefore the correction for additive bias is coordinate-dependent and it is this effect that causes the main discrepancies between the measured power spectra. Further descriptions of the impact of the additive shear bias correction can be found in Appendix 3.H.1.

The discrepancies between the cosmic shear power spectra measured from maps in Galactic and equatorial coordinates are still within the experimental uncertainties. We therefore choose to correct for the additive shear bias in equatorial coordinates, apply the rotation to the corrected shears and compute the cosmic shear power spectrum from the maps in Galactic coordinates. We note however, that these differences will become signiﬁcant for surveys measuring cosmic shear with higher precision. It is therefore important to develop coordinate-independent methods for shear bias correction when performing a joint analysis of different cosmological probes.

67 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing

3.5.4 CMB and galaxy overdensity cross-correlation

To compute the spherical harmonic cross-power spectrum between CMB temperature anisotropies and the galaxy overdensity, we use the maps and masks described in Sections 3.4.1 and 3.4.2.

We generally have two possibilities to compute cross-correlations between two maps with different angular masks. We can either compute the cross-correlation by keeping the respective mask for each probe, or we can compute a combined mask, which is the union of all pixels masked in at least one of the maps. When testing both these cases on Gaussian simulations, we observed a better recovery of the input power spectra when applying the combined mask to both maps. We therefore mask both maps with the combined mask, which covers a fraction of sky f 0.26. sky ∼ The spherical harmonic cross-power spectrum between CMB temperature anisotropies and galaxy overdensity is shown in the middle left panel of Fig. 3.4. We see that the ISW power spectrum is very noisy, which makes its detection significance small. Since the power spectrum uncertainties for the considered angular scales are mainly due to cosmic variance, we suspect that the low signal-to-noise is mainly due to the fraction of sky covered by the SDSS CMASS1-4 galaxies. Despite its low significance, we include the ISW power spectrum in our analysis, because we expect it to help break degeneracies between cosmological parameters. We check that the ISW power spectrum does not depend on the choice of foreground-reduced CMB map. We find that the results using the maps provided by the NILC, SEVEM and SMICA algorithms are virtually the same, as illustrated in Appendix 3.H (Fig. 3.16).

3.5.5 CMB and weak lensing shear cross-correlation

We estimate the spherical harmonic cross-power spectrum between CMB temperature anisotropies and the weak lensing shear E-mode ﬁeld from the maps and masks described in Sections 3.4.1 and 3.4.3. Both maps are masked with the combination of the masks, which covers a fraction of sky f 0.0065. sky ∼ The bottom left panel in Fig. 3.4 shows the spherical harmonic power spectrum between CMB temperature anisotropies and the weak lensing shear E-mode ﬁeld. As can be seen, the noise level is too high to allow for a detection of the ISW correlation between CMB temperature anisotropies and weak lensing shear. This is to be expected due to the small sky fraction covered by the SDSS Stripe 82 galaxies and the intrinsically low signal-to-noise of this cross- correlation. Nevertheless, we include the power spectrum into the joint analysis to provide an upper limit on the ISW from weak lensing. The measured power spectrum is unaffected by the choice of CMB mapmaking method, as illustrated in Fig. 3.16 in Appendix 3.H.

68 3.6. Covariance matrix

3.5.6 Galaxy overdensity and weak lensing shear cross-correlation

We compute the spherical harmonic cross-power spectrum between the galaxy overdensity and weak lensing shear E-mode field from the maps and masks described in Sections 3.4.2 and 3.4.3. We mask both maps with the combination of the two masks. The combined mask covers a sky fraction f 0.0053. sky ∼ The spherical harmonic cross-power spectrum between galaxy overdensity and weak lensing shear E-mode is shown in the bottom-middle panel of Fig. 3.4. We see that the signal-to-noise of the power spectrum is low at the angular scales considered. This is probably due to the small sky fraction covered by Stripe 82 galaxies. We nevertheless include this cross-correlation in our analysis to serve as an upper limit. In Appendix 3.H we show the comparison between the power spectra measured from the maps in Galactic and in equatorial coordinates. We find reasonable agreement between the two, even though the discrepancies are significantly enhanced compared to the effects on the galaxy overdensity power spectrum. As discussed in Section 3.5.3 this is probably due to the coordinate-dependence of the additive shear bias correction.

3.6 Covariance matrix

In order to obtain cosmological constraints from a joint analysis of CMB temperature anisotropies, galaxy clustering and weak lensing we need to estimate the joint covariance matrix of these cosmological probes. In this work we assume all the fields to be Gaussian random fields, i.e. we assume the covariance between all probes to be Gaussian and neglect any non-Gaussian contribution. This is appropriate for the CMB temperature field as well as the galaxy overdensity field at the scales considered but it is only an approximation for the weak lensing shear field [233]. For example, for a survey with source redshifts z 0.6, Sato et al. [233] s = found that neglecting non-Gaussian contributions leads to an underestimation of the diagonal terms in the cosmic shear covariance matrix by a factor of approximately 5 at multipoles ` 600. In our case the discrepancy may be more pronounced since our sample contains a ∼ significant number of galaxies with z 0.6. On the other hand we will be less sensitive to s < the non-Gaussian nature of the covariance matrix since the covariance for our galaxy sample is dominated by shape noise especially at the highest multipoles considered. We therefore decide to leave the introduction of non-Gaussian covariance matrices to future work.

In this work, we employ two different models for the joint Gaussian covariance matrix CG : the ﬁrst is a theoretical model and the second is based on simulations of correlated Gaussian realizations of the three cosmological probes. We use the theoretical covariance matrix to validate the covariance matrix obtained from the simulations.

69 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing

3.6.1 Theoretical covariance estimate

The covariance between cosmological spherical harmonic power spectra is composed of two parts: cosmic variance and noise. For spherical harmonic power spectra computed over the full sky, different ` modes are uncorrelated and the covariance matrix is diagonal. Partial sky coverage, i.e. f 1, has the effect to couple different ` modes and thus leads to a non- sky < diagonal covariance matrix. This covariance becomes approximately diagonal if it is binned into approximately uncorrelated bandpowers of width ∆` [42]. Cabré et al. [42] found the empirical relation ∆`fsky 2 to be a good approximation. In this case the covariance matrix ∼ i j i j between binned power spectra C and C 0 0 can be approximated as [133,42,76] ` `0 h i j i 0 j 0 i j i 0 j 0 δ``0 ii 0 ii 0 j j 0 j j 0 CovG (C ,C ) ∆C ∆C (C N )(C N ) ` `0 = 〈 ` `0 〉' (2` 1)∆`f ` + ` + + sky (3.6) i j ji i (C 0 N i j 0 )(C 0 N ji 0 ) , + ` + ` + where i, j, i 0, j 0 denote different cosmological probes; in our case i, j, i 0, j 0 {T,δ ,γ}. The ∈ g quantities N i j are the noise power spectra of the different probes, which vanish unless i j. = Given a cosmological model and survey speciﬁcations such as fractional sky coverage and noise level, we can approximate CG using Eq. 3.6 for each block covariance matrix. We choose a hybrid approach: we adopt a cosmological model to calculate the signal power spectra whereas we approximate N i j with the measured noise power spectra used to remove the noise bias in the data as described in Section 3.5.

3.6.2 Covariance estimate from Gaussian simulations

The theoretical covariance matrix estimate described above is expected to only yield accurate results for uncorrelated binned power spectra, since in this approximation the covariance matrix is fully diagonal. For this reason we also estimate the covariance matrix in an alternative way that does not rely on this approximation: we estimate an empirical covariance matrix from the sample variance of Gaussian simulations of the three cosmological probes. To this end, we simulate correlated realizations of both the two spin-0 fields, CMB temperature and galaxy overdensity, as well as the spin-2 weak lensing shear field. We follow the approach outlined in Giannantonio et al. [93] for simulating correlated maps of spin-0 fields and we make use of the polarization version of the HEALPix routine synfast to additionally simulate correlated maps of the spin-2 field. We estimate noise maps from the data and add these to the correlated signal maps. The details of the algorithm are outlined in Appendix 3.F.

In order to compute the power spectrum covariance matrix, we apply the masks used on the data to the simulated maps and calculate both the auto- and the cross-power spectra of all the probes using the same methodology and PolSpice settings as described in Section 3.5. We

70 3.7. Cosmological constraints

TT Tδg δgδg Tγ δgγ γγ 610

0.90 γγ 0.45 0.15 70 210 δ γ 30 g 610

Tγ

70 210 δgδg 0

` 10 210 Tδg 0.00 10 610

0.15 10 − 10 610 210 210 610 210 610 10 10 70 30 70 `

Figure 3.5: Correlation matrix for the spherical harmonic power spectra derived from the sample variance of the Gaussian simulations. The binning scheme and angular multipole range for each probe follow those outlined in Tab. 3.2.

generate Nsim random realizations and estimate the covariance matrix as

Nsim i j i 0 j 0 1 X h i j i j ih i 0 j 0 i 0 j 0 i CovG (C ,C ) C (`) C¯ (`) C (`0) C¯ (`0) , (3.7) ` `0 k k k k = Nsim 1 k 1 − − − = ¯i j where Ck (`) denotes the mean over all realizations.

The accuracy of the sample covariance estimate depends on the number of simulations. As described in Cabré et al. [42], N 1000 achieves better than 5% accuracy for estimating sim = the covariance matrix for the ISW effect from Gaussian simulations. We therefore follow Cabré et al. [42] and compute the covariance matrix from the sample variance of N 1000 sim = Gaussian realizations of the 4 maps or 6 spherical harmonic power spectra respectively.

The correlation matrix for the spherical harmonic power spectra derived from the Gaussian simulations for binning schemes and angular multipole ranges described in Section 3.5 is shown in Fig. 3.5. We see that the survey masks lead to signiﬁcant correlations between bandpowers.

3.7 Cosmological constraints

Each of the power spectra presented in Section 3.5 carries cosmological information with probe-speciﬁc sensitivities and degeneracies. An integrated combination of these cosmologi-

71 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing cal probes therefore helps break these parameter degeneracies. It further provides robust cosmological constraints since it is derived from a joint ﬁt to the auto- as well as cross-correlations of three cosmological probes.

In order to derive cosmological constraints from a joint ﬁt to the six spherical harmonic power spectra discussed in Section 3.5, we assume the joint likelihood to be Gaussian, i.e.

1 1 obs theor T 1 obs theor 2 (C` C` ) CG− (C` C` ) L (D θ) 1 e− − − , (3.8) d /2 | = [(2π) detCG ]

theor where CG denotes the Gaussian covariance matrix. C` denotes the theoretical prediction obs for the spherical harmonic power spectrum vector of dimension d and C` is the observed power spectrum vector, deﬁned as

³ δ T δ δ T γδ ´ Cobs C TT C g C g g C γ C g C γγ . (3.9) ` = ` ` ` ` ` ` obs A Gaussian likelihood is a justified assumption for both the CMB temperature anisotropy and galaxy clustering power spectra due to the central limit theorem. Since the weak lensing shear power spectrum receives significant contribution from non-linear structure formation, its likelihood will deviate from being purely Gaussian [110]. It has been shown however, that a Gaussian likelihood is a sensible approximation, especially when CMB data is added to weak lensing [234]. In our first implementation we will thus assume both a joint Gaussian likelihood and Gaussian single probe likelihoods.

We estimate the covariance matrix using both methods outlined in Section 3.6. In both cases we compute the covariance for a ΛCDM cosmological model, which we keep fixed in the joint fit. Note that the covariance matrices depend on the cosmological model and should therefore vary in the fitting procedure [75]. Following standard practice, (e.g. [8]), we approximate the covariance matrix to be constant and compute it for a ΛCDM cosmological model with parameter values {h, Ω , Ω , n , σ , τ , T } {0.7, 0.3, 0.049, 1.0, 0.88, 0.078, 2.275K}, m b s 8 reion CMB = where h is the dimensionless Hubble parameter, Ωm is the fractional matter density today, Ωb is the fractional baryon density today, ns denotes the scalar spectral index, σ8 is the r.m.s. of 1 matter fluctuations in spheres of comoving radius 8h− Mpc and τreion denotes the optical depth to reionization. We further set the linear, redshift-independent galaxy bias parameter to b 2. To obtain an unbiased estimate of the inverse of the covariance matrix derived from the = Gaussian simulations, we apply the correction derived in Kaufman [150], Hartlap et al. [109] and Anderson [19], i.e. we multiply the inverse covariance matrix by (N d 2)/(N 1). sim − − sim − The theoretical covariance matrix estimate does not suffer from this bias and is thus left unchanged.

From the likelihood given in Eq. 3.8, we derive constraints in the framework of a ﬂat ΛCDM cosmological model, where our ﬁducial model includes one massive neutrino eigenstate of mass

0.06 eV as in [212]. Our parameter set consists of the six ΛCDM parameters {h, Ωm, Ωb, ns, σ8, τreion}. We further marginalize over two additional parameters: a redshift independent, linear galaxy

72 3.7. Cosmological constraints

Table 3.3: Parameters varied in the MCMC with their respective priors and posterior means. The uncertainties denote the 68% c.l..

Parameter Prior Posterior mean h flat [0.2, 1.2] 0.699 0.018 ∈ ± Ω flat [0.1, 0.7] 0.278 0.019 m ∈ +0.020 Ω flat [0.01, 0.09] 0.0455 −0.0018 b ∈ ± n flat [0.1, 1.8] 0.975 0.019 s ∈ +0.018 σ flat [0.4, 1.5] 0.799 −0.029 8 ∈ ± τ Gaussian with µ 0.089, σ 0.02a 0.0792 0.0196 reion = = ± b flat [1., 3.] 2.13 0.06 ∈ ± 0.080 m Gaussian with µ 0.0, σ 0.1 0.142+0.081 = = − − a This corresponds to a WMAP9 [124] prior with increased variance to accommodate the Planck results. bias parameter b and a multiplicative bias parameter m for the weak lensing shear. The multiplicative bias parametrizes unaccounted calibration uncertainties affecting the weak lensing shear estimator γˆ and is defined as [120]

γˆ (1 m)γ. (3.10) = + We note that we do not include additional nuisance parameters such as additive weak lensing shear bias, stochastic and scale-dependent galaxy bias [269, 202, 68], photometric redshift uncertainties, intrinsic galaxy alignments (for reviews, see e.g. [273, 141]) or parameters describing the effect of unresolved point sources on the CMB temperature anisotropy power spectrum [208]. In this present work we restrict the analysis to angular scales where these effects are expected to be subdominant.

We sample the parameter space with a Monte Carlo Markov Chain (MCMC) using CosmoHammer [15]. The parameters sampled are summarized in Table 3.3 along with their priors. We choose

ﬂat, uniform priors for all parameters except for τreion and m. The optical depth to reionization can only be constrained with CMB polarization data. Since we do not include CMB polarization in this analysis, we apply a Gaussian prior with µ 0.089 and σ 0.02 on τ . = = reion This corresponds to a WMAP9 [124] prior with increased variance to accommodate the Planck 2015 results [212]. We further apply a Gaussian prior on the multiplicative bias m with mean µ 0 and σ 0.1. This is motivated by Hirata and Seljak [125], who found the multiplicative = = bias for the linear PSF correction method to lie in the range m [ 0.08,0.13] for the sample ∈ − considered in this analysis.

In our fiducial configuration presented below we use the covariance matrix derived from the Gaussian simulations as described in Section 3.6.2. We find that this choice does not influence our results since the constraints derived using the theoretical covariance are consistent. In order to further assess the impact of a cosmology-dependent covariance matrix, we perform

73 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing the equivalent analysis using a covariance matrix computed with a cosmological model with 5% lower σ . We ﬁnd that the derived parameter values change by at most 0.5σ. The width ∼ 8 of the contours is only marginally changed.

In addition to the joint analysis, we also derive parameter constraints from separate analyses of TT δg δg γγ the three auto-power spectra C` ,C` and C` . In all three cases we assume a Gaussian likelihood as in Eq. 3.8 and derive constraints on the base ΛCDM parameters {h, Ωm, Ωb, ns, σ8} as well as additional parameters constrained by each probe. These are τreion for the CMB temperature anisotropies, b for galaxy clustering and m for the cosmic shear.

Fig. 3.6 shows the constraints on the ΛCDM parameters {h, Ωm, Ωb, ns, σ8} derived from the joint analysis using the spherical harmonic power spectrum vector and likelihood deﬁned in

Equations 3.8 and 3.9. These have been marginalized over τreion, b and m. Also shown are TT δg δg the constraints derived from separate analyses of the three auto-power spectra C` ,C` γγ and C` , each of them marginalized over the respective nuisance parameter. As expected, we find that the constraints derived from the CMB anisotropies are the strongest, followed by the galaxy clustering and cosmic shear constraints, which both constrain the full ΛCDM model rather weakly. The constraints from the CMB temperature anisotropies are broader and have central values which differ from those derived in Planck Collaboration et al. [212]. The reason for these discrepancies is the limited angular multipole range ` [10, 610] employed in the ∈ CMB temperature analysis. This causes the CMB posterior to become broader, asymmetric and results in a shift of the parameter means. We have verified that the Planck likelihood and our analysis give consistent results when the latter is restricted to a similar `-range. If on the other hand, we increase the high multipole limit to ` 1000, we find significant differences max = between our analysis and the Planck likelihood. We therefore choose to be conservative and use ` 610 throughout this work. Comparing the single probe constraints to one another max = we see that they agree reasonably well, the only slight discrepancy being the low value of both

Ωm and σ8 derived from the cosmic shear analysis. This is similar to the results derived in Lin et al. [170] even though the values for Ωm and σ8 are even lower in our analysis. However, care must be taken since the amplitude of the cosmic shear auto-power spectrum appears to have a small dependence on the choice of the coordinate system as discussed in Appendix 3.H.

The potential of the joint analysis emerges when the three auto-power spectra are combined together with their three cross-power spectra. Due to the complementarity of the different probes the constraints tighten and the allowed parameter space volume is significantly reduced. This is especially true in our case, since the constraints from CMB temperature anisotropies are broadened due to the restricted multipole range that we employed. Including more CMB data would significantly reduce the impact of adding additional cosmological probes. The numerical values of the best-fit parameters and their 68% confidence limits (c.l.) derived from the joint analysis are given in Tab. 3.3.

Fig. 3.7 compares the constraints derived from the joint analysis to the constraints derived by the Planck Collaboration [212]. We show two versions of the Planck constraints: the

74 3.7. Cosmological constraints

SDSS Stripe 82 weak lensing SDSS DR8 galaxy clustering Planck 2015 CMB, this work joint analysis

60 0.

.45 m 0 Ω 30 0.

15 0.

08 0.

.06

b 0 Ω 04 0.

02 0.

6 1.

2 1. s n 8 0.

4 0.

25 1.

8 .00

σ 1 75 0. 50 0.

.4 .6 .8 .0 15 30 45 60 02 04 06 08 .4 .8 .2 .6 50 75 00 25 0 0 0 1 0. 0. 0. 0. 0. 0. 0. 0. 0 0 1 1 0. 0. 1. 1. h Ωm Ωb ns σ8

Figure 3.6: Cosmological parameter constraints derived from the joint analysis, marginalized over τreion, b and m and from the single probes. The single probe constraints have been marginalized over the respective nuisance parameters i.e. τreion for the CMB temperature anisotropies, b for galaxy clustering and m for the weak lensing shear. In each case the inner (outer) contour shows the 68% c.l. (95% c.l.). For clarity the cosmic shear 68% c.l. are solid while the 95% c.l. are dashed.

75 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing

joint analysis Planck Collaboration 2015, TT + lowP Planck Collaboration 2015, TT, TE, EE + lowP + lensing + ext

36 0.

32 0.

m 28 Ω 0.

24 0.

051 0.

.048 b 0 Ω 045 0. 042 0.

04 1.

00 1. s n 96 0.

92 0. 90 0.

85 0.

8 80 σ 0.

75 0.

70 0. 12 0. 09 0. reion τ 06 0. 03 0.

.60 .65 .70 .75 .80 .24 .28 .32 .36 042 045 048 051 .92 .96 .00 .04 .70 .75 .80 .85 .90 .03 .06 .09 .12 0 0 0 0 0 0 0 0 0 0. 0. 0. 0. 0 0 1 1 0 0 0 0 0 0 0 0 0 h Ωm Ωb ns σ8 τreion

Figure 3.7: Comparison between the parameter constraints derived from the joint analysis, marginalized over b and m and the constraints from Planck Collaboration et al. [212] using only CMB data (TT+lowP) or adding external data (TT,TE,EE+lowP+lensing+ext). The Planck constraints are marginalized over all nuisance parameters. In each case the inner (outer) contour shows the 68% c.l. (95% c.l.).

76 3.8. Conclusions constraints derived from the combination of CMB temperature anisotropies with the Planck low-` polarization likelihood (TT+lowP) and the ones derived from a combination of the latter with the Planck polarization power spectra, CMB lensing and external data sets (TT, TE, EE + lowP + lensing + BAO + JLA + H0). We see that the joint analysis prefers slightly lower values of the parameters Ωm and Ωb and a higher Hubble parameter h, but these differences are not signiﬁcant. Despite this fact we ﬁnd sensible overall agreement between the constraints derived in this work with both versions of the Planck constraints. While the constraints we derived in this analysis are broadened by the restricted multipole range we used, the results already demonstrate the power of integrated probe combination: the complementarity of different cosmological probes and their cross-correlations allows us to obtain reasonable constraints.

The measured power spectra together with the theoretical predictions for the best-fitting cosmological model derived from the joint analysis are shown in Fig. 3.4. The best-fit cosmology γδg γγ provides a rather good fit to all power spectra except C` and C` , whose measured values are generally lower than our best-fit model. This is mainly due to the assumed Gaussian prior on the multiplicative shear bias m, which does not allow for more negative values of m as would be preferred by the data. If we relax the prior to a Gaussian with standard deviation σ 0.2, we find a best-fit value for the multiplicative bias parameter of m 0.276 0.108. = γδg γγ = − ± This results in an improved fit to both C` and C` , but is in tension with the values derived for the multiplicative bias by Hirata and Seljak [125]. We therefore find evidence for a slight tension between CMB temperature anisotropy data and weak gravitational lensing, as already seen by e.g. [177, 101].

3.8 Conclusions

To further constrain our cosmological model and gain more information about the dark sector, it will be essential to combine the constraining power of different cosmological probes. This work presents a ﬁrst implementation of an integrated approach to combine cosmological probes into a common framework at the map level. In our ﬁrst implementation we combine CMB temperature anisotropies, galaxy clustering and weak lensing shear. We use CMB data from Planck 2015 [209], photometric galaxy data from the SDSS DR8 [13] and weak lensing data from SDSS Stripe 82 [20]. We take into account both the information contained in the separate maps as well as the information contained in the cross-correlation between the maps by measuring their spherical harmonic power spectra. This leads to a power spectrum matrix with associated covariance, which combines CMB temperature anisotropies, galaxy clustering, cosmic shear, galaxy-galaxy lensing and the ISW [232] effect with galaxy and weak lensing shear tracers.

From the power spectrum matrix we derive constraints in the framework of a ΛCDM cosmological model assuming both a Gaussian covariance as well as a Gaussian likelihood. We ﬁnd that the constraints derived from the combination of all probes are signiﬁcantly tightened

77 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing compared to the constraints derived from each of the three separate auto-power spectra. This is due to the complementary information carried by different cosmological probes. We further compare these constraints to existing ones derived by the Planck collaboration and ﬁnd reasonable agreement, even though the joint analysis slightly prefers lower values of both

Ωm and Ωb and a higher Hubble parameter h. For a joint analysis of three cosmological probes, the constraints derived are still relatively weak, which is mainly due to our conservative cuts in angular scales. Nevertheless this analysis already demonstrates the potential of integrated probe combination: the complementarity of different data sets, that alone yield rather weak constraints on the full ΛCDM parameter space, allows us to obtain robust constraints which are signiﬁcantly tighter than those obtained from probes taken individually. In addition, our analysis reveals challenges intrinsic to probe combination. Examples are the need for foreground-correction at the map as opposed to the power spectrum level and the need for coordinate-independent bias corrections.

In this first implementation we have made simplifying assumptions. We assume a Gaussian covariance matrix for all cosmological probes considered. This is justified for the CMB temperature anisotropies and the galaxy overdensity at large scales. The galaxy shears on the other hand exhibit non-linearities already at large scales and their covariance therefore receives significant non-Gaussian contributions [233]. Furthermore, we do not take into account the cosmology-dependence of the covariance matrix [75]. In addition we only include systematic uncertainties from a potential multiplicative bias in the weak lensing shear measurement and neglect effects from other sources. Finally we also used the Limber approximation for the theoretical predictions. We leave these extensions to future work but we do not expect them to have a significant impact on our results since we restrict the analysis to scales where the above effects are minimized.

In order to fully exploit the wealth of cosmological information contained in upcoming surveys, it will be essential to investigate ways in which to combine these experiments. It will be thus interesting to extend the framework presented here to include additional cosmological probes, 3-dimensional tomographic information and tests of cosmological models beyond ΛCDM.

78 3.A. Theoretical prediction for CMB and weak lensing shear cross-correlation

3.A Theoretical prediction for CMB and weak lensing shear cross- correlation

The CMB temperature anisotropies are correlated with the weak lensing shear due to the ISW effect. The anisotropies in the temperature ﬁeld generated by time-varying gravitational potentials Φ are given by (see e.g. [197]):

Z η0 ∂Φ ∆TISW(θ) TCMB δTISW 2TCMB dη , (3.11) = = ηr ∂η where η0 denotes the conformal time today and ηr is the conformal time at recombination. Note that we follow the conventions for the gravitational potential Φ as in Bartelmann [25]. These anisotropies can be decomposed into spherical harmonics with multipole coefﬁcients

Z η0 Z 3 ` d k d ∆TISW,`m 4πi 2TCMB dη 3 [Φ(k,z)]j`(kχ(z))Y`∗m(θk ). (3.12) = ηr (2π) dη

The multipole coefﬁcients of the weak lensing shear E-modes can be expressed through the lensing potential ψ and are given by [25]

s s 1 (` 2)! (` 2)! ` aE,`m + ψ`,m + 4πi = −2 (` 2)! = − (` 2)! − − Z Z 3 d k0 dχg(χ) Φ(k0,z)j`(k0χ(z))Y ∗ (θk ), (3.13) × (2π)3 `m 0 where

Z χh 1 χ(z0) χ(z) g(χ) dz0 − n(z0). (3.14) = χ(z) χ(z) χ(z0)

γT The spherical harmonic power spectrum C` between CMB temperature anisotropies and the weak lensing shear is deﬁned as

γT ∆TISW,`m a∗ C δ`` δmm . (3.15) 〈 E,`0m0 〉 = ` 0 0 Expressing the integrals in terms of redshift and interchanging the integration boundaries gives

s (` 2)! Z z Z d3k d 2 + ∗ ∆TISW,`m aE,∗ ` m (4π) 2TCMB dz 3 [D(z)(1 z)]Φ(k,z 0) 〈 0 0 〉 = (` 2)! 〈 0 (2π) dz + = − Z Z 3 c d k0 j (kχ(z))Y (θ ) dz g ¡χ(z )¢ Φ(k ,z )j (k χ(z ))Y (θ ) , (3.16) ` `∗m k 0 0 3 0 0 `0 0 0 `0m0 k0 × H(z0) (2π) 〉 where z? denotes the redshift at recombination. In order to derive Eq. 3.16 we have used that in linear perturbation theory the time- and scale-dependence of the gravitational potentials

79 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing can be separated i.e.:

Φ(k,z) Φ(k,z 0)D(z)(1 z), (3.17) = = + where D(z) denotes the linear growth factor. We further have that

3 lin Φ(k,z 0)Φ(k0,z0 0) (2π) P (k,z 0)δ(k k0), (3.18) 〈 = = 〉 = ΦΦ = − and therefore Eq. 3.16 reduces to

s (` 2)! Z z Z k2dk d γT 2 + ∗ C` (4π) 2TCMB dz 3 [D(z)(1 z)] = (` 2)! 0 (2π) dz + Z − c ¡ ¢ lin dz0 g χ(z0) D(z0)(1 z0)PΦΦ(k,z 0) j`(kχ(z)) j`(kχ(z0)). (3.19) × H(z0) + = Eq. 3.19 is the exact expression for the spherical harmonic cross-power spectrum between CMB temperature anisotropies and weak lensing shear. In order to speed up computations, it can be simpliﬁed by resorting to the Limber approximation [169, 147, 148] which gives

s (` 2)! Z z d γT + ∗ C` 2TCMB dz [D(z)(1 z)] = (` 2)! 0 dz + − ¡ ¢ µ 1 ¶ g χ(z) ` /2 D(z)(1 z) P lin k + ,z 0 . (3.20) × + χ2(z) ΦΦ = χ(z) =

The power spectrum of the gravitational potential at late times is related to the matter power spectrum through Poisson’s equation

µ ¶2 2 4 lin 3 ΩmH P (k,z 0) P lin (k,z 0) 0 δδ = . (3.21) ΦΦ = = 2 c4 k4

For large ` we can make the approximations

s (` 2)! + `2, (` 2)! ∼ (3.22) − 1 2 2 (` /2) ` . + ∼ Using Equations 3.21 and 3.22 we can write Eq. 3.20 as

2 Z µ 1 ¶ γT ΩmH TCMB 1 d ` /2 C 3 0 dz [D(z)(1 z)]D(z)W γ ¡χ(z)¢P lin k + ,0 , (3.23) ` 2 1 2 δδ = c (` /2) dz + = χ(z) + which is the expression given in Eq. 3.5.

80 3.B. Treatment of systematic uncertainties in galaxy clustering data

3.B Treatment of systematic uncertainties in galaxy clustering data

The number density of galaxies observed in SDSS DR8 photometric data is affected by various systematic uncertainties such as stellar density, Galactic extinction and PSF size variation [229, 129]. These effects remain even after masking and removal of the highest contamination regions. In order to obtain an unbiased galaxy overdensity map, we need to correct for the number density variation due to systematics. The SDSS recorded the values of several potential systematic uncertainties for the observed fields: airmass, Galactic extinction and seeing (as measured by the PSF FWHM) in all 5 SDSS bands for the field each galaxy has been observed in as well as sky emission at the position of the galaxy for all the 5 bands. These quantities can be queried for each galaxy position on the CAS17. In this work, we consider four different observational systematics: Galactic extinction in the r-band as well as FWHM of the PSF, airmass and sky emission in the i-band. A further potential systematic uncertainty is the presence of foreground stars. Ross et al. [229] show that the effects of foreground stars on galaxy number density are largely independent of the magnitude of the stars. We therefore follow Ho et al. [129] and investigate how the number density of stars with i-band magnitudes in the range 18.0 i 18.5 affects the number density of detected galaxies. ≤ < We pixelize all quantities onto HEALPix maps of resolution NSIDE 1024 and compute the = number density of galaxies relative to their mean number density as a function of the value of the systematic in the pixel. In order to correct for these systematic uncertainties, we fit a 3rd-order polynomial to the functional dependence of the relative galaxy number density on the systematic. Then we multiply the uncorrected number densities by the inverse of this function. Various potential systematics such as Galactic extinction and stellar density are spatially correlated to one another. When correcting for various systematics simultaneously, the order in which the corrections are applied could influence results [229]. In our sample we find that the corrections are both independent of ordering and SDSS band and correcting for the effect in one band simultaneously corrects for all the other bands. The results are shown in Fig. 3.8 and we use those to correct the galaxy maps from residual systematic uncertainties. We clip the systematics maps at the minimum and maximum systematics value shown in the figure and apply the fitted corrections to the galaxy number density. The galaxy clustering spherical harmonic power spectra before and after correcting for systematic uncertainties are discussed in Section 3.5.2 and shown in Appendix 3.H.

3.C PSF correction and construction of weak lensing shear maps

The galaxy shapes measured from images represent a convolution of the intrinsic galaxy shapes with the PSF of the telescope and the atmosphere. We therefore need to correct this effect using PSF estimates measured from the shapes of stars observed in the survey. As described in Annis et al. [20] and Lin et al. [170], the PSF model for SDSS Stripe 82 data is derived from weighted sums of shapes measured in the individual runs as opposed to the co-adds. Lin et al.

17See footnote9.

81 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing

1.03 1.02 1.010 1.02 1.01 1.005 1.01 1.00 1.00 0.99 1.000 gal 0.99 gal gal ¯ ¯ ¯ n n n

/ / 0.98 /

gal 0.98 gal gal n n 0.97 n 0.995 0.97 0.96 0.96 0.990 0.95 0.95 0.94 0.94 0.985 0 2 4 6 8 10 0.8 1.0 1.2 1.4 1.6 1.8 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1 stars [pixel− ] psffwhm i [arcsec] airmass i 1.02 1.06

1.01 1.04 1.02 1.00 ﬁt 1.00 uncorr. gal 0.99 gal ¯ ¯ n n

/ / 0.98 corr.

gal 0.98 gal n n 0.96 0.97 0.94 0.96 0.92 0.95 0.90 6 8 10 12 14 0.01 0.02 0.03 0.04 0.05 0.06 0.07 sky i [nanomaggies/arcsec2] extinction r [mag]

Figure 3.8: Galaxy number density ngal relative to mean galaxy density n¯gal as a function of potential systematic value. The ﬁgures show both the uncorrected data and the data corrected using a third order polynomial ﬁt to the uncorrected relation. The error bars assume Poisson noise and are thus likely underestimated due to the correlations between galaxy positions.

[170] found that this leads to biases that need to be removed prior to PSF correction. In order to correct for these effects, we follow the steps outlined in Lin et al. [170]. We select bright stars with i-band magnitudes in 16 i 17, which pass the clean photometry cuts18, and ﬁt < < polynomials to the residuals between their shapes measured from the co-adds and the PSF model for these stars. The residuals before and after subtraction of the polynomial ﬁt are shown in Fig. 3.9. We see that the correction introduced in Lin et al. [170] has considerably removed both an overall bias as well as discontinuities at the CCD camera column (camcol) edges.

Using the revised PSF model, we correct the measured shapes for the effect of the PSF.We use the linear PSF correction algorithm derived in Hirata and Seljak [125], which can be applied to the adaptive moment measurements from the SDSS PHOTO pipeline19.

In order to obtain a galaxy sample for reliable weak lensing shear measurement, we follow Lin et al. [170] and perform two additional selection cuts on the galaxies after PSF correction: we select galaxies with ellipticity components e ,e satisfying e 1.4 as well as e 1.4 and 1 2 | 1| < | 2| < photometric redshift uncertainties σ 0.15. This additional selection leaves a galaxy sample z < consisting of N 3322915 galaxies. gal = Lin et al. [170] found a camcol dependent additive bias in the PSF-corrected ellipticities. The 5 mean ellipticities for each camcol lie in the range e¯ [6 10− ,0.02] and e¯ [0.002,0.009], | 1| = × | 2| = which is larger than expected for a mean zero ﬁeld [170]. We therefore follow Lin et al. [170] and correct for the additive bias by subtracting the mean ellipticity for each camcol. We choose

18See footnote14. 19 Note that there is a typo in Hirata and Seljak [125]: The quantities Cg ,C f ,Dg ,D f in Eq. (B9) should be squared.

82 3.C. PSF correction and construction of weak lensing shear maps

0.10 0.10

0.05 0.05

mE1PSF 0.00 mE1PSF 0.00 − − 1 1

mE 0.05 mE 0.05 − −

0.10 0.10 − 1.5 1.0 0.5 0.0 0.5 1.0 1.5 − 1.5 1.0 0.5 0.0 0.5 1.0 1.5 − − − − − − DEC [deg] DEC [deg]

Figure 3.9: Residuals between ellipticity component e1 of a random sample of 10000 bright stars measured on the co-add images and PSF models for these objects as a function of declination (DEC) before (left panel) and after (right panel) applying the correction described in the text.

to perform this step prior to coordinate transformation (i.e. for ellipticities deﬁned relative to equatorial coordinates) as opposed to after rotation. We ﬁnd that removing the mean camcol ellipticity reduces PSF leakage to a level which is subdominant in our analysis.

Fig. 3.10 shows the distributions of the ellipticity components e1 and e2 defined relative to equatorial coordinates. They are averaged over HEALPix pixels of resolution NSIDE 512, = which corresponds to a pixel area of A 0.013 deg2. The figure displays the ellipticity pix ≈ histograms both prior to PSF correction and subtraction of additive bias as well as the final distributions obtained after applying both corrections. We see that the corrections have removed the effects of the PSF and the final histograms can be described by Gaussian distributions.

In the final step, these ellipticities need to be transformed to shear estimates by correcting for the shear resolution factor R, which is defined as ¿ À ∂γî R . (3.24) = ∂γi

The shear resolution factor R quantiﬁes the response of the estimated mean ellipticity to an applied shear. For the adaptive moment method described in Bernstein and Jarvis [32] it is given by R 2(1 e2 ), where e denotes the intrinsic r.m.s. ellipticity per component. We = − int int follow Lin et al. [170] and use e 0.37 as measured by Hirata et al. [127]. int = In order to construct the ﬁnal weak lensing shear maps we thus apply the resolution correction to the ellipticities and transform them from equatorial to Galactic coordinates.

83 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing

103 103 before PSF after PSF corr. corr. 102 102 ) ) 1 1 e e ( ( n n 101 101

100 100 0.25 0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20 0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20 − − − − − − − − − e1 e1 103 103 before PSF after PSF corr. corr. 102 102 ) ) 2 2 e e ( ( n n 101 101

100 100 0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20 0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20 − − − − − − − − e2 e2

Figure 3.10: Histograms of the ellipticity components e1 and e2 averaged over HEALPix pixels of resolution NSIDE 512. The left panels show the distributions before correction for the = PSF and subtraction of a camcol dependent additive bias, while the right panels show the distributions after the application of these corrections.

84 3.D. Transformation of weak lensing shear under rotation

3.D Transformation of weak lensing shear under rotation

We rotate the weak lensing galaxy shears from equatorial to Galactic coordinates following the implementation in HEALPix. The method is brieﬂy summarized below.

The rotation angle of the shears under a coordinate rotation as described by the rotation matrix R is equal to twice the rotation angle ψ of the coordinate axes with respect to which they are deﬁned. In HEALPix the x-axis is in the direction of eφ and the y-axis in the direction of eθ.

In order to derive ψ we deﬁne the following quantities: The position before rotation is denoted ³ ´ ³ ´ as r x y z and the position after rotation is r0 x y z . We further deﬁne the = = 0 0 0 vector towards the north pole in the unrotated coordinate system, which is given by p ³ ´ ³ ´ 1 = x0 y0 z0 0 0 1 . Under the inverse rotation R− the north polar vector is mapped to ³ = ´ p00 x y z . At the position r the unit vectors in θ- and φ-direction are given by = 000 000 000 p r eφ × , = p r | × | (3.25) (p r) r e × × . θ = (p r) r | × × | We have the following identities

1 R eφ\θ p eφ\θ R− p, · = · (3.26) 1 R− p r p R r. · = · Taking into account the left-handedness of the HEALPix coordinate system and inserting the explicit expressions, it follows that

c cosψ (zz0 z000), = p1 z2 − c− (3.27) sinψ (x y000 yx000), = p1 z2 − − where c is a constant, which we can remove by ensuring that sin2 ψ cos2 ψ 1. Under this + = rotation the weak lensing shear transforms as

γ10 cos2ψγ1 sin2ψγ2, = + (3.28) γ0 sin2ψγ cos2ψγ . 2 = − 1 + 2

3.E Choice of PolSpice parameter settings

In this section we illustrate the determination of the maximal angular scale θmax used to compute spherical harmonic power spectra on the example of the SDSS Stripe 82 mask. Fig. 3.11 shows the real space correlation function of this mask. It is non-zero for small angular

85 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing

0.007 0.006 0.005 0.004 ) θ

( 0.003 C 0.002 0.001 0.000 0.001 − 0 2 4 6 8 10 12 14 θ [deg]

Figure 3.11: Real space correlation function of the SDSS Stripe 82 mask. The dashed line denotes the value chosen for θmax.

scales, then starts to fall-off and approximately vanishes for large angular separations. From this figure we see that θmax is not a well-defined quantity. Our approach is thus to choose a maximal angular scale by eye and validate it on the Gaussian simulations. For the SDSS Stripe 82 mask, we choose θ 10 degrees and θ 5 degrees. When testing these PolSpice max = FWHM = settings on the simulations, we find a reasonable agreement between input and recovered power spectra.

3.F Correlated maps of spin-0 and spin-2 ﬁelds

Our analysis relies on Gaussian simulations both for validation of the data analysis pipeline and covariance matrix estimation. We thus need to generate correlated HEALPix maps of both spin-0 and spin-2 fields from input auto- and cross-power spectra. Cabré et al. [42] and Giannantonio et al. [93] describe an algorithm for generating correlated HEALPix maps of spin-0 fields. In order to consistently simulate the weak lensing shear field, we extend this algorithm to also include correlations between spin-0 and spin-2 fields.

These algorithms are all based on the HEALPix routine synfast, which generates HEALPix ii maps of realizations of input spherical harmonic power spectra C` . If the ﬁelds are additionally mean-subtracted, this is equivalent to requiring that the spherical harmonic coefﬁcients a`m of the maps satisfy

D i E a`m 0, = (3.29) D i i E ii a a ∗ C δ`` δmm . `m `0m0 = ` 0 0 In synfast these conditions are imposed by assigning a random phase ξ with mean 0, ξ 0, 〈 〉 = 86 3.F. Correlated maps of spin-0 and spin-2 ﬁelds

and unit variance, ξξ∗ 1, to each spherical harmonic mode ` and setting 〈 〉 = q ai C ii ξ. (3.30) `m = ` As derived in Giannantonio et al. [93], this method can be extended to correlated maps using more random phases. The simplest case is to create two correlated spin-0 zero maps with ii j j i j power spectra C` , C` and cross-power spectrum C` . This is the only case relevant for our work and it is achieved by choosing the amplitudes of the maps of the two probes i, j as [93]

q ai C ii ξ , `m = ` 1 v i j u i j 2 (3.31) j C u j j (C ) a ` tC ` . `m q ξ1 ` ξ2 = ii + − C ii C` `

As described in [93] this algorithm can be implemented using synfast by ﬁrst creating a ii map with power spectrum C` and a second map using the same seed with power spectrum i j (C )2 ii ` /C` . Finally the second map needs to be added to a third map, generated with a different

j j i j (C )2 ii random seed and with power spectrum C ` /C . This ensures the desired auto- and ` − ` cross-correlations.

To extend this algorithm to spin-2 fields, we make use of the polarization version of synfast, which allows us to generate correlated spin-0 and spin-2 maps consistent with input auto- 00 and cross-power spectra. Let 0 denote the spin-0 field. Then C` denotes the auto-power EE BB spectrum of the spin-0 field, C` , C` are the E- and B-mode power spectra of the spin-2 field 0E and C` is the cross-power spectrum between the spin-0 field and the spin-2 E mode. Given these input power spectra, the polarization mode of synfast generates a map of the spin-0 field and two maps of the spin-2 field with the desired auto- and cross-power spectra.

In order to obtain correlated maps mT,mδg ,mγ1 ,mγ2 of CMB temperature anisotropies, galaxy TT δg T overdensity and galaxy weak lensing shear with auto- and cross-power spectra C` ,C` , δg δg γT γδg γγ C` ,C` ,C` ,C` we therefore proceed as follows:

(i) We ﬁrst create three correlated HEALPix maps using synfast in polarization mode with the power spectra

C 00 C TT, ` = ` EE C γγ C ` /2, ` = C BB 0, ` = T C 0E C γ . ` = `

These maps are denoted m1, where i {T,γ ,γ }. i ∈ 1 2 87 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing

(ii) Following Eq. 3.31, we then create three maps with a new random seed and the power spectra

δ δ 00 g g δg T C C (C )2/C TT, ` = ` − ` ` EE C γγ C ` /2, ` = C BB 0, ` = γδ C 0E C g . ` = `

These maps are denoted m2, where i {δ ,γ ,γ }. i ∈ g 1 2 (iii) We create a spin-0 map generated with the same seed as used for m1 with the power spectrum

00 δg T C (C )2/C TT, ` = ` ` which is called m3.

(iv) Finally we combine the maps i.e.

m m1, T = T 2 3 mδ m m , g = δg + m m1 m2 , γ1 = γ1 + γ1 m m1 m2 . γ2 = γ2 + γ2

TT δg T This procedure yields four correlated maps with auto- and cross-power spectra C` ,C` , δg δg γT γδg γγ C` ,C` ,C` ,C` . The algorithm described above introduces an unwanted, additional correlation between mδg and mγ1 ,mγ2 . It can in principle be corrected for by adding counter- terms to the respective maps. Since the additional correlation is subdominant in the present case, we neglect these counter-terms.

In order to obtain realistic maps we need to account for the effects of HEALPix pixel and beam window function. The signal measured in each HEALPix pixel is a convolution of the underlying signal with the HEALPix window function. If further experimental beams are present, the signal is additionally convolved with the beam window function. Since a convolution in real space is equivalent to a multiplication in Fourier space, we account for these effects by multiplying the input power spectra by the power spectra of the respective window functions prior to generating the HEALPix maps.

To compute the covariance matrix as well as to validate the analysis pipeline we need to add realistic noise to the correlated Gaussian simulations. We choose to add the noise on the map level. For the CMB temperature anisotropies we add the Commander HMHD map provided by the Planck collaboration to each simulated temperature map. We do not randomize the

88 3.G. Validation of spherical harmonic power spectrum measurements noise map for each new realization since the HMHD map features significant correlations which would be lost by randomizing. Since we are adding the same noise map to each random realization we expect to slightly underestimate the noise using our simulations. However, we do not expect this to have a significant effect on our results, since the noise in the CMB temperature power spectrum is dominated by cosmic variance at the scales considered. For the galaxy overdensity field we create noise maps by randomizing the positions of the galaxies in our data inside the survey mask. We then pixelize those on a HEALPix map and add the noise map to the simulated map. The galaxy shear noise maps are created by rotating each galaxy shear by a random angle and repixelizing the rotated shears onto HEALPix maps. As before these noise maps are added to the signal maps to produce the Gaussian simulations including both signal and noise.

3.G Validation of spherical harmonic power spectrum measurements

We validate the spherical harmonic power spectrum measurement outlined in Section 3.5 using the correlated Gaussian simulations described in Appendix 3.F. We compute theoretical predictions for the six spherical harmonic power spectra considered in this work, i.e. TT δg T δg δg γT γδg γγ C` , C` , C` , C` , C` , C` for a ΛCDM cosmological model with parameters {h, Ωm, Ωb, Ω , n , σ , τ , T } {0.7, 0.3, 0.049, 1.0, 0.88, 0.078, 2.275K}. We further set the linear, Λ s 8 reion CMB = redshift-independent galaxy bias parameter to b 2. = Using the algorithm described above, we generate N 1000 Gaussian realizations of these sim = power spectra and add the noise maps determined from the data. We then apply angular masks equivalent to those in the data and compute the spherical harmonic power spectra from the masked maps using the exact same methodology and PolSpice settings as applied on the data. To estimate the noise bias we follow the same randomization approaches as outlined in Section 3.5.

Figures 3.12 and 3.13 show a comparison between the input power spectra for all the six probes and the means of the recovered realizations. The uncertainties are derived from the sample covariance of the Gaussian realizations. We see that the input power spectra are recovered rather well. Also shown are the χ2 values between the reconstruction and the theory. These are not rigorous measures for the goodness of the recovery since they strongly depend on binning and angular multipole range chosen.

3.H Spherical harmonic power spectrum robustness tests

This section summarizes the robustness tests performed for the spherical harmonic power spectra.

89 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing

6000 χ2/dof = 1.82

) 5000 π (2

/ 4000 TT ` theor.

C 3000 reconst.

+ 1) 2000 ` ( ` 1000 0 0 100 200 300 400 500 600 9 10− 4 2.0 × 10− χ2/dof = 1.15 χ2/dof = 0.64 1.5

1.0 g T δ g g δ δ ` ` C 0.5 C

5 0.0 10−

0.5 − 50 100 150 200 3 101 102 × 11 4 10− 10− 12 χ2/dof = 1.33 χ2/dof = 0.23 χ2/dof = 2.91 10−

13 10− 7

g 10− T γγ γ 14 ` 5 γδ ` 10− ` 10− C C C 15 10−

16 10−

17 8 6 10− 10− 10− 102 6 102 3 101 102 102 6 102 × × × ` ` `

Figure 3.12: Comparison between input power spectra and mean recovered power spectra as estimated from N 1000 Gaussian realizations generated using the algorithm outlined in sim = Appendix 3.F. The noise level of the Gaussian realizations is tuned to the data and the spherical harmonic power spectra have been computed using the same methodology and PolSpice settings as applied on the data. The angular multipole ranges and binning schemes for all power spectra are summarized in Table 3.2. Dashed lines denote negative spherical harmonic power spectrum values.

6 10− 0.4 × 0.2 0.0 0.2 BB ` − C 0.4 − 0.6 reconst. − 2 0.8 χ /dof = 1.33 − 1.0 − 100 200 300 400 500 600 `

Figure 3.13: The cosmic shear B-mode power spectrum reconstructed from N 1000 sim = Gaussian realizations generated using the algorithm outlined in Appendix 3.F. The angular multipole range and binning scheme is summarized in Table 3.2.

90 3.H. Spherical harmonic power spectrum robustness tests

4 10− δgδg g δ

g gal. δ `

C eq.

5 10− 3 101 102 ×5 10− 1.5 × γδg γγ

1.0 ) 5 π 10− (2 /

g 0.5 γγ ` γδ ` C

`C 0.0 6 10− + 1) ` ( 0.5 ` − 7 1.0 10− − 50 100 150 200 102 6 102 × ` `

Figure 3.14: Comparison between spherical harmonic power spectra computed from the maps in Galactic and equatorial coordinates.

3.H.1 Comparison between spherical harmonic power spectra in equatorial and Galactic coordinates

We test that the spherical harmonic power spectra involving maps which can be transformed between coordinate systems, i.e. galaxy overdensity and weak lensing shear maps, are unaffected by the rotation. The comparison between spherical harmonic power spectra computed from maps in Galactic and equatorial coordinates are shown in Fig. 3.14. We find good agree- δg δg γδg ment between the two power spectra for both C` and C` , while we find discrepancies for γγ C` . We attribute this to the additive bias correction applied to the galaxy shears as outlined in Section 3.5.3. The additive bias correction described in Appendix 3.C, causes an asymmetry between the galaxy shears in different coordinate systems, which is the cause for the large discrepancies detected. This can be seen from Fig. 3.15, which shows a comparison between the cosmic shear power spectra prior to noise removal as estimated from maps in Galactic and equatorial coordinates. The left panel shows the comparison when the additive bias correction is applied while in the right panel we do not apply any correction. As can be seen, we find discrepancies when we apply the additive bias correction in equatorial coordinates and then rotate the corrected shears to Galactic coordinates. Not applying any additive bias correction on the other hand, removes most of these effects.

91 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing

9 9 10− 10− × × gal. gal. 5 eq. eq. 4 γγ ` 3 C 2 1 0 100 200 300 400 500 600 100 200 300 400 500 600 ` `

Figure 3.15: Comparison between cosmic shear spherical harmonic power spectra prior to noise removal measured from the maps in Galactic and equatorial coordinates. The left hand panel shows the results when applying the correction for additive bias in equatorial coordinates and then rotating the shears to Galactic coordinates. The right hand panel shows the results when no additive bias correction is applied.

3.H.2 Comparison between spherical harmonic power spectra derived from different foreground-reduced CMB maps

We test that the spherical harmonic power spectra involving CMB data are unaffected by our choice of foreground-reduced map. The power spectra involving CMB data are shown in Fig. 3.16 for the foreground-reduced CMB maps derived using the component separation methods Commander, NILC, SEVEM and SMICA. As can be seen, the power spectra are virtually the same.

3.H.3 Impact of systematics correction on galaxy clustering power spectrum

We further investigate the effect of systematics correction on the galaxy clustering power spectrum. The galaxy clustering spherical harmonic power spectra before and after correcting for systematic uncertainties are shown in Fig. 3.17. Our systematics removal method slightly reduces the clustering amplitude at large angular scales, while leaving small angular scales almost unaffected. This is to be expected since Galactic foregrounds typically exhibit signiﬁcant large scale clustering.

92 3.H. Spherical harmonic power spectrum robustness tests

7 10− 6000 × ] 2 TT 0.8 δgT K 5000 µ

)[ 0.6

π 4000 [K] (2 0.4 / T

3000 g δ ` TT ` 0.2 C `C 2000 0.0 + 1) `

( 1000 0.2 ` − 0 0.4 0 100 200 300 400 500 600 − 50 100 150 200 9 10− ` 2.5 × 2.0 γT

1.5 com. smica

[K] 1.0 sevem nilc T γ ` 0.5 `C 0.0 0.5 − 1.0 − 100 200 300 400 500 600 `

TT δg T γT Figure 3.16: Comparison between spherical harmonic power spectra C` , C` , C` derived using the four different foreground-reduced CMB maps from Commander, NILC, SEVEM and SMICA.

4 10− gal. corr. gal. uncorr. g δ g δ ` C

5 10− 3 101 102 × `

Figure 3.17: Comparison between galaxy overdensity power spectra computed before and after systematics removal.

93 Chapter 3. Integrated approach to cosmology: combining CMB, large-scale structure and weak lensing

60◦

30◦

120◦ 60◦ 0◦ 60◦ 120◦ 0◦ − −

30◦ −

60◦ −

7.5 7.5 − 5 ∆T [K] 10− ×

Figure 3.18: Half-mission half-difference (HMHD) Commander CMB temperature anisotropy map. This map contains only noise and potential residual systematics.

5 10− 1.5 × 1.0 ) π

(2 0.5 / BB `

C 0.0

+ 1) 0.5 `

( − ` 1.0 − 1.5 − 100 200 300 400 500 600 `

Figure 3.19: Spherical harmonic power spectrum of cosmic shear B-modes computed from the SDSS Stripe 82 maps in Galactic coordinates. The angular multipole range and binning scheme is summarized in Table 3.2.

94 4 Integrated cosmological probes: extended analysis

E quindi uscimmo a riveder le stelle. — Dante, Inferno, Canto XXXIV

This chapter appeared in a similar form as Nicola, Réfrégier & Amara, 2017 [193].

4.1 Introduction

In Chapter3, we implemented an integrated approach to cosmology in which the cosmological probes are combined into a common framework at the map level. This has the advantage of taking full account of the correlations between the different probes which generally probe common survey volumes, to provide a stringent test of systematics through the test of the consistency between the probes and to yield a test of the validity of the cosmological model. We applied this framework to a combination of the CMB temperature from the Planck mission [209], galaxy clustering from the eighth data release of the Sloan Digital Sky Survey (SDSS DR8) [13] and weak lensing from SDSS Stripe 82 [20], making simplifying approximations but also conservative cuts on the data.

In the present work, we extend the integrated analysis of Chapter3 to also include CMB lensing maps from the Planck mission [211], the recent weak lensing measurement with the publicly available Dark Energy Survey (DES) Science Veriﬁcation (SV) data [138], SNe Ia data from the joint light curve analysis (JLA) [34] and constraints on the Hubble parameter from the Hubble Space Telescope (HST) [227, 74]. This yields 12 auto and cross power spectra which include the CMB temperature power spectrum, cosmic shear, galaxy clustering, galaxy- galaxy lensing CMB lensing cross-correlation along with other cross-correlations as well as background probes. Furthermore, we extend the treatment of systematic uncertainties and relax some of the approximations as compared to Chapter3. In particular, we study the impact of intrinsic alignments, baryonic corrections, residual foregrounds in the CMB temperature,

95 Chapter 4. Integrated cosmological probes: extended analysis and calibration factors for the different power spectra. This extended analysis allows us to derive more robust constraints on the ΛCDM cosmological model and a more thorough test of the consistency between the different probes. Other joint analyses of different sets of cosmological probes have been performed (see references in Chapter3 and Refs. [ 94, 92, 253]).

This chapter is organized as follows. We review the framework for cosmological probe combination employed in this work in Section 4.2. In Section 4.3, we describe the data used in this work and Section 4.4 describes the theoretical modeling of the cosmological observables. We detail the computation of spherical harmonic power spectra in Section 4.5, while Section 4.6 summarizes the systematic uncertainties considered in this work. The computation of the covariance matrix is described in Section 4.7. The method for parameter inference is described in Sec. 4.8 and our results on cosmological constraints are presented in Section 4.9. We conclude in Section 4.10. Robustness tests as well as implementation details are deferred to the Appendices.

4.2 Framework for integrated probe combination

The integrated probe combination framework is described in detail in Chapter3, but here we summarize the most important steps: ﬁrst, we create projected two-dimensional maps for the large-scale structure (LSS) and CMB probes. We then compute both the spherical harmonic auto-power spectra of these probes as well as the cross-power spectra for physically overlapping surveys. This yields a set of 12 spherical harmonic power spectra, which does not include the auto-power spectrum of the CMB lensing convergence but only its cross- correlations. We complement the observed power spectra with theoretical predictions and an estimate of their covariance matrix and combine these into a Gaussian likelihood. We combine the power spectrum likelihood with the likelihood of SNe Ia distance moduli and a constraint on the Hubble parameter, assuming these probes to be independent. In a last step we compute cosmological parameter constraints in a joint ﬁt to these data. The implementation details for the CMB lensing convergence, weak lensing data from DES SV, SNe Ia and the Hubble constant measurement are described below. For a description of the remaining data the reader is referred to Chapter3.

4.3 Data

The data used in this analysis is summarized in Table 4.1 and the footprints of the different surveys are illustrated in Figure 4.1 together with the background probes. We consider the data used in Chapter3 namely, the Planck 2015 foreground-reduced CMB temperature anisotropy map derived using the Commander algorithm [210], a map of the galaxy overdensity ﬁeld derived using the CMASS1-4 sample from SDSS DR8 [13, 129, 229] and a weak lensing shear map derived using SDSS Stripe 82 co-add data [20, 170]. In addition to the cosmological maps we also employ the binary survey masks presented in Chapter3. In this work, we complement

96 4.3. Data

46 44 42

) 40 z (

µ 38 36 34 32 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 z 0.14 0.12 0.10 )

0 0.08 H (

p 0.06 0.04 0.02 0.00 60 65 70 75 80 H0

Planck temp. Planck lensing SDSS DR8 SDSS Stripe 82 DES SV

Figure 4.1: Summary of the data used in this work. The left hand side shows an overlay of the footprints of the surveys used in this work: the CMB temperature and CMB lensing convergence from Planck, the galaxy density from SDSS DR8 and weak lensing from SDSS Stripe 82 and DES SV. The right hand side shows the background probes: SNe Ia from JLA and Hubble parameter data from HST. All footprints are shown in Galactic coordinates. The map is shown in Mollweide projection at a HEALPix resolution of NSIDE = 1024. (See Tab. 4.1 for references for these different surveys.)

these three maps with several data sets as described below.

4.3.1 DES weak lensing

We use publicly available data from the DES SV.1 The DES is an ongoing survey, imaging the sky in five photometric bandpasses (g,r,i,z,Y ) using DECam [83]. After its five-year duration the DES will have covered approximately 5000 deg2 of the southern sky to a limiting magnitude of about 24. The SV data were taken before the start of the main survey and they consist of more than 250 deg2 [63]. In our analysis we use the largest contiguous area in the DES SV data, which is part of the South Pole Telescope East (SPT-E) field and covers an area of approximately 139 deg2. The weak lensing shear for galaxies in the SPT-E region has been measured using two independent shape measurement codes, NGMIX [245] and IM3SHAPE [285]. Both are model-fitting shear measurement codes and are described in Ref. [138]. Photometric redshifts (photo-z) have been obtained using four different methods as described in Ref. [37]. The photometric redshift catalogs both provide the full photo-z probability distribution function (pdf) as well as an estimate of the mean of the pdf for each galaxy. We follow the choice of

1The data is available at: https : //des.ncsa.illinois.edu/releases/sva1.

97 Chapter 4. Integrated cosmological probes: extended analysis

Table 4.1: Summary of the data sets used in our analysis.

Survey: Planck 2015 [210] CMB temperature Fiducial foreground-reduced map: Commander Sky coverage: f 0.776 sky = Survey: SDSS DR8 [13] Sky coverage: f 0.27 sky = Galaxy density Galaxy sample: CMASS1-4 Number of galaxies: N 854063 gal = Chapter3 Photometric redshift range 0.45 z 0.65 ≤ phot < Survey: SDSS Stripe 82 co-add [20] Sky coverage: f 0.0069 SDSS sky = Number of galaxies: N 3322915 Stripe 82 gal = Photometric redshift range: 0.1 . zphot . 1.1 r.m.s. ellipticity per component: σ 0.43 Weak e ∼ lensing Survey: DES SV [138] Sky coverage: f 0.0039 sky = Number of galaxies: N 3279967 DES gal = Sec. 4.3.1 Photometric redshift range: 0.3 z 1.3 < phot < r.m.s. weighted ellipticity per component: σ 0.24 e ∼ Survey: Planck 2015 [211] CMB lensing Sec. 4.3.2 Sky coverage: f 0.67 sky = Compilation: JLA [34] SNe Type Ia Number of SNe: N 740 Sec. 4.3.3 SNe = Redshift range: 0.01 z 1.3 < < Distance anchor: NGC 4258 [135] Number of Cepheids: N 600 [227] Hubble parameter Ceph. = Sec. 4.3.4 Number of SNe: N 8 [227] SNe = Analysis: Efstathiou [74]

98 4.3. Data

ﬁducial catalog of Refs. [29, 8] and perform our analysis using the galaxy shapes measured by NGMIX and the photometric redshifts determined using SkyNet.

Our analysis closely follows the spherical harmonic power spectrum measurement described in Appendix A of Ref. [29]. We select objects passing the SVA1 and the NGMIX cuts deﬁned in Ref. [138] which fall into any of the three tomographic redshift bins described in Refs. [37, 138]. This selection yields N 3279967 galaxies. gal = In order to construct the weak lensing shear maps we weight each galaxy’s shear by its inverse variance weight described in Ref. [138]. The galaxy shapes given in the DES SV shear catalogues are biased estimators of the galaxy shears and the NGMIX shape estimates therefore need to be corrected for the sensitivity as described in Ref. [138]. Since this correction factor is a noisy estimate of the true correction it cannot be applied on single galaxies. In order to avoid introducing a bias caused by the noisy estimators of the sensitivity we therefore follow Ref. [29] and estimate the weighted average of the galaxy sensitivities in our sample and correct each galaxy shape with this mean correction.

We then rotate the galaxy shears from equatorial to Galactic coordinates2 and pixelize them onto HEALPix3 [99] pixelizations of the sphere choosing a resolution of NSIDE = 1024. This resolution corresponds to a pixel area of 11.8 arcmin2. We apply a binary mask constructed from the union of unobserved and empty pixels to both shear maps. The ﬁnal maps cover a fraction of sky f 0.0039. The mean number of galaxies per pixel is approximately given by sky = n 2 gal/pix 67, which corresponds to n 5.73 arcmin− . Figure 4.2 shows the map of the shear = gal = modulus together with a zoom-in region with overlaid whisker plot illustrating the direction of the shear.

We follow Refs. [37, 29] and estimate the redshift distribution of the galaxies from the sum of the individual galaxy pdfs, weighted by their weak lensing shear weights. The resulting redshift distribution together with the weak lensing window function is shown in the Appendix (Fig. 4.19).

4.3.2 CMB lensing convergence

CMB lensing causes statistical anisotropies in CMB maps and the lensing potential can be reconstructed from these maps using a quadratic estimator [195]. We use the CMB lensing potential estimate φˆCMB provided by the Planck Collaboration in their second data release [211]. This estimator has been derived from the foreground-reduced CMB temperature and polarization maps computed using the SMICA algorithm [211, 210]. The use of both CMB temperature as well as polarization data allows for several CMB lensing potential estimators

(φˆTT,φˆTE,φÊE,φÊB,φˆTB), which can be combined into a minimal-variance estimator. This estimate is given in the form of spherical harmonic coefficients of the CMB lensing conver-

2The exact rotation applied is given in Chapter3. 3http : //healpix.sourceforge.net.

99 Chapter 4. Integrated cosmological probes: extended analysis gence κ in the angular multipole range 8 ` 2048. These are related to the CMB lensing CMB ≤ ≤ potential φCMB through:

`(` 1) κ + φ . (4.1) CMB,`,m = 2 CMB,`,m

We use these spherical harmonic coefﬁcients to create a HEALPix map of resolution NSIDE = 1024 using the HEALPix routine alm2map. The analysis mask derived by the Planck Collabora- tion is provided as a HEALPix map of NSIDE = 2048. We downgrade this map to a resolution of NSIDE = 1024 following the procedure outlined in Ref. [210], which yields a binary analysis mask. We choose the CMB lensing convergence over the CMB lensing potential map since the lensing convergence is more local and should thus be less affected by masking effects arising when computing angular power spectra. The CMB lensing convergence map covers a fraction of sky f 0.67 and is shown in Fig. 4.2. sky =

4.3.3 Type Ia supernovae

We complement the CMB and LSS data with geometrical constraints on the homogeneous Universe from the distance-redshift relation measured from Type Ia supernovae. We use data from the JLA [34], which is a compilation of 740 SNe Ia comprising data from SDSS-II [88, 154, 254, 163, 46], the Supernova Legacy Survey (SNLS) [22, 258], the HST [226, 262] and several low-redshift experiments [34].4 The JLA data consist of SNe Ia light curve parameters which can be used to calculate observed distance moduli.

4.3.4 Hubble parameter

We also add a local H0 measurement from HST [227] to our analysis. We use the Hubble parameter estimate by Ref. [74], which is a revision of the measurement presented in Ref. [227]. Both measurements are derived from Cepheid-calibrated SNe Ia distance moduli but the former uses a revised distance to the anchor NGC 4258 [135] to calibrate the Cepheid distances. 1 1 This analysis constrains the Hubble parameter to be given by H 70.6 3.3 km s− Mpc− , 0 = ± where the uncertainties are 1σ and assumed to be Gaussian.

4.4 Model predictions

The auto- and cross-correlations of the CMB and LSS cosmological probes can be computed theoretically from the primordial power spectrum. In order to compute the power spectra of the cosmological ﬁelds δg , γ and κCMB we employ the Limber approximation [169, 147, 148] as in Chapter3. We further assume a ﬂat cosmological model, i.e. Ωk 0. With these i j = approximations the spherical harmonic power spectrum C` between cosmological probes

4The data can be found at: http : //supernovae.in2p3.fr/sdss_snls_jla/ReadMe.html.

100 4.4. Model predictions

31.0 CMB convergence 3.5 60

120 60 0 60 120

b 33.5 0

60 3.5 36.0 55.5 53.0 50.5 CMB [1] l

42.5 DES weak lensing 0.1000 60

120 60 0 60 120

b 40.0 0

60 0.0003 37.5 92.5 95.0 97.5 ˆ [1] l | |

Figure 4.2: New maps used in this analysis in addition to the CMB, galaxy clustering and SDSS weak lensing maps of Chapter3. The full-sky maps are in Galactic coordinates and are shown in Mollweide projection while the zoom-in versions are in Gnomonic projection. The CMB lensing convergence map derived from the foreground-removed CMB temperature and polarization anisotropy maps from SMICA is shown in the top panel. It is masked using the analysis mask provided by the Planck Collaboration. The zoom-in shows an enlarged version of the 5 5 deg2 region centered on (l,b) (53°, 33.5°) shown in the map. The bottom panel × = − shows the map of the weak lensing shear modulus γˆ derived from DES SV. Gray regions | | are masked because they are either unobserved or they do not contain any galaxies at our resolution. The zoom-in shows an enlarged version of the 5 5 deg2 region centered on (l,b) × ( 95°, 40°) shown in the map. It is overlaid with a whisker plot illustrating the direction of = − − the shear. Both maps are shown at a HEALPix resolution of NSIDE = 1024.

101 Chapter 4. Integrated cosmological probes: extended analysis i, j [δ ,γ,κ ] at angular multipole ` can be expressed as ∈ g CMB

Z i ¡ ¢ j ¡ ¢ µ 1 ¶ i j c W χ(z) W χ(z) ` /2 C dz P nl k + ,z , (4.2) ` = H(z) χ2(z) δδ = χ(z) where H(z) is the Hubble parameter, χ(z) the comoving distance and c denotes the speed of nl light. Furthermore, Pδδ (k,z) denotes the nonlinear matter power spectrum at redshift z and i 0 ¡ ¢ wave vector k and W χ(z) is the window function for probe i 0.

The window functions for δg and γ are given in Chapter3. Since the CMB lensing convergence is approximately sourced by a single-lens plane located at the last scattering surface with redshift z its window function can be expressed as the single-plane limit of the weak lensing ∗ shear window function. We therefore have

2 3 ΩmH χ(z) χ(z?) χ(z) κCMB ¡ ¢ 0 − W χ(z) 2 , (4.3) = 2 c a χ(z?) where Ωm is the fractional matter density today and a is the scale factor. In our calculations we set z 1090. ? = The power spectra involving CMB temperature anisotropies can also be related to the primordial density ﬂuctuations. The expression for the CMB temperature power spectrum is given in Chapter3. The observed CMB temperature anisotropies are further correlated to tracers of the LSS. For the galaxy overdensity and weak lensing shear this cross-correlation is mainly due to the integrated Sachs-Wolfe (ISW) [232] effect and the resulting cross-power spectra are given in Chapter3. The cross-correlation between the CMB temperature anisotropies and the CMB lensing convergence is dominated by the ISW but receives further contributions from Doppler effects arising from bulk velocities of electrons scattering the CMB photons and from the Sunyaev-Zel’dovich (SZ) [259] effect (for a description of these effects see e.g. Refs. [96, 60]). The cross-correlation due to the SZ effect is not observable using the foreground-reduced CMB temperature anisotropy maps from Ref. [210] but the remaining effects are observable. The cross-power spectrum between CMB temperature anisotropies and CMB lensing convergence can be computed from

a a C TκCMB δ δ , (4.4) 〈 T,`m κCMB,`0m0 〉 = ` ``0 mm0 where aT,`m denotes the spherical harmonic coefﬁcients of the CMB temperature anisotropies

∆T(θ) and aκCMB,`0m0 denotes the spherical harmonic coefﬁcients of the CMB lensing convergence deﬁned as

Z c κCMB ¡ ¢ κCMB(θ) dz W χ(z) δ(χ(z)θ,z). (4.5) = H(z)

While the observables discussed so far probe cosmic structure formation, SNe Ia mainly probe the background evolution through their distance moduli. The distance modulus µ of a Type Ia

102 4.5. Spherical harmonic power spectra

Table 4.2: Summary of spherical harmonic power spectrum parameters and angular multipole ranges used in this analysis. The ﬁrst six power spectra are described in Chapter3.

Power spectrum θmax [deg] θFWHM [deg] `-range ∆` TT C` 40 20 [10, 610] 30 δg δg C` 80 40 [30, 210] 30 γ1γ1 C` 10 5 [70, 610] 60 δg T C` 40 20 [30, 210] 30 γ1T C` 10 5 [70, 610] 60 γ1δg C` 10 5 [30, 210] 60

κT C` 40 20 [40, 400] 60 δg κ C` 80 40 [40, 190] 30 γ1κ C` 10 5 [70, 370] 60 γ2T C` 15 7.5 [70, 610] 60 γ2κ C` 15 7.5 [70, 370] 60 γ2γ2 C` 15 7.5 [70, 610] 60

supernova at redshift zSNe is given by µ ¶ dL(zSNe) µ(zSNe) 5log , (4.6) = 10 10[pc] where dL(zSNe) is the luminosity distance to redshift zSNe.

To compute theoretical predictions for all cosmological observables we follow Chapter3. We use the publicly available Boltzmann code CLASS5 [164] to compute the CMB temperature anisotropy power spectra and the cross-correlation between the CMB temperature anisotropies and the CMB lensing convergence. For the other observables we use PYCOSMO [221]. As in Chapter3, we calculate the linear matter power spectrum from the transfer function derived by Ref. [78]. In order to compute nonlinear matter power spectra we use the HALOFIT ﬁtting function [251] with the revisions of Ref. [267].

4.5 Spherical harmonic power spectra

Following Chapter3 we use PolSpice6 [265, 264, 57] to measure the demasked spherical harmonic power spectra from the maps. We calculate the auto-power spectrum of the DES SV weak lensing shear map as well as the cross-correlations between the maps discussed in Section 4.3 and in Chapter3 which have overlaps.

5http : //class-code.net. 6http://www2.iap.fr/users/hivon/software/PolSpice/.

103 Chapter 4. Integrated cosmological probes: extended analysis

We do not include the auto-power spectrum of the CMB lensing convergence in this analysis. This is due to the fact that the auto-power spectrum of the CMB lensing convergence estimator is a biased estimate of the CMB lensing convergence auto-power spectrum because it probes both the connected and the disconnected part of the 4-point function of the CMB temperature anisotropies [206]. In order to obtain the power spectrum of the CMB lensing convergence, the auto-power spectrum of the estimator thus needs to be corrected for this disconnected bias [206], which is beyond the scope of this chapter.

In order to compute the power spectra, we follow the method outlined in Chapter3 to estimate the values of the maximal angular scale used by PolSpice θmax and the apodization parameter θFWHM. We validate these settings using the Gaussian simulations described in Appendix 4.C. The demasking procedure used by PolSpice leads to biases in the recovered power spectra, as discussed in Chapter3. The kernels that relate average PolSpice estimates to the true power spectra can be computed analytically for each choice of θmax,θFWHM and we take them into account by convolving all theoretical predictions with these kernels. The choice of angular multipole ranges follows that described in Chapter3 for all power spectra already included in that analysis. The angular multipole range for power spectra involving the CMB lensing convergence follows the conservative choice described in Ref. [211]. The low-` limit for the power spectra is chosen to minimize the impact of mean ﬁeld corrections; the high-` limit is chosen because of mild evidence for systematic errors at higher multipoles [211]. The chosen bin widths largely follow the conservative binning outlined in Ref. [211](∆` 45) and the = binning scheme in Chapter3, which is chosen to roughly correspond to the width of the PolSpice kernels. Where we choose angular multipole bins broader than necessary this is mainly done to reduce the size of the spherical harmonic power spectrum vector. The binning schemes and PolSpice parameters used for all power spectra are summarized in Table 4.2.

All the spherical harmonic power spectra are computed from the maps of resolution NSIDE = 1024. They are further corrected for the effect of the HEALPix pixel window function and the power spectra involving the CMB temperature anisotropy map are further corrected for the Planck effective beam window function. The uncertainties are derived from the Gaussian simulations described in Appendix 4.C.

The power spectra computed in this work are described in more detail below; for a description of the remaining power spectra the reader is referred to Chapter3.

4.5.1 DES SV cosmic shear

We compute the cosmic shear power spectrum for DES SV using the map and mask described in Section 4.3.1. In order to estimate the contribution of shape noise, we follow Chapter3 and resort to simulations. We generate 100 noise maps by rotating the galaxy shears by a random angle. We then calculate the power spectra of these maps and our estimator of the shape noise power spectrum is given by the mean power spectrum of the noise maps.

104 4.5. Spherical harmonic power spectra

The weak lensing shear E-mode power spectrum for DES SV is shown in the 4,4-panel of Fig. 4.3 and the B-mode power spectrum is shown in the Appendix (Fig. 4.14). The noise level of DES SV is lower than the one from SDSS Stripe 82 as can be seen by comparing panels 4,4 and 3,3. This is to be expected from the higher galaxy number density and smaller measurement noise of DES SV data.

In Appendix 4.E we compare the DES SV cosmic shear power spectra computed from the maps in Galactic and equatorial coordinates. We ﬁnd discrepancies similar to those found for SDSS Stripe 82 (Chapter3), especially at small angular scales. Since the differences detected are within the uncertainties of the measurement, we use the cosmic shear power spectrum calculated from the maps in Galactic coordinates in our integrated analysis.

4.5.2 CMB temperature and DES SV weak lensing shear cross-correlation

We compute the cross-power spectrum between the DES SV weak lensing shear and the CMB temperature anisotropies using the maps and masks presented in Section 4.3.1 and Chapter 3. As discussed in Chapter3, we choose to compute cross-correlations using the combined masks of the respective probes rather than the single-probe masks. This is due to the fact that the former approach results in a better recovery of the input cross-power spectra in the Gaussian simulations described in Appendix 4.C. We therefore mask both maps with the combination of the single-probe masks, which covers a fraction of sky f 0.0035. sky ∼ The resulting spherical harmonic power spectrum is shown in the 4,0-panel of Fig. 4.3. As can be seen, the noise level is too high to allow for a detection of the ISW from DES SV weak lensing shear. This is in agreement with the results found for SDSS Stripe 82 in Chapter3. We nevertheless include the power spectrum in our analysis since it provides an upper limit to the ISW signal from weak lensing.

In Appendix 4.E we investigate the impact of our choice of ﬁducial foreground-reduced CMB temperature map by comparing the power spectra obtained using the four different foreground-reduction algorithms employed by Ref. [210]. As can be seen from Fig. 4.16 we ﬁnd the measured power spectra to be virtually the same.

4.5.3 CMB lensing convergence and galaxy overdensity cross-correlation

To compute the cross-power spectrum between the CMB lensing convergence and the SDSS DR8 galaxy overdensity we use the maps and masks described in Section 4.3.2 and Chapter3. We mask both maps with their combined mask, which covers a fraction of sky f 0.26. sky ∼ The spherical harmonic cross-power spectrum between the CMB lensing convergence and the galaxy overdensity is shown in the 2,1-panel in Fig. 4.3. We see that we clearly detect a nonzero correlation between the CMB lensing convergence and the galaxy overdensity.

105 Chapter 4. Integrated cosmological probes: extended analysis

4.5.4 CMB lensing convergence and CMB temperature cross-correlation

To compute the cross-power spectrum between the CMB lensing convergence and the CMB temperature anisotropies we use the maps and masks presented in Section 4.3.2 and Chapter 3. The combined mask of both probes covers a fraction of sky f 0.65 and we apply this sky ∼ mask to both maps.

The resulting spherical harmonic power spectrum is shown in the 2,0-panel of Fig. 4.3. Com- paring to the results derived in Ref. [211], we ﬁnd good overall agreement.

In Appendix 4.E, we again compare the power spectra obtained from the different foreground- reduced CMB temperature anisotropy maps and ﬁnd them to agree rather well.

4.5.5 CMB lensing convergence and SDSS Stripe 82 weak lensing shear cross-correlation

We estimate the cross-power spectrum between the CMB lensing convergence and the SDSS Stripe 82 weak lensing shear map using the maps and masks described in Section 4.3.2 and Chapter3. We mask both maps with their combined mask, which covers a fraction of sky f 0.0064. sky ∼ The spherical harmonic power spectrum is illustrated in the 3,2-panel in Fig. 4.3. We see that the obtained cross-power spectrum is rather noisy and it does not allow for a detection of the correlation between CMB lensing convergence and SDSS Stripe 82 weak lensing shear. This is probably due to the combined effect of a low fractional sky coverage of SDSS Stripe 82 data and signiﬁcant noise in both the CMB lensing convergence and SDSS Stripe 82 weak lensing shear. We nevertheless include this cross-correlation into our analysis to serve as an upper limit.

4.5.6 CMB lensing convergence and DES SV weak lensing shear cross-correlation

To compute the cross-power spectrum between the CMB lensing convergence and the DES SV weak lensing shear map we use the maps and masks presented in Sections 4.3.1 and 4.3.2. The combined mask of both maps covers a fraction of sky f 0.0037 and we apply it to both sky ∼ maps.

The 4,2-panel in Fig. 4.3 shows the resulting spherical harmonic power spectrum. We see that the signal-to-noise of the cross-correlation is low for the angular scales considered, which we attribute to both a small sky coverage of DES SV data and the noise level in both maps. Nevertheless we include the power spectrum in our analysis since it provides an upper limit to the cross-correlation of the CMB lensing convergence and the DES SV weak lensing shear ﬁeld.

106 4.5. Spherical harmonic power spectra

] 6000 2

K TT

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⇡ 4000 (2 / 3000 TT `

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T ( 0.8 g g g H ( µ 38 0.06 p 0.6 36 0.04 g [K]

0.4 g 34 0.02 T ` g ` 0.2 C 32 0.00

` C 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 60 65 70 75 80 0.0 z H0 0.2 5 10 0.4 50 100 150 200 3 101 102 ⇥ 9 5 10 10 ⇥ 7 ⇥ 5 T g 6 4 g [K] 3 5  ` T  ` 2 ` C 4 ` C 1 3 0 1 100 200 300 400 60 100 140 180

9 5 6 10 10 10 2.5 ⇥ 1.5 ⇥ 1.5 ⇥ 1T 1g 1 11 2.0 1.0 ) 1.0 ⇡ 5 10 1.5 (2

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 0.5 ` T C ` 0 ` C 0.0 ` C 0.5 + 1) ` 1 ( ` 6 1.0 10 2 1.5 100 200 300 400 500 600 100 200 300 102 6 102 ⇥ ` ` `

Figure 4.3: Measured auto and cross spherical harmonic power spectra along with background probes for this analysis. Starting from the top-left corner, the 0,0-panel shows the CMB temperature anisotropy power spectrum from Planck 2015. The 1,0-panel shows the cross-power spectrum between the galaxy overdensity from the SDSS DR8 CMASS sample and the CMB temperature anisotropies from Planck 2015. The 1,1-panel shows the galaxy clustering power spectrum from the SDSS DR8 CMASS sample. The 2,0-panel shows the cross-power spectrum between the CMB lensing convergence and the CMB temperature anisotropies from Planck 2015. The 2,1-panel shows the cross-power spectrum between the CMB lensing convergence from Planck 2015 and the galaxy overdensity from the SDSS DR8 CMASS sample. The 3,0-panel shows the cross-power spectrum between the weak lensing shear from SDSS Stripe 82 (γ1) and the CMB temperature anisotropies from Planck 2015. The 3,1-panel shows the cross-power spectrum between the weak lensing shear from SDSS Stripe 82 and the galaxy overdensity from the SDSS DR8 CMASS sample. The 3,2-panel shows the cross-power spectrum between the weak lensing shear from SDSS Stripe 82 and the CMB lensing convergence from Planck 2015. The 3,3-panel shows the cosmic shear power spectrum from SDSS Stripe 82. The 4,0-panel shows the cross-power spectrum between the weak lensing shear from DES SV (γ2) and the CMB temperature anisotropies from Planck 2015. The 4,2-panel shows the cross-power spectrum between the weak lensing shear from DES SV and the CMB lensing convergence from Planck 2015. The 4,4-panel shows the cosmic shear power spectrum from DES SV. The gray panel in the upper right corner shows the SNe Ia distance moduli and error bars from the JLA and the Hubble constant measurement from Ref. [74]. All the power spectra have been computed using the maps in Galactic coordinates. The solid lines show the theoretical predictions for the best-ﬁt cosmological model determined from the integrated analysis which is summarized in Tab. 4.3. The theoretical predictions for the power spectra have been convolved with the PolSpice kernels described in Chapter3 and Section 4.5. The error bars for the power spectra are derived from the Gaussian simulations described in Section 4.7 and Appendix 4.C. The angular multipole ranges and binning schemes used for all the power spectra are summarized in Tab. 4.2.

107 Chapter 4. Integrated cosmological probes: extended analysis

4.6 Systematics

Cosmological measurements are generally affected by systematics. We parametrize these using eight different nuisance parameters, which we simultaneously ﬁt with the cosmological parameters. A summary of these parameters can be found in Tab. 4.3 and they are described separately for each cosmological probe below.

4.6.1 CMB temperature anisotropies

The foreground-reduced CMB temperature anisotropy maps contain significant contamination from unresolved extragalactic sources, mainly dusty and radio galaxies [207]. Following Ref. [208], these can be modeled as additional, residual power spectra, which become significant at high angular multipoles. Ref. [208] include two different contributions: a contribution ps of an unclustered Poisson component C` and the contribution of a clustered component cl C` . We study the impact of including these power spectra and find that they do not have a significant impact on our results; see Appendix 4.F for further details. We therefore do not include them in our fiducial analysis.

4.6.2 Galaxy overdensity

The galaxy overdensity ﬁeld is a biased tracer of the underlying dark matter. We account for this uncertainty with a linear galaxy bias parameter b, which relates the galaxy overdensity δg to the dark matter overdensity δ i.e. we set

δ (k,z) b δ(k,z). (4.7) g = In addition, the observed galaxy overdensity is potentially affected by observational and sky systematics. As detailed in Chapter3 we correct the galaxy overdensity map for these systematics. We do not find any significant cross-correlation between the foreground-reduced galaxy overdensity map and systematics that could be common to other probes, such as the extinction map from Ref. [235] as well as a map of stars detected by SDSS with i-band magnitudes 18.0 i 18.5. We therefore do not include nuisance parameters accounting for ≤ < foreground contamination of the galaxy overdensity field into our analysis.

4.6.3 Weak lensing

The estimated weak lensing shear of galaxies γˆ is prone to multiplicative biases. We parametrize these potential unaccounted calibration uncertainties using a scalar multiplicative bias parameter deﬁned as

γˆ (1 mi )γ, (4.8) = + calib

108 4.6. Systematics where i [SDSS, DES]. ∈ A further potential contaminant to the observed weak lensing shear signal are intrinsic correlations between the unlensed shapes of galaxies (for reviews see e.g. [141, 273], for observational detections see e.g. [120, 180, 128, 80, 196, 140, 247]). The measured weak lensing shear in the presence of intrinsic alignments is given by

γobs γG γI , (4.9) = + where γG denotes the gravitational and γI the intrinsic part of the shear. To first order, the shapes of galaxies are linearly related to the tidal field in which they form [49]. This gives rise to the so-called linear alignment model [49, 126, 41]. The expected linear alignment signal in this model follows very closely the scale dependence of the weak lensing shear power spectrum as discussed in Appendix 4.A. We therefore choose to model intrinsic alignments as an additional contribution that modifies the amplitude of the weak lensing shear i.e.

γˆ (1 mi mi )γ (1 mi )γ, (4.10) = + calib + IA , + ? where i [SDSS, DES]. In order to reduce the number of free parameters we combine these ∈ i two amplitudes into an effective multiplicative calibration m?. We include a separate effective multiplicative bias parameter for each of the two weak lensing surveys considered in this work.

The dark matter power spectrum at small scales is affected by baryonic processes such as feedback from supernovae and active galactic nuclei (AGN) or gas cooling. We investigate the effects of baryonic feedback on the power spectra using the effective halo model prescription of Mead et al. [186] (for further details see Appendix 4.B). We ﬁnd the effect of baryon feedback to be smaller than the uncertainties on our measurement on all angular scales considered, which means that our data currently is not sensitive to baryonic feedback. Therefore we choose to model the nonlinear matter power spectrum using HALOFIT in this work and leave the investigation of baryonic feedback to future work.

4.6.4 CMB lensing convergence

The CMB lensing potential estimator described in Sec. 4.3.2 has a non-trivial, cosmology- dependent response to an input CMB lensing potential [206, 211]. Ref. [211] correct for this bias assuming a ﬁducial cosmological model [211]. Therefore, the normalization of the CMB lensing convergence estimator is cosmology-dependent and should be varied alongside the cosmological parameters in a sampling process. To reduce computation time we choose an alternative approach and do not take into account the cosmology-dependence of the CMB lensing convergence amplitude as e.g. Ref. [92]. We account for this normalization uncertainty by including a multiplicative bias parameter mκCMB such that

κˆ (1 m )κ , (4.11) CMB = + κCMB CMB

109 Chapter 4. Integrated cosmological probes: extended analysis and we allow it to vary independently from the cosmological parameters. We leave the introduction of a cosmology-dependent CMB lensing convergence amplitude to future work.

4.6.5 SNe Ia

Type Ia supernovae are not perfect standard candles since their absolute peak magnitude depends on the duration of the SNe explosion, as measured using the stretch parameter X1, the color C of the SNe and the properties of the host galaxy. In order to take these effects into account we parametrize the observed distance modulus following Ref. [34] as:

µ m∗ (M αX βC), (4.12) = B − B − 1 + where mB∗ denotes the observed peak magnitude in rest frame B-band. The parameters α,β and MB are nuisance parameters accounting for the uncertainties in the absolute peak SNe Ia magnitude. Both the absolute magnitude parameter MB and the parameter β were found to depend on the properties of the supernova’s host galaxy [257, 143]. In order to take these effects into account, we follow Ref. [34] and set  1 10 MB if Mstellar 10 M , MB < ¯ (4.13) = M 1 ∆M otherwise. B +

1 This parametrization thus ﬁnally gives rise to four different nuisance parameters α,β,MB and ∆M.

4.7 Covariance matrix

In order to compute constraints on cosmological parameters from the 12 power spectra described above, we need to estimate the joint covariance matrix of these probes. We follow Chapter3 and assume the covariance matrix to be Gaussian. This assumption is justified for the CMB lensing convergence field, as shown by e.g. Ref. [276]. As discussed in Chapter3 this is also appropriate for the CMB temperature anisotropy and galaxy density fields at the scales considered but it is only an approximation for the weak lensing shear field. We expect this to be a reasonable approximation since we do not include small angular scales in our analysis and our uncertainties on the cosmic shear power spectrum are dominated by shape noise. We therefore leave the issue of non-Gaussian covariance matrices to future work.

Following Chapter3 we compute the covariance matrix employing two different methods: the ﬁrst is based on a theoretical prediction of the covariance matrix while the second is an empirical method based on Gaussian simulations of the cosmological probes considered in this work. As in Chapter3, we use the empirical covariance matrix for our ﬁducial analysis and we compute it using 1000 Gaussian simulations, which are described in Appendix 4.C. We validate the empirical covariance matrix using the theoretical prediction. A more detailed

110 4.8. Parameter inference

TT Tδg δgδg Tκ δgκ Tγ1 γ1δg γ1κ γ1γ1 Tγ2 κγ2 γ2γ2 610

γ2γ2 0.90 70 370 0.45 κγ2 70 0.15 610

Tγ2

70 610

γ1γ1

70 370 γ1κ 70 210 30 γ1δg 610

Tγ1 0 ` 70 190 δgκ 0.00 40 400 Tκ 40 210

δgδg 30 210 Tδg 30 610

0.15 10 − 10 610 210 210 400 190 610 210 370 610 610 370 610 30 30 40 40 70 30 70 70 70 70 70 `

Figure 4.4: Correlation matrix of the spherical harmonic power spectra derived using the Gaus- sian simulations described in Sec. 4.7. Gray regions are set to zero because they correspond to the covariance between non-overlapping surveys. The binning scheme and angular multipole ranges for each probe follow those given in Tab. 4.2.

description of both methods can be found in Chapter3.

Figure 4.4 illustrates the correlation matrix derived from the sample variance of the 1000 Gaussian simulations. We explicitly set sub-covariance matrices of non-overlapping surveys to zero. These regions are marked in gray in the ﬁgure.

4.8 Parameter inference

To compute cosmological parameter constraints from the data presented in Sections 4.3 and 4.5 we follow Chapter3 and assume the joint likelihood of the 12 spherical harmonic power spectra to be Gaussian i.e.

1 1 obs theor T 1 obs theor 2 (C` C` ) CG− (C` C` ) L (D θ) 1 e− − − , (4.14) d /2 | = [(2π) detCG ]

theor where CG denotes the Gaussian covariance matrix described in Sec. 4.7. C` denotes the theoretical prediction for the spherical harmonic power spectrum vector of dimension d 92 obs = and C` is the observed power spectrum vector, deﬁned as

δ T δ δ κ δ T γ δ Cobs (C TT C g C g g C κCMBT C CMB g C γ1 C 1 g ` = ` ` ` ` ` ` ` γ1κCMB γ1γ1 γ2T γ2κCMB γ2γ2 C` C` C` C` C` )obs. (4.15)

111 Chapter 4. Integrated cosmological probes: extended analysis

As in Chapter3 we neglect the potential non-Gaussian nature of the weak lensing likelihood.

We estimate the joint covariance matrix CG both from simulations as well as analytically as described in Sec. 4.7. We compute it for a fiducial ΛCDM cosmological model with parameter values {h, Ω , Ω , n , σ , τ , T } {0.7, 0.3, 0.049, 1.0, 0.88, 0.078, 2.275K}, where m b s 8 reion CMB = h is the dimensionless Hubble parameter, Ωm is the fractional matter density today, Ωb is the fractional baryon density today, ns denotes the scalar spectral index, σ8 is the r.m.s. 1 of linear matter fluctuations in spheres of comoving radius 8h− Mpc, τreion denotes the optical depth to reionization and TCMB is the mean temperature of the CMB today. We assume no systematic uncertainties in our fiducial model except a linear galaxy bias i.e. SDSS DES {b, m , m , mκCMB , Aps, Acl} {2., 0., 0., 0., 0., 0.}. As described in Chapter3 we employ ∗ ∗ = the corrections described in Refs. [150, 19, 109] to debias the inverse of the empirical covariance matrix and we neglect the cosmology dependence of the covariance in our sampling process.

We further assume a Gaussian likelihood for the SNe Ia distance moduli µ i.e.

1 1 obs theor T 1 obs theor 2 (µ µ ) C − (µ µ ) L (D θ) 1 e− − − , (4.16) | = [(2π)d detC] /2 where d N , µobs is the vector of observed SNe Ia distance moduli and µtheor denotes the = SNe theoretical prediction. The covariance matrix C contains both statistical as well as systematic errors and is constructed following Ref. [34]7. We note that C depends on the values of the nuisance parameters α and β and thus needs to be reevaluated in each step when sampling parameters.

We combine the power spectra together with the SNe Ia and the Hubble constant measurement assuming them to be independent i.e. we multiply the likelihoods. We note that this is an approximation since the residuals of the SNe Ia distance moduli are affected by weak gravitational lensing and redshift space distortions. It will be interesting to investigate these correlations for future surveys (see e.g. [239]).

From the combined likelihood we compute cosmological parameter constraints in the framework of a ﬂat ΛCDM cosmological model. All our constraints are derived assuming a vanishing neutrino mass i.e. Pm 0.00 eV. We sample the likelihood in a Monte Carlo Markov ν = Chain (MCMC) with CosmoHammer [15] varying 14 different parameters. We vary the six cosmological parameters {h, Ωm, Ωb, ns, σ8, τreion} as well as the eight nuisance parameters SDSS DES 1 {b, m , m , mκCMB , α, β, MB, ∆M}, which are described in Sec. 4.6. We note that our ∗ ∗ ﬁducial constraints do not include the nuisance parameters Aps, Acl and we recall that we further neglect several sources of systematic uncertainties such as photometric redshift uncertainties, stochastic and scale-dependent galaxy bias [202, 269, 68] or baryonic effects on the matter power spectrum. Also, we do not model intrinsic alignments but include them as part of our effective multiplicative bias. The data combination considered weakly

7The covariance matrix can be found at: http : //supernovae.in2p3.fr/sdss_snls_jla/ReadMe.html.

112 4.8. Parameter inference constrains the optical depth to reionization and we therefore follow Chapter3 and apply a Gaussian prior of τ 0.089 0.02. As discussed in Chapter3 this corresponds to an reion = ± enlarged WMAP9 prior [124]. We further assume Gaussian priors on the effective multiplicative bias parameters mi , i [SDSS,DES], as mi 0. 0.22. The width of the prior is given 2 i 2 i 2∗ i∈ ∗ = ± i σ (m ) σ (mcalib) σ (mIA). Assuming a maximal calibration uncertainty of σ(mcalib) 0.1 ∗ =i + i i = and σ(mIA) 0.2 gives σ(m ) 0.22. The choice of σ(mcalib) is motivated by Ref. [125], who = ∗ ≈ found the multiplicative bias for the shape measurement method of SDSS Stripe 82 data to lie in the range mSDSS [ 0.08,0.13]. We choose a conservative approach and apply the same calib ∈ − uncertainty to the DES SV galaxy shears, even though the reported calibration uncertainty is σ(mDES ) 0.05 [138]. The choice for σ(mi ) follows from the discussion in Appendix 4.A, calib = IA from which we can see that the contribution of intrinsic alignments amounts to maximally 20% of the measured weak lensing signal in our fiducial model. We nevertheless choose to center the prior on mSDSS, mDES on 0 rather than on the value expected from our fiducial ∗ ∗ model for intrinsic alignments in order to confirm that the data moves the posterior. We allow for intrinsic alignments by broadening the prior as compared to the expectation from only multiplicative calibration uncertainties. For all other parameters we assume flat priors, which are summarized in Tab. 4.3.

We compute all parameter constraints using the covariance matrix derived from the Gaussian simulations. The results using the theoretical covariance matrix or a covariance matrix derived using a cosmological model with parameter values similar to the ones derived for our best-fit8 are shown in the Appendix (Fig. 4.17) and we find them to be consistent with our fiducial results. This confirms that the computation of the empirical covariance matrix has converged with 1000 simulations. It further verifies that our results are not sensitive to the slightly high value of σ8 chosen for our fiducial simulations.

In order to investigate the impact of residual foregrounds on our analysis, we compute constraints on cosmological parameters from CMB temperature data alone both including the parameters Aps, Acl and neglecting them. As can be seen from the Appendix (Fig. 4.18) we find no significant difference between the constraints derived from the extended parameter set and the ones derived ignoring these additional degrees of freedom. This suggests that the low-` CMB temperature anisotropy power spectrum is not affected by these foregrounds. We therefore choose to not include contamination from residual extragalactic point sources into our fiducial analysis.

We further validate our analysis of the CMB temperature anisotropies by comparing our fiducial CMB-only parameter constraints to those obtained from running the official Planck likelihood [212] with nuisance parameters fixed to the best-fit values derived by Ref. [212] (TT+lowP). Figure 4.18 shows the comparison between the CMB temperature constraints derived in this work with the constraints derived from the Planck likelihood for ` 610. We max ' show the results obtained from the Planck likelihood both with ` 10 and ` 30. This min = min =

8 {h, Ωm, Ωb, ns, σ8, τreion, TCMB} {0.699, 0.278, 0.0455, 0.975, 0.799, 0.0792, 2.275K} and SDSS DES = {b, m , m , mκCMB , Aps, Acl} {2.13, 0.142, 0., 0., 0., 0.}. ∗ ∗ = −

113 Chapter 4. Integrated cosmological probes: extended analysis is due to the fact that in the Planck likelihood the angular multipoles for ` 30 are unbinned < while for ` 30 the power spectrum is binned into bins of ∆` 21, which is more comparable ≥ = to our binning scheme of ∆` 30. As can be seen we ﬁnd reasonable agreement between the = derived parameter constraints. This is not the case when we extend the angular multipole range and we therefore use ` 610 as in Chapter3 for our ﬁducial analysis. max =

4.9 Cosmological constraint results

Figure 4.5 shows our integrated cosmological parameter constraints along with those from Chapter3. The associated means of the posterior distributions for all the parameters along with their 68% confidence limits (c.l.) are given in Tab. 4.3. As can be seen from the figure, our constraints with the new data and greater flexibility for systematics agree very well with our previous findings9, demonstrating the robustness of our results. In fact, we also find that the results are robust to removing random probes from our analysis. We note that the slight tension between CMB temperature anisotropies and weak lensing discussed in Chapter3is resolved when accounting for the expected effects of intrinsic alignments. This can be seen from the fact that broadening the prior on the multiplicative calibration bias parameter in Chapter3 recovers a value consistent with the expectation from intrinsic alignments and results in a better fit to the power spectra involving weak gravitational lensing, as discussed in Chapter3. Furthermore we see from Fig. 4.3 that there is no noticeable tension between CMB temperature anisotropies and weak lensing in the extended analysis.

In Fig. 4.5, we also compare our constraints to those derived by the Planck Collaboration [212]. For the latter, we show the constraints derived from the combination of CMB temperature anisotropies with the Planck low-` polarization likelihood (TT+lowP) and the constraints when also including the Planck polarization power spectra, CMB lensing and external data sets

(TT,TE,EE+lowP+lensing+BAO+JLA+H0). While we see a general broad agreement, a global tension is nevertheless apparent in most of the panels. Our analysis indeed continues to prefer a lower value of Ωm,Ωb and σ8 as well as a higher value of h. This tension is at similar levels to what has been found by other groups (e.g. [122] and references therein). Since our analysis includes a subset of the Planck data, the tension appears to be with the Planck data that we have not included, namely low-` polarization data, and CMB temperature anisotropy data for ` [2, 9] and ` [611, 2508], where the latter has the greatest impact. This parameter shift ∈ ∈ induced by the Planck high-` measurements has also been reported and studied by others, including [217] and [11].

Exploring this further, Fig. 4.6 shows the comparison of our integrated analysis with the constraints from WMAP9 [124] and WMAP9 combined with high-` data from the Atacama Cosmology Telescope (ACT) [87, 64] and the South Pole Telescope (SPT) [151, 222]. We also

9In Chapter3 we included massive neutrinos in our fiducial model in contrast to this work, but we find the constraints without massive neutrinos to be consistent. The constraints of Chapter3 with and without massive neutrinos differ at the percent-level in the best-fit value for σ8, with the constraints with massive neutrinos being lower in σ8.

114 4.10. Conclusions show the constraints from the Planck Collaboration [212]. We see that our results are in good agreement with WMAP9 and the combination of WMAP9, ACT and SPT and are consistent with the already highlighted tension with the Planck high-` measurement [11, 217].

In Figure 4.7 we focus on one of the panels, showing the Ω σ plane, and now include the m − 8 results from KiDS [122]. We also see that our results are in good agreement with KiDS.

The constraints on the nuisance parameters varied in our analysis are shown in Figures 4.8 and 4.9. From Tab. 4.3 we see that we find values of the effective weak lensing shear calibration SDSS DES 0.0953 parameter of m 0.229 0.113 and m 0.0708+0.0946. The obtained value for the ∗ = − ± ∗ = − − effective multiplicative bias of SDSS Stripe 82 is broadly consistent with an overall intrinsic alignment contribution of 15% and a calibration uncertainty of around 5%. The effective − multiplicative bias for DES SV is slightly lower than would be expected from our fiducial model, which would suggest a lower limit on mDES of around 10% coming solely from intrinsic ∗ − alignments. This discrepancy could be due to cancellations between the two components of the effective bias. We find a value of the multiplicative bias of the CMB lensing convergence of 0.0941 mκCMB 0.0598+0.0946, which is again consistent with the expectation of mκCMB 0. Finally, = − − = our constraint on the galaxy bias b for the SDSS CMASS1-4 sample is consistent with the mean of the tomographic bias parameters reported in Ref. [129]. We further find the constraints on different nuisance parameters to be only slightly correlated between one another. This shows that the combination of auto and cross correlations can be used for cross-calibration of different surveys and cosmological probes.

The theoretical predictions for the best-ﬁtting cosmological model together with the measured data are shown in Fig. 4.3. As can be seen, the theoretical predictions ﬁt the data rather well and there is no sign of a tension between different data sets.

4.10 Conclusions

In this work, we have extended our framework for integrated cosmological probe analysis presented in Chapter3. We have combined data from CMB temperature anisotropies, galaxy clustering, weak lensing and CMB lensing focusing on two-point statistics and taking into account the cross-correlations between the different probes. We used CMB temperatures from Planck 2015 [209], galaxy clustering data from SDSS DR8 [13], weak lensing data from both SDSS Stripe 82 [20] and DES SV [138] and CMB lensing data from Planck 2015 [211]. We have further included SNe Ia distance moduli from the JLA [34] and Hubble constant measurements from HST [227, 74]. For all the probes of the inhomogeneous Universe we have computed 12 spherical harmonic power spectra and we have combined them into a joint likelihood assuming both a Gaussian covariance matrix as well as a Gaussian likelihood. We have then combined this likelihood with the Gaussian SNe Ia likelihood assuming these two data sets to be independent. We have extended our treatment of systematic uncertainties and relaxed some of the approximations used in Chapter3. Furthermore, we have also studied the impact of intrinsic alignments, baryonic corrections, residual foregrounds in the CMB temperature,

115 Chapter 4. Integrated cosmological probes: extended analysis

Nicola et al., 2016 36 0. This work (integrated analysis)

32 Planck Collaboration 2015, TT + lowP 0. Planck Collaboration 2015, TT, TE, EE + lowP + lensing + ext m 28 Ω 0.

24 0.

051 0.

.048 b 0 Ω 045 0. 042 0.

04 1.

00 1. s n 96 0.

92 0. 90 0.

85 0.

8 80 σ 0.

75 0.

70 0. 12 0. 09 0. reion τ 06 0. 03 0.

.60 .65 .70 .75 .80 .24 .28 .32 .36 042 045 048 051 .92 .96 .00 .04 .70 .75 .80 .85 .90 .03 .06 .09 .12 0 0 0 0 0 0 0 0 0 0. 0. 0. 0. 0 0 1 1 0 0 0 0 0 0 0 0 0 h Ωm Ωb ns σ8 τreion

Figure 4.5: Comparison of the constraints obtained from the integrated analysis from Chapter3 to the constraints obtained in this work and the constraints obtained by the Planck Collaboration [212] using only CMB data (TT+lowP) or adding external data SDSS (TT,TE,EE+lowP+lensing+ext). The constraints from Chapter3 are marginalized over b,mcalib while the constraints from this work are marginalized over all nuisance parameters given in Tab. 4.3. The Planck constraints are marginalized over all nuisance parameters. In each case the inner (outer) contour shows the 68% c.l. (95% c.l.).

116 4.10. Conclusions

This work (integrated analysis) 36 0. Planck Collaboration 2015, TT + lowP

32 WMAP9 0. WMAP9 + SPT + ACT m 28 Ω 0.

24 0.

051 0.

.048 b 0 Ω 045 0. 042 0.

04 1.

00 1. s n 96 0.

92 0. 90 0.

85 0.

8 80 σ 0.

75 0.

70 0. 12 0. 09 0. reion τ 06 0. 03 0.

.60 .65 .70 .75 .80 .24 .28 .32 .36 042 045 048 051 .92 .96 .00 .04 .70 .75 .80 .85 .90 .03 .06 .09 .12 0 0 0 0 0 0 0 0 0 0. 0. 0. 0. 0 0 1 1 0 0 0 0 0 0 0 0 0 h Ωm Ωb ns σ8 τreion

Figure 4.6: Comparison of the constraints obtained in this work to the constraints obtained by the Planck Collaboration [212] using only CMB data (TT+lowP) and the constraints obtained by WMAP9 [124] both using multipoles ` 1200 (WMAP9) and combined with high-` data < from SPT and ACT (WMAP9+SPT+ACT). The constraints from this work are marginalized over all nuisance parameters given in Tab. 4.3. The Planck and the WMAP9 constraints are marginalized over all nuisance parameters. In each case the inner (outer) contour shows the 68% c.l. (95% c.l.).

117 Chapter 4. Integrated cosmological probes: extended analysis

This work (integrated analysis) 1.1 Planck Collaboration 2015, TT + lowP WMAP9 + SPT + ACT KiDS 450

1.0

0.9 8 σ

0.8

0.7

0.6 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Ωm

Figure 4.7: Comparison of the constraints on Ωm and σ8 obtained in this work and the constraints obtained by the Planck Collaboration [212] using only CMB data (TT+lowP), the constraints obtained by WMAP9 [124] combined with high-` data from SPT and ACT (WMAP9+SPT+ACT) and the constraints obtained from cosmic shear measurements from KiDS-450 [122]. The constraints from this work are marginalized over all nuisance parameters given in Tab. 4.3. The Planck, WMAP9+SPT+ACT and KiDS constraints are marginalized over all nuisance parameters. In each case the inner (outer) contour shows the 68% c.l. (95% c.l.).

118 4.10. Conclusions

25 0. 00 0.

SDSS ∗ 25 0. m − 50 0. −

2 0.

0 0. DES ∗

m 2 0. − 4 0. −

25 0.

CMB .00 κ 0 m 25 0. −

165 0.

150 0. α 135 0.

120 0.

4 3.

2 3. β

0 3.

8 2. .00 19 − .04

B 19

M − .08 19 − .12 19 − 03 0. − 06 0. M

∆ − 09 0. − 12 0. − .4 .2 .0 .2 .8 .0 .2 .4 .80 .95 .10 .25 .50 .25 .00 .25 0 0 0 0 .25 .00 .25 120 135 150 165 2 3 3 3 .12 .08 .04 .00 .12 .09 .06 .03 1 1 2 2 0 0 0 0 0 0 0 0. 0. 0. 0. 19 19 19 19 0 0 0 0 SDSS − − DES m b − m− m − κCMB α β − − M−B − − − ∆M− − ∗ ∗

Figure 4.8: Constraints on the nuisance parameters described in Tab. 4.3 obtained in this work. These have been marginalized over all cosmological parameters. The inner (outer) contour shows the 68% c.l. (95% c.l.).

119 Chapter 4. Integrated cosmological probes: extended analysis

16 0.

12 0.

reion 08 τ 0.

04 0.

25 0. 00 0.

SDSS ∗ 25 0. m − 50 0. −

2 0.

0 0. DES ∗

m 2 0. − 4 0. −

25 0.

CMB .00 κ 0 m 25 0. −

80 95 10 25 04 08 12 16 50 25 00 25 .4 .2 .0 .2 25 00 25 1. 1. 2. 2. 0. 0. 0. 0. 0. 0. 0. 0. 0 0 0 0 0. 0. 0. SDSS − − DES b τreion − m− m − mκCMB ∗ ∗

Figure 4.9: Constraints on the nuisance parameters described in Tab. 4.3 affecting the amplitude of the power spectra as well as τreion obtained in this work. These have been marginalized over all remaining parameters. For the linear galaxy bias parameter b, the dashed line and the gray band show the mean and 1σ uncertainty of the tomographic galaxy bias derived in Ref. [129] for the CMASS1-4 sample. The dashed line and the gray band for τreion show the mean and 1σ uncertainty of the adopted prior. While for mSDSS and mDES, the dashed lines and gray bands show the effective multiplicative bias expected∗ from intrinsic∗ alignments together with the 1σ calibration uncertainties for both SDSS Stripe 82 and DES SV. For mκCMB ﬁnally, the dashed line is centered on m 0. The inner (outer) contour shows the 68% c.l. κCMB = (95% c.l.).

120 4.10. Conclusions

Table 4.3: Parameters varied in the MCMC with their respective priors and posterior means. The uncertainties denote the 68% c.l..

Parameter Prior Posterior mean h flat [0.2, 1.2] 0.700 0.014 ∈ ± Ω flat [0.1, 0.7] 0.279 0.015 m ∈ ± Ω flat [0.01, 0.09] 0.0458 0.0015 b ∈ ± n flat [0.1, 1.8] 0.974 0.018 s ∈ +0.017 σ flat [0.4, 1.5] 0.819 −0.029 8 ∈ ± τ Gaussian with µ 0.089, σ 0.02a 0.0787 0.0200 reion = = +0.0199 b flat [1., 3.] 2.09 −0.06 ∈ ± mSDSS Gaussian with µ 0.0, σ 0.22 0.229 0.113 = = − ± m∗DES Gaussian with µ 0.0, σ 0.22 0.0708 0.0953 = = − +0.0946 m ∗ flat [ 0.5, 0.5] 0.0598−0.0941 κCMB ∈ − − +0.0946 α flat [0.1, 0.2] 0.142 −0.007 ∈ ± β flat [2., 4.] 3.11 0.08 1 ∈ ± MB flat [ 25., 10.] 19.06 0.02 ∈ − − − ±0.0230 ∆M flat [ 0.13, 0.01] 0.0711+0.0227 ∈ − − − − a This corresponds to a WMAP9 [124] prior with increased variance to accommodate the Planck results.

and calibration factors for the different power spectra. This extended analysis allows us to derive more robust constraints on the cosmological model and a more thorough test of the consistency between the different probes.

From this analysis, we have computed cosmological parameter constraints for a flat ΛCDM cosmological model marginalizing over eight nuisance parameters. We have made several simplifying approximations throughout this analysis: We have assumed the joint covariance matrix of the considered cosmological probes to be Gaussian. Further, we have only included systematic uncertainties from linear galaxy bias, multiplicative biases due to either calibration or intrinsic alignments in the weak lensing shear, uncertainties in the SNe Ia intrinsic luminosities and multiplicative calibration uncertainties in the CMB lensing convergence. We have not taken into account uncertainties due to photometric redshift errors or additive weak lensing calibration bias and we have not included intrinsic alignment modeling in our analysis. Since we have found our data to be insensitive to baryonic effects on the matter power spectrum and residual foregrounds in the CMB temperature anisotropy maps, we have not included these effects into our analysis. Finally our theoretical predictions are all computed using the Limber approximation. Due to the conservative cuts we have applied on the data we do not believe these effects to significantly affect our conclusions. We find results that are consistent with those presented in Chapter3, which, given our enlarged data set and systematics treatment, confirms the robustness of our analysis and results. Furthermore, we find our data to be well fit by our best-fit cosmological model and we do not see any sign of tension between the data

121 Chapter 4. Integrated cosmological probes: extended analysis sets considered in our analysis.

We also find that our constraints are consistent with those derived by other analyses such as the joint analysis of the WMAP9, SPT and ACT CMB experiments and the KiDS weak lensing survey [122]. Comparing the obtained constraints to those from the Planck Collaboration [212], we find a broad agreement but also tensions in the marginalized constraints in most pairs of cosmological parameters. In particular, we find a lower value of Ωm,Ωb and σ8 as well as a higher value of h. Since our analysis includes low-` Planck CMB temperature data (` [10,610]), the tension appears to arise between the Planck high-` modes and the other ∈ measurements.

We further ﬁnd the constraints on the probe calibration parameters to be largely independent and in agreement with expectation. This shows that these data sets are mutually consistent. This also yields a conﬁrmation of the amplitude calibration of the weak lensing measurements from SDSS, DES SV and Planck CMB lensing from our integrated analysis.

Future cosmological surveys will provide data with unprecedented precision. It will thus be interesting to extend the framework presented here to models beyond ΛCDM as well as extend this framework to include three-dimensional tomographic information, further data or higher-order statistics.

122 4.A. Impact of intrinsic alignments

4.A Impact of intrinsic alignments

The shapes of unlensed galaxies have been found to exhibit signiﬁcant correlations, which are called intrinsic alignments [120, 180, 128, 80, 196, 140, 247]. Since weak lensing shear assumes all correlations between galaxy shapes to be due to gravitational lensing, any measurement of weak lensing will be biased by the presence of intrinsic alignments. These affect both the cosmic shear power spectrum as well as any cross-correlation between LSS probes and the weak lensing shear.

The observed cosmic shear power spectrum in the presence of intrinsic alignments can be written as

,obs C γγ C γγ 2C GI C II, (4.17) ` = ` + ` + ` γγ GI where C` is the cosmic shear power spectrum. C` denotes correlations between the intrinsic alignments of foreground galaxies and the weak lensing shear of background galaxies [126]. These arise because the gravitational field causing the intrinsic alignments is the same as that giving rise to the weak lensing shear and they are called gravitational-intrinsic (GI) correlations. II Finally C` denotes correlations between shapes of neighboring galaxies which arise because these form under the influence of similar tidal gravitational fields [49]. These correlations are termed intrinsic-intrinsic (II) galaxy alignments.

Since intrinsic galaxy alignments are due to the large-scale gravitational ﬁeld in which galaxies form, these will be correlated to any tracer of the LSS. Therefore we should expect intrinsic alignment contributions to the cross-correlation between the weak lensing shear and both the galaxy overdensity and the CMB lensing convergence. The observed cross-correlation between the galaxy overdensity and the weak lensing shear is given by

γδ ,obs γδ Iδ C g C g C g , (4.18) ` = ` + `

γδg Iδg where C` denotes the contribution due to weak lensing and C` is due to intrinsic galaxy alignments.

The cross-correlation between the CMB lensing convergence and the weak lensing shear can be written as

,obs C κCMBγ C κCMBγ C κCMBI, (4.19) ` = ` + `

κCMBγ κCMBI where again C` is the signal coming from weak gravitational lensing and C` denotes the intrinsic alignment contribution.

In order to model these effects and investigate the impact of intrinsic alignments on the spherical harmonic power spectra used in our analysis we follow Refs. [8, 122] and adopt the nonlinear alignment model [126, 41]. In this model the nonlinear intrinsic alignment power

123 Chapter 4. Integrated cosmological probes: extended analysis spectra are parametrized as

2 nl PII F (z)P (k,z), = δδ (4.20) P F (z)P nl (k,z). δI = δδ The function F (z) parametrizes the response of a galaxy shape to an external gravitational tidal ﬁeld and is given by

Ωm F (z) AIAC1ρcrit , (4.21) = − D(z) where we have neglected any redshift- or luminosity dependence of intrinsic alignments. ρcrit is the critical density of the Universe at z 0, D(z) denotes the linear growth factor normalized 14 2 1= 3 to unity today and C1 5 10− h− M − Mpc is a normalization constant. The amplitude of = × ¯ the intrinsic alignments is determined by the free parameter AIA.

In this model the intrinsic alignment spherical harmonic power spectra for the cosmic shear become (see e.g. [139])

Z γ γ µ 1 ¶ II H(z) n (z) n (z) ` /2 C dz PII k + ,z , ` = c χ2(z) = χ(z) (4.22) Z γ ¡ ¢ γ µ 1 ¶ GI W χ(z) n (z) ` /2 C dz PδI k + ,z , ` = χ2(z) = χ(z) where nγ(z) denotes the normalized redshift selection function of the weak lensing survey under consideration.

The intrinsic alignment contribution to the cross-correlation of the galaxy overdensity and the weak lensing shear can be written as [139]

Z δg γ µ 1 ¶ Iδg H(z) n (z) n (z) ` /2 C dz b PδI k + ,z . (4.23) ` = c χ2(z) = χ(z)

Analogously the contribution of intrinsic alignments to the cross-correlation between the CMB lensing convergence and weak lensing shear is given by [105, 272]

κCMB ¡ ¢ γ 1 Z W χ(z) n (z) µ 2 ¶ κCMBI ` / C dz PδI k + ,z . (4.24) ` = χ2(z) = χ(z)

γγ,obs γδg ,obs κCMBγ,obs Figure 4.10 shows the linear alignment contribution to C` ,C` and C` for SDSS Stripe 82 weak lensing shear evaluated for our best-ﬁt cosmological model given in Tab. 4.3 and assuming an intrinsic alignment amplitude A 1 and using HALOFIT. Since the magnitude IA = of intrinsic alignments does not depend on calibration uncertainties or galaxy bias we further assume no systematic uncertainties. The choice of intrinsic alignment amplitude is consistent with the recent results of Ref. [122]. As can be seen from the ﬁgure, the observed spherical

124 4.B. Impact of baryonic processes on the dark matter power spectrum

6 4.0 10− γγ × C` 3.5 γγ GI II ) C` + 2C` + C`

π 3.0 5

(2 10−

/ 2.5 γ γγ g ` δ `

C 2.0 `C 1.5 + 1)

` δgγ

( C

` 1.0 `

δgγ δgI 0.5 C` + C` 6 10− 0.0 102 6 102 50 100 150 200 × ` ` 7 10− 7 × CκCMBγ 6 ` CκCMBγ + CκCMBI 5 ` ` γ 4 CMB κ ` 3 `C 2 1 0 100 200 300 `

Figure 4.10: Impact of intrinsic galaxy alignments on the theoretical spherical harmonic power spectra involving weak lensing data from SDSS Stripe 82 as estimated using the nonlinear alignment model.

harmonic power spectra are signiﬁcantly reduced in the presence of intrinsic alignments. γδg The effect is most pronounced for C` for which the clustering amplitude is reduced by approximately 20%. We can also see that the scale-dependence of the intrinsic alignment power spectra closely follows the weak lensing shear power spectra. In order to reduce computation time for our parameter estimation we therefore choose to model the intrinsic alignments as an additional multiplicative degree of freedom as described in Section 4.6.

We note that the results for DES SV are similar but we choose to show the results for SDSS Stripe 82 because the latter shows a more signiﬁcant impact of intrinsic alignments.

4.B Impact of baryonic processes on the dark matter power spectrum

Numerical simulations have shown that baryonic processes such as gas cooling and feedback from both AGN and supernovae have the potential to signiﬁcantly alter the dark matter power spectrum on small scales (e.g. [275]). These processes are still poorly understood which leads to systematic uncertainties on the matter power spectrum at small scales [275]. Depending on the chosen model for baryonic physics, the matter power spectrum can be reduced by around 1 10% at scales of k 1 Mpc− , which could signiﬁcantly alter the weak lensing power spectrum = 125 Chapter 4. Integrated cosmological probes: extended analysis

) 5 ) π 10− π (2 (2

/ / 5 HALOFIT 1 2 10 γ γ − 1 2

γ γ HM + AGN ` ` C C 6 10− + 1) + 1) ` ` ( ( ` ` 6 10− 7 10− 102 6 102 102 6 102 × × ` `

Figure 4.11: Comparison of the theoretical predictions for the cosmic shear power spectra obtained taking into account baryonic corrections to the matter power spectrum as well as without taking them into account. The left hand panel shows the results for SDSS Stripe 82 while the right hand panel shows the results for DES SV. The theoretical predictions for the power spectra have been convolved with the PolSpice kernels described in Chapter3 and Section 4.5. The error bars are derived from the Gaussian simulations described in Section 4.7 and Appendix 4.C. The angular multipole range and binning scheme is summarized in Tab. 4.2.

due to the broadness of the redshift kernel W γ ¡χ(z)¢.

In order to investigate the impact of baryonic corrections on the power spectra involving weak lensing shear considered in this work we adopt the effective halo model described in Refs. [186, 187]. This optimized halo model has been shown to accurately reproduce the results from the COSMIC EMU dark matter-only simulations [117, 118]. Baryonic effects can be incorporated by adjusting the internal structure of the halos. The two free parameters of the model are the amplitude A of the mass-concentration relation and a halo bloating parameter η0 which controls the density profile of the halo. In order to investigate the impact of baryonic corrections on our analysis we compute the matter power spectrum using the best-fit halo model parameters described in Refs. [186, 187]. We incorporate baryonic effects using the parameters of the AGN model of Ref. [186], which is the model that shows the largest deviations from the dark matter-only results. We then compute the cosmic shear power spectra for the surveys considered in this work and compare them to those obtained when neglecting baryonic effects. The results are shown in Fig. 4.11 for a cosmological model defined by SDSS DES {h, Ωm, Ωb, ns, σ8} {0.71, 0.27, 0.045, 0.97, 0.8} and {m , m } { 0.11, 0.02}. As can = ∗ ∗ = − − be seen from the figure, the amplitude of the cosmic shear power spectra is significantly reduced for small angular scales. On the smallest angular scales considered, the suppression reaches approximately 15%. Comparing the magnitude of this effect to the measured data we see that it is significantly smaller than the size of our error bars and our data is therefore currently not sensitive to baryonic effects. We therefore do not include baryonic corrections into our fiducial model and leave an investigation of those to future work.

126 4.C. Correlated spin-0 and spin-2 ﬁelds

4.C Correlated spin-0 and spin-2 ﬁelds

To validate our analysis as well as to obtain an estimate of the covariance matrix, we need to generate correlated Gaussian realizations of CMB temperature anisotropies T, galaxy overdensity δg , CMB lensing convergence κCMB and two weak lensing shear ﬁelds γ1,γ2 with TT δg T δg δg κCMBT κCMBδg γ1T γ1δg γ1κCMB γ1γ1 auto- and cross-power spectra {C` ,C` ,C` ,C` ,C` ,C` ,C` ,C` ,C` , γ2T γ2κCMB γ2γ2 C` ,C` ,C` }. To this end we extend the algorithm presented in Chapter3, which is based on Refs. [93, 42]. We further improve the algorithm by including several terms we neglected due to their low amplitude in Chapter3.

Due to the sky coverage of the surveys considered in this work, the DES SV weak lensing shear field is not correlated to both the SDSS galaxy overdensity and the SDSS Stripe 82 weak lensing shear field. To take this into account we simulate two separate sets of maps for each weak lensing survey. Both these sets consist of the weak lensing shear maps as well as all the spin-0 maps correlated to those. We then add the spin-0 maps obtained from these two sets of maps to create the final Gaussian realizations.

The ﬁrst set of maps consists of a correlated realization of CMB temperature anisotropies, galaxy overdensity, CMB lensing convergence and the SDSS weak lensing shear ﬁeld and is constructed as follows:

(i) We ﬁrst create three correlated HEALPix maps using synfast in polarization mode with the power spectra

00 C TT C ` /2, ` = EE C γ1 γ1 C ` /3, ` = C BB 0, ` = T C 0E C γ1 . ` = `

These maps are denoted m01, where i {T,γ1,γ2}. i ∈ 1 1

(ii) Following Chapter3 and Refs. [ 93, 42] we then create three maps with a new random seed and the power spectra

δ δ 00 g g δg T C C 2(C )2/C TT, ` = ` − ` ` EE C γ1 γ1 C ` /3, ` = C BB 0, ` = γ δ 0E 1 g δg T γ T C C 2C C 1 /C TT. ` = ` − ` ` ` The second term in the last equation removes unwanted cross-correlations between the spin-0 and spin-2 ﬁelds, which would otherwise arise in this process. These maps are denoted m02, where i {δ ,γ1,γ2}. i ∈ g 1 1 127 Chapter 4. Integrated cosmological probes: extended analysis

(iii) In a next step we create three additional maps with another random seed and power spectra

κ κ 00 C CMB CMB (C κCMB T)2 TT C ` /2 ` /(2C ) ` = − ` − κCMB δg δ δ δg T κCMB T TT 2 g g δg T (C C C /C ) 2(C )2 C TT ` ` ` /(C ` / ` ), ` − ` −

EE C γ1 γ1 C ` /3, ` = C BB 0, ` = 0E γ1κCMB γ1 T κ T C C CMB TT C C ` ` /C ` = ` − ` − γ1 δg κCMB δg δ δ g g δg T C C 2(C )2 C TT /[C ` / ` ]. ` ` ` −

As before, unwanted spin-0/spin-2 correlations are removed by adding the second and third term in the last equation. These maps are denoted m03, where i {κ ,γ1,γ2}. i ∈ CMB 1 1 (iv) We create two spin-0 maps generated with the same seed as used for m01 with the power spectra

00 δg T C 2(C )2/C TT, ` = ` ` 00 (C κCMB T)2 TT C ` /(2C ). ` = `

The map corresponding to the ﬁrst power spectrum is denoted m04 , while the one δg corresponding to the second is called m04 . κCMB

(v) We create a spin-0 map generated with the same seed as used for m02 with the power spectrum

00 κCMB δg δ δ δg T κCMB T TT 2 g g δg T (C C C /C ) 2(C )2 C TT C ` ` ` /(C ` / ` ), ` = ` − ` −

This map is called m05 . κCMB

(vi) Finally we combine the maps i.e.

m0 m01, T = T m0 m02 m04 , δg = δg + δg m0 m03 m04 m05 , κCMB = κCMB + κCMB + κCMB 0 01 02 03 m 1 m 1 m 1 m 1 , γ1 = γ1 + γ1 + γ1 0 01 02 03 m 2 m 2 m 2 m 2 . γ1 = γ1 + γ1 + γ1

We construct a correlated realization of CMB temperature anisotropies, CMB lensing convergence and the DES SV weak lensing shear ﬁeld analogously:

1. We ﬁrst create three correlated HEALPix maps using synfast in polarization mode with

128 4.C. Correlated spin-0 and spin-2 ﬁelds

the power spectra

00 C TT C ` /2, ` = EE C γ2 γ2 C ` /2, ` = C BB 0, ` = T C 0E C γ2 . ` = ` These maps are denoted m˜ 1, where i {T,γ1,γ2}. i ∈ 2 2

2. Following Chapter3 and Refs. [ 93, 42] we then create three maps with a new random seed and the power spectra

κ κ 00 C CMB CMB (C κCMB T)2 TT C ` /2 ` /(2C ), ` = − ` EE C γ2 γ2 C ` /2, ` = C BB 0, ` = 0E γ2κCMB κ T γ2 T C CMB C TT C C ` ` /C . ` = ` − ` These maps are denoted m˜ 2, where i {κ ,γ1,γ2}. i ∈ CMB 2 2

3. We create a spin-0 map generated with the same seed as used for m˜ 1 with the power spectrum

00 (C κCMB T)2 TT C ` /(2C ). ` = ` The map is denoted m˜ 3 . κCMB

4. Finally we combine the maps i.e.

m˜ m1, T = T m˜ m˜ 2 m˜ 3 , κCMB = κCMB + κCMB 1 2 m˜ γ1 m˜ 1 m˜ 1 , 2 = γ2 + γ2 1 2 m˜ γ2 m˜ 2 m˜ 2 . 2 = γ2 + γ2

129 Chapter 4. Integrated cosmological probes: extended analysis

In a last step we combine both sets of maps i.e.

m m0 m˜ , T = T + T 0 mδ m , g = δg m m0 m˜ , κCMB = κCMB + κCMB 0 mγ1 m 1 , 1 = γ1 0 mγ2 m 2 , 1 = γ1

mγ1 m˜ γ1 , 2 = 2 mγ2 m˜ γ2 . 2 = 2

This algorithm yields a set of seven correlated HEALPix maps {mT,mδ ,mκ ,m 1 ,m 2 ,m 1 ,m 2 } g CMB γ1 γ1 γ2 γ2 TT δg T δg δg κCMBT κCMBδg γ1T γ1δg with auto- and cross-power spectra given by {C` ,C` ,C` ,C` ,C` ,C` ,C` , γ1κCMB γ1γ1 γ2T γ2κCMB γ2γ2 C` ,C` ,C` ,C` ,C` }. As in Chapter3, we include the HEALPix window function and CMB temperature beam window function into the spherical harmonic power spectra prior to creating the maps.

Note that it is not possible to create maps with negative power spectra using HEALPix. This means that we always need to make sure that all cross-terms remain positive. This can usually be achieved by applying the counter-terms to the power spectra with larger amplitude.

In order to obtain simulations of observed maps, we need to add realistic noise to the signal- only maps. To this end we follow the implementation in Chapter3 for the CMB temperature anisotropy, the galaxy overdensity and the weak lensing shear maps. As described in more detail in Chapter3, we add noise to the simulations on the map level. For the CMB temperature anisotropies we add the Commander half-mission half-difference (HMHD) map to the signal. The noise maps for the galaxy overdensity consist of randomized maps of the galaxy positions in the data while the weak lensing noise maps are created by rotating the galaxy shears in the data by a random angle. We create noise-only maps for the CMB lensing convergence using the simulated data provided by the Planck Collaboration in Ref. [211]. These data consist of a set of 100 simulations of observed spherical harmonic coefficients of the CMB lensing convergence as well as the spherical harmonic coefficients of the input CMB lensing convergence to each simulation. To compute noise-only maps we first compute the difference between observed and true spherical harmonic coefficients. Using those we create HEALPix maps of the CMB lensing convergence reconstruction noise at resolution NSIDE = 1024. This approach is an approximation for two reasons. Firstly, it is only exactly valid in the limit of linearity in signal and noise. This is not true in this case and we therefore expect to see differences between the estimated noise power spectra and the true one. Comparing these two we find differences of at most 3.5%, which is an acceptable accuracy for covariance matrix computations. Secondly, the CMB lensing noise power spectrum is cosmology dependent since the main source of noise is the disconnected part of the CMB temperature 4-point function, which depends on

130 4.D. Validation of spherical harmonic power spectrum measurement the cosmological model assumed in the simulations. Therefore we should strictly only use these maps for the ﬁducial cosmological model adopted in Ref. [211]. Since we only include cross-power spectra involving the CMB lensing convergence we believe the errors due to assuming a different cosmological model for the theory and the noise to be subdominant. We therefore leave the reﬁnement of the CMB lensing convergence noise estimate to future work.

4.D Validation of spherical harmonic power spectrum measurement

As in Chapter3 we validate our power spectrum measurement using the Gaussian simulations described in Sec. 4.C. We compute N 1000 realizations of signal and noise for our sim = fiducial cosmological model defined by {h, Ωm, Ωb, ns, σ8, τreion, TCMB} {0.7, 0.3, 0.049, 1.0, SDSS DES = 0.88, 0.078, 2.275K} and {b, m , m , mκCMB , Aps, Acl} {2., 0., 0., 0., 0., 0.}. The noise level ∗ ∗ = in the simulations is consistent with the data and we apply the survey masks determined from the data to each simulated map. The spherical harmonic power spectra are computed with the same PolSpice settings as used on the data and the noise bias correction is performed as described in Sec. 4.5. Figures 4.12 and 4.13 show the comparison between the mean of the recovered power spectra and the input theoretical power spectra. The error bars denote the error on the mean of the power spectra and are derived using the Gaussian simulations. We find that the input power spectra are generally recovered rather well, which validates our power spectrum measurement.

4.E Spherical harmonic power spectrum robustness tests

In this section we summarize the robustness tests performed for the measured spherical harmonic power spectra.

4.E.1 Comparison between spherical harmonic power spectra in equatorial and Galactic coordinates

We test that the measured power spectra are independent of the chosen coordinate system by comparing the results obtained from the maps in equatorial and Galactic coordinates. The only power spectrum that can be transformed between coordinate systems is the cosmic shear power spectrum for DES SV and the results are shown in Fig. 4.15. We ﬁnd discrepancies between the two power spectra, which are comparable to the discrepancies found for SDSS Stripe 82 in Chapter3. This suggests that coordinate-dependent bias corrections are not the only explanation for differences between the power spectra but that these differences are partly also due to different shape noise properties in different coordinate systems. Since the discrepancies detected are well within our measurement uncertainties, we do not investigate this issue further and include the spherical harmonic power spectrum measured from the maps in Galactic coordinates into our analysis.

131 Chapter 4. Integrated cosmological probes: extended analysis

] 6000 2 K

µ 5000 )[

π 4000 (2 / 3000 TT `

C 2000

+ 1) 1000 ` ( ` 0 0 100 200 300 400 500 600

8 10− 4 5 × 10− 4 g [K] 3 δ g T δ ` g δ ` 2 C `C 1 5 10− 0 50 100 150 200 3 101 102 × 9 5 10− 10− 5 × 7.0 × 6.5 4 6.0

g 5.5

[K] 3 κδ ` T 5.0 theor. κ ` 2 `C 4.5 reconst. `C 1 4.0 3.5 0 100 200 300 400 60 100 140 180

5 7 9 10− 10− 10− × 8.0 × 0.9 10 7.5 ) 10− 0.8 π

7.0 (2 / 1

11 g

γ 5 κ

[K] 6.5

δ 0.7

10 1 − 1 10

1 − γ ` γ ` γ T `

1 6.0 C γ ` 12 0.6 `C 10− `C 5.5 `C

0.5 + 1) 5.0 13 `

10− ( 0.4 4.5 ` 14 6 10− 0.3 4.0 10− 102 6 102 50 100 150 200 100 200 300 102 6 102 × × ` 6 ` 9 10− 10− 1.2 ×

10 ) 1.1 π 10− (2 /

11 1.0 2 γ κ [K]

10 2

− 2 γ ` γ ` T

2 0.9 5 C γ ` 12 10− 10− `C

`C 0.8 + 1)

13 `

10− (

0.7 ` 14 10− 0.6 102 6 102 100 200 300 102 6 102 × × ` ` `

Figure 4.12: Comparison between input theoretical power spectra and the mean reconstruction derived from N 1000 Gaussian simulations as described in Sec. 4.D. The angular sim = sky coverage and the noise level of the simulations closely match the data. The PolSpice settings used to compute these power spectra are identical to those applied on the data and the angular multipole range as well as binning schemes match those described in Tab. 4.2. Dashed lines denote negative spherical harmonic power spectrum values. The input power spectra have been convolved with the PolSpice kernels described in Chapter3 and Section 4.5. The error bars are derived from the Gaussian simulations described in Section 4.7 and Appendix 4.C.

132 4.E. Spherical harmonic power spectrum robustness tests

7 6 10− 10− 6 × 1.2 × 5 1.0 )

π 4 0.8 (2

/ 3 0.6 BB `

C 2 0.4 reconst. 1 0.2 + 1) `

( 0 0.0 ` 1 0.2 − − 2 0.4 − 100 200 300 400 500 600 − 100 200 300 400 500 600 ` `

Figure 4.13: Mean of the reconstructed cosmic shear B-modes derived from N 1000 sim = Gaussian simulations as described in Sec. 4.D. The angular sky coverage and the noise level of the simulations closely match the data. The PolSpice settings used are identical to those applied on the data and the angular multipole range as well as binning schemes match those described in Tab. 4.2. The left hand panel shows the results for the SDSS Stripe 82 simulations, while the right hand panel shows the results for the DES SV simulations. The error bars are derived from the Gaussian simulations described in Section 4.7 and Appendix 4.C.

5 10− 2.0 × 1.5 )

π 1.0 (2

/ 0.5 BB `

C 0.0 0.5 + 1) − `

( 1.0 ` − 1.5 − 2.0 − 100 200 300 400 500 600 `

Figure 4.14: Spherical harmonic B-mode power spectrum of DES SV. The error bars are derived from the Gaussian simulations described in Section 4.7 and Appendix 4.C. The angular multipole range and binning scheme used is summarized in Tab. 4.2.

133 Chapter 4. Integrated cosmological probes: extended analysis

gal.

) eq. π (2

/ 5 2 10− γ 2 γ ` C + 1) ` ( `

6 10− 102 6 102 × `

Figure 4.15: Comparison of cosmic shear power spectrum from DES SV as measured using the maps in Galactic and equatorial coordinates. The theoretical prediction for the power spectrum has been convolved with the PolSpice kernels described in Chapter3 and Section 4.5. The error bars are derived from the Gaussian simulations described in Section 4.7 and Appendix 4.C. The angular multipole range and binning scheme used is summarized in Tab. 4.2.

4.E.2 Comparison between spherical harmonic power spectra measured from different foreground-reduced CMB temperature maps

We investigate the impact of our choice of ﬁducial foreground-reduced CMB temperature anisotropy map by comparing the power spectra involving CMB temperature data obtained using the four foreground reduction algorithms employed by the Planck Collaboration [210]. γ2T κCMBT These are Commander, NILC, SEVEM and SMICA and the results for C` and C` are shown in Fig. 4.16. As can be seen, we ﬁnd the derived power spectra to be well consistent with each other.

4.F Impact of unresolved foregrounds on CMB temperature anisotropies

As described in Sec. 4.6, the foreground-reduced CMB temperature anisotropy maps are contaminated by unresolved extragalactic sources [207]. Following Ref. [208], the power spectra of these foregrounds can be modeled using a contribution of an unclustered Poisson ps cl component C` and the contribution of a clustered component C` [208]. These become signiﬁcant at high angular multipoles. The two power spectra are deﬁned in terms of their amplitudes Aps and Acl as:

ps Aps C 2π , ` = ` (` 1) p p + µ ¶0.8 (4.25) Acl ` C cl 2π , ` = `(` 1) ` + p 134 4.F. Impact of unresolved foregrounds on CMB temperature anisotropies

9 9 10− 10− 3 × × γ2T 5 κCMBT 2 4

[K] com. 1 [K] T 3 sevem T γ ` CMB

κ 2 0 ` smica `C

`C 1 nilc 1 − 0 2 1 − 100 200 300 400 500 600 − 100 200 300 400 ` `

γ2T κCMBT Figure 4.16: Comparison of the spherical harmonic power spectra C` ,C` obtained from the foreground-reduced CMB temperature anisotropy maps derived using the four different algorithms Commander, NILC, SEVEM and SMICA. The theoretical predictions for the power spectra have been convolved with the PolSpice kernels described in Chapter3 and Section 4.5. The error bars are derived from the Gaussian simulations described in Section 4.7 and Appendix 4.C. The angular multipole ranges and binning schemes used for all the power spectra are summarized in Tab. 4.2.

where the Pivot angular multipole is deﬁned to be `p 3000 and both amplitudes have units of 2 `(` 1)= ps cl µK . The normalization ensures that for D` + C` we have D Aps and D Acl. We = 2π `p = `p = investigate the impact of these residual foregrounds by comparing the cosmological parameter constraints obtained from CMB temperature data alone both including these two additional degrees of freedom and neglecting them. The constraints obtained in both cases are virtually the same as can be seen from Fig. 4.18, which means that the low-` temperature anisotropy power spectrum is insensitive to residual foregrounds. We therefore do not include these additional degrees of freedom in our ﬁducial analysis.

135 Chapter 4. Integrated cosmological probes: extended analysis

This work (integrated analysis), ﬁd. This work (integrated analysis), theor. cov. This work (integrated analysis), cosm. 1

33 0.

.30

m 0 Ω 27 0. 24 0.

051 0.

.048 b 0 Ω 045 0. 042 0.

04 1.

00 1. s n 96 0.

92 0. 90 0.

85 0. 8 σ 80 0.

75 0.

12 0. 09 0. reion τ 06 0. 03 0.

.66 .69 .72 .75 .24 .27 .30 .33 042 045 048 051 .92 .96 .00 .04 .75 .80 .85 .90 .03 .06 .09 .12 0 0 0 0 0 0 0 0 0. 0. 0. 0. 0 0 1 1 0 0 0 0 0 0 0 0 h Ωm Ωb ns σ8 τreion

Figure 4.17: Comparison of the constraints obtained from the ﬁducial joint analysis to the constraints obtained using the theoretical covariance matrix and the constraints obtained assuming a different cosmological model to estimate the empirical covariance. The constraints are marginalized over all nuisance parameters given in Tab. 4.3. In each case the inner (outer) contour shows the 68% c.l. (95% c.l.).

136 4.F. Impact of unresolved foregrounds on CMB temperature anisotropies

Planck 2015 CMB (this work) no nuisance params. Planck 2015 CMB (this work) nuisance params. .5 0 Planck 2015 likelihood, `min = 30

.4 Planck 2015 likelihood, `min = 10 m 0 Ω 3 0.

2 0.

07 0. 06 0. b

Ω 05 0. 04 0. 03 0. 04 1.

.00

s 1 n 96 0.

92 0.

05 1. 8

σ 90 0.

75 0.

16 0.

12 0.

reion .08 τ 0

04 0.

60 65 70 75 80 .2 .3 .4 .5 03 04 05 06 07 92 96 00 04 75 90 05 04 08 12 16 0. 0. 0. 0. 0. 0 0 0 0 0. 0. 0. 0. 0. 0. 0. 1. 1. 0. 0. 1. 0. 0. 0. 0. h Ωm Ωb ns σ8 τreion

Figure 4.18: Comparison of the constraints obtained from CMB temperature anisotropies for angular multipoles ` [10,610] in this work both including the additional nuisance parameters ∈ Aps, Acl and neglecting them. Also shown are the constraints obtained from the Planck likelihood with ` 610 and ` 10 as well as ` 30. In each case the inner (outer) max ' min = min = contour shows the 68% c.l. (95% c.l.).

137 Chapter 4. Integrated cosmological probes: extended analysis

1.6 W γ(χ(z)) γ 1.2 n (z) ) z

( 0.8 n

0.4

0.0 0.0 0.3 0.6 0.9 1.2 1.5 1.8 z

Figure 4.19: Redshift selection function and weak lensing shear window function for the DES SV galaxies. The weak lensing window function has been rescaled relative to the redshift selection function. The window function has been derived using our best-ﬁt cosmological parameters.

138 5 Integrated cosmological probes: concordance quantiﬁed

A wise man proportions his beliefs to the evidence. — David Hume, An enquiry concerning human understanding

This chapter appeared in a similar form as Nicola, Amara & Réfrégier, 2017 [192].

5.1 Introduction

In recent years, many large cosmological surveys have been conducted and many more are currently in the planning [7, 5, 1, 2, 3, 4, 6]. These surveys have allowed us to constrain our cosmological model, leading to the establishment of ΛCDM (see e.g. Ref. [212] and references therein). An important way to further test our cosmological model is to assess if different cosmological probes are consistent within the same model. Inconsistencies might be a sign for tensions within the model assumed and systematics in the measurements considered.

In Chapters3 and4, we computed constraints on cosmological parameters for a ΛCDM cosmological model from an integrated analysis (IA) of Cosmic Microwave Background (CMB) temperature anisotropies and CMB lensing maps from the Planck mission [209], galaxy clustering from the eighth data release of the Sloan Digital Sky Survey (SDSS DR8) [13], weak lensing from both SDSS Stripe 82 [20] and the Dark Energy Survey (DES) Science Veriﬁcation (SV) data [138] as well as Type Ia supernovae (SNe Ia) data from the joint light curve analysis (JLA) [34], and constraints on the Hubble parameter from the Hubble Space Telescope (HST) [227, 74]1. Comparing these results to those obtained by the Planck Collaboration [212] and

1We note that the latter two analyses are not completely independent since they both contain low redshift SNe Ia from CfAIII [121]. In this work however, we do not take into account this correlation and treat the two analyses as independent. We thank our referee for pointing this out.

139 Chapter 5. Integrated cosmological probes: concordance quantiﬁed those from WMAP9 [124], we found a good agreement between the IA constraints and those from WMAP9 while the marginalized 2D contours suggested hints of tensions with the results from Planck 2015. Impressions gained through 2D contours however, can be misleading as shown by e.g. Ref. [241]. In this work, we therefore extend our previous analysis (see Chapters 3 and4) by quantifying the concordance between the results derived from these different data sets. Since for consistency with Ref. [212], the IA does not include the latest Hubble parameter measurement presented in Ref. [228], we further investigate the impact of our choice of Hubble parameter measurement on the concordance between the data sets considered.

Several different concordance measures have been proposed in the literature (see e.g. [130, 161, 220, 54, 171]). In the present work, we employ the Surprise statistic, which was introduced in Ref. [240] and is based on the Kullback-Leibler divergence [160] or relative entropy. The Surprise has been applied to data from the CMB, Large Scale Structure (LSS) as well as background probes [240, 241, 101, 100, 284] and has been shown to be a robust measure for concordance in cosmology. In the framework of a ΛCDM cosmological model we use the Surprise to compare the constraints obtained from the integrated analysis in Chapter4 to several other data sets under the assumption of Gaussianity of all posterior distributions. We consider the constraints derived by the Planck Collaboration [212], those derived from the combination of WMAP9 data [124] with data from the Atacama Cosmology Telescope (ACT) [87, 64] and the South Pole Telescope (SPT) [151, 222] as well as the recent update to the WMAP9+SPT+ACT constraints from Ref. [45].

This chapter is organized as follows. We review the Surprise statistic in Section 5.2 while Section 5.3 describes the data sets used in this work. Our results are presented in Section 5.4 and we conclude in Section 5.5. Robustness tests are deferred to the Appendix.

5.2 Assessing concordance

In order to quantify the concordance between different cosmological parameter constraints we employ the Surprise statistic [240]. The Surprise S is based on the Kullback-Leibler (KL) divergence or relative entropy [160] DKL of two probability distribution functions (pdf) p1(θ), p2(θ), which is given by Z p2(θ) DKL(p2 p1) dθ p2(θ)log . (5.1) || = p1(θ)

The KL-divergence gives a measure of information gain from a prior distribution p1(θ) to a posterior distribution p2(θ) in a Bayesian framework [61]. The Surprise is then deﬁned as the difference between the measured relative entropy D and its expectation value D , KL 〈 KL〉 i.e. S D D . The quantity D represents the expectation for the posterior based = KL − 〈 KL〉 〈 KL〉 on the prior and the likelihood for the new experiment; for a more detailed description, see Ref. [240]. By construction, the Surprise is expected to scatter around zero. A large positive Surprise suggests that the posterior is more different from the prior than expected a priori

140 5.2. Assessing concordance while a negative Surprise suggests an agreement that is better than expected. When the base of the logarithm in Eq. 5.1 is 2, the relative entropy and the Surprise are measured in bits.

In cosmology, there are two main situations in which to update prior knowledge. The ﬁrst case consists of an update of the parameter constraints using uncorrelated or weakly correlated data. In this case we can obtain the updated distribution by simply multiplying the prior distribution with the likelihood of the new data. Following Ref. [240] we denote this case as ‘add’. The second case consists in updating parameter constraints using correlated data. Such data sets need to be combined in a joint analysis, which takes the cross-correlations of the data into account. However, if two data sets are strongly correlated and the posterior is superior to the prior, it can be more efﬁcient to replace the prior constraints with the posterior constraints. As in Ref. [240] we refer to this second case as ‘replace’. In these two cases, Ref. [240] showed that the Surprise can be computed analytically from the moments of samples of the pdfs when assuming all distributions to be Gaussian, and the model to be linear in the parameters. As shown in Ref. [240], these are reasonable approximations for CMB data sets. In these limits, the Surprise for an update from a prior p (θ) N(θ,θ ,Σ ) to a posterior p (θ) N(θ,θ ,Σ ) 1 = 1 1 2 = 2 2 is given by [240]

1 T 1 1 1 S DKL DKL (θ2 θ1) Σ1− (θ2 θ1) [tr(Σ1− Σ2) d] add, = − 〈 〉 = 2 − − + 2 − (5.2) 1 T 1 1 1 S DKL DKL (θ2 θ1) Σ− (θ2 θ1) [tr(Σ− Σ2) d] replace, = − 〈 〉 = 2 − 1 − − 2 1 + where N(θ,θi ,Σi ) denotes a d-dimensional Gaussian distribution with mean θi and covariance matrix Σ . The relative entropy D is expected to ﬂuctuate around D with a variance i KL 〈 KL〉 given by [240]

2 1 1 2 σ (DKL) tr((Σ− Σ2 1) ), (5.3) = 2 1 ± where the holds when replacing correlated data, while the holds when adding comple- + − 2 mentary data. The relative entropy DKL follows a generalized chi-squared (χ ) distribution, which allows us to compute the signiﬁcance of the obtained Surprise as shown in Ref. [240].

In order for Eq. 5.2 to be a reasonable approximation to the true Surprise, the distributions considered need to be well-approximated by multivariate Gaussians. Assessing Gaussian- ity can be complicated in high-dimensional parameter space and we therefore choose to Gaussianize all samples before computing the Surprise with Eq. 5.2. A common Gaussian- ization method is the so-called Box-Cox transformation [39]. In this work, we consider the one-parameter Box-Cox transformation, which for a sample of a d-dimensional parameter set θ (θ ), i [1,n ], k [1,d] is deﬁned as [39]: = ik ∈ samp ∈  1 λk λ− (θ 1) if λk 0, 0(λk ) k ik − 6= θik (5.4) = log(θ ) if λ 0. ik k =

141 Chapter 5. Integrated cosmological probes: concordance quantiﬁed

This transformation is deﬁned only for θik 0. The parameters λk denote the Box-Cox pa- > 0(λk ) rameters that render the 1D distribution of the transformed θik approximately Gaussian. To perform all Box-Cox transformations, we use the python library scipy2. In this implementation, the optimal parameters λk are determined by maximizing the likelihood for the transformed sample to be Gaussian. For a more detailed discussion of Box-Cox transformations the reader is referred to Refs. [39, 277, 140, 236].

Since even the optimal Box-Cox transformation does not always result in sufﬁciently Gaussian distributions we follow a similar approach to Refs. [236, 100] and iteratively Gaussianize the chains. We compute the Surprise for an update from the prior p1(θ) to the posterior p (θ) from the respective pdf samples θ1 (θ1 ), i [1,n ], j [1,d] and θ2 (θ2 ), i 2 = i j ∈ samp,1 ∈ = i j ∈ [1,n ], j [1,d] with the procedure outlined in Algorithm1. samp,2 ∈ Algorithm 1 Algorithm used to compute the relative entropy. for i N do ≤ it if i 1 then > Determine the linear transformation ψ that standardizes the prior sample θ1. θ10 ψ(θ1) ← Apply ψ to the posterior sample θ2. θ20 ψ(θ2) ← end if Transform to positive parameter values for k d do ≤ 10 20 ak minll (θ ,θ ) = 0 lk l 0k if a 0 then k < θ10 θ10 a , l [1,n ] lk ← lk − k ∈ samp,1 θ20 θ20 a , l [1,n ] lk ← lk − k ∈ samp,2 end if end for Find the optimal one-parameter Box-Cox transformation φ for θ10 (Eq. 5.4). θ100 φ(θ10 ) ← Apply the transformation φ to θ20 θ200 φ(θ20 ) ← Compute D , D ,S,σ(D ) for θ100 ,θ200 . KL 〈 KL〉 KL end for

In the ﬁrst step, we linearly transform θ1 into the eigenbasis of its correlation matrix, which yields the linear transformation ψ. The transformed samples θ10 are therefore approximately uncorrelated and have zero mean and unit standard deviation (see e.g. Ref. [241]). In the second step we apply the same transformation ψ to the samples θ2; the results of this transformation are denoted as θ20 3. As noted above, the Box-Cox transformation is only deﬁned for

2Specifically, we use the scipy method scipy.stats.boxcox. 3We do not perform the standardization step in the first iteration, since it can result in significantly non- Gaussian constraints. This can cause difficulties in the convergence of our algorithm or it can even cause it to fail. Nevertheless, in order to obtain numerically stable results for the update CMB set P15 in the ΛCDM →

142 5.3. Data positive values. In step 3, we therefore shift the new parameters such that each value is positive. We repeat this procedure for N iterations while monitoring the values of D , D ,S and it KL 〈 KL〉 σ(DKL). We terminate the iteration as soon as the entropy values reach a steady state. In order to compute the entropy values we use the publicly available Surprise package4 described in Ref. [241]. We monitor the Gaussianity of the chains in each iteration by testing if the Mahalanobis distance [178] of the samples is distributed as a χ2 variable [183, 95, 281], which is a good approximation for the sample sizes used in this work [142]. In the Appendix we show an example of the performed Gaussianity tests.

Using the above prescription, we compute the Surprise between the different data sets for a 2 2 ﬁducial parametrization of the ΛCDM cosmological model given by {h, Ωch , Ωbh , ns, As, τreion}. In this notation, h is the dimensionless Hubble parameter, Ωc is the fractional cold dark matter density today, Ωb is the fractional baryon density today, ns denotes the scalar spectral index, As is the primordial power spectrum amplitude at a pivot scale of k0 0.05 1 = Mpc− and τreion denotes the optical depth to reionization. In order to assess the stability of our results, we also repeat the above computations for the following two reparametrizations of

ΛCDM: {h, Ωm, Ωb, ns, σ8, τreion}, {h, Ωm, Ωb, ns, As, τreion}, where Ωm denotes the fractional 5 matter density today and σ8 is the r.m.s. of linear matter ﬂuctuations in spheres of comoving 1 radius 8h− Mpc. This allows us to test the independence of the Surprise statistic on the parametrization of the cosmological model [160, 240] as well as to test if the assumptions of Gaussianity of the pdf samples and linearity of the model parameters are appropriate. An additional way to test the Gaussianity of the CMB constraints is to make use of the so-called CMB Gaussian parameters derived in Ref. [58]: these parameters exhibit an approximately Gaussian likelihood and are obtained through a nonlinear transformation of the cosmological

2 2 2τreion 1 p 2 ns 1 parameters. Explicitly, they are given by {Ωch , Ωbh , θ?, Ase− ,zreion, / Ωbh 2 − }, where zreion is the reionization redshift and θ? denotes the angular size of the sound horizon at the redshift z? for which the optical depth in the absence of reionization equals unity. Since these parameters are approximately Gaussian distributed for CMB data [58], the relative entropies in this parameter set should be well approximated by Eq. 5.2. When comparing CMB data sets we therefore additionally compute the relative entropies for the CMB Gaussian parameters6.

5.3 Data

In our analysis, we consider the constraints derived from the integrated analysis of Chapter 4, the constraints derived by the Planck Collaboration [212] (TT+lowP) in their second data release as well as those derived from the combination of WMAP9 data [124] with high-`

parametrization with σ8 we do perform the standardization step in the ﬁrst iteration, because the algorithm fails otherwise. We will come back to this case in Sec. 5.4. 4https://github.com/seeh/surprise 5 The parameters Ωm and Ωc are related by Ωm Ωc Ω Ω , where Ω is the fractional non-cold = + b + ncdm ncdm dark matter density today. 6 We note that these parameters were not available to us for P15∗ and we therefore did not perform this additional test for CMB comparisons involving P15∗.

143 Chapter 5. Integrated cosmological probes: concordance quantiﬁed data from ACT [87, 64] and SPT [151, 222]. We further include the recent update to the WMAP9+SPT+ACT constraints from Ref. [45]. For all these data sets except the IA, we use the publicly available parameter chains from the PLANCK LEGACY ARCHIVE7 and LAMBDA8.

We further complement these data sets by computing cosmological parameter constraints in the framework of a ΛCDM cosmological model from the publicly available Planck CMB temperature likelihood for ` [2,2508]. We sample the likelihood in a Monte Carlo Markov ∈ Chain (MCMC) with CosmoHammer [15] and vary the six cosmological parameters {h, Ωc, Ωb, ns, As, τreion}. In addition to the cosmological parameters, we also vary the 15 nuisance parameters employed by the Planck Collaboration [212] along with the respective Gaussian priors. Since the optical depth to reionization τreion cannot be constrained from only CMB temperature data, we follow Ref. [45] and include the recent Planck measurement of τreion [216] by assuming a Gaussian prior of τ 0.06 0.01. For the computation of theoretical reion = ± CMB power spectra, we use the Boltzmann code class9 [36, 164]. Following Ref. [212], our fiducial model contains two massless neutrinos and one massive neutrino with the fixed minimal mass of Pm 0.06 eV. ν = In order to assess the consistency between the constraints derived in Chapter4 and those from the Planck Collaboration [212], we further compute constraints from the IA combined with Planck CMB temperature data for ` [802,2508]. Since the IA contains Planck CMB tem- ∈ perature data up to a maximal angular multipole ` 610 and cross-correlations between max = CMB temperature anisotropies and other cosmological probes are mostly confined to large angular scales, we assume these two datasets to be independent and perform importance sampling to derive the joint constraints. In practice, we first compute the value of the Planck likelihood for all the points in the IA chains, fixing the nuisance parameters to their best-fit values derived in Ref. [212] (TT+lowP). We then use these likelihood values as weights for the IA chains, which results in chains weighted according to the product of both the IA and the high-` Planck likelihood.

All these constraints are derived assuming ﬂat priors on all cosmological parameters except the optical depth to reionization τreion. These priors are not identical for different data sets but they are all signiﬁcantly broader than the obtained posterior constraints and they can therefore be approximated as equal over the support of the posterior distributions. A summary of the data sets used in this work can be found in Tab. 5.1.

5.4 Results

Using the procedure outlined in Sec. 5.2, we compute the estimates of the relative entropy for the different data sets considered. We compute all quantities from moments of the Monte 2 2 Carlo Markov chain samples, choosing a ﬁducial ΛCDM parametrization of {h, Ωch , Ωbh ,

7http://pla.esac.esa.int/pla/#cosmology 8https://lambda.gsfc.nasa.gov/product/ 9http : //class-code.net.

144 5.4. Results

Table 5.1: Overview of the data sets considered in this work.

Acronym Description Reference Integrated analysis with WMAP9 τreion-prior, Chapters3 and IA τ 0.089 0.02 (Pm 0.0 eV) 4 reion = ± ν = Integrated analysis with Hubble parameter Chapters3 and IA* measurement from R16, 1 1 4,[228] H 73.24 1.74 km s− Mpc− 0 = ± Constraints from Planck Collaboration 2015, P15 [212] TT+lowP (Pm 0.06 eV) ν = Constraints from WMAP9+SPT+ACT (Pm 0.0 CMB set ν = [124] eV)

Updated constraints from WMAP9+SPT+ACT CMB set∗ with Planck HFI τreion-prior, [45, 216] τ 0.06 0.01 (Pm 0.06 eV). reion = ± ν = Constraints from Planck TT likelihood P15∗ for ` [2,2508] and Planck HFI τ -prior, [212, 216] ∈ reion τ 0.06 0.01 (Pm 0.06 eV). reion = ± ν = This chapter, IA+P15 hi-`, Constraints from combination of IA with Planck TT ∗∗ Chapters3 and massless ν likelihood for ` [802,2508] (Pm 0.0 eV). ∈ ν = 4,[212]

This chapter, IA+P15 hi-`, Constraints from combination of IA with Planck TT ∗∗ Chapters3 and massive ν likelihood for ` [802,2508] (Pm 0.06 eV). ∈ ν = 4,[212]

145 Chapter 5. Integrated cosmological probes: concordance quantiﬁed

ns, As, τreion}. Where possible, we make use of the publicly available parameter chains. The numerical values for the estimated entropies are given in Tab. 5.2, where we show the entropies computed in our ﬁducial ΛCDM parametrization together with the results for {h, Ωm, Ωb, ns, σ8, τreion} in parentheses. For the ﬁrst two comparisons in the table we further show the results

2 2 2τreion 1 p 2 ns 1 obtained using the CMB Gaussian parameters {Ωch , Ωbh , θ?, Ase− ,zreion, / Ωbh 2 − } [58] without performing any Gaussianization. The constraints for a subset of the data sets considered are shown in Figures 5.1 and 5.2 for the former two parametrizations of the ΛCDM cosmological model.

We ﬁrst consider the relative entropy between WMAP9+SPT+ACT (denoted CMB set) [124] and Planck 2015 (TT+lowP) (denoted P15) [212]. These two data sets are strongly correlated with one another, especially on large angular scales. We therefore replace the WMAP9+SPT+ACT constraints with those from Planck 2015 (TT+lowP) when computing the relative entropy. As can be seen from Tab. 5.2 we ﬁnd a negative Surprise (S 2.3, σ(D ) 5.5), suggesting a = − KL = slightly better agreement between the two data sets than expected a priori. This was also seen in Ref. [241] for a replacement of WMAP9 with Planck 2015 temperature and polarization data.

When computing the relative entropy in the ΛCDM parametrization with σ8 on the other hand, we find significantly different results (S 5.1, σ(D ) 4.5). This appears surprising = KL = at first, since the relative entropy is invariant under transformations of the model parameters. Furthermore, the computation of the Surprise obtained with σ8 becomes numerically unstable. This behavior is due to the differences in the fiducial neutrino model assumed by WMAP9+SPT+ACT and Planck 2015: the WMAP9+SPT+ACT constraints are derived assuming massless neutrinos, whereas the fiducial model used by the Planck Collaboration includes two massless and one massive neutrino with Pm 0.06 eV. Being a parameter of the early ν = Universe, the constraints on As are not significantly affected by a different neutrino model. The constraints on σ8 on the other hand are sensitive to the mapping between early and late Universe and are therefore significantly affected by differences in the neutrino model. When we compare the constraints from WMAP9+SPT+ACT and Planck 2015 in σ8, we therefore compare two different fiducial models with each other and this is the reason for the discussed instabilities in the relative entropy (see also Sec. 5.2). In other words, the relative entropy is not invariant under inconsistent transformations of the considered prior and posterior and is therefore sensitive to inconsistencies in the models that are used in different parts of the analysis. As discussed in Sec. 5.2, we additionally compute the relative entropies using the CMB Gaussian parameters [58] in order to test the Gaussianization method. The results obtained are similar to those obtained in our fiducial ΛCDM parameterization, thus suggesting that the procedure described in Sec. 5.2 yields chains sufficiently Gaussian for our purpose.

Ref. [45] recently computed updated WMAP9+SPT+ACT constraints (denoted CMB set∗) assuming a single family of massive neutrinos with Pm 0.06 eV.We compare these constraints ν = to those from Planck 2015 (TT+lowP) by computing the relative entropy when replacing the revised WMAP9+SPT+ACT constraints with those from Planck 2015. As can be seen from Tab. 5.2, we ﬁnd these two data sets to be in good agreement (S 1.0, σ(D ) 6.4), consistent with the = KL = results presented in Ref. [45]. Furthermore, the results are insensitive to the parametrization

146 5.5. Conclusions of the ΛCDM cosmological model. This confirms that the discrepancies discussed before are due to the different neutrino models assumed and additionally confirms the Gaussianization procedure chosen in this work. We finally compare the updated WMAP9+SPT+ACT constraints from Ref. [45] to the constraints obtained from the Planck TT likelihood including the recent

Planck High Frequency Instrument (HFI) τreion measurement (P15∗). When computing the relative entropy for a replacement of the constraints from Ref. [45] with the Planck TT likelihood constraints, we ﬁnd a minor improvement in the agreement between the two data sets (S 0.1, σ(D ) 4.7). These results are independent of the ΛCDM parametrization, as = − KL = expected.

Finally, we also assess the consistency between the constraints derived in the IA with those derived by the Planck Collaboration [212]. To this end, we estimate the relative entropy between the IA constraints (Chapters3 and4) and the constraints obtained after adding

Planck high-` data to the IA through importance sampling (denoted IA+P15∗∗ hi-`). In order to investigate the impact of the neutrino model on the consistency between these two data sets, we once weight the IA chains by the Planck likelihood values computed assuming massless neutrinos (as we did in the IA), and once with the Planck likelihood values computed assuming the fiducial Planck neutrino model. When we consistently assume massless neutrinos we find a good agreement between the constraints from these two data sets (S 0.1, σ(D ) 1.7). = KL = Furthermore, we find the results to be independent of the parametrization of the cosmological model, as can be seen from Tab. 5.2. In the case in which we assume different neutrino models, we still find consistency between the two data sets in our fiducial parametrization (S 0.7, = σ(D ) 1.7), but we cannot obtain stable results for the σ parametrization of ΛCDM. Similar KL = 8 to before, we thus find that the Surprise depends on the parametrization of the cosmological model only for inconsistent comparisons. We stress that constraints derived using different fiducial models should not be compared and the cases considered here should therefore serve as a warning illustrating the effects of inconsistent comparisons.

In the IA we have used the value of the Hubble parameter derived by Ref. [74] (denoted E14), which is a reanalysis of the measurement by Ref. [227]. This measurement has been updated with a more precise value in Ref. [228] (denoted R16), which has been found to be in tension with the results from Ref. [212] (see e.g. Refs. [100, 114]). In order to investigate the impact of this choice on the consistency between the constraints derived in the IA and those derived by the Planck Collaboration [212], we additionally compute the relative entropy between the IA combined with R16 (denoted IA*) and the constraints obtained when combining the IA* with Planck high-` data. As can be seen from Tab. 5.2 we ﬁnd the two data sets to be in agreement with each other, albeit with a slightly lower p-value.

5.5 Conclusions

In this work, we have used the Surprise statistic introduced in Ref. [240] to quantify the concordance between different data sets in the framework of a ΛCDM cosmological model.

147 Chapter 5. Integrated cosmological probes: concordance quantiﬁed

Table 5.2: Values of the relative entropy D , expected relative entropy D , standard devia- KL 〈 KL〉 tion of the relative entropy σ(DKL) and Surprise S for the parameter updates considered in this work. All the values are given in units of bits. The p-value denotes the probability of observing a value of the Surprise that is greater or equal (less or equal) than S if S is greater (smaller) than zero when assuming consistency between the two data sets. The ﬁducial entropy val- 2 2 ues are computed for the ΛCDM parametrization {h, Ωch , Ωbh , ns, As, τreion}, while the 2 2 values in parentheses show the results for {h, Ωm, Ωb, ns, σ8, τreion} (top) and {Ωch , Ωbh ,

2τreion 1 p 2 ns 1 θ?, Ase− ,zreion, / Ωbh 2 − } (bottom) respectively.

Updating Data combination D D S σ(D ) p-value scheme KL 〈 KL〉 KL 9.1 11.5 -2.3 5.5 0.4 CMB set P15 replace (17.5) (12.4) (5.1) (4.5) (0.1) → (8.3) (11.6) (-3.3) (5.5) (0.3) 13.7 12.6 1.1 6.5 0.3 CMB set∗ P15 replace (13.1) (13.4) (-0.4) (7.1) (0.6) → (12.6) (12.1) (0.5) (6.1) (0.4) 9.6 9.7 -0.1 4.7 0.6 CMB set∗ P15∗ replace → (10.1) (10.4) (-0.3) (5.2) (0.6) IA+P15 hi-`, 8.3 8.2 0.1 1.7 0.4 IA ∗∗ add → massless ν (8.3) (8.1) (0.2) (1.7) (0.4) IA*+P15 hi-`, 10.7 9.3 1.5 1.8 0.2 IA* ∗∗ add → massless ν (10.6) (9.5) (1.2) (1.9) (0.2) IA+P15 hi-`, 8.9 8.2 0.7 1.7 0.3 IA ∗∗ add → massive ν (-) (-) (-) (-) (-)

148 5.5. Conclusions

Nicola et al., 2016 WMAP9 + SPT + ACT WMAP9 + SPT + ACT, Calabrese et al., 2017 Planck Collaboration 2015, TT + lowP Planck Collaboration 2015, TT + HFI τreion prior 128 − 0.

120 2 0. h c

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.66 .69 .72 .75 104 112 120 128 .93 .96 .99 .02 .95 .10 .25 .40 .04 .08 .12 .16 0 0 0 0 0. 0. 0. 0. .0195 .0210 .0225 .0240 0 0 0 1 1 2 2 2 0 0 0 0 2 0 0 0 2 0 9 h Ωch Ωbh ns 10 As τreion

Figure 5.1: Comparison of the constraints obtained from the integrated analysis to the constraints obtained by the Planck Collaboration (TT+lowP), the constraints obtained from the Planck TT likelihood with the HFI τreion-prior, the constraints obtained by WMAP9+SPT+ACT and the updated constraints from WMAP9+SPT+ACT [45] in the ΛCDM parametrization 2 2 {h, Ωch , Ωbh , ns, As, τreion}. All constraints are marginalized over the respective nuisance parameters. In each case the inner (outer) contour shows the 68% c.l. (95% c.l.).

149 Chapter 5. Integrated cosmological probes: concordance quantiﬁed

Nicola et al., 2016 WMAP9 + SPT + ACT WMAP9 + SPT + ACT, Calabrese et al., 2017 Planck Collaboration 2015, TT + lowP Planck Collaboration 2015, TT + HFI τreion prior 36 − 0.

32 0. m

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.66 .69 .72 .75 .24 .28 .32 .36 042 045 048 051 054 .93 .96 .99 .02 .75 .80 .85 .04 .08 .12 .16 0 0 0 0 0 0 0 0 0. 0. 0. 0. 0. 0 0 0 1 0 0 0 0 0 0 0 h Ωm Ωb ns σ8 τreion

Figure 5.2: Same as Fig. 5.1 but showing the constraints on the parameters 2 2 {h, Ωm, Ωb, ns, σ8, τreion} instead of {h, Ωch , Ωbh , ns, As, τreion}.

150 5.5. Conclusions

We have compared the constraints obtained from the integrated analysis presented in Chapters 3 and4 to those obtained by the Planck Collaboration [ 212]. We have further quantiﬁed the agreement between the constraints obtained from CMB data by the Planck Collaboration [212], WMAP9+SPT+ACT [124] and the recently updated WMAP9+SPT+ACT constraints [45].

We find good agreement between the different data sets considered, i.e. we find that all the considered CMB data sets yield constraints consistent at a level of less than 1σ and we furthermore find that the constraints obtained from the IA are in good agreement with those obtained by the Planck Collaboration. The only apparent discrepancies detected are due to differences in the fiducial neutrino model employed to compute the Planck 2015 and the WMAP9+SPT+ACT constraints and they only appear in specific parametrizations of the cosmological model. Therefore, the relative entropy and the Surprise present a reliable measure of concordance between data sets, as they are sensitive to subtle effects such as differences in neutrino models. These results show that the data sets considered in this work are all well fit by ΛCDM and do not provide evidence for tensions.

151 Chapter 5. Integrated cosmological probes: concordance quantiﬁed

0.14 χ2, dof = 6 0.12 WMAP + SPT + ACT Calabrese Planck Collab. 2015, TT + lowP 0.10 0.08 0.06 0.04 0.02 0.00 0 5 10 15 20 25 30 dMahalanobis

Figure 5.3: Distribution of Mahalanobis distances for Gaussianized CMB set∗ and P15∗ chains compared to the theoretically expected χ2 distribution for 6 degrees of freedom.

5.A Gaussianity tests

As discussed in Sec. 5.2, we assess the Gaussianity of the chains during and after the Box-Cox iteration by testing if the Mahalanobis distances [178] of the samples follow a χ2 distribution

[183, 95, 281]. In this Appendix we consider the Bayesian update CMB set∗ P15∗ as an ex- → ample. Fig. 5.3 shows the distribution of the Mahalanobis distances for the ﬁnal Gaussianized chains, while in Fig. 5.4 we compare these chains to their Gaussian approximations. As can be seen from both of these ﬁgures, the chains are well-approximated by normal distributions, thus justifying the use of the Gaussian approximation to compute relative entropies.

152 5.A. Gaussianity tests

Planck Collab. 2015, TT + lowP, G. Planck Collab. 2015, TT + lowP WMAP + SPT + ACT Calabrese, G. WMAP + SPT + ACT Calabrese

9 2

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6 6

0 2 4 6 8 3 6 9 12 15 0 2 4 6 .5 .0 .5 .0 .0 .5 .0 .5 0 3 6 9 12 2 5 7 10 0 2 5 7 1 2 3 4 5 6

Figure 5.4: Comparison of the Gausssianized contours for CMB set∗ and P15∗ to their Gaussian approximations in the transformed parameter space. In each case the ﬁrst (second, third) contour shows the 68% c.l. (95% c.l., 99% c.l.).

153

6 Consistency tests in cosmology using relative entropy

If we knew what it was we were doing, it would not be called research, would it? — Unknown, often attributed to Albert Einstein

This chapter will appear in a similar form in Nicola et al., in prep.

6.1 Introduction

The Λ Cold Dark Matter (ΛCDM) cosmological model has been remarkably successful in describing cosmological data from many different probes across a wide range of length scales and to date has thus passed all the tests it has been subject to. Despite its success, several of its key ingredients are still poorly understood, such as dark matter, dark energy and inﬂation. In future, many cosmological experiments, such as the Dark Energy Spectroscopic Instrument (DESI1), the Large Synoptic Survey Telescope (LSST2), Euclid3, the Wide Field Infrared Survey Telescope (WFIRST4) and the Simons Observatory5 are going to deliver data at an unprecedented precision, thus yielding ever more stringent tests of ΛCDM. In order to interpret the results from these experiments, we require consistency measures as well as model comparison methods, which allow us to assess consistency both between these high-precision data and with ΛCDM.

Several model selection methods have been developed in the mathematical and statistical literature, examples include the Bayesian evidence (e.g [158]), the Akaike Information Crite-

1http://desi.lbl.gov. 2http://www.lsst.org. 3http://sci.esa.int/euclid. 4http://wfirst.gsfc.nasa.gov. 5https://simonsobservatory.org.

155 Chapter 6. Consistency tests in cosmology using relative entropy rion (AIC) [14], the Bayesian Information Criterion (BIC) [237] and the Deviance Information Criterion (DIC) [255]. All these methods have been successfully applied to cosmology (e.g. [145, 114, 69]). In this work, we introduce a novel model selection method based on the relative entropy or Kullback-Leibler (KL) divergence [160] and the posterior predictive distribution (PPD). Both these quantities have already been employed in cosmological applications: Refs. [240, 241, 101, 100, 284] have used the relative entropy to assess consistency between several different data sets and Ref. [81] have used the posterior predictive distribution to quantify tensions in the value of the Hubble parameter. In this Chapter, we build on Refs. [240, 241] and extend the relative entropy into a model testing framework by combining it with the PPD. It is legitimate to ask: why do we need another method for model selection? We believe that the proposed model selection method has a couple of properties that make it a useful addition to existing model testing approaches. We demonstrate the method on a series of toy models and apply it to Type Ia supernovae (SNe Ia) data from the JLA [34] and CMB constraints [214] in the framework of six different cosmological models. In addition, we also revisit the relative entropy as a consistency measure and investigate some of its properties in more detail.

This Chapter is organized as follows. In Section 6.2, we review the notion of data and parameter space in Bayesian inference. Section 6.3 revisits the relative entropy as a consistency measure while Section 6.4 introduces our novel model selection method and describes an application to cosmological data. Finally, we conclude in Section 6.5. Robustness tests and implementation details are deferred to the Appendix.

6.2 Data and parameter space

In any application of relative entropy or any Bayesian inference problem, there are two fundamental spaces involved: the space of all possible parameter values θi of a given model M, i.e. model space SM , and the space of all possible outcomes yi of a given experiment E, i.e. data space SD . These two spaces are connected through an underlying, data-generating model, which we can use alongside Bayes’ theorem to transform probability distributions between data and model space or vice versa.

In connection to these distinct spaces, we can ask two separate questions in Bayesian inference: (i) Are two different data sets consistent with each other within a given model (data set consistency)?, (ii) Given a set of different data-generating models, which one is preferred by a given data set (model selection)? Or more generally, does a given model fit the data? Question (i) amounts to fixing a particular model and comparing model parameter constraints in SM obtained with different data set combinations. If the differences seen are consistent with measurement uncertainties, we can conclude that the data are consistent within this particular model. Significant differences on the other hand, indicate tensions between the data sets being considered. In order to answer question (ii), we need to compare models and thus different model spaces while the data remains the same. Therefore, the comparison cannot easily be studied in SM as in case (i), but we rather have to compare the respective

156 6.3. Data set consistency: ﬁxed model and different data

models in data space SD . In order to answer the questions above, we thus need to develop methods that can be applied to these two different spaces.

In this work, we address questions (i) and (ii) using the relative entropy or KL divergence [160].

The relative entropy DKL between a prior p1(θ) and a posterior distribution p2(θ) is given by [160] Z p2(θ) DKL(p2 p1) dθ p2(θ)log . (6.1) || = p1(θ)

The relative entropy is a measure of the information gain for a Bayesian update from p1(θ) to p2(θ) and has units of bits, provided the logarithm in Eq. 6.1 is taken to the base 2.

Building on previous work [240, 241], in Sec. 6.3 we consider the question of data set consistency and revisit a number of properties of the KL divergence. In Sec. 6.4, we introduce a novel model selection method, which is also based on the relative entropy and compares different models in data space.

6.3 Data set consistency: ﬁxed model and different data

A discussed above, the relative entropy is a measure for the information gain for a Bayesian update from a probability distribution function (pdf) p1(θ) to p2(θ). By itself however, the KL divergence does not allow us to assess consistency between different data sets. In order to extend the relative entropy into a consistency measure, Ref. [240] defined the Surprise statistic S as the difference between the observed D and the expected relative entropy D , i.e. KL 〈 KL〉 S D D . By definition, the Surprise is expected to scatter around zero and significant = KL − 〈 KL〉 deviations from this behavior allow us to detect inconsistencies: a large, positive value of S suggests a tension between the considered data sets, as the they are more different than expected. If on the other hand S 0, the pdfs are more similar that expected. < The Surprise statistic has been applied to a wide variety of data sets (e.g. [240, 241, 101, 100, 284], Chap.5) and has proven to be a powerful consistency measure for cosmological applications. In the course of these works, it has however become clear that several properties of the relative entropy and the Surprise require further investigation and we aim to address these in the following.

6.3.1 Sequential experiments

From Eq. 6.1, we see that the relative entropy and thus the Surprise statistic are not symmetric upon exchange of prior and posterior6. When applying the KL divergence to testing the consistency of data sets y1 and y2, obtained in two different experiments, we thus need to decide on a choice of prior and posterior. There exist two common cases in which this choice is

6This implies that the relative entropy is not a metric, as deﬁned in mathematics.

157 Chapter 6. Consistency tests in cosmology using relative entropy rather clear: (i) when comparing an earlier experiment to a current one, it is natural to choose the distribution derived from the current experiment as posterior in DKL and the distribution from the earlier experiment as the prior, and (ii) when comparing a more constraining to a less constraining experiment, it is equally natural to set the posterior to the distribution of the more constraining experiment and the prior to the less constraining one. In other situations, however, the choice of prior and posterior is not obvious and the asymmetry of the relative entropy may appear as a problem. However, as we will discuss below, this asymmetry is a feature of the relative entropy, rather than a shortcoming.

In order to investigate cases that do not fall in any of the two categories described above, we consider two uncorrelated experiments with comparable constraining power that cannot easily be arranged chronologically. The setup chosen is illustrated in Fig. 6.1. We assume that an identical model parameter prior p(θ) is used throughout the analysis, and denote the experimental outcomes as y1 and y2 respectively. The posterior derived from the ﬁrst experiment is thus given by

p(θ)p(y1 θ) p1(θ) p(θ y1) | , (6.2) ≡ | = p(y1) where p(y θ) denotes the likelihood of the ﬁrst experiment, p(θ) is the prior and the evidence 1| is given by p(y1). The posterior of the second experiment is analogously given by

p(θ)p(y2 θ) p2(θ) p(θ y2) | , (6.3) ≡ | = p(y2) where again p(y θ) denotes the likelihood of the second experiment and the evidence is given 2| by p(y2). The information gains for a Bayesian update from prior p(θ) to posteriors p1(θ), p2(θ) respectively, are

Z p1(θ) DKL,1 DKL(p1(θ) p(θ)) dθp1(θ)log , (6.4) ≡ || = p(θ) Z p2(θ) DKL,2 DKL(p2(θ) p(θ)) dθp2(θ)log . (6.5) ≡ || = p(θ)

We can further compute the information gains from y2 and y1 with respect to p1(θ) and p2(θ) taken as priors, respectively (see Fig. 6.1). As the two data sets are uncorrelated, the combined posterior is given by the product of the prior and the likelihood of the second experiment, which gives (for both cases)

p(θ,y1,y2) p(θ y1)p(y2 θ,y1) p(θ y2)p(y1 θ,y2) p12(θ) p(θ y1,y2) | | | | . (6.6) ≡ | = p(y ,y ) = p(y y ) = p(y y ) 1 2 2| 1 1| 2

158 6.3. Data set consistency: ﬁxed model and different data

Therefore, the associated relative entropies become

Z p12(θ) DKL,3 DKL(p12(θ) p1(θ)) dθ p12(θ)log , (6.7) ≡ || = p1(θ) Z p12(θ) DKL,4 DKL(p12(θ) p2(θ)) dθ p12(θ)log . (6.8) ≡ || = p2(θ)

In general, the relative entropies DKL,3 and DKL,4 are not equal, as the KL divergence depends both on the prior and the speciﬁc outcome of an experiment, yi . This makes sense intuitively: if the prior is a delta function, i.e. we know the true value of the parameters, then no experiment can be informative [174]. Likewise, depending on the outcome of an experiment, we can gain more or less information; for example if the experiment returns a very unlikely case, then the uncertainty on the value of the parameter might be larger after the experiment has been performed [174, 50]. This implies that the relative entropy depends on the path taken to reach a given result and it is therefore not additive for sequential updates, i.e. in our case we have (see Fig. 6.1)

D D (p (θ) p(θ)) D (p (θ) p(θ)) D (p (θ) p (θ)), (6.9) KL,0 ≡ KL 12 || 6= KL 1 || + KL 12 || 1 D D (p (θ) p(θ)) D (p (θ) p(θ)) D (p (θ) p (θ)). (6.10) KL,0 ≡ KL 12 || 6= KL 2 || + KL 12 || 2

We can eliminate the dependence of the KL divergence on yi by averaging over all possible experimental outcomes. This was done in Ref. [240] to derive the expected relative entropy, which for DKL,1 as an example, is given by: Z D (p (θ) p(θ)) dy p(y )D (p (θ) p(θ)), (6.11) 〈 KL 1 || 〉 = 1 1 KL 1 ||

7 where p(y1) is the prior predictive distribution for y1 , deﬁned as Z p(y ) dθp(y θ)p(θ). (6.12) 1 = 1|

Eq. 6.11 can be rewritten using Eq. 6.4 and Bayes’ Theorem to obtain

Ï p(θ,y1) DKL(p1(θ) p(θ)) dy1dθ p(θ,y1)log . (6.13) 〈 || 〉 = p(θ)p(y1)

This is known as the mutual information I (y1;θ) between y1 and θ and is further equivalent to Lindley’s information measure for a given experiment [174]. The mutual information quantiﬁes the reduction of uncertainty in y1 due to the knowledge of θ [61]. It is symmetric, i.e. I (y ;θ) I (θ;y ) and satisﬁes a chain rule for a sequence of experiments with outcomes 1 = 1

7The prior predictive distribution, evaluated for the observed data, is equivalent to the evidence.

159 Chapter 6. Consistency tests in cosmology using relative entropy

common single-experiment combined prior posteriors posterior

p(✓ y1) DKL,1 | DKL,3 D p(✓) KL,0 p(✓ y , y ) | 1 2

DKL,2 DKL,4 p(✓ y2) | consistency

Figure 6.1: Illustration of method to assess consistency of two experiments within a given model. The quantities y1, y2 denote the outcomes of the experiments and the KL divergences DKL,1,DKL,2 are the information gains for updates from a common prior p(θ) to the posteriors of the respective experiments. We quantify the consistency between the two data sets by computing the relative entropies for updates from the single-experiment posteriors p1(θ),p2(θ) to the combined posterior, i.e. DKL,3,DKL,4 and comparing them to expectations. If we ﬁnd a signiﬁcant difference between observed and expected relative entropy in either of these cases, we reject the null hypothesis that the data sets are consistent.

160 6.3. Data set consistency: ﬁxed model and different data

yi [61]

X I (y1,y2, ,yn;θ) I (yi ;θ yi 1,yi 2, ,y1). (6.14) ··· = i | − − ···

Here I (yi ;θ yi 1,yi 2, ,y1) denotes the mutual information between yi and θ conditional | − − ··· on yi 1,yi 2, ,y1. The case illustrated in Fig. 6.1 corresponds to n 2 and thus Eq. 6.14 − − ··· = reduces to

I (y ,y ;θ) I (y ;θ) I (y ;θ y ). (6.15) 1 2 = 1 + 2 | 1 Applying the chain rule for mutual information to the expected relative entropy, we obtain:

D (p (θ) p(θ)) D (p (θ) p(θ)) D (p (θ) p (θ)) , (6.16) 〈 KL 12 || 〉 = 〈 KL 1 || 〉 + 〈 KL 12 || 1 〉 where we have deﬁned8 Ñ p(θ,y2 y1) DKL(p12(θ) p1(θ)) dθdy1dy2 p(θ,y1,y2)log | . (6.17) 〈 || 〉 = p(y y )p (θ) 2| 1 1

If we equivalently consider y2 as the ﬁrst and y1 as the second experiment, we obtain

D (p (θ) p(θ)) D (p (θ) p(θ)) D (p (θ) p (θ)) . (6.18) 〈 KL 12 || 〉 = 〈 KL 2 || 〉 + 〈 KL 12 || 2 〉 This means that the expected information gain from a prior to a posterior is additive for sequential updates, because it does not depend on the speciﬁc outcomes of the experiments. The observed KL divergence and thus the Surprise on the other hand, do not add and it is thus important to explicitly take the path into account when computing information gains.

Returning to the question of assessing consistency between two arbitrary experiments with outcomes y1 and y2, we thus propose to compute both DKL,3 and DKL,4, as well as the associated Surprise values, S3, S4. If both S3 and S4 are compatible with 0, the two data sets are consistent. In the case in which one or both Surprise values are signiﬁcantly larger than expected, we propose to reject the null hypothesis of consistency between y1 and y2, as the path dependence of the KL divergence can lead to inconsistencies only being detectable in one of the two updates. In the following, we illustrate this on a toy model. For an alternative discussion of the case in which S3 and S4 give inconsistent results, the reader in referred to Appendix 6.A9.

8We note that this deﬁnition is an extension of the expected relative entropy given in Ref. [240], as here we average over realizations of both y1 and y2. 9 We note that analogous considerations can also be made if y1 and y2 are correlated, with the only difference that the combined posterior in Eq. 6.6 is given by the joint analysis of the two data sets

161 Chapter 6. Consistency tests in cosmology using relative entropy

7 7

6 6

5 5 p(θ)

2 p1(θ)

θ 4 4 p2(θ)

3 3 p12(θ) 2 2

1 1 2 3 4 5 6 7 2 3 4 5 6 7 θ1 θ1

Figure 6.2: Illustration of the probability distributions in parameter space considered in the toy models of Sec. 6.3.2; on the left, toy model I; on the right, toy model II. In each case the inner (outer) contour shows the 68% c.l. (95% c.l.).

6.3.2 Toy model

In order to illustrate the concepts discussed above, we consider one of the toy models introduced in Ref. [171]. This toy model consists of two uncorrelated, contemporaneous experiments with likelihoods p(y θ) and p(y θ). We complement this toy model with a common 1| 2| prior p(θ) and the conﬁguration (toy model I) is illustrated in the left panel of Fig. 6.2. Both likelihoods p(y θ) and p(y θ) are assumed to be Gaussian with means and covariances 1| 2|

Ã1 ! /4 0 p(y θ): µ ¡4,4¢, C , (6.19) 1 1 1 1 | = = 0 /128

Ã1 ! /128 0 p(y θ): µ ¡5,4¢, C . (6.20) 2 2 2 1 | = = 0 /4

Furthermore, we assume a wide, Gaussian prior p(θ), given by Ã ! ¡ ¢ 1 0 p(θ): µp 4.5,4 , Cp . (6.21) = = 0 1

We quantify the consistency between y1 and y2 by computing the relative entropy and the Surprise between the separately derived posteriors and their combination (also see Fig. 6.1). As all distributions involved are Gaussian, we compute the KL divergences using the analytic expressions given in Ref. [240] using the publicly available Surprise package10, described in Ref. [241]. The obtained results are given in Tab. 6.1.

From Tab. 6.1, we first of all see that the observed relative entropies are not additive, while the expected information gains are, which is expected from the discussion in Sec. 6.3.1. Further- more, we find that the Surprises S3,S4 obtained for DKL,3 and DKL,4 differ in both magnitude and sign. The first case shows a positive, marginally significant Surprise, while the second case

10The package can be found at https://github.com/seeh/surprise.

162 6.3. Data set consistency: ﬁxed model and different data

Table 6.1: Values of the relative entropy D , expected relative entropy D , standard de- KL 〈 KL〉 viation of the relative entropy σ(DKL) and Surprise S for the parameter updates considered in the toy model. All values are given in units of bits. The p-values denote the probability of observing a Surprise value greater or equal (less or equal) than S if S is greater (smaller) than zero when assuming consistency between the two data sets.

Toy model Data combination D D S σ(D ) p-value KL 〈 KL〉 KL DKL,1 : p(θ) p1(θ) 3.5 4.7 -1.2 1.3 0.08 → DKL,2 : p(θ) p2(θ) 3.6 4.7 -1.1 1.3 0.13 → I DKL,3 : p1(θ) p12(θ) 4.4 2.4 2.0 1.0 0.05 → DKL,4 : p2(θ) p12(θ) 1.8 2.4 -0.6 1.0 0.25 → DKL,0 : p(θ) p12(θ) 5.8 7.1 -1.3 1.4 0.1 → DKL,1 : p(θ) p1(θ) 3.5 4.7 -1.2 1.3 0.08 → DKL,2 : p(θ) p2(θ) 5.0 4.7 0.3 1.3 0.29 → 5 II DKL,3 : p1(θ) p12(θ) 13.8 2.4 11.4 1.0 3 10− → × DKL,4 : p2(θ) p12(θ) 2.0 2.4 -0.4 1.0 0.50 → DKL,0 : p(θ) p12(θ) 7.1 7.1 0.0 1.4 0.6 →

shows a negative Surprise, suggesting that p1(θ) and p12(θ) are slightly more different than expected, while p2(θ) and p12(θ) are more similar than expected. This is also evident from the left panel of Fig. 6.2: the means of the combined posterior and p1(θ) differ by more than one standard deviation of the latter distribution, while the means of p2(θ) and p12(θ) are very similar. This asymmetry thus explains the difference in observed relative entropies.

However, as neither DKL,3 or DKL,4 suggest a significant tension between the two data sets, we 11 conclude that y1 and y2 are consistent . We can construct a more discrepant case (toy model II) by moving the mean of p(y θ) to µ ¡6,4¢, as shown in the right panel of Fig. 6.2. In this 2| 2 = case we find a significant Surprise for D (S 11.4, σ(D ) 1.0 bits), while p (θ) and KL,3 3 = KL,3 = 2 p12(θ) are still consistent (see Tab. 6.1). This is equivalent to the results obtained for toy model I except that the discrepancy in DKL,3 is more pronounced. We thus see that inconsistencies in parameters in which the constraining power of the two distributions is different, tend to only appear in the update from the weaker constraints to the combined distribution. The reason being that the combined mean lies close to the mean of the distribution with the stronger constraining power and thus the latter two will be consistent even though the original distributions are not12. Therefore, a significant discrepancy in at least one of the updates is a sign for tensions between the distributions being combined. In addition, we see that the asymmetry of the relative entropy allows us to determine the distribution that is most likely to drive the discrepancy (in this case the more constraining one).

11We use the convention of p-value 0.05 as our deﬁnition of signiﬁcant discrepancy. < 12This will cease to be true for large discrepancies, as then the mean of the more constraining distribution will shift as well.

163 Chapter 6. Consistency tests in cosmology using relative entropy

6.4 Model rejection: ﬁxed data and different models

Relative entropy is a measure for the information gain from a prior distribution to a posterior and thus has a wide range of applications. As discussed in Sec. 6.2, model testing involves different model spaces SM and therefore different models are best compared in data space SD . The ﬁrst step towards model testing with the relative entropy is thus to identify appropriate priors and posteriors in SD .

A central quantity in Bayesian model testing is the posterior predictive distribution (PPD, see e.g. [91, 158]). The PPD quantifies the probability of future data y’ conditioned on the current observation Y y, given an underlying model M that links data and model space (for = an application of the PPD in cosmology, see [81]). Essentially, the PPD is the average of the likelihood of the new data under the posterior of the model parameters, i.e. Z p(y’ y,C ,C ,M) dθp(y’ θ,C )p(θ y,C ), (6.22) | y y’ = | y’ | y where θ denotes the vector of model parameters of M and Cy,Cy’ are the covariance matrices of current and future data respectively. Furthermore, p(θ y,C ) denotes the posterior dis- | y tribution for the parameters θ and p(y’ θ,C ) is the likelihood of the new data, which does | y’ not necessarily equal the likelihood for the current data. In Eq. 6.22 we explicitly condition on a model M. We can however equally define a PPD conditioned on current data without assuming a model. This latter quantity is given by Z p(y’ y,C ,C ) dyˆ p(y’ yˆ ,C )p(yˆ y,C ), (6.23) | y y’ = t | t y’ t| y where yˆt denotes the expectation for the underlying true data point values, i.e. the data points that would be measured in an infinite precision experiment. The quantity p(yˆ y,C ) is the t| y sampling distribution of yˆ given current data and p(y’ yˆ ,C ) is the likelihood of future data t | t y’ conditioned on yˆt. The sampling distribution of yˆt can be thought of as the expectation for yˆt given observation y. As an example, consider the case in which the observation consists of m draws from a Gaussian N(q,σ ) with known σ , which are denoted as q ,i 1,...,m. Then q q i = σ p(yˆ y,C ) will be the distribution of the mean of these observations, i.e. N(q¯, q/pm), where t| y 1 Pm q¯ /m i 0 qi . = = Eq. 6.23 expresses the distribution of future data as predicted using only current data without assuming a model. We call this the PPD from the data while we denote Eq. 6.22 as PPD from data and model.

If model M captures the relevant features in the data, we expect the two distributions in Equations 6.22 and 6.23 to be similar, as the observed data y represents a typical draw from the underlying model. If on the other hand, M is not a good ﬁt to the data, we expect to see signiﬁcant differences, especially in the means of the distributions.

In order to assess how well model M describes the data, we therefore propose to quantify the

164 6.4. Model rejection: ﬁxed data and different models

Data space Model space SD SM

p(y’ y,C ,C ,M) p(✓ y,C ) | y y’ | y j j ✓ y

p(y’ y,C ,C ) | y y’

yi ✓i

Figure 6.3: Illustration of model selection framework proposed in this work. The left panel shows data space SD , while model space SM is illustrated in the right panel. To quantify how well a model fits the data, we compute the relative entropy between the PPD derived from data and model p(y’ y,C ,C ,M) S and the PPD derived only from the data p(y’ y,C ,C ) S . | y y’ ∈ D | y y’ ∈ D A relative entropy significantly larger than expected signifies that the model is ruled out, while consistency between expected and observed KL divergence means that the data cannot rule out the model. The solid lines in the left panel show the distributions needed to compute the PPDs: the sampling distribution of y , p(yˆ y,C ) (c.f. Eq. 6.23) and the data distribution t t| y derived from sampling the posterior of the model parameters p(θ y,C ) and transforming to | y data space (c.f. Eq. 6.22).

difference between p(y’ y,C ,C ) and p(y’ y,C ,C ,M) through their relative entropy D , by | y y’ | y y’ KL considering

Z p(y’ y,Cy,Cy’,M) DKL(p(y’ y,Cy,Cy’,M) p(y’ y,Cy,Cy’)) dy’p(y’ y,Cy,Cy’,M)log | , (6.24) | || | = | p(y’ y,C ,C ) | y y’ where we have set the prior to the PPD from the data and the posterior to the PPD from the data and model. Note that this is similar to the model breaking ﬁgure of merit introduced in Ref. [18]. With this choice, Eq. 6.24 can be interpreted as the information gain coming from the assumption of a given model M. However, the value of DKL by itself does not allow us to quantify the consistency between data and model M. In analogy to the Surprise statistic [240, 241], we need to compare the observed relative entropy to the relative entropy expected under the null hypothesis, which we denote by D . In the present case, the null hypothesis 〈 KL〉 is that the data are drawn from model M and we can thus deﬁne the expected relative entropy as

D (p(y’(M) y(M),C ,C ,M) p(y’(M) y(M),C ,C )) 〈 KL | y y’ || | y y’ 〉 = Ï p(y’(M) y(M),Cy,Cy’,M) dy(M)dy’(M)p(y’(M) y(M),Cy,Cy’,M)log | , (6.25) | p(y’(M) y(M),C ,C ) | y y’

165 Chapter 6. Consistency tests in cosmology using relative entropy where y(M) implies that y is drawn from model M. To simplify the notation, we further deﬁne:

D (y,M y) D (p(y’ y,C ,C ,M) p(y’ y,C ,C )), (6.26) KL || ≡ KL | y y’ || | y y’ D (y(M),M y(M)) D (p(y’(M) y(M),C ,C ,M) p(y’(M) y(M),C ,C )) . 〈 KL || 〉 ≡ 〈 KL | y y’ || | y y’ 〉 (6.27)

These two quantities, combined with the standard deviation of the expected relative entropy, σ( D (y(M),M y(M)) ), allow us to assess how well M describes the data: a relative entropy 〈 KL || 〉 D (y,M y) significantly larger than D (y(M),M y(M)) suggests that M is not a good fit to KL || 〈 KL || 〉 the data, as the two PPDs are significantly different. Consistency between D (y,M y) and KL || D (y(M),M y(M)) on the other hand, means that we cannot rule out the null hypothesis 〈 KL || 〉 and model M thus provides a good fit to the data.

In this work, we compute the quantities D (y,M y), D (y(M),M y(M)) and KL || 〈 KL || 〉 σ( D (y(M),M y(M)) ) assuming all distributions considered to be well-approximated by 〈 KL || 〉 Gaussians, which allows us to compute all relative entropies analytically using the expressions given in Ref. [240]. This approximation is trivially justiﬁed for the toy model considered in Sec. 6.4.2. In Sec. 6.4.3, we apply the model selection method to cosmological data from SNe Ia and CMB and we test the Gaussianity of the relevant PPDs in Appendix 6.B. We ﬁnd them to be well-approximated by normal distributions, which is due to the fact that PPDs are generally close to Gaussian if the data is normally distributed, regardless of the posteriors.

In our discussion so far, we have omitted the issue of model complexity: increasing the number of degrees of freedom of a model M, will enable us to fit an increasing number of features in the data. Thus, growing model complexity generally leads to better fits for a given data set. The proposed model selection method as is, does not have a built-in notion of model complexity. We include this by invoking Occam’s razor: when comparing models of different complexity that give relative entropies consistent with the null hypothesis, we invoke the principle of parsimony and choose the model with the smallest complexity as the one preferred by the data13. In other words, our method allows us to identify models that fail to fit the data but it also enables us to detect which models cannot be distinguished (or ruled out) given current uncertainties.

6.4.1 Implementation

We estimate all quantities discussed in the previous section using simple Monte Carlo simulations. For a given, data-generating model M and observed data y with likelihood p(y θ,C ), | y we ﬁrst derive the posterior distribution for the model parameters p(θ y,C ) and the sam- | y pling distribution of yˆ , p(yˆ y,C ). The method for computing the observed relative entropy t t| y

13Here complexity stands for any measure for the number of degrees of freedom in a model (e.g. [161]), and is not limited to counting the number of free parameters. In the present work however, we are going to identify the complexity of a model with its number of free parameters for simplicity.

166 6.4. Model rejection: ﬁxed data and different models

(Eq. 6.24) using the integral expressions in Equations 6.22 and 6.23, can then be summarized as:

(i) Eq. 6.22: For i 1,...,N, ﬁrst sample θ from p(θ y,C ), then sample a realization y’ = i | y i of length N¯ from p(y’ θ,C ). Marginalizing over θ, i.e. combining all the samples | y’ y’ ,i 1,...,N gives us a sample from p(y’ y,C ,C ,M). i = | y y’ (ii) Eq. 6.23: For i 1,...,N, ﬁrst sample y from p(yˆ y,C ), then sample a realization = t,i t| y y’ of length N¯ from p(y’ yˆ ,C ). Marginalizing over yˆ , i.e. combining all the samples i | t y’ t y’ ,i 1,...,N gives us a sample from p(y’ y,C ,C ). i = | y y’

In a last step we use these two combined samples to compute the relative entropy deﬁned in Eq. 6.24. For an alternative method, which can be applied if we can sample from the PPDs directly, see Appendix 6.C.

Recall that we have defined the expected relative entropy in Eq. 6.25 as the information gain from the model M conditional on the data being drawn from that same model. Instead of attempting to analytically predict this quantity, we again resort to a Monte Carlo approach. For each model M considered in our analysis we first derive the best-fit model parameters from the data. Using these best-fit parameters we generate a mock true data set yt. We then sample a realization y from the true data distribution p(y y ,C ), compute the best-fit | t yt model parameters for M and the corresponding posterior p(θ y,C ). For this simulated data | y set we now compute the relative entropy D (y,M y) using the procedure outlined above. KL || Repeating this experiment N times for different realizations y of yt finally gives us a sample of D (y(M),M y(M)) and we estimate D (y(M),M y(M)) and σ( D (y(M),M y(M)) ) as KL || 〈 KL || 〉 〈 KL || 〉 the mean and the standard deviation of this distribution respectively.

The three quantities D (y,M y), D (y(M),M y(M)) and σ( D (y(M),M y(M)) ) allow KL || 〈 KL || 〉 〈 KL || 〉 us to assess the consistency of each model M with the observed data. In practice we use the full distribution of D (y,M y) to compute the p-value, i.e. the probability of observing a KL || value greater or equal than D (y,M y) under the null hypothesis that the data is drawn from KL || model M.

6.4.2 Toy model

We illustrate the methods described in the previous section by applying them to a toy model in which we consider fitting data with a polynomial of varying order. For illustration, we consider polynomials of order n 0...7, where n 0 denotes a linear polynomial with y-intercept = = set to zero. For each polynomial of degree n we choose fiducial coefficients and generate simulated data drawn from this model with covariance matrix Cy. We then fit this data with polynomials of varying degree nˆ and assess the consistency of model and data by computing the KL divergences discussed above. In our calculations we further make the assumption that C C , which amounts to assuming that the first and second experiment are identical, an y’ = y 167 Chapter 6. Consistency tests in cosmology using relative entropy

102

0.8 101 0.6 [bits] value − KL 0.4 p D 100 0.2

1 0.0 10− 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 α α

Figure 6.4: Illustration of the relative entropy and the p-value for the null hypothesis obtained for ﬁtting a simulated data set with n 5 with a polynomial of the same degree as a function = of covariance scaling parameter α. Results for other combinations of n,nˆ are similar. The dashed, horizontal line denotes a p-value of 0.05.

approximation that we will revisit later in this section. For each ﬁducial polynomial model of degree n, the procedure can thus be summarized as follows:

1. Draw a data realization y from a ﬁducial polynomial p P , where P denotes the n ∈ n n vector space of all polynomials of degree n. We denote this polynomial as model Mn.

2. Fit y with a polynomial p ˆ P ˆ with nˆ [0,n ], where nˆ not necessarily equal to n n ∈ n ∈ max and determine the corresponding posterior p(θ y,C ). We denote this polynomial as | y model Mnˆ .

3. Compute the observed relative entropy D (y,M ˆ y), the expected relative entropy KL n|| D (y(M ˆ ),M ˆ y(M ˆ )) and the standard deviation of the relative entropy 〈 KL n n|| n 〉 σ( D (y(M ˆ ),M ˆ y(M ˆ )) ). Note that the expected relative entropy has to be com- 〈 KL n n|| n 〉 puted for each model Mnˆ separately.

4. Compute the p-value of the observed D (y,M ˆ y) for the pdf of the expected relative KL n|| entropy.

In our implementation, we assume the data to be drawn from a Gaussian distribution with covariance matrix C σ21 and we generate N 100 realizations of length N¯ 1000 each y = = = to compute the KL divergences D (y,M ˆ y) and D (y(M ˆ ),M ˆ y(M ˆ )) . For a detailed KL n|| 〈 KL n n|| n 〉 description of how we determine the fiducial polynomial coefficients and the specific implementation of the toy model, the reader is referred to Appendix 6.D.

As outlined above, we assume the ﬁrst and second experiment to be equivalent in our analysis. Intuitively, this assumption is justiﬁed as we would like to quantify the consistency of data and model based on our current knowledge. In other words, we would like to understand

168 6.4. Model rejection: ﬁxed data and different models which models can be ruled out with current observational uncertainties and thus assuming a repetition of the current observation is a natural choice. Nevertheless, this choice is arbitrary and it is thus instructive to investigate the dependence of our results on the covariance matrix of the second experiment. To this end, we introduce a covariance matrix scaling parameter

α as C 0 αC and compute the relative entropies and p-values obtained as a function y’ = y’ of α [0,2]. In Fig. 6.4 we show the results for fitting the data with a polynomial of degree ∈ nˆ n 5, but results for other combinations of polynomials are similar. We see that the relative = = entropy increases as α decreases, which is expected as the information gain from the model is maximized when measurement uncertainties are minimized. The p-value on the other hand stays constant as we vary α. This means that both D (y,M ˆ y) and D (y(M ˆ ),M ˆ y(M ˆ )) KL n|| 〈 KL n n|| n 〉 increase similarly as we make the second experiment more constraining. It therefore appears that the model selection results are stable for a wide range of α parameters. The only exception, which is not shown in the figure, is that we find an instability as we approach α 0 due to finite = machine precision. At α 0, the covariance matrix of the PPD from data and model exhibits = large off-diagonal elements and all calculations become numerically unstable, meaning that we cannot compute the information gain for an infinite precision experiment. However, since all experiments in cosmology have a fundamental error floor (e.g. cosmic variance, limited number counts in observable Universe, etc.), this limit is never reached in practice and we can thus safely perform all computations at the fiducial point C C . y’ = y In Fig. 6.5, we show the results obtained for data generated from a polynomial of fiducial order n 5, which we fit with polynomials of degrees nˆ [0,7]. The results for all other com- = ∈ binations of polynomial degrees are similar. As can be seen from comparing D (y,M ˆ y), KL n|| D (y(M ˆ ),M ˆ y(M ˆ )) and σ( D (y(M ˆ ),M ˆ y(M ˆ )) ) in the left panel of Fig. 6.5, all poly- 〈 KL n n|| n 〉 〈 KL n n|| n 〉 nomials of degrees nˆ n are ruled out with high significance as they result in relative entropies < significantly larger than expected for the true, data-generating model. All models with nˆ n ≥ on the other hand, give relative entropies consistent with expectations from the true model, suggesting that these models provide a good fit to the data. Following Sec. 6.4 we select the simplest model that fits the data as the preferred one. From Fig. 6.5, we thus see that the preferred model is the one that generated the data in the first place, i.e. the polynomial with order n 5. Finally, in the right panel of Fig. 6.5 we show the p-values corresponding to = these relative entropies and we can see that the p-values reflect the results discussed for

D (y,M ˆ y). KL n||

6.4.3 Application to cosmological data

We test the method described above on cosmological data by applying it to the SNe Ia sample from the Joint Lightcurve Analysis (JLA) [34], which comprises data from SDSS-II [88, 154, 254, 163, 46], the Supernova Legacy Survey (SNLS) [22, 258], the HST [226, 262] and additional low-redshift experiments [34].14 The JLA data consists of light curve parameters for N 740 SNe =

14The data can be found at: http : //supernovae.in2p3.fr/sdss_snls_jla/ReadMe.html.

169 Chapter 6. Consistency tests in cosmology using relative entropy

7 10 n (ﬁd.) 106 0.8 105 104 0.6 [bits] 103 value KL − 0.4 D 102 p 101 0.2 0 10 0.0 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 nˆ nˆ

Figure 6.5: Illustration of D (y,M ˆ y) and the p-value of D (y,M ˆ y) as a function of the KL n|| KL n|| degree nˆ of the polynomial chosen to ﬁt the data for the toy model described in Sec. 6.4.2. The gray bands show the 1σ and 3σ uncertainties on the information gain from the true model, i.e. D (y(M ˆ ),M ˆ y(M ˆ )) . The dashed, horizontal line denotes a p-value of 0.05. 〈 KL n n|| n 〉

SNe Ia, which can be used to calculate distance moduli µ(zSNe), as well as their covariance.

The distance modulus of an SNe Ia at redshift zSNe is given by µ ¶ dL(zSNe) µ(zSNe) 5log , (6.28) = 10 10[pc] where dL(zSNe) denotes the luminosity distance. The observed supernovae absolute peak magnitudes and thus their distance moduli have been found to depend on several SNe Ia and host galaxy properties. We therefore follow Ref. [34] and parametrize the observed distance moduli as

µ m∗ (M αX βC), (6.29) obs = B − B − 1 + where mB∗ denotes the observed peak magnitude of the SNe in rest frame B-band, C is the color of the SNe and X1 is the so-called stretch parameter, which quantiﬁes the duration of the SNe explosion. The parameters α,β and MB are nuisance parameters and both MB and β were found to depend on supernova host galaxy properties [257, 143]. In order to take these effects into account, we follow Ref. [34] and set  1 10 MB if Mstellar 10 M , MB < ¯ (6.30) = M 1 ∆M otherwise, B +

1 where ∆M is a nuisance parameter additional to α,β and MB.

We apply the model selection method described above to quantifying the consistency between the JLA data and several cosmological models. In this analysis, we consider six models: (i)

CDM, (ii) curved CDM, (iii) ΛCDM, (iv) curved ΛCDM, (v) w0CDM and (vi) w0waCDM. For

170 6.4. Model rejection: ﬁxed data and different models

Table 6.2: Summary of the cosmological models considered in this work.

Parameters w/o Parameters w/ Model no. Cosmology Density constraints CMB CMB h,Ω ,α,β,M 1, 0 CDM Ω 0,Ω 0 α,β,M 1,∆M b B Λ = K = B ∆M 1 Ωm,α,β,MB, h,Ωm,Ωb,α,β, 1 curved CDM ΩΛ 0 1 = ∆M MB,∆M 1 Ωm,α,β,MB, h,Ωm,Ωb,α,β, 2 ΛCDM ΩK 0 1 = ∆M MB,∆M Ωm,ΩΛ,α,β, h,Ωm,ΩΛ,Ωb,α, 3 curved ΛCDM - 1 1 MB,∆M β,MB,∆M Ωm,w0,α,β, h,Ωm,w0,Ωb,α, 4 w0CDM ΩK 0 1 1 = MB,∆M β,MB,∆M Ωm,w0,wa,α, h,Ωm,w0,wa, 5 w0waCDM ΩK 0 1 1 = β,MB,∆M Ωb,α,β,MB,∆M

each model, we first compute posterior distributions for all model parameters (cosmological and nuisance) in a Monte Carlo Markov Chain (MCMC) using the publicly-available code CosmoHammer15 [15]. The parameters varied for each considered model are given in Tab. 6.2, where h is the dimensionless Hubble parameter, Ωm is the fractional matter density today, Ωb is the fractional baryon density today, ΩK is the fractional curvature density today, ΩΛ is the fractional dark energy density today and w0,wa parametrize the dark energy equation of state parameter as w(a) w w (1 a)[56, 173]. All theoretical predictions are com- = 0 + a − puted using PyCosmo [221] and we give the posterior means derived for each cosmological model in Appendix 6.E. We use these posteriors and the SNe Ia likelihood given in Ref. [34] for computing the relative entropies defined above. Following the discussion in Sec. 6.4.2, we assume repetition of an identical experiment when computing all PPDs. In order to reduce the dimensionality of the PPDs, we bin the SNe Ia data into 30 equally log-spaced bins in the redshift range 0.01 z 1.3. We then follow Sec. 6.4.1 and compute the observed rel- < < ative entropy D (µ,M µ), the expected relative entropy D (µ(M ),M µ(M )) and the KL i || 〈 KL i i || i 〉 standard deviation of the expected relative entropy σ( D (µ(M ),M µ(M ) )) for all models 〈 KL i i || i 〉 M ,i 0,...,5 considered. However, the algorithm for computing D (µ(M ),M µ(M )) i = 〈 KL i i || i 〉 and σ( D (µ(M ),M µ(M )) ) requires us to determine the best-fit model parameter values 〈 KL i i || i 〉 for each simulated data set, which is prohibitively expensive for the present case. Therefore, we resort to an alternative method that avoids this step: instead of first sampling a realization of the data distribution p(µ µ ,C ), and then determining the posterior distribution of the | t µt parameters, we first sample a realization of model parameters from the posterior distribution determined from the observed data, p(θ µ,C ), and then use these values to compute a | µ

15CosmoHammer is based on emcee [86] and the code can be found at: http://cosmo-docs.phys.ethz.ch/cosmoHammer/.

171 Chapter 6. Consistency tests in cosmology using relative entropy

102 0.8

0.6 [bits] value KL − 0.4 D p 1 10 0.2

0.0 0 1 2 3 4 5 0 1 2 3 4 5 model model

Figure 6.6: Illustration of D (µ,M µ) and the p-value of D (µ,M µ) obtained for a KL i || KL i || ﬁt to JLA SNe Ia as a function of the cosmological model chosen to ﬁt the data. The gray bands show the 1σ and 3σ errors on the information gain from the true model, i.e. D (µ(M ),M µ(M )) . The dashed, horizontal line denotes a p-value of 0.05. The corre- 〈 KL i i || i 〉 spondence between model number and cosmological model is given in Tab. 6.2.

corresponding data vector. These two approaches are equivalent, provided we account for residual noise when generating data from model parameters. In this work, we assume a simple

(NSNe Np ) error model and set the covariance of the residuals to C − /NSNeC , i.e. we account for res = µ the reduction in the covariance caused by using the data to fit Np model parameters. This is a rather crude approximation but we have found it to provide acceptable results in our toy model. Once we have created corresponding model parameter and data distributions, we proceed to compute D (µ(M ),M µ(M )) and σ( D (µ(M ),M µ(M )) ) as described 〈 KL i i || i 〉 〈 KL i i || i 〉 in Sec. 6.4.1, using N 1000 realizations of length N¯ 1000 each. = = The relative entropies and associated p-values for each cosmological model are shown in Fig. 6.6. As we can see, the CDM model is clearly ruled out by JLA supernovae data with a p-value smaller than 0.0001, whereas the curved CDM model cannot be ruled out using supernovae data alone, as it provides an acceptable fit with p-value p 0.09. This is similar = to the results found in Refs. [194, 184], albeit at a different significance, which is probably due to the different methodologies employed. Finally, the ΛCDM, curved ΛCDM, w0CDM and w0waCDM models all provide good fits to the data with probabilities to obtain a relative entropy as large or larger of at least 0.6. In order to test the stability of the choice of binning scheme discussed above, we repeat this analysis for several cases: (i) 30 bins with constant number of SNe Ia, and (ii) n 65,100,200,300 equally log-spaced bins. For all cases, bin = we find our results to not change significantly. However, when we use the unbinned SNe likelihood to compute the KL divergences, our results do change and we find that none of the cosmological models can be excluded with high significance. This is not unexpected due to known instabilities in χ2 analyses to binning and discrete data. We thus conclude that the proposed method yields stable results for a wide range of binning schemes, but more work is needed to investigate if the method is applicable to unbinned data.

172 6.4. Model rejection: ﬁxed data and different models

0.8 102 0.6 [bits] value KL − 0.4 D p

101 0.2 0.0 0 1 2 3 4 5 0 1 2 3 4 5 model model

Figure 6.7: Illustration of D (µ,M µ) and the p-value of D (µ,M µ) obtained for a ﬁt to KL i || KL i || JLA SNe Ia and the CMB shift parameter R as a function of the cosmological model chosen to ﬁt the data. The gray bands show the 1σ and 3σ errors on the information gain from the true model, i.e. D (µ(M ),M µ(M )) . The dashed, horizontal line denotes a p-value of 0.05. 〈 KL i i || i 〉 The correspondence between model number and cosmological model is given in Tab. 6.2.

In order to break this potential degeneracy between curvature and cosmological constant, we complement the SNe data with constraints on ΩK in form of the CMB shift parameter R, which is deﬁned as q 2 D A(z?) R ΩmH . (6.31) = 0 c

Here D A(z?) denotes the comoving angular diameter distance to the redshift of decoupling z?. In our analysis, we include the constraint on R obtained by the Planck Collaboration [214] in their second data release. With this additional constraint, we repeat the above analysis, additionally varying the Hubble parameter h and the fractional baryon density Ωb (see Tab. 6.2). As we only include one additional data point for the CMB, we do not distinguish between residual and measurement uncertainty as we did for the SNe sample.

Fig. 6.7 shows the obtained relative entropies and associated p-values for the cosmological models considered. We see that the inclusion of the CMB constraint has the effect that besides CDM, curved CDM is also ruled out with very high signiﬁcance. The remaining models, i.e. ΛCDM, curved ΛCDM, w0CDM and w0waCDM, still provide good ﬁts to the data. This shows that SNe Ia data combined with a minimal constraint on the matter density and the curvature of the Universe are able to clearly rule out both non-accelerating cosmological models considered in this work. All accelerating models on the other hand are consistent with the data. Invoking the principle of parsimony, we therefore conclude that the data combination considered in this work does not show any preference for models beyond ΛCDM.

We again compare different binning schemes and ﬁnd our results to be unaffected in all the considered cases (even in the unbinned case). This is probably due to the fact that the disagreement between the non-accelerated models and the data is so clear that it is insensitive

173 Chapter 6. Consistency tests in cosmology using relative entropy to any analysis choice .

These results are consistent with earlier works that have also found SNe Ia data combined with CMB or BAOs to rule out non-accelerated models to a very high signiﬁcance (e.g. [231, 108, 194, 184]). The above analysis thus demonstrates the applicability of the methodology described in Section 6.4.1 to cosmological data sets.

6.5 Conclusions

In this work, we have revisited the relative entropy as a consistency measure between constraints obtained from different data sets and have investigated some of its key properties, such as asymmetry and path dependence. We have extended the Surprise statistic, developed in Ref. [240], to be applicable to assessing consistency between data sets for which the choice of prior and posterior is not obvious, taking into account both the asymmetry and path dependence of the KL divergence. Finally, we have illustrated these concepts in a simple toy model.

Furthermore, we have proposed a novel model selection method based on combining the KL divergence and the posterior predictive distribution. In our algorithm, we assess consistency between data and model by computing the relative entropy between the PPD derived from the data alone and that derived from data and model. This allows us to perform model selection as well as assess the goodness of fit of any given model. Some of the advantages of this method are that the PPDs are quite close to Gaussian in most cases, it is reasonably fast to implement and inconsistencies can be easily quantified with a p-value. We have demonstrated the method in a series of toy models. In order to test the applicability of the method to cosmological data, we have further applied it to SNe Ia data from the JLA [34] and CMB constraints from Ref. [214] in form of the shift parameter R. We have tested six cosmological models, i.e. (i) CDM, (ii) curved CDM, (iii) ΛCDM, (iv) curved ΛCDM, (v) w0CDM and (vi) w0waCDM and we found that SNe Ia data alone rule out CDM with high significance but they cannot unambiguously distinguish between curved CDM and accelerated models. When we add the CMB constraint to the JLA data, we find that both the CDM and the curved CDM model are clearly ruled out, whereas all other models provide a good fit to the data. Invoking the principle of parsimony, we conclude that the combination of SNe Ia data from JLA and CMB constraints from Planck 2015 is well-fit by ΛCDM and does not provide any evidence for model extensions. This result is consistent with previous works (e.g.[231, 108, 194, 184]), thus demonstrating that model selection based on relative entropy is applicable to cosmological data. Our framework thus provides a promising method for model testing in cosmology, which we aim to explore further and apply to additional data in future work.

174 6.A. Assessing consistency for arbitrary updates

6.A Assessing consistency for arbitrary updates

In Sec. 6.3, we have argued that the path dependence of the relative entropy can cause data set inconsistencies to only be detectable in one of the two updates deﬁned in Equations 6.7 and 6.8. We can alternatively understand this case through the Shannon and the cross-entropy. To this end, we can rewrite Equations 6.7 and 6.8 as

D H(p (θ),p (θ)) H(p (θ)), (6.32) KL,3 = 12 1 − 12 D H(p (θ),p (θ)) H(p (θ)). (6.33) KL,4 = 12 2 − 12

The quantity H(q(θ)) denotes the generalization of the Shannon entropy for continuous random variables, also called differential entropy, which for a pdf q, is deﬁned as [61] Z H(q(θ)) dθ q(θ)logq(θ). (6.34) = −

Furthermore, H(q1(θ),q2(θ)) is the cross-entropy between distributions q1(θ) and q2(θ), i.e. [98, 153] Z H(q (θ),q (θ)) dθ q (θ)logq (θ). (6.35) 1 2 = − 1 2

The cross-entropy is a similarity measure between a pdf q1(θ) and its approximation q2(θ) and is minimized when q (θ) q (θ)[98, 153]. 1 = 2 A relative entropy between the combined distribution and one of the single-experiment distributions that is significantly larger than expected thus suggests a large cross-entropy between the two. This means that the two distributions are significantly different and it is thus advisable to reject the null hypothesis of consistency if either one of the relative entropies in Equations 6.7 and 6.8 is significantly larger than expected a priori.

6.B Gaussianity tests

In order to compute relative entropies using the normality approximation derived in Ref. [240], we need to test that the relevant distributions are well-approximated by Gausssians. In this work we perform two different tests: (i) we visually check that the Mahalanobis distances [178] of the distributions follow a χ2 distribution with number of degrees of freedom equal to the data vector dimension and (ii) we compare the marginalized PPD constraints to their Gaussian approximations. Both these tests show good agreement with Gaussianity. In Fig. 6.8 we show the distribution of the Mahalanobis distances for the PPD derived from data and model and its Gaussian approximation for SNe and CMB data in curved CDM, as an example. From the ﬁgure we see that the Mahalanobis distances of the PPD indeed follow a χ2 distribution with 31

175 Chapter 6. Consistency tests in cosmology using relative entropy

0.06 χ2, dof = 31 0.05 data Gaussian approx. 0.04

0.03

0.02

0.01

0.00 0 10 20 30 40 50 60 70 dMahalanobis

Figure 6.8: Distribution of Mahalanobis distances for the PPD derived from data and model and its Gaussian approximation using data from SNe and CMB in the framework of curved CDM. We also show the theoretically expected χ2 distribution for 31 degrees of freedom. For this plot, we have set the number of realizations to N 200. =

of degrees of freedom, as expected for Gaussian data16. The results for all other distributions are similar and we thus do not show them here.

6.C Alternative method for computing the relative entropy

As discussed in Sec. 6.4.1, there exists an alternative method for computing the relative entropies in the case in which we can directly sample from the PPDs. We illustrate the method for the computation of D (y,M y), but all other quantities can be obtained similarly. In a KL || ﬁrst step we directly create a sample of length N N¯ from each of the PPDs p(y’ y,C ,C ,M) × | y y’ and p(y’ y,C ,C ). We then compute the relative entropy D (y,M y) between prior and | y y’ KL || posterior from these samples. This method is signiﬁcantly faster than the one described in Sec. 6.4.1 and is thus preferable in cases in which direct sampling from the PPDs is possible.

6.D Toy model implementation details

6.D.1 Implementation choices

In the toy model described in Sec. 6.4.2 we consider ﬁtting data to polynomials of varying degree nˆ. For each polynomial of degree n, we assume the underlying true model to be given by

n X k yi ck xi . (6.36) = k 0 =

16We do not show the marginalized PPD contours as the parameter space is 30-dimensional but a visual inspection also shows very good agreement between the original and the Gaussian samples.

176 6.E. SNe Ia and CMB posterior means

The data yd,i is further assumed to be normally distributed around the truth yi with a constant standard deviation σ, i.e.

y y ² , i 1,...,m, (6.37) d,i = i + i = where ²i is drawn from a Gaussian with mean zero and standard deviation σ, i.e. N(0,σ). In the speciﬁc implementation described in Sec. 6.4.2, we choose the dimension of the data vector as m 10, the x-values, x , linearly spaced in [0.1, 2.2] and the standard deviation of = i the data as σ 0.2. =

6.D.2 Choice of ﬁducial polynomial coefﬁcients

We choose the fiducial polynomial coefficients in a similar way we would do for real data: we first determine fiducial coefficients for an 8-degree polynomial, which we choose to be given by c ¡3.,1.5,1.,0.5,1.4,0.1,0.7,0.3,3.¢17. We then generate a realization y thereof with = d covariance matrix Cy. This represents the observed data. We then set the fiducial polynomial coefficients for each degree n 0,...,7 considered in our analysis to the best-fit coefficients = for this particular model and yd . This procedure ensures that the choice of fiducial polynomial in our analysis is driven by the data, which is the case in any application of this method to real data.

6.E SNe Ia and CMB posterior means

In this section, we give the posterior means obtained for the cosmological analyses performed in this work: we show the constraints obtained for the analysis of JLA SNe Ia data alone in Tab. 6.3 and the constraints for the combined analysis of JLA data, and the CMB shift parameter R are shown in Tab. 6.4. Where we can compare, our results agree well with those given in Ref. [34].

17These coefficients are not completely random, as we need to make sure that the different fiducial models result in sufficiently different data compared to the uncertainties such that we have the statistical power to distinguish the considered models.

177 Chapter 6. Consistency tests in cosmology using relative entropy 0.023 0.023 0.023 0.023 0.023 0.023 ± ± ± ± ± ± M ∆ 0.070 0.087 0.074 0.070 0.070 0.071 − − − − − − 0.87 a ± - w 0.02 0.02 0.02 0.03 0.03 0.03 1 ± ± ± ± ± ± B 0.57 − M 19.05 18.80 19.00 19.04 19.04 19.04 − − − − − − 0.07 0.06 0.17 + − 0 ± w 0.08 0.08 0.08 0.08 0.08 0.08 1.00 0.91 − ± ± ± ± ± ± − 3.10 0.028 - - Λ ------± Ω 0.007 0.006 0.007 3.16 0.007 3.09 0.007 3.11 0.007 3.10 0.007 3.10 + − α β ± ± ± ± ± 0.701 0.141 0.012 0.013 0.012 0.013 0.027 0.028 0.028 0.027 0.027 - 0.027 - - - b + − + − + − + − 0.98 1.03 ± ± + − Ω a w 0.30 0.050 − 0.20 - 0.141 0.22 0.017 0.018 0.0010.037 0.074 0.020 0.050 0.050 0.026 0.050 0 m + − ± ± - - 0.141 ± ± ± ± Ω w 0.93 0.95 0.297 − − 0.297 0.303 0.154 0.156 Λ + − 0.35 0.34 0.33 0.34 0.31 0.30 0.34 0.298 0.0020.002 0.999 - 0.075 - - - 0.139 - - + − + − + − Ω h ± ± ± 0.563 0.017 0.026 0.107 0.102 0.092 0.097 0.095 0.100 0.034 - - - 0.141 m + − + − + − + − ± CDM 0.69 Ω CDM 0.69 a CDM 0.71 CDM 0.75 0 Λ CDM 0.202 w Model Λ Table 6.3: Summary of SNe Ia posterior means for all cosmological models considered in this work. 0 c. CDM 0.202 w c. w Table 6.4: Summary of SNe Ia and CMB posterior means for all cosmological models considered in this work. CDM 0.280 CDM 0.206 a CDM 0.254 CDM 0.297 0 Λ CDM - - - - 0.137 w Model Λ 0 c. CDM 0.030 w c. w

178 6.E. SNe Ia and CMB posterior means ). continuation 0.022 0.023 0.023 0.022 0.022 0.022 0.023 + − ± ± ± ± ± M ∆ 0.070 0.070 0.087 0.086 0.070 0.070 − − − − − − 0.02 0.02 0.02 0.03 0.02 0.02 1 ± ± ± ± ± ± B M 19.04 19.05 19.05 19.05 18.80 18.80 − − − − − − 0.08 0.08 0.08 0.08 0.08 0.08 ± ± ± ± ± ± 0.007 3.10 0.007 3.15 0.007 3.16 0.007 3.10 0.007 3.11 0.007 3.11 α β ± ± ± ± ± ± CDM 0.141 CDM 0.141 a CDM 0.141 CDM 0.141 0 Λ CDM 0.137 w Model Λ 0 c. CDM 0.136 w c. w Table 6.4: Summary of SNe Ia and CMB posterior means for all cosmological models considered in this work (

179

7 Conclusions

Habe nun, ach! Philosophie, Juristerei und Medizin, Und leider auch Theologie! Durchaus studiert, mit heißem Bemühn. Da steh ich nun, ich armer Tor! Und bin so klug als wie zuvor; — Johann Wolfgang von Goethe, Faust, Der Tragödie erster Teil

Current large-scale cosmological observations are well-fit by the standard model of cosmology, ΛCDM, which relies on several pillars, such as dark matter, dark energy and inflation. Despite the observational success of ΛCDM, the nature of dark matter and dark energy are currently not understood and inflation still faces conceptual issues. Current and future surveys have the potential to constrain the properties of these key ingredients and thus improve our fundamental understanding of cosmology through observations of cosmological probes. The information contained in these probes is highly complementary and their combination will allow us to derive the tightest cosmological constraints to date. In order to optimally combine the information from different surveys, we however need to go beyond the notion of single, isolated probes towards a more holistic view of cosmology and the Universe.

The core of this thesis consists of implementing ﬁrst steps in this direction by developing an integrated scheme for cosmological probe combination.

In Chap.2 we studied the analysis of future spectroscopic galaxy surveys. These surveys will be both wide and deep and will therefore require an analysis in three-dimensional spherical geometry. We compared the constraining power of a future survey when analyzed using two different analysis methods, which both explicitly take the curvature of the sky into account: the spherical harmonic tomography (SHT) power spectrum and the spherical Fourier-Bessel

181 Chapter 7. Conclusions

(SFB) power spectrum. In a Fisher matrix analysis we ﬁnd that the SHT power spectrum appears appropriate for surveys focusing mainly on the power spectrum shape, because it can be tailored to be sensitive to modes at the redshift boundary of the survey. The SFB power spectrum on the other hand appears suited for speciﬁc applications such as Baryonic Acoustic Oscillations, as these tend to be washed out in SHT power spectrum analyses.

In Chap.3 we developed an integrated scheme for cosmological probe combination in which the probes are combined into a common framework starting at the map level. We applied this framework to CMB temperature anisotropy data from Planck 2015 [209], galaxy clustering from SDSS DR8 photometric galaxies [13] and weak lensing from SDSS Stripe 82 [20].

In Chap.4 we extended the analysis to also include CMB lensing from Planck [ 209], weak lensing from DES SV data [138] as well as SNe Ia data from the Joint Lightcurve Analysis [34] and constraints on the Hubble parameter [227, 74]. This latter analysis yields 12 auto- and cross-power spectra, including amongst others the CMB temperature power spectrum, cosmic shear, galaxy clustering, galaxy-galaxy lensing and CMB lensing cross-correlations as well as background probes. We have derived constraints on cosmological parameters in a joint fit to these data and find that their complementarity allows us to constrain both cosmological and probe-specific nuisance parameters. The constraints derived from our latter work agree well with those derived in our earlier analysis and the data is well-fit by the best-fit cosmological model. In summary, the analyses in Chapters3 and4 have led to self-consistent and competitive constraints on cosmological parameters and have provided a confirmation of ΛCDM through the consistency of different probes.

Assessing concordance between cosmological constraints in high-dimensional parameter space from marginalized two-dimensional contours can be misleading. In Chap.5, we have therefore extended the analysis described in Chapters3 and4 by investigating possible tensions between the derived cosmological parameter constraints and other existing results using relative entropy [160] and the Surprise statistic [240]. We have considered several CMB data sets [124, 212, 45] and found all of them to be consistent in the framework of a ΛCDM cosmological model.

Finally, in Chapter6 we have revisited the relative entropy or KL divergence [ 160] as a consistency measure and discussed some of its key properties in more detail. Furthermore, we have proposed a novel model testing framework, which combines relative entropy and posterior predictive distributions (PPD): in our method, we quantify the consistency between data and model by computing the KL divergence between the PPD derived from the data alone and the PPD derived from data and model. Comparing the observed relative entropy to expectations under the null hypothesis of consistency, allows us to both perform model selection and assess the goodness of ﬁt of any given model. Through a suite of toy models and applications to cosmological data from SNe Ia [34] and CMB [214], we have shown that the proposed method performs rather well and yields results consistent with expectations. We therefore conclude that our framework provides a promising new avenue for model testing in cosmology.

182 The future of cosmology is exciting; a large number of experiments are going to deliver data at an unprecedented precision and will allow us to observe the Universe in yet unexplored wavelength ranges. The large amount of high-precision data delivered by these surveys will require novel analysis methods that extract the maximal amount of information while yielding unbiased results, and joint cosmological probe analyses will thus play an increasingly important role. The methods and results presented in this thesis provide a ﬁrst step towards integrated analysis methods suitable for future cosmological surveys.

183

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206 List of symbols

a Scale factor.

1 A Primordial power spectrum amplitude at a pivot scale of k 0.05 Mpc− . s 0 = b Linear galaxy bias. c Speed of light.

χ Comoving distance.

C` Spherical harmonic power spectrum at angular multipole `.

C`(k,k0) Spherical Fourier-Bessel power spectrum at angular multipole `. cs Speed of sound.

D1(a) Linear growth rate at scale factor a.

δ Dark matter overdensity.

δb Baryon overdensity.

δg Galaxy overdensity.

δH Scalar power spectrum amplitude.

δm Total matter overdensity.

DKL Kullback-Leibler divergence or relative entropy.

² Galaxy ellipticity.

η Conformal time.

Fαβ Fisher matrix. fsky Fractional sky coverage.

G Newton’s gravitational constant.

207 List of symbols

γ Weak lensing shear. h Dimensionless Hubble parameter at z 0. = H Hubble parameter at z 0. 0 = H Hubble parameter. j` Spherical Bessel function of order `. k Wave vector amplitude.

κ Lensing convergence.

K Spatial curvature parameter. m Multiplicative weak lensing shear bias parameter. mν Neutrino mass.

µ(z) Distance modulus at redshift z.

I Mutual information. ns Power spectrum scalar spectral index.

Ωb Fractional baryon density today.

ΩK Fractional curvature density today.

ΩΛ Fractional dark energy density today.

Ωm Fractional matter density today.

Φ Newtonian potential perturbation.

P(k) Three-dimensional power spectrum.

P` Legendre polynomial of order `.

Ψ Curvature potential perturbation.

R Baryon-to-photon ratio.

ρcrit Critical density of the Universe. rs Sound horizon.

S Surprise statistic.

1 σ8 r.m.s. of matter ﬂuctuations in spheres of comoving radius 8h− Mpc.

τreion Optical depth to reionization.

208 List of symbols

TCMB CMB temperature today.

Θ Photon temperature perturbations

Θν Neutrino temperature perturbations.

ΘP Photon polarization perturbations.

T (k) Matter transfer function for wave vector k. vb Baryon velocity. v Dark matter velocity. w Equation of state parameter.

ξ Two-point correlation function. z Redshift. zreion Reionization redshift. z? Redshift of recombination.

209

List of abbreviations

ΛCDM Λ cold dark matter.

BAO Baryon acoustic oscillations.

CDM Cold dark matter.

CMB Cosmic Microwave Background.

DM Dark matter.

EFE Einstein ﬁeld equations.

FLRW Friedmann-Lemaître-Robertson-Walker metric.

FM Fisher matrix.

GR General Relativity.

ISW Integrated Sachs-Wolfe effect.

LSS Large-scale structure.

MCMC Monte Carlo Markov Chain.

PPD Posterior predictive distribution.

PSF Point spread function.

RSD Redshift space distortions.

SFB Spherical Fourier-Bessel power spectrum.

SHT Spherical harmonic tomography power spectrum.

SNe Ia Type Ia supernovae.

211

Acknowledgements

Life is what we make of it. Travel is the traveler. What we see isn’t what we see but what we are. — Fernando Pessoa, The Book of Disquiet

When, on a starry night in summer, while lying on the damp grass, I was told that the Universe is expanding, I was puzzled and even more so, when I tried to imagine how this actually works. Now, a great many years later, the moment has come to write the acknowledgments for my PhD thesis. Nevertheless, I still remember that moment and to be honest, the expansion of the Universe is still confusing to me, I just got older and have become better at ignoring this feeling.

Even though I still can’t believe the time has come to say thanks, let me start by ﬁrst of all, thanking my two supervisors, Alex and Adam, for giving me the opportunity to do a PhD in their group and for being fantastic advisors. Thank you very much for all the support and encouragement and for your open ears for personal matters, too. Thank you very much also for teaching me so much about cosmology and doing research in general, and for giving me guidance as well as a lot of freedom to pursue my own ideas.

I would also like to thank all the members, past and present, of the cosmology group at ETH. First of all I would like to thank the astrohackers. Thanks guys for accepting me in your WhatsApp chat :-). Thank you very much Claudio, Simon, Sebastian and Joël, for your friendship and the great time we spent together. I cannot imagine having better PhD friends than you, guys. Thank you so much, Claudio, for your constant support, your always open ears and the many late night conversations about random things, which sometimes even included science ;-). Thank you also, Simon, for all the encouragements and for always calming me down when I was freaking out. I would also like to thank you, Chihway, for being a fantastic friend and for the advice and encouragement throughout these years. I am very much looking forward to ﬁnally sharing the same continent with you again. Thank you very much, Annie, for being a great friend and for supporting me throughout these years. Thank you also very

213 Acknowledgements much, Milena, for so quickly becoming a sports buddy and a friend.

I would also like to thank all the other PhD students, past and present, for making these ﬁve years a really great time. Especially, I would like to thank Anna for always being there for me, for listening and for being a wonderful friend. Thank you also very much for going through the application period with me; I really cannot imagine a better person walking that path alongside me. Thank you also very much, Lia, for being a great friend and ﬂatmate and for always cheering me up when I was sad.

Furthermore, I would like to thank all the members of the institute for the many nice dis- cussions and encouragements. I would especially like to thank Benny and Kevin for all their advice, encouragement and support with my job applications.

The institute would not be complete without the many project students. First of all, I would like to thank all the students I had the pleasure to supervise: Michael, Dominik, Eliane, Julian, Raphaël and Adrian. Furthermore, I would like to thank you, Mahra, for your friendship, the many “sports-suicides” and everything else. Thank you very much also, Lorenza and Lukas, for your friendship and coming to sports with me.

Apart from all the people I met at the institute, I would also like to thank all the other people who have made these past years so fantastic. Thank you so much, Rossella, Martina, Maria, Verena, Simon, Franziska, Chantal, Daniel, Ingo, Alan, Markus, Alison, Zohar, Francesco and everyone else I might have forgotten, for your friendship, support and the uncountably many Superkondi (aka body attack). Huge thanks go to ASVZ for Superkondi, body combat and muscle pump! Oh, I cannot imagine how I would have survived without that :-). Thank you so much, Zita and Simona, for your amazing friendship and for being great ﬂatmates - I will miss Jungholzstrasse 35! Thank you very much also, my study friends, Tessa, Nina, Oliver and Giada. Finally, big thanks also go to Bonny and Chitra for all the very nice conversations.

During my PhD and especially during the last year, I have been lucky to travel to many places and meet a lot of people. I would hereby like to thank all the people at the places that I visited for hosting me and for making me immediately feel welcome. Special thanks go to Jochen, Kerstin, Nico and Steffen at LMU, Shirley, Emmanuel, Elena and Joanne at UC Berkeley, Dragan at UMich, Daniel at Stanford, Elisabeth and Tim at JPL, David at Princeton, Stephen at CCA, Zoltán at Columbia, Eva-Maria at Portsmouth, and Klaus and Ami at OSU.

Especially, I would also like to thank Eiichiro for being my external examiner.

I think it is an amazing privilege to have been able to spend the last ﬁve years working on what I really like, essentially doing my hobby. I would therefore very much like to thank the Swiss public for ﬁnancing ETH and the SNF.

Finally, I would like to thank my mum and dad. Thank you so much Mec and Cifo for always being there for me, for going with me through the highs and the very lows and for never giving up on me. I would not be writing these acknowledgments without your constant support and

214 Acknowledgements encouragement.

I would like to thank Eric Hivon for his valuable help with PolSpice. I would also like to thank Duncan Hanson for his help with the Planck CMB lensing maps and Donnacha Kirk for result comparison. I also thank all my anonymous referees and Stephen Feeney, for the many suggestions that helped me improve the quality and clarity of the manuscripts presented in this thesis.

Some of the results in this thesis have been derived using the HEALPix (Górski et al. [99]) package. Icons used in Fig. 3.1 made by Freepik from www.flaticon.com. The color palettes employed in these works are taken from http : //colorpalettes.net and http : //flatuicolors.com. I further acknowledge the use of the color map provided by [209]. The contour plots have been created using corner.py [85]. Fig. 4.1 has been created using the footprint package1. The work in this thesis would not have been possible without the use of supercomputing resources and open-source software. I would thus like to thank the people at CSCS, the IT services at ETH and ISG (especially Claude and Christian) for dealing with my questions and problems. I would also like to thank all the python developers and all people contributing to the python packages NumPy2, SciPy [144], Matplotlib [136], jupyter [157], Ipython [204], healpy3 and astropy [23, 271]. Finally, I would also like to acknowledge the extensive use of the SAO/NASA Astrophysical Data System4 and the arXiv5 preprint server.

Funding for the SDSS and SDSS-II has been provided by the Alfred P.Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.

The SDSS is managed by the Astrophysical Research Consortium for the Participating Institu- tions. The Participating Institutions are the American Museum of Natural History, Astrophys- ical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.

1See https://github.com/nasa-lambda/cmb_footprint. 2See http://www.numpy.org. 3See http://healpy.readthedocs.io/en/latest. 4See http://adsabs.harvard.edu. 5See https://arxiv.org.

215 Acknowledgements

Funding for SDSS-III has been provided by the Alfred P.Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Ofﬁce of Science. The SDSS-III web site is http://www.sdss3.org/.

SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institu- tions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Partic- ipation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astroﬁsica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck In- stitute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, Univer- sity of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.

This project used public archival data from the Dark Energy Survey (DES). Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for Eng- land, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cientíﬁco e Tecnológico and the Ministério da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft and the Collab- orating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Enérgeticas, Medioambientales y Tecnológicas-Madrid, the Uni- versity of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi National Accel- erator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciències de l’Espai (IEEC/CSIC), the Institut de Física d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universität München and the associated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, the Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University.

Based on observations obtained with Planck (http://www.esa.int/Planck), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada.

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