Linking Large-Scale Structure and Peculiar Velocities in the Low-Redshift Universe

Caitlin Adams

Presented in fulﬁllment of the requirements of the degree of Doctor of Philosophy

January 2019

Faculty of Science, Engineering and Technology Swinburne University

i Abstract

The standard cosmological model has been thoroughly tested over the past two decades, but we still remain in the dark about the underlying cause of accelerating expansion. Interestingly, the two primary classes of viable models that explain this behaviour, dark energy and modified gravity, can be distinguished by measurements of the growth rate of structure in the low-redshift universe. Given the smaller cosmological volume probed by low-redshift surveys in contrast to high-redshift surveys, it is critical that methods are developed to extract as much information about the growth rate as possible from low-redshift data. In this thesis, we present a significant contribution to low-redshift cosmology through the development and application of a method that utilises the information provided by redshift-space distortions, peculiar velocities and their cross-correlation. This method is the first of its kind to self-consistently model all three sources of information and use them to simultaneously constrain the growth rate of structure. Throughout the thesis, we consistently show that the inclusion of the cross-correlation allows higher-precision constraints from low-redshift surveys than using redshift-space distortions or peculiar velocities alone, which is due to their highly complementary nature. To begin, we develop the theoretical models that underpin all of the analysis presented in this thesis. Chapter 2 focusses on the construction of fully self-consistent models for the galaxy overdensity auto-correlation function, peculiar velocity auto-correlation function, and the cross-correlation function. These can be used to construct a covariance matrix that is a function of the growth rate fσ8, allowing this parameter to be constrained through a maximum likelihood approach. Throughout this chapter, we take care to model relevant observational effects, making the model highly physically accurate. In Chapter 3, we present the first application of our method to a subsample of the 6-degree Field Galaxy Survey (6dFGS). This survey is one of the best low-redshift samples, especially since it provides the largest single sample of peculiar velocities to date. In this first analysis, we focus on constraining the growth rate from the peculiar velocity information, which helps to highlight the benefits of including cross-correlation informa- +0.087 tion. In the absence of the cross-covariance, we find fσ8 = 0.461 0.079. We find that − the statistical uncertainty is reduced by 20% when including the cross-covariance, giv- +0.067 ing fσ8 = 0.424 0.064. Our constraint is consistent with other results from 6dFGS, and − makes a significant addition to the literature on the use of multiple cosmological probes to improve cosmological parameter constraints. In addition to our constraints, we also find 1 evidence of the cross-correlation signal in the 6dFGS data on scales up to 50 h− Mpc. ii

In Chapter 4, we extend the analysis performed in Chapter 3 by implementing a complete model for redshift-space distortions and by using a larger galaxy redshift sample from 6dFGS. The inclusion of redshift-space distortions adds additional information on the growth rate of structure from the galaxy distribution, which is complementary to the information from peculiar velocities. We also conduct thorough tests of the underlying model assumptions by applying our method to sophisticated mock catalogues of 6dFGS. By identifying the key systematic effects of our model, we provide a systematic uncertainty for our growth rate constraint in addition to the statistical uncertainty. Using our complete covariance model, we find fσ = 0.384 0.052(stat) 0.061(sys), which has a 64% reduction 8 in statistical uncertainty compared to only using the galaxy overdensity information, and a 50% reduction in statistical uncertainty compared to only using the peculiar velocity information. We find that the improved modelling and larger galaxy redshift sample leads to an 18% reduction in the statistical uncertainty of our constraint from Chapter 3, demonstrating the power of our improved model. Finally, in Chapter 5, we consider our method in the context of the Taipan Galaxy Survey, which represents the next generation in low-redshift large-scale structure surveys. Taipan is set to significantly extend 6dFGS, measuring up to 2 million galaxy redshifts and 50,000 peculiar velocities over the next 4 years. We apply our method to mock catalogues for the survey and run an equivalent Fisher matrix forecast, allowing us to make a direct comparison between these two methods. We find that the Fisher matrix typically underestimates the uncertainties obtained with our method: the uncertainty in the growth rate constraint from our covariance matrix method is 1.55 times that of the forecast. We use this difference to estimate the percentage constraints that could be obtained by applying our method to Taipan, based on existing forecasts for the survey. We find that our method could yield a 4.2% constraint on the growth rate of structure by the end of the survey, which is in line with one of Taipan’s key cosmological goals of a growth rate constraint of less than 5%. This demonstrates the success of our method in harnessing cosmological information about the growth rate of structure at low redshifts and highlights the value of our approach for the next generation of low-redshift observations. iii iv Acknowledgements

I count myself incredibly fortunate to have had the unconditional support of so many people while completing my PhD. Firstly, I’d like to extend a huge thank you to Chris Blake, my primary supervisor. I am grateful to you for always being excited about the science we were working on; for providing such a supportive environment, which helped me flourish as a researcher; for always being kind when I was hard on myself; and for always encouraging me, no matter how ambitious I got (and, of course, for helping me handle things when it turned out to be a little too ambitious). I feel so privileged to have worked with you. I’d also like to thank David Parkinson for his supporting supervision. Thank you for all of your help, feedback and advice over the years – I treasured all of it. You are largely responsible for my love of cosmology and Bayesian statistics, for which I am incredibly grateful. Finally, thank you for always being one of my strongest advocates; knowing how highly you think of me has always provided me great comfort and confidence. During my PhD, I was lucky to be mentored and supported by Emma Ryan-Weber and Cath Trott. Thank you to both of you – your advice and encouragement at various critical junctures made so much difference. I am also extremely fortunate to have had the support of an entire research centre throughout my PhD. CAS has been such a wonderful place to work, and I’m incredibly grateful to all its staff and students for making my experience such a positive one. I would particularly like to thank Andrew Johnson, Adam Stevens, Emily Petroff, Rebecca Allen, Steph Pointon, Geoff Bryan, Michelle Cluver, Ned Taylor, Rossana Ruggeri, Sara Webb, Simon Goode, Jennifer Piscionere and Manodeep Sinha. A special thanks goes to my two closest friends, Sabine Bellstedt and Leonie Chevalier. Both of you have been wonderful friends to me and I am so lucky to have met you. Thank you for all the cups of tea, board games and chats, without which I surely would have gone mildly mad. Throughout my PhD, I have been incredibly grateful for the financial support I received from both the Australian Government Research Training Program and the ARC Centre of Excellence for All-sky Astrophysics (CAASTRO). I would also like to thank Athol Whitten, Simone Stuckey and Michael Smith for taking me on as an intern, and then employee, at Mezo Research. It was such a joy to work with all of you. Thank you for always being so respectful of me, especially when I needed some time away from work to finish writing my thesis. v

I’d like to express a huge amount of gratitude to my parents, Liz and Lewis Adams, for all of their love and support. Your encouragement and advice has helped me so much throughout the ups and downs of my PhD. Finally, my thanks go to my wonderful partner, Jacob Shearer. Thank you for all that you have done for me during the past few years: for making me laugh whenever I despaired; for always celebrating my milestones and achievements; and for encouraging me to keep at it, even when it all felt insurmountable. Your constant companionship has meant (and continues to mean) the world to me. vi vii Declaration

The work presented in this thesis has been carried out in the Centre for Astrophysics & Supercomputing at Swinburne University of Technology between 2015 and 2019. This thesis contains no material that has been accepted for the award of any other degree or diploma. To the best of my knowledge, this thesis contains no material previously published or written by another author, except where due reference is made in the text of the thesis. The content of the chapters listed below has appeared in refereed journals. Minor alterations have been made to the published papers in order to maintain argument continuity and consistency of spelling and style.

The majority of Chapter 2 (all aside from Section 2.3.4) and all of Chapter 3 has • been published as Improving constraints on the growth rate of structure by modelling the density-velocity cross-correlation in the 6dF Galaxy Survey Adams, C., & Blake, C. 2017, Monthly Notices of the Royal Astronomical Society, 471, 839

My contribution to this paper was the model development, data analysis and writing, accounting for 90 per cent of the final manuscript. My co-author contributed scientific expertise and editing of the text, accounting for 10 per cent of the final manuscript.

Caitlin Adams Melbourne, Victoria, Australia 2019

Contents

Abstract i

Acknowledgements iii

Declaration vi

1 Introduction 1 1.1 Theoretical Cosmology ...... 1 1.1.1 General Relativity ...... 1 1.1.2 FLRW Metric ...... 2 1.1.3 Evolution of Matter and Energy ...... 5 1.1.4 Redshift ...... 6 1.1.5 Dynamics ...... 7 1.1.6 Distance Measures ...... 9 1.1.7 Structure Formation ...... 12 1.1.8 Correlation Functions and Power Spectra ...... 15 1.2 Observational Cosmology ...... 17 1.2.1 The Cosmic Microwave Background ...... 17 1.2.2 Baryon Acoustic Oscillations ...... 19 1.2.3 Type Ia Supernovae ...... 21 1.2.4 Galaxy Bias and Redshift-Space Distortions ...... 23 1.2.5 Peculiar Velocities ...... 27 1.3 Explanations for Accelerating Expansion ...... 29 1.3.1 Dark Energy ...... 30 1.3.2 Modiﬁed Gravity ...... 31 1.4 Multiple-Tracer Analyses ...... 32 1.4.1 Statistical Perspective ...... 33 1.4.2 Physical Perspective ...... 35 1.5 Low-Redshift Galaxy Surveys ...... 36 1.5.1 6dFGS ...... 37 1.5.2 Taipan ...... 38 1.6 Thesis Motivation and Overview ...... 38

2 Theory 41 2.1 Overview ...... 41

ix x Contents

2.1.1 Deﬁnitions and Conventions ...... 41 2.2 Likelihood Formalism ...... 42 2.2.1 Bayesian Inference ...... 42 2.2.2 Modelling the Likelihood ...... 43 2.2.3 Modelling the Covariance ...... 44 2.2.4 Fiducial Power Spectra ...... 54 2.3 Data Modiﬁcations ...... 56 2.3.1 Modelling the Logarithmic Distance Ratio ...... 56 2.3.2 Gridding to Obtain the Data Vector ...... 58 2.3.3 Inclusion of Error Terms ...... 61 2.3.4 Modelling Redshift-Space Distortions ...... 62

3 The Density-Velocity Cross-Correlation in 6dFGS 71 3.1 Overview ...... 71 3.2 Data and Simulations ...... 72 3.2.1 6dFGS ...... 72 3.2.2 Simulation ...... 75 3.3 Theory and Methodology ...... 76 3.3.1 Eﬀect of Redshift-Space Distortions ...... 77 3.3.2 Generation of the Fiducial Power Spectra ...... 78 3.3.3 Integration Bounds ...... 78 3.3.4 Evaluating the Likelihood Function ...... 79 3.4 Simulation Results ...... 80 3.4.1 Parameter Constraints ...... 80 3.4.2 Additional Bias ...... 82 3.5 Data Results ...... 84 3.5.1 Parameter Constraints ...... 84 3.5.2 Additional Bias ...... 86 3.5.3 Direct Evidence of the Cross-Covariance ...... 88 3.5.4 Comparison to Existing Literature ...... 89 3.6 Summary ...... 93

4 Joint Growth Rate Measurements from Peculiar Velocities and RSD 95 4.1 Overview ...... 95 4.2 Data and Simulations ...... 96 4.2.1 6dFGS ...... 96 Contents xi

4.2.2 Simulation ...... 98 4.3 Theory and Methodology ...... 99 4.3.1 Generation of the Fiducial Power Spectra ...... 100 4.3.2 Integration Bounds ...... 101 4.3.3 Accounting for Diﬀerent Galaxy Bias Values ...... 101 4.3.4 MCMC Sampling ...... 102 4.3.5 Evaluating the Likelihood Function ...... 103 4.4 Simulation Results ...... 104 4.4.1 Galaxy Overdensity Auto-Covariance ...... 104 4.4.2 Logarithmic Distance Ratio Auto-Covariance ...... 113 4.4.3 Complete Covariance ...... 117 4.4.4 Summary of Fiducial Model Results ...... 123 4.5 Data Results ...... 124 4.5.1 Systematics ...... 127 4.5.2 Comparison to Existing Literature ...... 132 4.6 Summary ...... 138

5 Forecasts for the Taipan Galaxy Survey 139 5.1 Overview ...... 139 5.2 The Taipan Galaxy Survey ...... 140 5.2.1 Taipan Mock Catalogues ...... 141 5.2.2 Sample for Analysis ...... 143 5.3 Theory and Methodology ...... 143 5.3.1 Fisher Matrix Forecasts ...... 143 5.3.2 Fisher Matrix Set-Up ...... 145 5.3.3 Covariance Matrix Model ...... 147 5.4 Fisher Matrix Results ...... 147 5.5 Model Covariance Results ...... 148 5.5.1 Galaxy Overdensity Auto-Covariance ...... 148 5.5.2 Logarithmic Distance Ratio Auto-Covariance ...... 149 5.5.3 Complete Covariance ...... 150 5.6 Discussion of Results ...... 152 5.6.1 Comparison of Fisher Matrix Forecasts to Model Covariance . . . . 152 5.6.2 Comparison to Similar Analyses ...... 154 5.6.3 Comparison to 6dFGS Results ...... 155 5.6.4 Implications for Taipan ...... 155 xii Contents

5.7 Summary ...... 156

6 Conclusions and Future Work 157 6.1 Conclusions ...... 157 6.2 Future Work ...... 159 6.2.1 Model and Method Improvements ...... 160 6.2.2 Research Opportunities ...... 162

Bibliography 175 1 Introduction

Cosmology, the study of the Universe as a whole, encompasses a broad range of topics by nature. In this introduction, we aim to give the reader an overview of the key concepts that feature in this thesis. We begin with the theoretical perspective in Section 1.1, with a focus on how large-scale structure came to form and how we measure it statistically. We then cover the observations that have both validated our theoretical understanding and posed new questions in Section 1.2. In Section 1.3, we review theories that might address one of cosmology’s key outstanding questions: the cause of the Universe’s accelerating expansion. Next, in Section 1.4, we look at how we may better test the cosmological model by combining multiple cosmological probes. This leads us to overview the available data in the local universe in Section 1.5. Finally, in Section 1.6, we discuss the motivation for this thesis and an overview of the work undertaken.

1.1 Theoretical Cosmology

1.1.1 General Relativity

When it comes to the immense physical scales of our Universe, gravity plays a prominent role. Consequently, an accurate model of gravity serves as a necessary platform for our understanding of the Universe. Modern cosmology has been thoroughly shaped by Einstein’s general theory of relativity (GR), which came together over a number of years (Einstein, 1916). In the past 100 years, there has been a substantial amount of evidence to support the theory, most notably laboratory tests that measured the eﬀect of gravitational redshifting (e.g. Pound & Rebka, 1960), and indirect evidence of gravitational waves through the orbital decay of binary pulsars (e.g. Taylor, 1975). More recently, gravitational wave interferometry has achieved the necessary sensitivity to directly detect gravitational waves.

1 2 Chapter 1. Introduction

Heralded by the measurement of gravitational waves from two merging black holes by Ab- bott et al. (2016), gravitational wave astronomy marks a new era for how we explore the Universe. Where Newton originally posited that gravity acted as an attractive force between massive objects, Einstein pursued the vision that gravity can instead be thought of as a consequence of the geometric properties of space itself, or more correctly, space-time. This relationship is captured through the Einstein ﬁeld equations:

8πG G = T . (1.1) µν c4 µν

Although this may look like a single equation, the indices µ and ν run over the different µ 0 1 2 3 coordinates of space-time: X = (X ,X ,X ,X ) = (ct, x, y, z). Gµν is the Einstein tensor, which captures the geometry of space-time, and Tµν is the energy-momentum tensor, which captures the properties of mass and energy in various forms. G is Newton’s gravitational constant and c is the speed of light. By considering the different indices, one may extract multiple differential equations that describe how space-time is influenced by energy and vice-versa. Given the mathematical properties of the field equations, Einstein noted that an additional term may be inserted:

8πG G + g Λ = T , (1.2) µν µν c4 µν where gµν is the metric tensor, which deﬁnes the separation of two events in space-time, and Λ is a constant. This additional term has two possible physical interpretations: on the left-hand side of the equation, it represents the curvature of empty space; on the right-hand side, it acts as an energy-momentum tensor for the vacuum of space. It is this second interpretation that is of most interest to cosmology: if Λ is non-zero, the vacuum would have negative pressure, acting against the gravitational collapse of all other matter. Einstein originally introduced this term as a way to balance the attractive behaviour of gravity and keep the Universe static.

1.1.2 FLRW Metric

In order to have any success in understanding the Universe, we must have a way to describe the relative positions of events in space-time. In any mathematical formalism, such a description is known as a metric: an equation that describes the distances between pairs of elements in a set. In cosmology, the elements are space-time events and are described 1.1. Theoretical Cosmology 3 by their position, x = (x, y, z), and the time they occurred, t. Within the framework of general relativity, the generic form of the metric is

2 µ ν ds = gµνdX dX , (1.3)

2 where ds indicates the interval between two space-time events and gµν is the metric tensor (as seen in Eq. 1.2). Here and throughout, we use the Einstein summation convention for brevity: repeated indices indicate a sum over all values of the index. The most commonly used metric in cosmology today is the Friedmann-Lemaˆıtre- Robertson-Walker (FLRW) metric, named for those who contributed to its development throughout the 1920s and 30s (Friedman, 1922; Lemaˆıtre,1931; Robertson, 1936; Walker, 1937). While the metric is a solution of Einstein’s ﬁeld equations, it is generally derived from geometric arguments. The primary assumption is that the cosmological principle is valid; it states that the Universe is homogeneous (it has uniform density) and isotropic (it is the same in all directions). The assumption of these attributes is important as it introduces symmetries that simplify the form of the metric and ﬁeld equations. Conse- quently, it was included in the cosmological model long before there was observational evidence to support it. In the last twenty years, however, we have begun to directly test the assumption of isotropy through measurements of the cosmic microwave background (CMB; Bennett et al., 1996), and the assumption of homogeneity through the distribution of large-scale structure (Hogg et al., 2004; Scrimgeour et al., 2012). If one separates the time and spatial components of the metric, we have the generic form:

ds2 = c2dt2 d`2, (1.4) − where d`2 represents the separation between the spatial components of two space-time events. The cosmological principle leads us to use a three-dimensional space that is max- imally symmetric, with constant curvature. There are three possible options that satisfy these criteria:

Euclidean (ﬂat) • spherical (positively curved) • hyperbolic (negatively curved) • Conveniently, we may express the metric for each of these options with a single equation, which is achieved by parametrising the curvature through the dimensionless parameter K. 4 Chapter 1. Introduction

Noting that curvature only aﬀects the spatial component:

 0 for Euclidean dχ2  d`2 = R2(t) + χ2dθ2 + χ2 sin2(θ)dφ2 , where K 1 for spherical , 1 Kχ2 ≡ −   1 for hyperbolic − (1.5) where we have used the dimensionless spherical coordinates (χ, θ, φ), taking advantage of the symmetric nature of the metric. Dimensional analysis tells us that d` should have units of length; this is encoded in the R(t) term, which is known as the scale factor. To reduce the metric to a more compact form, we make the substitution

dχ2 dr˜2 = , (1.6) 1 Kχ2 − such that  χ for K = 0  2 2 2 2 2 2 2  d` = R (t) dr˜ + S (˜r)[dθ + sin (θ)dφ ] , where SK (˜r) = sin(χ) for K = 1 . { K }  sinh(χ) for K = 1 − (1.7)

Finally, it is common to rewrite the metric using the dimensionless form of the scale factor a(t) R(t)/R , where R = R(t ) is the value of the scale factor today, making ≡ 0 0 0 a(t0) = 1. We note that a(t) describes the size of the Universe at time t relative to its size today, parametrizing the Universe’s expansion. In addition to this, we may also reassign dimensions of length to our radial coordinate by deﬁning r = R0r˜ such that the metric takes the form

ds2 = c2dt2 a2(t) dr2 + R2S2(r/R )[dθ2 + sin2(θ)dφ2] . (1.8) − 0 k 0

The metric coordinates (r, θ, φ) are known as comoving coordinates: the distance between pairs of comoving coordinates remains constant with the expansion of the Universe, unlike the physical distance, which will grow with expansion. We will cover the various distance measures used in cosmology later in this chapter. 1.1. Theoretical Cosmology 5

1.1.3 Evolution of Matter and Energy

With the FLRW metric in hand, we may derive the time evolution of the dimensionless scale factor a(t) using Einstein’s field equations. Before starting, however, we need to take a closer look at the energy-momentum tensor and how the various components of the Universe change with time as a function of the scale factor. Under the conditions of homogeneity and isotropy, matter and energy are treated as perfect fluids: they can be perfectly described in terms of their energy density and their pressure P . We note that the energy density is related to the mass density ρ through = ρc2. We can write the energy-momentum tensor as   0 0 0     0 P 0 0  µ µλ  −  Tν = g Tλν =   . (1.9)   0 0 P 0   −  0 0 0 P − To understand how perfect fluids evolve, we may impose the conservation of energy and momentum, which states that the rate of change of the energy-momentum tensor with respect to the metric coordinates is zero. Working through the mathematics for each index, one may write the conservation equation for a perfect fluid as

a˙ ˙ + 3 ( + P ) = 0, (1.10) a where the overdot represents a derivative with respect to time, i.e.a ˙ da/dt. ≡ We now wish to apply this equation to the various forms of matter and energy that wish to model:

Matter: • this encompasses both the matter we can see (baryons), and the matter we cannot (dark matter). Both can be modelled as ﬂuids where the pressure is much lower than the energy density: P . This allows us to simplify the conservation equation | | by setting P = 0. Solving the conservation equation then gives

3 a− . (1.11) m ∝

Radiation: • this covers all relativistic particles, such as photons and neutrinos. These are mod- 6 Chapter 1. Introduction

elled as having pressure that is around one-third of the energy density, so we may set P = /3, giving

4 a− . (1.12) r ∝

Dark Energy: • if we take the vacuum energy interpretation of the Λ term in Einstein’s field equations (Eq. 1.2), we may posit that the vacuum acts as a fluid with negative pressure: P = . More generally, a fluid with this behaviour is known as dark energy, and − solving the conservation equation gives

a0. (1.13) Λ ∝

Given that the scale factor captures the expansion of the Universe, we can see that the dilution of m scales with the volume of the Universe. r is diluted through both expansion and an additional eﬀect where the energy scales as E a 1 (this is known as redshifting ∝ − and is discussed below). Finally, Λ remains constant as the Universe expands, consistent with the interpretation that this is the energy of the vacuum (if there is more vacuum, there is more vacuum energy). We note that the solutions to the conservation equation for these ﬂuids may be parametrized in terms of a new parameter, known as the equation of state w = P/, such that

 0 for matter  3(1+w)  a− , where w = 1/3 for radiation . (1.14) ∝   1 for dark energy −

1.1.4 Redshift

In the previous section, we noted that the energy density of radiation is not only diluted by the expansion of the Universe, but that the radiation itself also loses energy. It is important to consider how this might aﬀect our observations, given that the majority of our understanding of the Universe is inferred from electromagnetic radiation (it’s still early days for gravitational wave astronomy). We may consider how the Universe’s expansion aﬀects our observations by looking at the energy of a photon

hc E = , (1.15) λ 1.1. Theoretical Cosmology 7 where h is Planck’s constant, c is the speed of light and λ is the wavelength of the photon. 1 Given that the energy scales as a− , we know the wavelength must scale with a. This means that light emitted at time te with wavelength λe will be observed at some later time to with wavelength

a(to) λo = λe. (1.16) a(te)

For an expanding universe, a(to) > a(te), so the wavelength of light increases, moving from the blue end of the spectrum to the red end; an eﬀect known as redshifting.

We can measure the shift in the light from its emitted wavelength as the redshift z, deﬁned as

(λ λ ) z o − e . (1.17) ≡ λe

This relation can also be expressed purely in terms of the scale factor. Given that we observe light today (to = t0) and that a(t0) = 1, the relationship between redshift and the scale factor becomes

1 1 + z = . (1.18) a(te)

This makes it clear that observing the redshift of distant objects can reveal a huge amount about the Universe’s expansion history.

1.1.5 Dynamics

In Section 1.1.3 we covered the right-hand side of the Einstein ﬁeld equations (Eq. 1.2), but to get the full picture about the evolution of the scale factor, and hence the dynamics of the universe as a whole, we also need to compute the various components of the Einstein tensor, µ µλ Gµν. As with the energy-momentum tensor, many of the components of Gν = g Gλν vanish under the conditions of homogeneity and isotropy, leaving

" 2 # 0 a˙ K G0 = 3 + 2 2 , (1.19) ac R0a 2 i a¨ a˙ K Gi = 2 2 + + 2 2 . (1.20) ac ac R0a 8 Chapter 1. Introduction

By substituting this form of the Einstein tensor and the energy-momentum tensor (Eq. 1.9) into the Einstein field equations, we find two differential equations for the scale factor:

a˙ 2 8πG Kc2 = 2 2 2 , (1.21) a 3 c − R0a a¨ 4πG = ( + 3P ), (1.22) a − 3c2 where, in these equations, and P represent the sum of all the contributions to the energy density (as discussed in Section 1.1.3). Together, these are known as the Friedmann equations.

The Hubble parameter is deﬁned as H a/a˙ . When we measure the Hubble parameter ≡ today, we have the Hubble constant H0, where a subscript of 0 serves as a short-hand for the value of various parameters today. The Hubble constant represents the current expansion rate of the Universe. It may also be linked to an object’s velocity v located at some physical distance `, which can be written as ` = a(t)d, where d is a comoving distance. The velocity of an object that moves with the expansion (such that d is constant) is

d` da a˙ v = = d = ad = H`. (1.23) dt dt a

Within the local universe, this is known as Hubble’s law, which states that the recession velocity of objects moving with expansion is proportional to their distance from us by a factor of H0. The prevalence of the Hubble constant in both evolution and distance equations has prompted the following convention:

1 1 H0 = 100 h km s− Mpc− , (1.24) such that the value (and our uncertainty) of the Hubble constant is parametrised by h and can be easily propagated through all equations. We note that the Hubble parameter has units of inverse seconds, and that a Megaparsec (Mpc) is a unit of distance. We can see from Eq. 1.23 that this unit choice allows us to work with cosmological distances in 1 Mpc and with velocities in km s− .

It is now useful to rewrite Eq. 1.21 in terms of the Hubble parameter; we also note that /c2 is just the total summed mass density, ρ:

2 2 8πG Kc H = ρ 2 2 . (1.25) 3 − R0a 1.1. Theoretical Cosmology 9

This makes it clear that expansion, mass density, and curvature are all linked. To that end, we may ask what the mass density of the Universe must be in order to be ﬂat. Known as the critical density, it is obtained by setting the curvature parameter K to 0:

3H2 ρ = . (1.26) crit 8πG

This serves as a useful quantity in that we can make the mass density of each individual component dimensionless:

ρi Ωi = . (1.27) ρcrit

For further simpliﬁcation, it is also common to write the curvature term in Eq. 1.25 as a mass density: ρ 3Kc2/8πGR2a2. The dimensionless mass densities are K ≡ − 0 2 2 8πGρm 8πGρr Λc Kc Ωm = 2 , Ωr = 2 , ΩΛ = 2 , ΩK = 2 2 2 . (1.28) 3H 3H 3H −H R0a

By deﬁnition, the sum of the dimensionless mass densities is equal to one:

Ωm + Ωr + ΩΛ + ΩK = 1. (1.29)

Recalling how each energy density (and consequently mass density) changes with the scale factor (Eq. 1.14), and noting that Ω a 2, it is possible to re-write the ﬁrst Friedmann K ∝ − equation in terms of how each dimensionless mass density (as measured today) scales with expansion:

2 2 3 4 2 H = H0 (Ωm,0a− + Ωr,0a− + ΩΛ,0 + ΩK,0a− ). (1.30)

1.1.6 Distance Measures

We now cover the key distance measures used in cosmology, and refer the reader to Hogg (1999) for a useful overview of other distance measures. Throughout this section, we often parametrize distances in terms of redshift rather than the scale factor, as redshift is the primary observable quantity.

1 1 Previously, we noted that the units of the Hubble parameter are km s− Mpc− . Its inverse naturally yields a characteristic distance, known as the Hubble distance, when 10 Chapter 1. Introduction multiplied by the speed of light:

c DH . (1.31) ≡ H0

Another commonly used distance measure is the comoving distance. Recall that we have already become familiar with comoving coordinates (discussed at the end Section 1.1.2). It follows that the comoving distance is the distance between two fundamental observers that are comoving with the Universe’s expansion. Typically, this is used when we wish to discuss the distance between us and a distant space-time event at redshift z. Given that the light will travel along a radial geodesic (ds = 0, dθ = dφ = 0) we may deduce from the metric (Eq. 1.8) that

dχ c R0 p = dt, (1.32) 1 Kχ2 a(t) − where χ is the dimensionless radial comoving coordinate. We may then integrate both sides from the time the light was emitted te to today t0, noting that at te the object is at redshift z and the light is at χ = 0, and at t0 the object is at redshift 0 and the light is at

χ = DC /R0, corresponding to a comoving distance of DC . For the right-hand side:

Z t0 Z 0 Z z Z z c c dt da c dz0 dt = dz0 = dz0 = DH , (1.33) te a(t) z a(t) da dz0 0 H(z0) 0 E(z0) where

q E(z) Ω (1 + z)3 + Ω (1 + z)4 + Ω + Ω (1 + z)2 (1.34) ≡ m,0 r,0 Λ,0 K,0 is deﬁned from Eq. 1.30.

For the left-hand side, we may use Eq. 1.28 to recast the result in terms of ΩK :

 h i DH 1 DC  sinh− √ΩK for ΩK > 0  √ΩK DH Z DC /R0 dχ  R0 = . (1.35) p 2 DC for ΩK = 0 0 1 Kχ  h i −  DH 1 DC  sin− √ Ωk for ΩK < 0 √ ΩK DH − − It is convenient to note that the comoving distance in a ﬂat universe from Eq. 1.35 can be simply equated to Eq. 1.33:

Z z dz0 DC,ﬂat = DH . (1.36) 0 E(z0) 1.1. Theoretical Cosmology 11

The remaining equations can be recast in terms of this, making the comoving distance a function of the dimensionless curvature density:

 h i DH DC,flat  sinh √ΩK for ΩK > 0  √ΩK DH  DC = DC,flat for ΩK = 0 . (1.37)  h i  DH DC,flat  sin √ ΩK for ΩK < 0 √ ΩK DH − − If we wish to consider the distance between two space-time events that are at the same redshift but are separated on the sky by some angle δθ, the distance between them is the comoving distance multiplied by the separation angle: DC δθ. For objects, like galaxies, that have a measurable angular size on the sky, we can define the angular diameter distance as the ratio between the object’s physical size s and the angle it subtends on the sky θ:

s D D = C . (1.38) A ≡ θ 1 + z

This distance is commonly used in analyses that involve standard rulers: objects that have a constant physical size at all redshifts; such objects allow us to convert a measure of angular size into a measure of distance. Similar to the angular diameter distance, we may also infer a distance based on how bright an object is, compared to how much energy we receive from it. In Euclidean geometry, light from a point source with luminosity L spreads out according to a spherical distribution, such that the ﬂux at some distance d is

L F = . (1.39) 4πd2

We may apply the same concept to the FLRW model, noting that the distance the light travels will be aﬀected by the expansion:

L F = 2 , (1.40) 4πDL where we deﬁne the luminosity distance as

D D (1 + z). (1.41) L ≡ C

Similar to the concept of standard rulers, we may also deﬁne standard candles as objects that have a constant luminosity at all redshifts; such objects allow us to convert a measure 12 Chapter 1. Introduction of ﬂux into a measure of distance.

1.1.7 Structure Formation

Much of what we have covered in this chapter pertains to the Big Bang model of cosmology. This model is able to explain a number of important observations, including the expansion of the Universe, the existence of the cosmic microwave background, and the formation of light elements (known as nucleosynthesis). It describes the very early universe as being in a state of high temperature and density, which rapidly expanded and cooled. This period of rapid expansion is known as inflation; it was originally proposed by Guth (1981) as a way of solving the flatness and horizon problems, which were consequences of the proposed initial conditions for the Big Bang model. For those interested in a discussion of these problems and the physics of inflation, Liddle & Lyth (2000) provide an in-depth review.

In the intervening years, inflation has gained increasing importance in the study of structure formation. It provides a method through which the Universe transitions from a homogeneous distribution of gas to the more clumpy structure we see today. Quantum fluctuations in the field that caused inflation translate to small inhomogeneities in the distribution of matter, which then grow through gravitational instability. We will now describe the model for this growth, following the prescription set out in section 4.4 of Liddle & Lyth (2000).

We continue to model mass as a perfect fluid, as outlined in Section 1.1.3. Within this section, we define the relation between the physical position r, and the comoving position x, as r(t) = a(t)x(t). As opposed to the derivation of Hubble’s law in Eq. 1.23, we have now allowed the comoving position to be a function of time. This reflects the fact that an object may move under gravitational attraction to other structures. The expression for an object’s velocity then becomes:

dr v(x, t) = = H(t)r + v (x, t). (1.42) dt pec

The additional contribution, vpec(x), is known as the peculiar velocity and is considered to be any motion from causes other than the Universe’s expansion.

The evolution of the mass ﬂuctuations that arise from inﬂation can be described using perturbation theory under Newtonian gravity. Three equations are required: the Euler equation relates the acceleration of an object to the gravitational potential, Φgr, as well 1.1. Theoretical Cosmology 13 as density and pressure:

dv 1 = P Φ ; (1.43) dt −ρ∇ − ∇ gr the gravitational potential satisﬁes the Poisson equation:

2Φ = 4πGρ; (1.44) ∇ gr and ﬁnally, since mass is conserved, we have the continuity equation:

dρ(x, t) = 3H(x, t)ρ(x, t). (1.45) dt −

One then perturbs each quantity, for example ρ(x, t) = ρ(t) + δρ(x, t), and determines the linear equations that the various perturbations satisfy:

1 δH = v , (1.46) 3∇ · pec 1 v˙ + Hv = δΦ δP, (1.47) pec pec −∇ gr − ρ∇ 2δΦ = 4πGδρ, (1.48) ∇ gr δρ˙ = 3ρδH 3Hδρ. (1.49) − −

These allow us to write a diﬀerential equation for the perturbations in the matter density, δρ. For convenience, we will now work with the density contrast δ δρ/ρ. From the ≡ linear equations, it is possible to show that

δ˙ = 3δH, (1.50) − 4πG 1 δH˙ = 2HδH δρ 2δP, (1.51) − − 3 − 3ρ∇ which can be combined to give a diﬀerential equation for the evolution of the density contrast:

1 δ¨ = 2Hδ˙ 4πGρδ 2P. (1.52) − − ρ∇

This relation shows us that the growth of the density contrast is damped by the Universe’s expansion (through the Hubble parameter), and grows when gravitational attraction over- comes pressure support.

The most general solution to Eq. 1.52 is that the density contrast can be written as 14 Chapter 1. Introduction the sum of a growing mode D1(t) and a decaying mode D2(t):

δ(x, t) = δ(x)D1(t) + δ(x)D2(t). (1.53)

When using this solution, it is common to only consider the growing mode, which dominates at late times.

It is possible to relate velocity to density by noting that the continuity equation (Eq. 1.45) may also be written as

dρ(x, t) = vρ(x, t). (1.54) dt −∇ ·

From the perturbed equations and Eq. 1.53, we may rewrite this in terms of the density contrast:

1 δ˙ = δ(x)D˙ (t) = v. (1.55) 1 −a∇ ·

At this point, we may take the opportunity to deﬁne a useful cosmological parameter, known as the linear growth rate of structure:

d ln(D (a)) 1 dD (a) f 1 = 1 , (1.56) ≡ d ln(a) D1(a) d ln(a) such that

v = aHfδ(x, a). (1.57) ∇ · −

This is an important relation, as it shows how the velocity of an object is related to the density contrast at a given position. We may also express this relation in physical coordinates today by integrating over all positions:

H f Z δ(r0)(r0 r) v(r) = 0 d3r0 − . (1.58) 4π r0 r 3 | − | These relations are an important part of the theoretical framework for this thesis, which aims to improve constraints on the growth rate of structure by utilising cross-correlations between the velocity and density ﬁelds. 1.1. Theoretical Cosmology 15

1.1.8 Correlation Functions and Power Spectra

The above model of structure formation includes the prescription that the initial matter fluctuations seeded by inflation can be modelled as a Gaussian random field. A consequence of this assumption is that the statistical properties of the continuous density contrast field (also known as the matter overdensity field) δ(x) can be described using two-point statistics, namely the autocorrelation function and its Fourier transform the power spectrum.

Conﬁguration and Fourier space

In the previous section, we modelled the overdensity as a function of position x; when working with positions, it is common practice to say that we are working in configuration space. This clarification is important because modelling the overdensity fluctuations as a Gaussian random field allows us to also work in Fourier space. The Fourier analogue of position is the wavevector k, which can be thought of as a spatial frequency. Using Fourier space allows us to decompose the overdensity field into a series of periodic overdensities with different amplitudes. For a given wavenumber k = k , the distance between the | | peak of the periodic overdensities is given by the wavelength λ such that k = 2π/λ. This 1 shows that the wavenumber has units of inverse distance; the units of k are h Mpc− , 1 corresponding to the unit of distance being h− Mpc (see Eq. 1.24 for the definition of h). We use the following Fourier transform convention throughout this thesis: Z V ik x 3 δ(x) = e− · δ˜(k)d k, (1.59) (2π)3 Z 1 ik x 3 δ˜(k) = e · δ(x)d x, (1.60) V where i is the imaginary unit, δ˜(k) is the Fourier transform of δ(x), and V is the Fourier volume. We note that δ˜(k) is a complex-valued function such that δ(x) is real.

Autocorrelation function

Commonly referred to as the correlation function, ξ(r) is the two-point statistic when working in configuration space. It defines the correlation between two points in a continuous field, separated by some distance r:

ξ(r) δ(x)δ(x + r)∗ , (1.61) ≡ h i 16 Chapter 1. Introduction where the average indicated by the angle brackets is over all positions x and the ∗ indicates complex-conjugation. Isotropy allows us to deﬁne the correlation function only in terms of the absolute separation r, rather than directional separation r. The correlation function can be used to assess how often overdensities are separated by a given distance compared to what would be expected from random chance.

Power spectrum

The power spectral density (most commonly referred to as the power spectrum) P (k) is the Fourier transform of the correlation function. In this context, it represents the amplitude of the overdensity ﬂuctuations as a function of wavenumber, and is deﬁned through the relation

0 P (k) 3 0 δ˜(k)δ˜(k )∗ = δ (k k ), (1.62) h i V D −

3 where the angle brackets represent the average over all wavevectors and δD is the Dirac delta function (it is one when k = k0 and zero otherwise). As with the correlation function, isotropy allows us to deﬁne the power spectrum in terms of the wavenumber k, rather than the wavevector k. The power spectrum is deﬁned to have volume units such that it takes the same value irrespective of the volume over which it is measured. The Fourier volume then serves to make the above relation unitless.

Converting between the two

Throughout this thesis, we will rely on both representations, and commonly, the Fourier transform that allows us to convert from one to the other. This conversion relies on the shifting property of the Dirac delta function:

Z 3 3 3 (2π) δ (k k0)f˜(k)d k = f˜(k0), (1.63) D − V where f˜(k) is the Fourier transform of an arbitrary function f(x), and the factor of (2π)3 arises from our normalisation convention for the Fourier transform. 1.2. Observational Cosmology 17

For r = x0 x: −

0 ξ(r) = δ(x)δ(x )∗ hZ i Z V ik x V ik0 x0 0 3 3 0 = e− · e · δ˜(k)δ˜(k )∗ d kd k (2π)3 (2π)3 h i Z V ik r P (k) 3 = e · d k (2π)3 V Z Z π Z 2π 1 ∞ 2 1 ik r = 2 P (k)k e · sin(θ)dφdθdk 2π 0 0 0 4π Z 1 ∞ 2 = 2 P (k)k j0(kr)dk. (1.64) 2π 0

Here, the zeroth-order spherical Bessel function j0(kr) = sin(kr)/kr comes from writing the exponential in terms of spherical harmonics and utilising the orthogonality condition for these functions. A more detailed example is given in Section 2.2.3.

1.2 Observational Cosmology

The standard cosmological model, commonly called ΛCDM, represents our current understanding of the Universe, which has been formed from both theory and observations. Consequently, we review it in more depth at the beginning of the next section, and focus on the observational evidence in this section. Readers may refer to Weinberg et al. (2013) for more detail.

1.2.1 The Cosmic Microwave Background

The cosmic microwave background (CMB) is a prediction from the Big Bang model, which underpins the standard cosmological model. The conditions of the early universe are thought to have been so hot and dense that matter and photons were bound together in a plasma that existed in a state of thermal equilibrium. In particular, electrons and photons were tightly coupled via Thomson scattering, which meant that light could not travel large distances. At around redshift z 1100, the Universe had expanded enough ≈ that the plasma had cooled sufficiently and formed neutral Hydrogen, allowing the photons to decouple from the baryons and travel freely. These photons constitute the CMB. The properties of the plasma make it a black-body: it absorbs all radiation and re- emits it over a range of frequencies that are determined by its temperature. Consequently, the CMB photons should have a black-body spectrum. Even though the frequency of these photons will have changed through redshifting, we should still observe a black-body 18 Chapter 1. Introduction spectrum today. The CMB was directly observed by Penzias & Wilson (1965) when they discovered a source of noise in their radio antenna that was isotropic with an effective temperature of about 3K. More recently, the CMB has been measured to have a frequency spectrum consistent with a black-body at 2.725K (Fixsen, 2009). Given that the CMB is a direct prediction of the Big Bang model, it serves as an incredibly important cosmological probe of the early universe. Today, it is the slight deviations in the temperature of the CMB that are of most interest to cosmologists. Sachs & Wolfe (1967) originally posited that the small density perturbations from inflation would introduce small deviations from the expected black- body spectrum through gravitational redshift. That is, photons that left the slightly over-dense regions of the plasma would experience a slight redshift, appearing cooler than those from regions where the density was average. These small temperature fluctuations are known as the CMB anisotropies. They were first observed by the Cosmic Background Explorer (COBE) satellite; launched in 1989, it was designed to search for the anisotropies on large scales. The anisotropies were discovered and reported by Smoot et al. (1992) who found that they were consistent with predictions from inflationary cosmology models. A range of ground-based and balloon programs went on to measure the anisotropies on smaller scales; the BOOMERanG experiment showed that the anisotropies were consistent with a flat universe (De Bernardis et al., 2000) and that they could be used to constrain cosmological parameters (Lange et al., 2001). The CMB anisotropies have continued to be measured to increasingly high precision through two satellite missions: The Wilkinson Microwave Anisotropy Probe (WMAP; Bennett et al., 2013) and the Planck satellite (Planck Collaboration et al., 2015a). Constraints on cosmological parameters are obtained by fitting a model to the temperature anisotropy angular power spectrum TT . Given that the anisotropies are well D` described by linear perturbation theory, it is possible to generate accurate model power spectra from codes such as CAMB (Lewis & Challinor, 2011). The angular power spectrum can be obtained by expressing the anisotropy signal as a sum over spherical harmonic functions. This behaves similarly to the power spectrum described in Section 1.1.8, but is measured as a function of angular scale ` rather than wavenumber k. The angular power spectrum measured by Planck is given in Figure 1.1. The location of the first peak constrains the geometry of the Universe, the height of the second peak constrains the relative amount of baryonic matter to dark matter, and the height of the third peak constrains the relative amount of dark matter to radiation. Although the primary anisotropies of the 1.2. Observational Cosmology 19

Figure 1.1 The upper panel shows the angular power spectrum for temperature anisotropies as measured by Planck (blue circles with uncertainties) with the best-ﬁtting ΛCDM model power spectrum (red line). The lower panel shows the residuals with respect to the model. (Credit: Planck Collaboration et al., 2015b).

CMB don’t constrain the amount of dark energy, they do tightly constrain the geometry of the Universe and amount of matter, which has implications for dark energy constraints.

1.2.2 Baryon Acoustic Oscillations

The conditions that are responsible for the CMB also generate another useful cosmological probe known as baryon acoustic oscillations (BAO). The dark matter ﬂuctuations generated by inﬂation are able to grow during the early universe, since dark matter is not coupled to radiation. Under gravitational attraction to these overdensities, the electron- photon plasma becomes compressed, resulting in a higher photon density and greater radiation pressure. This radiation pressure acts on the plasma to produce an acoustic wave, which displaces the plasma in a radial shell around the dark matter overdensity. When the photons decouple, the spherical shell of matter remains, and over time, dark matter and baryonic matter accumulate both at the position of the original overdensity and in the spherical shell, seeding galaxy formation. At late times, this produces a preferred clustering scale in the distribution of galaxies. The conditions of the plasma are well constrained by the CMB, and consequently, the distance that the spherical shell can travel before photons decouple is estimated to be 150 20 Chapter 1. Introduction

Figure 1.2 The correlation function of the SDSS Luminous Red Galaxy sample (black squares) with various models (coloured lines). Most importantly, the magenta line represents a model without baryons, showing that the BAO peak only appears in models that 1 include a baryon component. The detection of the peak at 100 h− Mpc is statistically signiﬁcant. (Credit: Eisenstein et al., 2005).

Mpc. This makes BAO a perfect candidate for a standard ruler, as discussed in Section 1.1.6, since their true size is known. BAO are known as a statistical standard ruler: the preferred separation can only be recovered when looking at many pairs of galaxies, and appears as a peak in the correlation function (Eisenstein & Hu, 1997). Despite being discussed as a possible standard ruler at the time (e.g. Eisenstein et al., 1998; Blake & Glazebrook, 2003), it wasn’t until 2005 that the BAO peak was detected in two large galaxy surveys: the Sloan Digital Sky Survey (SDSS; Eisenstein et al., 2005) and the 2- degree Field Galaxy Redshift Survey (2dFGRS; Cole et al., 2005). The BAO peak in the correlation function as measured by Eisenstein et al. (2005) is shown in Figure 1.2.

Since these first measurements, galaxy surveys have become larger and have aimed to measure the BAO peak at multiple different redshifts; each of these can then be used to measure the relationship between angular diameter distance and redshift, which sheds light on the Universe’s expansion history. More specifically, the preferred separation can 1.2. Observational Cosmology 21 be measured radially or tangentially; the radial measurement corresponds to a preferred redshift separation between galaxies, which constrains the inverse of the Hubble parameter 1 H(z)− , and the tangential measurement corresponds to a preferred angular separation between galaxies, which constrains the angular diameter distance DA(z). Frequently, surveys will measure a degenerate combination of both:

cz 1/3 D (z) = (1 + z)2D2 (z) . (1.65) V A H(z)

Surveys that have recently measured the BAO include the 6-degree Field Galaxy Survey (6dFGS; Beutler et al., 2011), the WiggleZ Dark Energy Survey (Blake et al., 2011a), the Sloan Digital Sky Survey’s Baryon Oscillation Spectroscopic Survey (using both galaxies and quasars) (SDSS-III BOSS; Alam et al., 2017; Bautista et al., 2017), and the extended Baryon Oscillation Spectroscopic Survey (SDSS-IV eBOSS; Ata et al., 2018). All of these surveys have been able to measure the BAO at various redshifts below z = 2.5, with recent studies measuring DV (z) at 1-2% precision and H(z) at 3-5% precision, depending on the eﬀective redshift of the sample. Given they are measured at low redshifts, BAO are a useful complement to the high-redshift information from the CMB and are consistent with a ﬂat ΛCDM model.

1.2.3 Type Ia Supernovae

Type Ia supernovae (SNe Ia) are a complementary probe to BAO as they independently measure the expansion history of the Universe over a similar redshift range. This is because they can be used as a standard candle, as discussed in Section 1.1.6, since they have an intrinsic brightness that is linked to their explosion mechanism: a white dwarf star exceeding the Chandrasekhar mass limit. While the physics is not perfectly understood and there are multiple progenitor scenarios, small variances in intrinsic brightness can be accounted for, making SNe Ia a standardisable candle. An example of this is the discovery that the brighter the supernova, the slower it fades (Phillips, 1993). SNe Ia became cemented as a cosmological probe through the discovery that the Uni- verse is undergoing a period of accelerating expansion by the High-z Supernova Search Team (Riess et al., 1998) and the Supernova Cosmology Project (Perlmutter et al., 1998). The two teams discovered that, for their measured redshifts, the SNe Ia had luminosity distances that were greater than expected for a universe that only contained matter, and their results were consistent with the presence of a positive cosmological constant. Just as BAO can be used to map the angular diameter distance as a function of redshift, 22 Chapter 1. Introduction

Figure 1.3 The upper panel shows the distance modulus-redshift relation (also known as the Hubble diagram) for a recent compilation of type Ia supernovae, from low-redshift surveys (blue), the Sloan Digital Sky Survey (SDSS, green), the Supernova Legacy Survey (SNLS, orange) and Hubble Space Telescope observations (HST, red), with the best-ﬁtting ΛCDM model (black line). The lower panel shows the residuals with respect to the model; the weighted average residuals for a set of redshift bins are given by the black dots. (Credit: Betoule et al., 2014).

SNe Ia can be used to map the luminosity distance as a function of redshift. This is done though the distance modulus

µ m M = 5 log [D (z)] + 25, (1.66) ≡ − 10 L where m is the observed apparent magnitude (corresponding to measured ﬂux), and M is the absolute magnitude (corresponding to intrinsic brightness). This relationship for a compilation of SNe Ia can be seen in Figure 1.3. The latest compilation of SNe Ia is the Pantheon sample, which contains 1048 supernovae in the redshift range 0.01 < z < 2.3 (Scolnic et al., 2017). The sample was used to constrain the dark energy equation of state, with the authors ﬁnding w = 1.090 0.220 when accounting for systematics. Dark − energy constraints from SNe Ia are consistent with those from BAO. 1.2. Observational Cosmology 23

1.2.4 Galaxy Bias and Redshift-Space Distortions

The distribution of galaxies may also be used as an observational probe of cosmology, in addition to the baryon acoustic oscillations (BAO) imprinted within it. Speciﬁcally, galaxy clustering tells us about the composition of the Universe and how gravity works on large scales; this information may be extracted from the correlation function or power spectrum (see Section 1.1.8 for deﬁnitions), or the corresponding higher-order statistics. Conveniently, both two-point statistics can be measured from the same large-scale structure surveys that are used to measure the BAO feature.

We note that there are a few additional steps that must be taken to link the galaxies we observe with the structure formation theory developed in Section 1.1.7. In that section, we parametrized the distribution of matter through the matter density contrast δm, which is also known as the matter overdensity (we’ve added the subscript for clarity in this section). Unfortunately, the majority of matter in the Universe is dark matter, which we cannot directly observe. Instead, we parametrize the distribution of galaxies through the galaxy overdensity:

ng(x) δg(x) 1, (1.67) ≡ n¯g − where ng(x) is the number density of galaxies at position x andn ¯g is the average number density in the absence of clustering given the survey selection. One may then estimate the power spectrum and correlation function for the galaxy distribution from the overdensity.

However, research in the 70s and early 80s revealed that diﬀerent populations of galaxies exhibit diﬀerent clustering behaviour. For example, Kaiser (1984) found that galaxy clusters were biased with respect to all galaxies: their correlation function had a larger amplitude than that for all galaxies. This was then generalised to the idea that baryonic matter in the form of galaxies is biased with respect to the underlying distribution of dark matter (e.g. Bardeen et al., 1986; Peacock & Heavens, 1985), which led to the introduction of the linear galaxy bias parameter b:

δg(x) = bδm(x), (1.68) such that the two-point statistics for the galaxy overdensity ﬁeld are related to those for 24 Chapter 1. Introduction the matter overdensity through

2 ξg(r) = b ξm(r), (1.69) 2 Pg(k) = b Pm(k). (1.70)

Critically, the galaxy bias allows us to link our observations of the galaxy distribution to our model of structure formation. We note that the interaction between dark and baryonic matter is more complex than implied by the linear galaxy bias parameter. Speciﬁcally, the galaxy bias is expected to be non-linear, stochastic, non-local and scale-dependent. However, the linear bias factor has been found to be a suitable approximation on large scales (e.g. Scherrer & Weinberg, 1997) and is the model we implement in this thesis. Large-scale galaxy surveys typically determine the location of galaxies through their redshift, which is readily available from spectroscopic observations. This is because true distance measures (such as the luminosity or angular diameter distance; see Section 1.1.6) require a distance estimator such as a standard candle or standard ruler, which are not available for the majority of galaxies. For galaxy redshift surveys, the extended form of Hubble’s law (Eq. 1.42) gives the expression for how the observed redshift z relates to the redshift from Hubble ﬂow z and from peculiar velocities z v /c: H pec ≡ pec

(1 + z) = (1 + zH )(1 + zpec). (1.71)

At low redshifts, this relationship can be approximated as

cz cz + cz ≈ H pec = H(z)D + vpec. (1.72)

We note that this approximation only holds below redshift z 0.3 (see ﬁgure 1 of Davis ∼ & Lineweaver, 2001). We have written the equation in terms of the line-of-sight peculiar velocity (v = v rˆ) because we are only able to observe this component when pec pec · measuring redshift. It is clear that if we ignored peculiar velocities, we would incorrectly estimate the distance from the measured redshift. This leads us to introduce the distinction between redshift space and real space (which is separate from the distinction between conﬁguration space and Fourier space introduced in Section 1.1.8). Given that a galaxy’s peculiar velocity will distort its apparent distance relative to its true distance, we can state that 1.2. Observational Cosmology 25 the redshift-space position s is related to the real-space position r by

v (r) s = r + pec rˆ (1.73) aH(z) = r + u(r)rˆ, (1.74) where we employ the convention that positive peculiar velocities are away from the observer, such that a positive peculiar velocity results in a redshift-space position greater than the corresponding real-space position.

Given that all of our theoretical modelling is in terms of the overdensity in real space, it’s important to specify the model in redshift space so that we can directly compare it to observations. This was ﬁrst put forward by Kaiser (1987), who noted that the Jacobian for transforming from real space to redshift space is

u(r)2 ∂u(r) d3s = d3r 1 + 1 + . (1.75) r ∂r

This can be simpliﬁed by noting that the distance modiﬁcation introduced by the peculiar velocity should be much smaller than the distance to the galaxy, such that u(r) r. Under this assumption, [1 + u(r)/r]2 goes to 1, such that

∂u(r) d3s = d3r 1 + . (1.76) ∂r

The relation between overdensities in real and redshift space is then obtained by combining the Jacobian with the fact that the the galaxy number density must be conserved:

ns(s)d3s = nr(r)d3r, (1.77) which leads to

1 ∂u(r)− δs = (1 + δr) 1 + 1 g g ∂r − ∂u(r) δr , (1.78) ≈ g − ∂r where we applied a Taylor expansion around u = 0 to the partial derivative term, and neglected the product of δr and ∂u(r)/∂r, given both are much less than 1. In order to obtain this relation in Fourier space, we utilise the Fourier space expression for the 26 Chapter 1. Introduction peculiar velocity ﬁeld (derived later in Section 2.2.3):

k v˜ (k) = iaHf θ˜(k), (1.79) p − k2 where f is the growth rate of structure (see Eq. 1.56) and θ is the velocity divergence ﬁeld, deﬁned as

v (r) θ(r) ∇ · p , (1.80) ≡ − aHf which is equal to the matter density ﬁeld δm(r) in linear theory. Taking the Fourier transform of Eq. 1.78, we ﬁnd

˜s ˜r 2 ˜ δg(k) = δg(k) + fµ θ(k), (1.81) where µ kˆ rˆ. The dependence on µ tells us that the measured overdensity in redshift ≡ · space is now anisotropic: it becomes enhanced along the line-of-sight rˆ. This enhancement is known as redshift-space distortion, and its presence in our measurements of large-scale structure can be used to constrain the growth rate of structure f. Recalling that we’re dealing with galaxies, rather than matter directly, the ﬁnal step is to write the redshift-space galaxy overdensity in terms of the real-space matter overdensity using the galaxy bias:

˜s ˜r 2 ˜ δg(k) = bδm(k) + fµ θ(k) ˜r 2 ˜ = b δm(k) + βµ θ(k) , (1.82) where β f/b is the redshift-space distortion parameter. This leads to the expression for ≡ the redshift-space anisotropic galaxy power spectrum:

s 2 2 2 4 Pgg(k, µ) = b [Pmm(k) + 2βµ Pmθ(k) + β µ Pθθ(k)], (1.83) where we have split k into its radial (k) and angular (µ) components. While we have chosen to focus on the Fourier space derivation here, one may also work with the redshift-space correlation function, originally presented by Hamilton (1992). Redshift-space distortions (RSD) have been measured and used to constrain both β and f by multiple large-scale structure surveys including the 2-degree Field Galaxy Redshift Survey (2dFGRS; Peacock et al., 2001; Hawkins et al., 2003; Percival et al., 2004), the VIMOS-VLT Deep Survey (VVDS; Guzzo et al., 2008), the WiggleZ Dark Energy Survey 1.2. Observational Cosmology 27

(Blake et al., 2011b), the 6-degree Field Galaxy Survey (6dFGS; Beutler et al., 2012; Achitouv et al., 2016; Blake et al., 2018), and multiple iterations of the Sloan Digital Sky Survey (SDSS; Samushia et al., 2012; Oka et al., 2014; Beutler et al., 2014; Ruggeri et al., 2019; Gil-Mar´ınet al., 2018). We note that RSD analyses may be conducted in either Fourier space or configuration space: Fourier space involves modelling the power spectrum (e.g. Blake et al., 2018), and configuration space involves modelling the correlation function (e.g. Beutler et al., 2012). In both types of analysis, the growth rate of structure is typically constrained by fitting the modelled two-point statistic to that measured from the data. Redshift-space distortion studies have been able to measure the growth rate of structure at the level of 10% for redshifts z < 1 and the results are in agreement with those for the matter density predicted by the CMB.

1.2.5 Peculiar Velocities

Redshift-space distortions constrain the growth rate of structure by tracing how correlated peculiar velocities aﬀect the two-point clustering statistics of galaxies, but peculiar velocities may also be used as a cosmological probe in their own right. Eq. 1.72 implies that it is possible to measure the line-of-sight peculiar velocity of a galaxy through

v = v rˆ cz H(z)D (1.84) pec pec · ≈ − but only if its true distance D is known, in addition to its redshift. While distances can be predicted from redshifts using a cosmological model (see Section 1.1.6), we require a redshift-independent estimate of the distance if we wish to use peculiar velocities to test cosmological models. Such distances can be obtained from either a standard candle (which measures the luminosity distance) or a standard ruler (which measures the angular diameter distance). The two examples we’ve covered so far are not particularly suitable when it comes to measuring large numbers of peculiar velocities: Type Ia Supernovae give accurate distances, but are not particularly numerous, and aren’t always present in the galaxies measured by large-scale structure surveys; baryon acoustic oscillations are a statistical standard ruler, so can give the distance to a particular redshift, but not a specific galaxy. Fortunately, there are two other methods for obtaining distances to large numbers of galaxies, both of which rely on empirical correlations of galaxy properties. The first is the Tully-Fisher relation, put forward by Tully & Fisher (1977), which relies upon a correlation between the rotational velocity and luminosity of spiral galaxies. The two quantities are 28 Chapter 1. Introduction believed to be linked through the virial theorem (the more massive the galaxy, the faster it spins) and the proportionality of luminosity to mass. Once the relationship is calibrated by galaxies with known luminosities, the luminosity of a spiral galaxy may be inferred from its measured rotational velocity, making the Tully-Fisher relation a standard candle. The second is the Fundamental Plane relation, which serves as a distance-estimator for elliptical galaxies. Early work by Faber & Jackson (1976) showed a correlation between the central velocity dispersion and luminosity, which came to be known as the Faber-Jackson relation. This work was then advanced by Dressler et al. (1987) and Djorgovski & Davis (1987), who introduced the size of the elliptical galaxy as a third parameter, making the relation a plane in three-dimensional space. This addition proved to be most useful when projecting the “fundamental plane” into two dimensions by plotting size against a combination of the velocity dispersion and surface brightness, reducing the scatter compared to that from the Faber-Jackson relation. The use of size as a parameter makes the Fundamental Plane a standard ruler: by measuring the angular size of a galaxy on the sky, one may estimate the distance to the galaxy through trigonometry. While both methods are useful in that they allow us to measure the true distance to galaxies, they suffer from large uncertainties due to the scatter in the empirical correlations. Worse, these uncertainties grow with distance (see figure 4 of Scrimgeour et al., 2016), and are typically anywhere between 20-30% of the distance being measured. In order to have reasonable uncertainties, measurements of peculiar velocities through these methods are restricted to the low-redshift universe (z . 0.1).

There are several key surveys that have measured peculiar velocities including the Spiral Field I-Band peculiar velocity sample (SFI; Haynes et al., 1999), the SFI Superset (SFI++; Springob et al., 2009), the Two-Micron All-Sky Survey (2MASS) Tully-Fisher Sample (2MTF; Masters et al., 2008), and the 6-degree Field Galaxy Survey peculiar velocity sample (6dFGSv; Springob et al., 2014), where each dataset contains a few thousand peculiar velocities. To obtain more uniform sky-coverage, there are also compilations of peculiar velocities from Tully-Fisher and Fundamental Plane studies, as well as those from Type Ia Supernovae (e.g. Tully et al., 2016). We note that combining peculiar velocity samples in this manner can be difficult due to differences in zero-point offsets (the size or magnitude of an object with no peculiar velocity), so compilation samples should be used with care.

Peculiar velocities are a complementary probe to redshift-space distortions in that each probes the growth rate of structure on different physical scales. This is highlighted by the fact that the peculiar velocity power spectrum is proportional to f 2/k2, whereas 1.3. Explanations for Accelerating Expansion 29 the redshift-space distortion power spectrum is contains both fµ2 and f 2µ4 terms. Con- sequently, the peculiar velocity power spectrum traces density modes on the largest scales (k 0.01 h Mpc 1, corresponding to hundreds of h 1 Mpc), whereas the signal in the RSD ∼ − − power spectrum is dominated by smaller scales (k 0.1 h Mpc 1). There are numerous ∼ − ways to constrain cosmology from peculiar velocities, but in particular, we highlight the construction of the model covariance between pairs of peculiar velocities, which can be constructed as a function of the power spectrum (Jaffe & Kaiser, 1995; Abate et al., 2008; Macaulay et al., 2012; Johnson et al., 2014; Huterer et al., 2017; Howlett et al., 2016). We also note that peculiar velocities can be used to measure the amplitude and overall direction of any bulk flow (e.g. Scrimgeour et al., 2016).

1.3 Explanations for Accelerating Expansion

The combination of theory and observations covered in the previous two sections has led to the standard cosmological model: assuming a Friedmann-Lemaˆıtre-Robertson-Walker metric and that gravity is described by general relativity, the Universe has a flat geometry with 5% of the energy density in the form of baryonic matter, 25% in the form ∼ ∼ of cold dark matter and 70% in the form of the vacuum energy (which is the physical ∼ interpretation of Einstein’s cosmological constant Λ). Within this framework, accelerating expansion is explained by the properties of the vacuum energy, specifically that it has negative pressure, increasing the distance between galaxies despite gravitational attraction. This acceleration is only a relatively recent phenomenon, occurring after matter-Λ energy density equality (z 0.4 for the standard model). ∼ While many observations are consistent with the interpretation of the cosmological constant as the vacuum energy, it suffers from a few issues. The first is that the observed value of the cosmological constant is on order of 120 magnitudes smaller than what the standard model of particle physics predicts for the vacuum energy (known as the cosmological constant problem). Secondly, the contribution of matter and the cosmological constant to the total energy budget are at the same order of magnitude today, which is unexpected for quantities that scale so differently with the scale factor of the Universe (known as the coincidence problem). Given these issues, there has been significant theorisation and investigation of possible alternative models that could provide a better physical understanding of what drives accelerating expansion. Models to explain accelerating expansion may be broken up into two categories: dark energy models, which add an energy density with negative pressure to drive accelerating expansion; and modified gravity models, which modify general relativity 30 Chapter 1. Introduction to produce accelerating expansion without introducing an additional energy density. Dark energy models can be thought of as those that modify the right-hand side of the Einstein equation (Eq. 1.2), whereas modified gravity models can be thought of as those that modify the left-hand side. We now provide a high-level summary, and refer the reader to Joyce et al. (2016) for a comprehensive overview of each category.

1.3.1 Dark Energy

Dark energy as an energy density can be considered as the broader interpretation of the cosmological constant as a dynamical scalar field. Scalar fields are already used to successfully explain forces in other areas of physics, such as the electric potential field from electrostatics.

Models that generate dark energy from a scalar field are generally known as quintessence models. Originally introduced by Peebles & Ratra (1988), the term quintessence was coined by Caldwell et al. (1998). The key idea is that the scalar field φ(t) has a conden- sate that behaves as a perfect fluid with the equation of state

P 1 φ˙2 V (φ) w = = 2 − , (1.85) 1 ˙2 2 φ + V (φ) where P is the pressure of the fluid, is its energy density, and V (φ) is the potential of the scalar field. The primary difference to the cosmological constant is then that the equation of state may vary with time, as opposed to being constant for all values of the scale factor (see Section 1.1.3). Given that the equation of state for dark energy is constrained to be close to -1 from observations, quintessence models can generally be broken up into “thawing” models (where w is moving away from -1) and “freezing” models (where w is moving toward -1). There are also further generalisations of the quintessence model that introduce additional behaviours; one example is k-essence models, where the scalar field may have self-interactions (e.g. Chiba et al., 2000).

An interesting feature of quintessence models is “tracking”, where the energy density of the scalar ﬁeld closely follows that of dark matter until relatively recent times, which could explain the coincidence problem. However, the cosmological constant problem has a direct analogue in this framework, given that the scalar ﬁeld’s potential today must be equivalent to the inferred dark energy density, which is still at odds with predictions from the standard model of particle physics. 1.3. Explanations for Accelerating Expansion 31

1.3.2 Modiﬁed Gravity

There are several broad classes of modiﬁed gravity (MG) theories, which use diﬀerent mechanisms to elicit accelerating expansion in the absence of dark energy. We now summarise a few key examples and refer the reader to Clifton et al. (2012) for an in-depth discussion of each.

Scalar-Tensor Theories

In general relativity, the gravitational force that influences matter can be interpreted as the physical manifestation of a rank-2 tensor field. Scalar-tensor theories, which are often cited as the best studied MG theories in the literature, go beyond this by adding a scalar field that also couples to matter, exerting an additional force. This construction often allows exact analytic solutions to be found, which is considered as one of the key benefits of this formalism. There are several distinct types of model that fit within this class. Generally considered to be the simplest modification, Brans-Dicke theory sets the coupling between the scalar field and matter to be a constant (Brans & Dicke, 1961). There are also f(R) theories, which modify the Einstein-Hilbert action to contain higher-order curvature terms through the Ricci scalar R gµκR , where R is the Ricci curvature tensor (a mathematical ≡ µκ µκ object from Riemannian geometry that represents how volumes are affected by curved space-time). Although they don’t appear to be adding a scalar field at first glance, f(R) theories can typically be represented as scalar-tensor theories, hence their inclusion in this category. There are several realistic f(R) models that produce accelerating expansion without a cosmological constant, such as those by Starobinsky (2007), Hu & Sawicki (2007) and Appleby & Battye (2007). Finally, galileon theories introduce scalar fields that are invariant under the Galilean transformation; this imbues them with useful properties such as being free of higher-order derivatives (Nicolis et al., 2009). Interestingly, they naturally arise when considering the decoupling limits of both massive gravity and Dvali- Gabadadze-Porrati theories, which we discuss below.

Massive Gravity Theories

When considering general relativity from the perspective of quantum field theory, the gravitational force is mediated by a massless spin-2 particle, known as the graviton. This is similar to how the electromagnetic field is mediated by the photon. Massive gravity theories instead posit that the graviton is a massive particle. Accelerating expansion may 32 Chapter 1. Introduction be caused be a weakening of gravity on large scales, which is a natural consequence of a massive force carrier. Modern work in this field is described by de Rham et al. (2011).

Braneworld Theories

Another commonly discussed theory is the Dvali-Gabadadze-Porrati model, where matter exists on a 4-dimensional brane, which is part of a larger 5-dimensional space-time (Dvali et al., 2000). In this framework, gravity “leaks” into the higher dimension at large scales, appearing to weaken. As with massive gravity theories, this weakening on large scales is what causes accelerating expansion.

Viability and Validation

Given the precise tests of general relativity in both laboratories and the solar-system (see Section 1.1.1), viable modified gravity models require screening mechanisms to recover general relativity on these small scales. Well known mechanisms include the Chameleon mechanism, where the strength of the coupling of the introduced scalar field to matter depends on the local matter density (Khoury & Weltman, 2004), and the Vainshtein mechanism, which becomes active in regions of large curvature due to nonlinearities in the higher derivatives of the considered scalar field (Vainshtein, 1972). The Chameleon mechanism is often employed by f(R) models, whereas the Vainshtein mechanism is commonly used by massive gravity and braneworld models. There are numerous avenues for testing these theories, which we will not cover in detail. However, we note that large-scale structure in the low-redshift universe is a particularly promising avenue: modified gravity models that recover accelerating expansion typically alter the growth rate of structure. For example, Baker et al. (2014) showed that MG theories exhibit observational signatures such as a scale-dependent growth rate of structure. Johnson et al. (2014) has already provided scale-dependent constraints on the growth rate of structure from the 6-degree Field Galaxy Survey, which were then used to constrain phe- nomenological descriptions of modified gravity (Johnson et al., 2016). Coupled with the fact that accelerating expansion begins around redshift z 0.4, the low-redshift universe ∼ is a promising place to test alternative explanations of accelerating expansion.

1.4 Multiple-Tracer Analyses

If we are to discriminate between competing models of accelerating expansion, we require highly-precise measurements of both the expansion history and the growth rate of struc- 1.4. Multiple-Tracer Analyses 33 ture. As highlighted in the previous section, modiﬁed gravity theories typically alter the predicted value and behaviour of the growth rate of structure, especially at low redshift. The expansion history, parametrized by the Hubble constant, has already been measured with accuracy on the order of a few percent in the local universe (e.g. Riess et al., 2016). However, the growth rate of structure has only been constrained at the level of 10-20% in the local universe through redshift-space distortions (RSD; e.g. Beutler et al., 2012) and peculiar velocities (e.g. Johnson et al., 2014).

The somewhat limited statistical power of low-redshift surveys is directly linked to their small volume, which is especially evident when considering modes in Fourier space. For large-scale structure analyses, the number density of Fourier-space modes is proportional to 3 the survey volume through ρk = V/(2π) , such that a small volume leads to fewer Fourier modes for a given k-range. This contributes statistical uncertainty to power spectrum measurements in the form of sample variance, which can only be beaten by extending the sample volume. This is especially relevant for peculiar velocities, since these objects are limited to low redshifts given the proportionality of their measured uncertainties to distance (see Section 1.2.5). We note that this is partially mitigated by the fact that peculiar velocities are correlated with galaxy overdensities on large scales, which may make them sensitive to larger Fourier modes than those contained within the survey.

Given our desire for high-precision constraints, there has been signiﬁcant research into methods that maximise the amount of information extracted from low-redshift large-scale structure and peculiar velocity surveys. Two schools of thought have emerged on this topic, which we discuss below.

1.4.1 Statistical Perspective

While statistical noise can be improved by increasing the number of tracers, there is still a sample variance limit that is tied to the number of Fourier modes within a volume. In particular, the sample variance contribution to the power spectrum uncertainty is p σP (k)/P (k) = 2/Nm(k) for all observable modes Nm at given wavenumber k. However, Seljak (2009) proposed a method for constraining certain cosmological parameters beyond the sample variance limit, meaning the precision is dominated by the observational uncertainty, which can be reduced. Since sample variance is actually tied to the underlying matter overdensity field δm, rather than tracers of that field (galaxies and their peculiar velocities), the ratio of tracers with different biases will be independent of δm, evading 34 Chapter 1. Introduction sample variance:

δ b δ b g1 = 1 m = 1 . (1.86) δg2 b2δm b2

McDonald & Seljak (2009) developed this idea, presenting a Fisher matrix analysis (a statistical method used for forecasting relative errors) for applying the multi-tracer approach to redshift-space distortion analyses. They confirmed that the ratio of the bias values of different tracers could be determined to higher precision than either of the individual bias values. Since these two studies, there have been numerous others that have reached similar conclusions for different cosmological tracers (e.g. Gil-Mar´ınet al., 2010; Bernstein & Cai, 2011). In addition to these, Abramo & Leonard (2013) showed that multi-tracer analyses may also access the cross-power spectrum of the two tracers, which provides additional information about each individual tracer, on top of the tighter constraints of their ratio. This has been confirmed by Fisher matrix forecasts for joint analyses of redshift-space distortions and peculiar velocities, which provide tighter constraints on the growth rate of structure f and the redshift-space distortion parameter β than can be obtained from only taking the ratio of the tracers (Koda et al., 2014; Howlett et al., 2016). For example, Koda et al. (2014) forecast that combining peculiar velocities and redshift-space distortions could improve the growth rate of structure constraint by up to a factor of 5 ∼ compared to a redshift-space distortion only analysis for the 6-degree Field Galaxy Survey (6dFGS).

These effects have also been demonstrated for observational data. Blake et al. (2013) presented the first application of the multiple-tracer approach to redshift-space distortions. They selected two galaxy samples with different galaxy bias values from the Galaxy and Mass Assembly Survey (GAMA; Driver et al., 2011). Rather than taking the ratio of two tracers, they modelled the auto-power spectrum of each tracer, as well as the cross-power spectrum. They found that they were able to constrain the ratio of the two galaxy bias values to higher precision than if they combined the individual estimates with propagation of uncertainty, confirming the original conclusion from Seljak (2009). They also found that their constraints on the growth rate of structure improved by 10-20% when using both galaxy samples, highlighting that multi-tracer analyses are a useful method for obtaining better measurements of the growth rate of structure. 1.4. Multiple-Tracer Analyses 35

1.4.2 Physical Perspective

Prior to the development of the statistical approach, the potential power of combining galaxy overdensities and peculiar velocities was highlighted by a key equation from gravitational instability theory (see Section 1.1.7):

H f Z δ (r0)(r0 r) v(r) = 0 d3r0 m − , (1.87) 4π r0 r 3 | − | which describes how peculiar velocities respond to the underlying matter distribution. This line of thinking arose in the early 1960s and was formally established by Peebles

(1976), where he derived Equation (1.87) under the assumption that δm can be treated as linear. The initial use of this equation was to attempt to estimate f and constrain the 0.6 dimensionless matter density parameter Ωm through f = Ωm . Peebles was unable to constrain Ωm tightly, reporting that both Ωm = 0.1 and Ωm = 1 were allowed. At the time, it was not yet known that galaxies are actually biased tracers of matter. If you can correct for redshift-space distortions, it’s possible to re-write this relation in terms of the galaxy overdensity:

H β Z δ (r0)(r0 r) v(r) = 0 d3r0 g − , (1.88) 4π r0 r 3 | − | such that it now constrains β = f/b, recalling that δg = bδm in the absence of redshift- space distortions. In the early 1980s, the Infrared Astronomical Satellite (IRAS) provided the first all-sky survey in the infrared (Neugebauer et al., 1984). Redshift information from this survey was used in multiple analyses in the early 1990s including work by Kaiser et al. (1991) and Strauss et al. (1992) to construct the velocity field from Equation (1.88). In the late 90s, more peculiar velocity measurements became available and this relationship was frequently applied. In particular, the Mark III catalogue provided a homogeneous compilation of peculiar velocities from smaller surveys (Willick et al., 1997). At the time, two competing approaches emerged: VELMOD (Willick & Strauss, 1998) compared constructed and observed velocities (velocity-velocity comparison), whereas POTENT (Dekel et al., 1999) compared constructed and observed overdensities (density-density comparison). Willick & Strauss (1998) noted that POTENT tended to give higher estimates of β compared to VELMOD. For example, Hudson et al. (1995) used POTENT and found β = 0.74 0.13 whereas Willick & Strauss (1998) found β = 0.50 0.04. In the past 20 years, the field has shifted again with the release of multiple redshift and peculiar velocity surveys, such as those mentioned throughout Sections 1.2.4 and 1.2.5. POTENT has largely 36 Chapter 1. Introduction been dropped, and velocity-velocity comparisons are favoured in the form of simple χ2 (e.g. Springob et al., 2015), VELMOD (e.g. Carrick et al., 2015) and Inverse Tully-Fisher analyses (e.g. Davis et al., 2011). While these recent analyses rest on the same underlying theory, there is significant variation in approach, underlying assumptions and data. All require data manipulation to obtain a model velocity field that can be compared to the observed field. Consequently, systematic and statistical error analysis is often inconsistent, making it difficult to compare the various analyses. We note that these studies may only constrain the growth rate of structure with an estimate of the galaxy bias, which requires additional analysis outside of the chosen velocity- velocity comparison method. Given that galaxy bias differs between galaxy samples, the growth rate provides a fairer point of comparison than β. For example, Carrick et al. (2015) found fσ = 0.40 0.02, which agrees with the result from Pike & Hudson (2005) 8 of fσ = 0.44 0.06. While these methods are likely providing tight constraints on 8 β by taking advantage of the lowered sample variance for multiple tracers discussed in the previous subsection, it’s unclear whether these methods are taking advantage of cross- correlation information as well, which would also provide tighter constraints on the growth rate. This is because the constraints on the growth rate come from mutliplying β and the galaxy bias and propagating uncertainties, rather than constraining f and β in tandem, as was done by Blake et al. (2018).

1.5 Low-Redshift Galaxy Surveys

Although much effort over the past few decades has been expended in constraining cosmology at higher redshifts, low-redshift surveys can offer unique insights about our universe. As mentioned in Section 1.3, low-redshift measurements of the growth rate of structure could be used to test different explanations for accelerating expansion, particularly modified gravity theories. In addition to this, low-redshift galaxy surveys are able to achieve very high number density (since it is easier to see low-luminosity galaxies), which can be useful for testing the density-dependent screening mechanisms of modified gravity models (e.g. Achitouv et al., 2016). They also provide a useful cosmological probe in the form of peculiar velocities, which can only really be used at low redshift because of how their errors scale with distance (see Section 1.2.5). Low-redshift surveys are also unaffected by Alcock-Paczynski distortion, which affects the certainty in the distance-redshift relation; in higher-redshift surveys, the accuracy of the growth rate is limited by this distortion. Finally, low-redshift measurements of large-scale structure can also provide highly com- 1.5. Low-Redshift Galaxy Surveys 37 petitive measurements of the Hubble constant H0, which is of particular interest due to recent tensions seen between low-redshift measurements and extrapolations from the cosmic microwave background (e.g. Riess et al., 2016). Given that this thesis will work solely with low-redshift surveys, it is useful to review the available and upcoming data. Specifically, we review the existing 6-degree Field Galaxy Survey (6dFGS) and the upcoming Taipan Galaxy Survey (Taipan), noting that both redshift surveys include large peculiar velocity samples.

1.5.1 6dFGS

The 6-degree Field Galaxy Survey is a large redshift survey of the southern hemisphere that was conducted on the UK Schmidt Telescope from May of 2001 to January of 2006. The original survey design and first data release was presented by Jones et al. (2004); 6dFGS was designed to overcome limitations of existing large-scale structure surveys such as the 2-degree Field Galaxy Redshift Survey (2dFGS) and the Sloan Digital Sky Survey (SDSS), both of which were biased toward current star-forming regions due to optical selection. The 6dFGS team achieved this by performing their galaxy selection in the near- infrared (JKH bands), which allowed the survey to track older stellar populations and provided a more accurate estimate of each galaxy’s total stellar mass. 6dFGS was selected from the Two Micron All-Sky Survey Extended Source Catalogue (2MASS-XSC Jarrett et al., 2000), and used photometry from SuperCOSMOS (Hambly et al., 2001a,b). As the survey progressed, there were two additional data releases (Jones et al., 2005, 2009). 6dFGS boasts some impressive statistics. It covered 17, 000 degrees2 of the southern ∼ sky, excluding 10 degrees either side of the galactic plane, with an average redshift completeness of 92%. The final redshift sample (6dFGSz) contains 125,071 redshifts in the redshift range z . 0.23, with a median redshift of z = 0.053. The final peculiar velocity sample (6dFGSv) contains Fundamental Plane measurements for 9794 galaxies in the redshift range z 0.055, with distance errors of around 26% (Magoulas et al., 2012). More ≤ than 10 years after the survey’s conclusion, 6dFGSv is still the largest uniform peculiar velocity sample, making up nearly half of the largest peculiar velocity compilation sample, Cosmicflows-3 (Tully et al., 2016). 6dFGS has been utilised in several key large-scale structure and peculiar velocity analyses, including measurements of the galaxy luminosity function (Jones et al., 2006), baryon acoustic oscillations (Beutler et al., 2011; Carter et al., 2018), redshift-space distortions (Beutler et al., 2012; Achitouv et al., 2016; Blake et al., 2018), stellar-mass dependence in galaxy clustering (Beutler et al., 2013), the amplitude of the peculiar velocity power 38 Chapter 1. Introduction spectrum (Johnson et al., 2014; Huterer et al., 2017) and the peculiar velocity bulk flow (Scrimgeour et al., 2016). The recentness of several of these results indicates the relevance of 6dFGS as a low-redshift survey.

1.5.2 Taipan

Given the success of 6dFGS, it is only natural that a successor has appeared in the form of the Taipan Galaxy Survey. Designed to extend 6dFGS signiﬁcantly, Taipan is to be conducted on the UK Schmidt Telescope (with an upgraded spectrograph), beginning in 2019 and running for 4 years (da Cunha et al., 2017). It has several key cosmological goals: to constrain the local Hubble parameter H0 to the 1% level using baryon acoustic oscillations (BAO), the growth rate to at least 5% through redshift-space distortions and peculiar velocities, and provide the largest homogeneous peculiar velocity sample to date. Taipan will be broken into two phases: Phase 1 and Final, where Phase 1 largely focusses on measuring baryon acoustic oscillations. The survey targets will be selected in a similar manner to 6dFGS, using near-infrared selection from the 2MASS Extended Source Catalogue (Jarrett et al., 2000) and Point Source Catalogue (Cutri et al., 2003) for Phase 1, and then moving to SkyMapper for the Final phase (Keller et al., 2007). Phase 1 aims to measure 300,000 redshifts for the redshift range z . 0.2 and 33,000 peculiar velocities for redshift range z . 0.1. By the end of the survey, this will be boosted to 2 million redshifts for the redshift range z . 0.4, and 50,000 peculiar velocities in the same redshift range as Phase 1.

1.6 Thesis Motivation and Overview

Low-redshift surveys can provide useful cosmological constraints, both from redshift-space distortions and peculiar velocities. This is especially true for the growth rate of structure, which is a key cosmological parameter when it comes to distinguishing between viable explanations of accelerating expansion. However, their comparatively small volume compared to high-redshift surveys leads to measurements of the growth rate being statistically limited, both by the number of galaxies measured and the presence of sample variance on large scales. These limitations can be mitigated by combining the information from correlated tracers of the matter overdensity ﬁeld. This has largely been demonstrated in the form of theoretical Fisher matrix forecasts, but has also been shown to apply to data when measuring redshift-space distortions from two galaxy overdensity samples with diﬀerent bias 1.6. Thesis Motivation and Overview 39 values. Although Fisher matrix forecasts have predicted that utilising the cross-correlation between peculiar velocities and redshift-space distortions will provide better constraints on the growth rate of structure than either probe independently, it has never been demonstrated with data. This is in part because a complete formalism that takes advantage of both tracers and their cross-correlation has not been developed. In fact, the development of such a formalism is critical to the upcoming Taipan Galaxy Survey; the realisation of their goal of at least a 5% constraint on the growth rate is entirely contingent on the availability of a method to utilise the cross-correlation between redshift-space distortions and peculiar velocities.

The overarching goal of this thesis is to directly address this need. We aim to build a fully self-consistent maximum likelihood approach for constraining the growth rate of structure while taking advantage of the cross-correlation between redshift-space distortions and peculiar velocities. Not only this, but we also aim to demonstrate its eﬀectiveness by applying it directly to data.

In Chapter 2, we present the theoretical modelling that underpins the results of this thesis. Specifically, we cover our self-consistent models for the three correlation functions used in the analysis: the galaxy overdensity auto-correlation, the peculiar velocity auto-correlation and the cross-correlation. We highlight that our analysis is conducted in configuration space, but that the correlation function models are obtained by taking the Fourier transform of power spectrum models. Our maximum likelihood method revolves around constructing the theoretical covariance matrix from our correlation function models, which are functions of cosmological parameters, namely the growth rate of structure f and the redshift-space distortion parameter β = f/b. This differs from standard maximum likelihood analyses in that we model the covariance rather than the data. While conduct- ing this work, we have applied the principle that the data should be manipulated as little as possible and that the model should directly account for any observational effects. We address many of these points in detail after constructing an idealised version of the model covariance. Our method can then be directly applied to data, which is the focus of the rest of the thesis.

We present the first application of our method to data in Chapter 3. We begin by validating our method on a mock galaxy survey constructed from the GiggleZ simulation (Poole et al., 2015), before applying it to data from the 6-degree Field Galaxy Survey (6dFGS). This study is the first of its kind to utilise information from both peculiar velocities and galaxy overdensities, as well as their cross-correlation, in an entirely self-consistent way. It also makes a significant contribution to existing literature on the statistical ben- 40 Chapter 1. Introduction efits of utilising cross-correlations, especially in low-redshift surveys. We note that in this analysis we account for the effects of redshift-space distortions, but do not explicitly use the information they provide on the growth rate of structure. We do this to highlight the strength of peculiar velocities as a cosmological probe and note that we apply a full redshift-space distortion model in Chapter 4. We measure the growth rate under two conditions: we start by treating the peculiar velocities and galaxy overdensities as independent and then move to including the cross-correlation information. This allows us to directly demonstrate the statistical advantage that comes from utilising the cross- correlation. As a part of our method, we also provide direct evidence for the existence of the cross-correlation on large scales. Finally, we compare our results to various results from the literature, including Fisher matrix forecasts, other work on 6dFGS and more traditional density-velocity comparison analyses. As mentioned above, we upgrade our model considerably in Chapter 4 by adding redshift-space distortions to our self-consistent correlation function models. However, this is not the only advance we make. We also validate our method against a suite of sophisticated 6dFGS mocks from Carter et al. (2018), allowing us to test various model choices before working with the data. We also expand our galaxy overdensity sample compared to that used in Chapter 3. Finally, we thoroughly investigate various systematics in our analysis and quantify how they affect our growth rate of structure constraint by providing a systematic uncertainty in addition to the statistical uncertainty from our maximum likelihood analysis. As with our analysis in Chapter 3, we demonstrate the advantage that comes from utilising the cross-correlation and compare our results to those from the literature. In Chapter 5, we look to the future by applying our method to a suite of mocks for the Taipan Galaxy Survey. This allows us to evaluate the performance of our method when working with a larger sample of peculiar velocities and galaxy redshifts. We also perform a Fisher matrix forecast, which we match to the sample and model specifications for our analysis of the mocks. This allows us to make a fair comparison between the forecast and the performance of our method, which we then use to comment on the expected performance for Taipan at various stages of the survey. Finally, we present the conclusions of this thesis in Chapter 6 and take the opportunity to reflect on the future work motivated by this thesis. 2 Theory

2.1 Overview

The aim of this chapter is to develop the theoretical model that underpins the analysis presented in this thesis. This primarily involves developing an expression for the auto- correlation and cross-correlation functions between peculiar velocities and galaxy overdensities. Similar work has been conducted by a number of studies, both for peculiar velocities (e.g Jaffe & Kaiser, 1995; Silberman et al., 2001; Macaulay et al., 2012; Johnson et al., 2014) and for galaxy overdensities (e.g. Heavens & Taylor, 1995; Ballinger et al., 1995). We begin by stating the key definitions and conventions that will be used throughout the derivations below. We then move on to a description of Bayesian inference and the definition of the likelihood and covariance functions used in our analysis in Section 2.2. This is followed by any modifications to our underlying theory to account for data in Section 2.3.

2.1.1 Deﬁnitions and Conventions

When dealing with both auto- and cross-correlation functions, we are expressing the correlation between two positions in configuration space. Typically, we consider a pair of observations located at positions xs = (xsx , xsy , xsz ) and xt = (xtx , xty , xtz ), with the observer located at the origin. We define the vector separation between the two observations as r = x x and the line-of-sight distance to the pair as d = 1 (x + x ). t − s 2 s t The angle between xs and xt is α, and the angle between the separation vector and the line-of-sight distance is γ, such that the dot-product is given by dˆ rˆ = cos(γ). These · definitions are represented pictorially in Figure 2.1. Throughout this thesis we will work 1 with comoving distances, with units measured in h− Mpc, where the Hubble constant at 1 1 redshift z = 0 is H0 = 100 h km s− Mpc− , and the reduced Hubble constant h will be

41 42 Chapter 2. Theory

Figure 2.1 Pictorial representation of relevant positions and angles in conﬁguration space. speciﬁed as part of choosing an underlying cosmology. We note that the majority of the modelling presented below is performed in real space, although we develop the model in redshift space when directly modelling redshift-space distortions in Section 2.3.4.

The basic observations that we’ll be modelling are the galaxy overdensity δg and the radial peculiar velocity vp. Since we’ll be considering the correlation between each possible pair, we typically work with a vector containing N overdensities δ = (δ , δ , ..., δ ), δ g g1 g2 gNδ v and a vector containing Nv peculiar velocities p = (vp1 , vp2 , ..., vpNv ). We will discuss the process for generating the data vectors from observations in Section 2.3.2.

2.2 Likelihood Formalism

2.2.1 Bayesian Inference

One of the most important tools for modern cosmology is Bayesian inference; we may use it to constrain our models of the Universe from the data that we observe. The process is based on Bayes’ theorem, which is merely the statement of conditional probability. Here, we express it in terms of some set of observed data ∆ and a set of parameters φ that are used to model the distribution from which the data is drawn:

P (φ)P (∆ φ) P (φ ∆) = | . (2.1) | P (∆) 2.2. Likelihood Formalism 43

In the Bayesian framework, probabilities P may represent our level of belief in particular outcomes. P (φ ∆) is the posterior, which represents our belief in the values of the param- | eters given the data we have observed. P (φ) is the prior, which captures our knowledge of the parameter values before seeing the data. P (∆ φ) is the likelihood, which represents the | probability of drawing the data for given values of the parameters. P (∆) is the evidence, which acts as a normalisation constant through the deﬁnition P (∆) = R P (∆ φ)P (φ)dφ. | The evidence is useful when one wishes to evaluate the performance of various models, known as Bayesian model selection. In this thesis, we will only work with the ΛCDM model and will absorb the evidence as a normalisation constant. Given these deﬁnitions, we may interpret Bayes’ theorem as a method for updating our knowledge of the possible values of the model parameters from the prior to the posterior through the likelihood.

2.2.2 Modelling the Likelihood

To utilise Eq. 2.1, we require a deﬁnition of the likelihood function. In this work, the data

∆ consists of the measured galaxy overdensities δg and peculiar velocities vp. As part of the model we will develop, we assume that each set of observations has been drawn from an underlying multivariate Gaussian distribution with a mean of zero. These assumptions allow us to construct a theoretical likelihood function:

= P (∆ φ) L | 1 1 T 1 = p exp ∆ C(φ)− ∆ , (2.2) (2π)N C(φ) −2 | | where ∆ is the data vector that contains the measured galaxy overdensities and peculiar velocities with length N = Nv + Nδ, C is the covariance matrix between each element of ∆, and φ is a vector containing the parameters of the model we wish to constrain. In this approach, we construct a theoretical model for the covariance matrix in terms of φ. Since our model assumes that the mean values of the overdensity ﬁeld and peculiar velocity ﬁeld are zero, we do not explicitly include the subtraction of the mean from the data in the likelihood function. The covariance has the same dimension as the data, and the structure of the data vector and covariance is     δg Cδgδg Cδgvp ∆ =   , C =   . (2.3) vp Cvpδg Cvpvp

Throughout this thesis, we assess goodness-of-ﬁt through the χ2 statistic, deﬁned for 44 Chapter 2. Theory our model as

2 T 1 χ ∆ C(φ)− ∆. (2.4) ≡

In the context of parameter estimation, the χ2 statistic allows us to capture how well the data are fitting the predictions of the model, given the uncertainty captured in the covariance matrix. More specifically, we use the χ2 per degree-of-freedom (χ2/dof ≡ χ2/N ), where the number of degrees of freedom is given as N = N N ; dof dof data − parameters this statistic allows us to compare data probability for different models.

2.2.3 Modelling the Covariance

Our goal is to model the four submatrices that make up the covariance matrix as a function of key cosmological parameters. We start by considering the covariance to be given by the correlation function between any two data points. We may do this by utilising the Fourier expressions for the galaxy overdensity and peculiar velocity ﬁelds to evaluate the deﬁnition of the correlation function, as done to arrive at Eq. 1.64.

Fourier Space Expressions

We begin by recalling the key definitions for the galaxy overdensity field (see Section 1.2.4) and the divergence of the peculiar velocity field (see Section 1.1.7) in configuration space:

δg(x) = bδm(x), (2.5) v (x) = aHfδ (x). (2.6) ∇ · p − m

The galaxy bias b and the growth rate of structure f are two of the key parameters that we will attempt to constrain with this method. As a reminder, a is the scale factor and H 1 1 is the Hubble constant measured in units of h km s− Mpc− . For the following derivations in Fourier space, we recall that k is the wavevector, k = k is the wavenumber in units of | | 1 h Mpc− , and i is the imaginary unit. The Fourier transform for the galaxy overdensity ﬁeld is straightforward, giving the Fourier expression

δ˜g(k) = bδ˜m(k). (2.7)

The transformation for the velocity divergence is slightly more complex, so we derive it here. The velocity field is a vector field, and the Helmholtz decomposition theorem states 2.2. Likelihood Formalism 45 that a vector field can be written as the sum of the gradient of a scalar field and the curl of a vector field. That is

v (x) = φ(x) + w(x). (2.8) p −∇ ∇ ×

Assuming the ﬁeld is irrotational, we can remove the curl term. We note that this is a long-standing assumption, see section 7.5 of Strauss & Willick (1995). This leaves

v (x) = φ(x). (2.9) p −∇

We may now evaluate the velocity ﬁeld in Fourier space as the Fourier transform of Eq. 2.9: ZZZ 1 ik x 3 v˜ (k) = [ φ(x)]e · d x. (2.10) p − V ∇

ik x Letting g(k, x) = e · , we can rewrite the integrand using the chain rule:

(gφ) = ( g)φ + g( φ) (2.11) ∇ ∇ ∇ g( φ) = ( g)φ (gφ). (2.12) − ∇ ∇ − ∇

Substituting Eq. 2.12 into Eq. 2.10 we ﬁnd ZZZ ZZZ 1 ik x 3 1 ik x 3 v˜ (k) = e · φ(x)d x e · φ(x) d x. (2.13) p V ∇ − V ∇

We note that a variant of Gauss’ theorem can be used to rewrite the second term. For a scalar ﬁeld, f(x): ZZZ ZZ f(x)d3x = f(x)nˆdS, (2.14) V ∇ S where n is the vector normal to the surface. Eq. 2.13 then becomes ZZZ ZZ 1 ik x 3 1 ik x v˜p(k) = e · φ(x)d x e · φ(x)nˆdS. (2.15) V ∇ − S V

The surface integral term will converge to zero so long as φ(x)nˆ and the partial derivatives of φ(x) are integrable. Physically, this is guaranteed by the integrability of the peculiar 46 Chapter 2. Theory velocity ﬁeld and the fact that it is uncorrelated on large scales. This leaves ZZZ 1 ik x 3 v˜ (k) = e · φ(x)d x. (2.16) p V ∇

The gradient of the exponential function evaluates to

ik x ik x e · = ike · , (2.17) ∇ leaving ZZZ 1 ik x 3 v˜ (k) = ik φ(x)e · d x p V = ikφ˜(k). (2.18)

Returning briefly to configuration space, even though the continuity equation allows us to write the peculiar velocity in terms of the matter overdensity field (Eq. 2.6), it is appropriate to define the velocity divergence field as

v (x) θ(x) ∇ · p , (2.19) ≡ − aHf such that θ(x) = δm(x) to linear order. Combining Eq. 2.9 and Eq. 2.19:

( φ(x)) = aHfθ(x). (2.20) ∇ · −∇ −

We can then express each ﬁeld in terms of its Fourier transform:

2 ZZZ ZZZ ik x 3 aHf ik x 3 −∇ V φ˜(k)e− · d k = − V θ˜(k)e− · d k. (2.21) (2π)3 (2π)3

The gradient operator then only applies to the exponential, which gives ZZZ ZZZ 2 ik x 3 ik x 3 V k φ˜(k)e− · d k = aHf V θ˜(k)e− · d k. (2.22) −

Equating the integrands gives

k2φ˜(k) = aHfθ˜(k), (2.23) − and in conjunction with Eq. 2.18 we obtain the Fourier expression for the peculiar velocity 2.2. Likelihood Formalism 47

ﬁeld:

k v˜ (k) = iaHf θ˜(k). (2.24) p − k2

Conventions and Set Up

We will now derive expressions for the correlation between any two data points at arbitrary positions, such that the elements of the covariance matrix are given as C∆i∆j = ∆ (x )∆ (x ) . We note that the method presented here follows that by Ma et al. (2011), h i s j t ∗i but for a more general deﬁnition of vector positions. Throughout, we use the following position conventions:

x = (x , x , x ), x = x , (2.25) s sx sy sz | s| s x = (x , x , x ), x = x , (2.26) t tx ty tz | t| t r = x x = (r , r , r ), r = r, (2.27) t − s x y z | | kˆ = (sin θ cos φ, sin θ sin φ, cos θ), (2.28) where the conﬁguration space vectors are the same as those presented in Figure 2.1. The expression of the wavevector in spherical coordinates allows us to write the exponential from the Fourier transform (Eq. 1.59) as a sum of spherical waves:

ik r X ` e · = i (2` + 1)j (kr)L (kˆ rˆ), (2.29) ` ` · ` where j` are the spherical Bessel functions and L` are the Legendre polynomials. It is useful to note that any function of θ and φ may be expressed as a linear sum of spherical harmonic functions:

` X∞ X f(θ, φ) = f`mY`,m(θ, φ), (2.30) `=0 m= ` − and that the coeﬃcients can be directly calculated through

Z π Z 2π f`m = f(θ, φ)Y`,m∗ (θ, φ) sin(θ)dθdφ. (2.31) θ=0 φ=0

We also note that our normalisation convention for spherical harmonics is such that we 48 Chapter 2. Theory may deﬁne the complex conjugate of Y`,m(θ, φ) to be

m Y`,m∗ (θ, φ) = ( 1) Y`, m(θ, φ), (2.32) − − and that the following orthonormal condition holds:

Z π Z 2π Y`,m(θ, φ)Y`∗0,m0 (θ, φ) sin(θ)dφdθ = δ`,`0 δm,m0 , (2.33) θ=0 φ=0 where δ`,`0 and δm,m0 are Kronecker delta functions, which evaluate to 1 if the subscripts are equal and 0 otherwise.

Finally, we note that the spherical harmonic addition theorem is useful when working with Legendre polynomials where the argument is a dot-product of unit vectors:

` 4π X L (kˆ rˆ) = Y (kˆ)Y (rˆ)∗ ` · (2` + 1) `m `m m= ` − ` 4π X = Y (kˆ)∗Y (rˆ). (2.34) (2` + 1) `m `m m= ` −

Galaxy Overdensity Auto-Covariance

Working in real-space and neglecting redshift-space distortions, the expression for the galaxy overdensity auto-covariance is

C (x , x ) = δ (x )δ (x )∗ δgδg s t h g s g t i 2 = b δ (x )δ (x )∗ h m s m t i Z Z 0 2 V ik xs V ik xt 3 3 = b e− · e · δ˜ (k0)δ˜ (k)∗ d k0d k (2π)3 (2π)3 h m m i Z 2 1 ik (xt xs) 3 = b e · − P (k)d k (2π)3 mm 2 Z Z π Z 2π b ∞ 2 1 ik r = 2 Pmm(k)k e · sin(θ)dφdθdk 2π 0 0 0 4π 2 Z b ∞ 2 = 2 Pmm(k)k (2.35) 2π 0 Z π Z 2π 1 X i`(2` + 1)j (kr)L (kˆ rˆ) sin(θ)dφdθdk. 4π ` ` · 0 0 ` 2.2. Likelihood Formalism 49

For ease, let us further evaluate only the angular component of this equation:

Z π Z 2π 1 X A i`(2` + 1)j (kr)L (kˆ rˆ) sin(θ)dφdθ. (2.36) ≡ 4π ` ` · 0 0 `

Given that we wish to utilise Eq. 2.33, we need to identify what value of ` will result in a non-zero evaluation of the Kronecker delta δ`,`0 . Since sin(θ) already forms part of the orthogonality condition, only the Legendre polynomial can be expressed in terms of spherical harmonics. In order to have a spherical harmonic to match against, we may note that the lowest order spherical harmonic does not have any angular dependence:

r 1 1 Y (θ, φ) = , (2.37) 0,0 2 π and can be directly inserted into the angular component by multiplying the integrand by 1:

Z π Z 2π 2√π X A = i`(2` + 1)j (kr)L (kˆ rˆ)Y (θ, φ) sin(θ)dφdθ. (2.38) 4π ` ` · 0,0 0 0 `

The presence of the Y0,0(θ, φ) term means that only ` = 0 will produce a non-zero evaluation of the Kronecker delta function. Noting that

L (kˆ rˆ) = 1 = 2√π Y (θ, φ), (2.39) 0 · 0,0 evaluating the angular integral produces

A = j0(kr), (2.40) such that

b2 Z C (x , x ) = P (k)k2j (kr)dk. (2.41) δgδg s t 2π2 mm 0

We note that the galaxy overdensity covariance is isotropic (depending only on r), as expected for the case where galaxies trace the underlying matter overdensity ﬁeld through the linear galaxy bias. This expression is consistent with our derivation of the matter correlation function (Eq. 1.64). 50 Chapter 2. Theory

Peculiar Velocity Auto-Covariance

Recalling that we only observe the line-of-sight component of the peculiar velocity ﬁeld v (x) v (x) xˆ, the expression for the peculiar velocity auto-covariance is p ≡ p ·

C (x , x ) = v (x )v (x )∗ vpvp s t h p s p t i = (v (x ) xˆ )(v (x )∗ xˆ ) h p s · s p t · t i Z ˆ 2 V k xˆs ik xs = (aHf) · e− · (2π)3 k

Z V (kˆ xˆ ) 0 0 t ik xt ˜ k ˜ k 3k 3k 3 · e · θ( 0)θ( )∗ d 0d (2π) k0 h i Z ˆ ˆ 2 1 (k xˆs)(k xˆt) ik r 3 = (aHf) · · e · P (k)d k (2π)3 k2 θθ (aHf)2 Z Z π Z 2π 1 ∞ kˆ x kˆ x ik r = 2 Pθθ(k) ( ˆs)( ˆt)e · sin(θ)dφdθdk 2π 0 0 0 4π · · 2 Z (aHf) ∞ = 2 Pθθ(k) (2.42) 2π 0 Z π Z 2π 1 (kˆ xˆs)(kˆ xˆt) 0 0 4π · · X i`(2` + 1)j (kr)L (kˆ rˆ) sin(θ)dφdθdk. ` ` · `

Again, we consider just the angular component:

Z π Z 2π 1 X A (kˆ xˆ )(kˆ xˆ ) i`(2` + 1)j (kr)L (kˆ rˆ) sin(θ)dφdθ. (2.43) ≡ 4π · s · t ` ` · 0 0 `

In order to use the orthonormal condition, we must express (kˆ xˆ )(kˆ xˆ ) as a linear · s · t combination of spherical harmonics. The expanded form in terms of θ and φ is

ˆ ˆ 1 2 2 2 (k xˆs)(k xˆt) = [xsx xtx sin (θ) cos (φ) + xsx xty sin (θ) cos(φ) sin(φ)+ (2.44) · · xsxt 2 xsx xtz sin(θ) cos(θ) cos(φ) + xsy xtx sin (θ) cos(φ) sin(φ)+ 2 2 xsy xty sin (θ) sin (φ) + xsy xtz sin(θ) cos(θ) sin(φ)+

xsz xtx sin(θ) cos(θ) cos(φ) + xsz xty sin(θ) cos(θ) sin(φ)+ 2 xsz xtz cos (θ)]. 2.2. Likelihood Formalism 51

Each function within the above expression may then be expressed as the sum of spherical harmonic functions using Eq. 2.31:

r r 2 2 2π 2 π 2√π sin (θ) cos (φ) = (Y2, 2 + Y2,2) Y2,0 + Y0,0 (2.45) 15 − − 3 5 3 r 2 2π sin (θ) cos(φ) sin(φ) = i (Y2, 2 Y2,2) (2.46) 15 − − r 2π sin(θ) cos(θ) cos(φ) = (Y2, 1 Y2,1) (2.47) 15 − − r r 2 2 2π 2 π 2√π sin (θ) sin (φ) = (Y2, 2 + Y2,2) Y2,0 + Y0,0 (2.48) − 15 − − 3 5 3 r 2π sin(θ) cos(θ) sin(φ) = i (Y2, 1 + Y2,1) (2.49) 15 − 1 rπ cos2(θ) = 4 Y + 2√πY (2.50) 3 5 2,0 0,0

From these, we can see that we will require the ` = 0, 2 terms for the plane wave expansion, as all other terms will vanish through the orthonormal condition. Consequently,

Z π Z 2π 1 X A = (kˆ xˆ )(kˆ xˆ ) i`(2` + 1)j (kr)L (kˆ rˆ) sin(θ)dφdθ. (2.51) 4π · s · t ` ` · 0 0 `=0,2

The zeroth-order Legendre polynomial is given in Eq. 2.39, and the second-order Legendre polynomial is

1 L (kˆ rˆ) = (3(kˆ rˆ)2 1). (2.52) 2 · 2 · −

The presence of the squared dot product means that the second-order Legendre polynomial will have a similar spherical harmonic expansion to Eq. 2.44, so we do not directly expand it here.

Due to the sheer number of terms when working with the spherical harmonic expansion, we evaluate Eq. 2.51 in Mathematica. With some additional simpliﬁcation we ﬁnd

1 A = (xˆ xˆ )[j (kr) + j (kr)] (rˆ xˆ )(rˆ xˆ )j (kr). (2.53) 3 s · t 0 2 − · s · t 2

We note that this is a more general expression than that from Ma et al. (2011), but the 52 Chapter 2. Theory two are functionally equivalent. The ﬁnal form for the covariance is

(aHf)2 Z 1 C (x , x ) = P (k) (xˆ xˆ )[j (kr) + j (kr)] (rˆ xˆ )(rˆ xˆ )j (kr) dk. vpvp s t 2π2 θθ 3 s · t 0 2 − · s · t 2 (2.54)

Cross-Covariance

The expression for the cross-covariance of a galaxy overdensity located at xs and peculiar velocity located at xt is

C (x , x ) = δ (x )v (x )∗ δgvp s t h g s p t i = δ (x )(v(x )∗ xˆ ) h g s t · t i Z V Z V (kˆ xˆ ) 0 ik xs 0 t ik xt ˜ k ˜ k 3k 3k = iaHfb 3 e− · 3 · e · δm( )θ( 0)∗ d 0d (2π) (2π) k0 h i Z 1 1 ik (xt xs) 3 = iaHfb P (k)(kˆ xˆ )e · − d k (2π)3 k mθ · t iaHfb Z Z π Z 2π 1 ∞ kˆ x ik r = 2 Pmθ(k)k ( ˆt)e · sin(θ) dφdθdk 2π 0 0 0 4π · iaHfb Z Z π Z 2π 1 ∞ kˆ x = 2 Pmθ(k)k ( ˆt) (2.55) 2π 0 0 0 4π · X i`(2` + 1)j (kr)L (kˆ rˆ) sin(θ) dφdθdk. ` ` · `

Again, we consider just the angular component

Z π Z 2π 1 X A (kˆ xˆ ) i`(2` + 1)j (kr)L (kˆ rˆ) sin(θ) dφdθ. (2.56) ≡ 4π · t ` ` · 0 0 `

Similarly to the velocity auto-covariance, we must express (kˆ xˆ ) as a linear combination · t of spherical harmonics in order to determine the order for the plane wave expansion:

ˆ 1 (k xˆt) = xtx sin θ cos φ + xty sin θ sin φ + xtz cos θ (2.57) · xt r r ! 1 2π 2π rπ = xtx (Y1, 1 Y1,1) + i xty (Y1, 1 + Y1,1) + 2 xtz Y1,0 . (2.58) xt 3 − − 3 − 3

From this, we can see that only ` = 1 remains through the orthonormal condition, leaving

Z π Z 2π 3i A = (kˆ xˆt)j1(kr)L1(kˆ rˆ) sin(θ) dφdθ. (2.59) 0 0 4π · · 2.2. Likelihood Formalism 53

Noting that

L (kˆ rˆ) = kˆ rˆ 1 · · r r ! 1 2π 2π rπ = rx(Y1, 1 Y1,1) + i ry(Y1, 1 + Y1,1) + 2 rzY1,0 , (2.60) r 3 − − 3 − 3 evaluating the angular integral produces

A = i(xˆ rˆ)j (kr), (2.61) t · 1 such that

aHfb Z C (x , x ) = − P (k)k(xˆ rˆ)j (kr)dk. (2.62) δgvp s t 2π2 mθ t · 1

It is important to note that the cross-covariance is not symmetric; if we instead evaluate the correlation function for a galaxy overdensity at xt and peculiar velocity at xs:

aHfb Z C (x , x ) = P (k)k(xˆ rˆ)j (kr)dk. (2.63) vpδg s t 2π2 θm s · 1

By examining these equations, we see that it is always the peculiar velocity position that appears in the dot product with the vector separation of the two galaxies, r. The diﬀerence in sign between Eq. 2.62 and 2.63 is illustrated by Figure 2.2, in which we show two example galaxies along the line of sight. If the galaxy overdensity increases, we expect the peculiar velocity as seen by the observer to increase in the direction of the overdensity. In the top panel, where the galaxy overdensity is closest to the observer, the peculiar velocity will increase towards the observer, giving negative peculiar velocity and leading us to expect a negative correlation (overdensity increases and peculiar velocity decreases). This situation corresponds to Eq. 2.62 and the analytic covariance is negative, as expected. Switching the positions in the bottom panel, we expect a positive correlation, as the peculiar velocity will increase away from the observer towards the overdensity. In this case, Eq. 2.63 applies and the analytic covariance is positive, as expected.

The symmetry of the two cross-covariance expressions is broken because the covariance relies on the definition of r, which we have defined as x x throughout the derivation. t − s We have found that the clearest way to express the cross-covariance is to define it in terms of the galaxy overdensity and peculiar velocity positions, rather than xs and xt. This means that a single equation can be used for both covariances, without needing to change the sign. For a galaxy overdensity measurement at xδ and a peculiar velocity measurement 54 Chapter 2. Theory

Figure 2.2 The sign of the covariance between a peculiar velocity and a galaxy overdensity is affected by their positions relative to the observer, as the direction of positive velocity is away from the observer. Here, we examine the effect of increasing the overdensity. In the top panel, the peculiar velocity increases towards the observer (becomes more negative), and in the bottom panel, it increases away from the observer (becomes more positive). at x , we define r x x , giving v ≡ δ − v aHfb Z C (x , x ) = C (x , x ) = P (k)k(xˆ rˆ)j (kr)dk. (2.64) δgvp δ v vpδg v δ 2π2 mθ v · 1

We note that this is similar to the expression for the cross-covariance previously presented by Fisher (1995), although we have now provided a complete derivation.

2.2.4 Fiducial Power Spectra

To evaluate the covariance equations, we require the power spectrum that appears in each equation: either Pmm(k), Pθm(k) = Pmθ(k), or Pθθ(k). Power spectra are typically generated by codes that are able to numerically evolve the initial perturbations of the early universe through to the present day. Within this thesis, we rely on two such codes: the Code for Anisotropies in the Microwave Background (CAMB, Lewis et al. 2000; Lewis & Challinor 2011) and velMPTbreeze, which is an extension of MPTbreeze (Crocce et al. 2012). CAMB uses linear perturbation theory to generate the linear matter power spectrum; the non-linear matter power spectrum can be obtained by applying corrections through Halofit (Smith et al., 2003), which has been directly incorporated into CAMB. velMPTbreeze, like MPTbreeze, generates non-linear power spectra through renormalized perturbation theory, as described by Crocce & Scoccimarro (2006). It is useful for us 2.2. Likelihood Formalism 55

CAMB Pmm velMPTbreeze Pmθ velMPTbreeze Pθθ ] 3 104 Mpc) 1 − h ) [( k ( P

2 1 10− 10− 1 k [h Mpc− ]

Figure 2.3 The three power spectra used in this thesis. In the realm of linear theory, all three power spectra agree; they begin to diverge at k 0.04 h Mpc 1. ≈ − because it provides both Pθm(k) and Pθθ(k), which cannot be obtained from CAMB. The three power spectra are shown in Figure 2.3. Both CAMB and velMPTbreeze require a basic set of cosmological parameters to produce 2 power spectra. These are the physical baryon density Ωbh ; physical dark matter density 2 Ωch ; reduced Hubble constant h; scalar spectral index ns; scalar amplitude As; and reionization optical depth τ. Throughout this thesis, we choose to hold these parameters fixed and generate what we refer to as the fiducial power spectra (see Section 3.3.2 for the values we use). Given the form of the covariance equations (Eq. 2.41, 2.54, 2.64), it’s clear that the cosmological parameters we’re interested in (the growth rate of structure f and the galaxy bias b) are degenerate with the amplitude of the power spectrum. The exact amplitude of each power spectrum is a function of the cosmological parameters used to generate it, and is parametrized by σ8, which is the root-mean-square amplitude of linear matter 1 fluctuations in spheres of 8 h− Mpc. Since we have fixed the fiducial power spectra, we can fid divide out by the fiducial amplitude σ8 and instead constrain the parameter combinations fσ8 and bσ8. For Eq. 2.41, 2.54, 2.64, we substitute:

f fσ /σﬁd, (2.65) → 8 8 b bσ /σﬁd. (2.66) → 8 8 56 Chapter 2. Theory

2.3 Data Modiﬁcations

The model we have presented thus far is for perfect measurements of galaxy overdensities and peculiar velocities in real space. This is far from the data we will test the model against, so we now make additional changes to account for various physical factors.

2.3.1 Modelling the Logarithmic Distance Ratio

Although linear theory describes peculiar velocities as being drawn from a multivariate Gaussian with a mean of zero, this is not the case when observational errors are introduced. Springob et al. (2014) has shown that peculiar velocities from the 6-degree Field Galaxy Survey (6dFGS) have uncertainty distributions that are log-normal rather than Gaussian. We can account for this by updating our model for the covariance. For 6dFGS, peculiar velocities were obtained using the Fundamental Plane relation, which links the eﬀective radius of an elliptical galaxy, its eﬀective surface brightness, and its central velocity dispersion. It can be used to derive peculiar velocities because it acts as a redshift-independent distance estimator, which can break the degeneracy between the redshift from peculiar velocity, zvp = vp/c, and the redshift from expansion, zH , that make up the observed redshift of an object:

(1 + zobs) = (1 + zvp )(1 + zH ). (2.67)

As discussed by Magoulas et al. (2012), the Fundamental Plane fit for each galaxy provides the probability distribution of the quantity log10[R(zobs)/R(zH )], which is the logarithmic ratio of a galaxy’s observed effective radius (measured by the survey) to the effective radius due purely to Hubble flow (inferred from the Fundamental Plane fit). However, peculiar velocities are determined from the logarithmic ratio of the comoving distance inferred from the observed redshift, D(zobs), to the true comoving distance, D(zH ), which can be calculated from R(zobs) and R(zH ). Springob et al. (2014) derived the probability distribution for the logarithmic distance ratio η log [D(z )/D(z )] for each galaxy, ≡ 10 obs H and Johnson et al. (2014) showed that the transformation between p(η) and p(vp) is non- linear, resulting in a skewed distribution for p(vp) (see figure 5 in that work). Since η has a Gaussian distribution, we rewrite the likelihood in terms of this parameter. This requires a conversion factor from our modelled peculiar velocity (Eq. 2.24) to η. Such modelling has already been performed for how peculiar velocities affect supernova magnitudes (e.g. Hui & Greene, 2006). We now cover the analytic relationship between the radial peculiar velocity, vp, and the observed logarithmic distance ratio, η. We begin 2.3. Data Modifications 57 with the definition of η: D(zobs) η = log10 , (2.68) D(zH )

1 where D is the comoving distance in h− Mpc calculated at the observed redshift zobs, and expansion redshift zH . If the Hubble constant is known as a function of redshift (generally assumed as a part of the model), we can express the comoving distance as

Z z dz D(z) = c 0 . (2.69) 0 H(z0)

Since we do not know zH we cannot directly evaluate Eq. 2.68. We can however perform a Taylor expansion of D(zobs) around zH :

c D(zH ) = D(zobs) + (zH zobs). (2.70) H(zobs) −

Then, using the relationship between the observed, expansion, and peculiar velocity redshifts: (1 + zobs) = (1 + zH )(1 + zvp ) (where zvp = vp/c), we can express the ratio between the comoving distances as

D(zH ) c = 1 + (zH zobs) D(zobs) D(zobs)H(zobs) − c(1 + z ) 1 = 1 + obs 1 . (2.71) D(zobs)H(zobs) (1 + vp/c) −

Applying a Taylor series to the bracketed term around vp/c = 0,

D(zH ) (1 + zobs) = 1 vp. (2.72) D(zobs) − D(zobs)H(zobs)

We can then calculate η as a function of the radial peculiar velocity by combining Eq. 2.68 and Eq. 2.72: D(zH ) η = log10 − D(zobs) (1 + zobs) = log10 1 vp − − D(zobs)H(zobs) 1 D(z ) = ln H −ln(10) D(zobs) 1 (1 + zobs) = ln 1 vp . (2.73) −ln(10) − D(zobs)H(zobs) 58 Chapter 2. Theory

This can then be simpliﬁed further from the ﬁrst order of the Maclaurin series ln(1 x) − ≈ x: −

1 (1 + zobs) η = vp (2.74) ln(10) D(zobs)H(zobs)

= κvp (2.75)

We note that this ﬁnal approximation will fail at very low redshift.

The equations for the logarithmic distance ratio auto-covariance and the cross-covariances then become

2 Cηη(xs, xt) = κ Cvpvp (xs, xt), (2.76)

Cδgη(xs, xt) = κCδgvp (xs, xt), (2.77)

Cηδg (xs, xt) = κCvpδg (xs, xt), (2.78) where κ is evaluated at the redshift of the peculiar velocity measurement. This means that the κ2 term is actually two diﬀerent values of κ, one for each position, but we use κ2 for notational ease throughout the thesis.

2.3.2 Gridding to Obtain the Data Vector

An important step in our analysis is constructing the data vector ∆ from the observed data. For an observed galaxy, redshift surveys return the position in terms of right as- cension RA, declination dec, and redshift z. The same goes for a peculiar velocity survey, with the addition of a measurement of the logarithmic distance ratio η. We begin by converting from angular coordinates (RA, dec, z) to Cartesian coordinates (x, y, z). The ﬁrst step is to estimate the comoving distance to the galaxy from the redshift and assumed cosmological parameters (see Section 1.1.6), noting that we will assume that the Universe is ﬂat throughout this thesis:

Z z dz0 DC = DH , (2.79) 0 E(z0) where

q E(z) Ω (1 + z)3 + Ω (1 + z)4 + Ω . (2.80) ≡ m,0 r,0 Λ,0 2.3. Data Modiﬁcations 59

The conversions between angular and Cartesian coordinates are

x = DC cos(dec) cos(RA), (2.81)

y = DC cos(dec) sin(RA), (2.82)

z = DC sin(dec). (2.83)

The next step is to estimate the galaxy overdensity from the measured positions. We do this by gridding the sample into cubic cells, such that the overdensity can be calculated as the number of galaxies in each cell compared to the number expected for that cell:

Ncell δg = 1, (2.84) Nexp − where the expected number is determined from the selection function for the survey (see Section 3.2 and corresponding sections in later chapters). In addition to calculating the galaxy overdensity, gridding also allows us to smooth over non-linear eﬀects and reduce the computation time for evaluating the likelihood function (as it lowers the dimensionality of the covariance matrix and data vector). We refer the reader to Abate et al. (2008) and Johnson et al. (2014) for additional discussion on these beneﬁts. Given these advantages, we also grid the logarithmic distance ratio sample; the logarithmic distance ratio value for each cell is then calculated to be the average of all measurements in the cell.

Most importantly, using a gridded sample changes the modelling in two key ways. The ﬁrst is that the samples become smoothed when gridded, reducing small-scale power. We account for this in our model covariance functions by multiplying the power spectra by a window function that characterises the eﬀect of gridding. We use a cubic gridding 1 approach, where each grid cell has length L in h− Mpc, corresponding to a sinc function in Fourier space. In three dimensions, the window function is

* L L L + 8 sin kx 2 sin ky 2 sin kz 2 Γ(k, L) = 3 , (2.85) L kx ky kz k k ∈ where the average is applied to all k vectors that have magnitude k. Figure 2.4 shows the window function for diﬀerent values of the grid cell length.

Since we may use different gridding sizes for the logarithmic distance ratios and galaxy overdensities, we define Γδg (k) = Γ(k, Lδg ) and Γη(k) = Γ(k, Lη). Our modified covariance 60 Chapter 2. Theory

1.0 1 L = 10 h− Mpc 1 L = 20 h− Mpc 1 0.8 L = 30 h− Mpc

0.6 ) k Γ( 0.4

0.2

0.0

2 1 0 10− 10− 10 1 k [h Mpc− ]

Figure 2.4 The window function in Fourier space for different gridding lengths. Larger gridding scales produce a stronger smoothing effect. Recalling that large k corresponds to small scales in configuration space, we see that the gridding window function damps the small-scale, non-linear physics. equations then become

(bσ )2 Z P (k) C (x , x ) = 8 mm k2j (kr)Γ2 (k)dk, (2.86) δ0gδg s t 2 fid 2 0 δg 2π (σ8 ) (κaHfσ )2 Z P (k)1 C (x , x ) = 8 θθ (xˆ xˆ )[j (kr) + j (kr)] (2.87) ηη0 s t 2 fid 2 s t 0 2 2π (σ8 ) 3 · − (rˆ xˆ )(rˆ xˆ )j (kr) Γ2(k)dk, · s · t 2 η κaHfbσ2 Z P (k) C (x , x ) = 8 mθ k(xˆ rˆ)j (kr)Γ (k)Γ (k)dk, (2.88) δ0gη s t − 2 fid 2 t 1 δg η 2π (σ8 ) · κaHfbσ2 Z P (k) C (x , x ) = 8 θm k(xˆ rˆ)j (kr)Γ (k)Γ (k)dk. (2.89) ηδ0 g s t 2 fid 2 s 1 η δg 2π (σ8 ) ·

After gridding, the position of the galaxy overdensity and average logarithmic distance ratio for each cell is set to be the centre of that cell.

The second eﬀect is that a correction must be applied to the diagonal elements of the logarithmic distance ratio auto-covariance to model the varying shot noise that arises from taking the average value. Abate et al. (2008) proposed that for Ns values of η observed in 2.3. Data Modiﬁcations 61 a cell located at xs the shot noise contribution is

2 Cηη(xs, xt) Cηη0 (xs, xt) σsn(xs, xt) = − δst, (2.90) Ns

2 where δst is the Kronecker delta, such that σsn only appears on the diagonal. This term behaves similarly to an error term, so we incorporate it into the model in the next section.

2.3.3 Inclusion of Error Terms

We now discuss the inclusion of errors; assuming that the errors on the data points are independent, these appear along the diagonal of the covariance matrix.

For an observed value of η at position xi, we include the observed error in η from the Fundamental Plane σobs(xi) (as detailed in Springob et al. 2014), the shot noise from averaging the value of η in each grid cell σsn (see Eq. 2.90), and a stochastic velocity term to account for the breakdown of linear theory σv. This takes the same value for all diagonal entries of the logarithmic distance ratio auto-covariance. σv is typically quoted in velocity units, so requires a modiﬁcation factor of κ in order to be incorporated into the logarithmic distance ratio auto-covariance. Given that κ is a function of redshift, we note that the σv term will not have the same value for every entry of the logarithmic distance ratio auto-covariance as it does for the peculiar velocity auto-covariance. The contribution to the logarithmic distance ratio auto-covariance will then be

2 2 2 2 σηη(xs, xt) = σobs(xs)σobs(xt)δst + σsn(xs, xt) + κ σvδst, (2.91) where δst is the Kronecker delta, ensuring that error terms only aﬀect the diagonal of the covariance matrix. The logarithmic distance ratio auto-covariance becomes

err 2 Cηη (xs, xt) = Cηη0 (xs, xt) + σηη(xs, xt). (2.92)

For an observed value of δg at position xi, we include the shot noise contribution,

σδg (xi), assuming that the galaxy counts are drawn from a Poisson distribution, such that the shot noise variance is equal to the inverse of the expected number count for a given grid cell (see Section 3.2.1). The contribution to the galaxy overdensity auto-covariance will then be

2 σδgδg (xs, xt) = σδg (xs)σδg (xt)δst. (2.93) 62 Chapter 2. Theory

The galaxy overdensity auto-covariance becomes

err 2 Cδgδg (xs, xt) = Cδ0gδg (xs, xt) + σδgδg (xs, xt). (2.94)

Given that the assumed errors are independent, there are no error terms for the cross- covariance matrices.

2.3.4 Modelling Redshift-Space Distortions

As discussed in Section 1.2.4, redshift-space distortions modify the galaxy overdensity in relation to the underlying matter overdensity:

˜s ˜ 2 ˜ δg(k) = bδm(k) + fµ θ(k), (2.95) where µ kˆ dˆ is the cosine of the angle between the wavevector and the line of sight. As ≡ · is typical for redshift-space distortion models, we adopt the local parallel-plane approximation, which assumes that the position vectors are parallel. This implies that

kˆ xˆ kˆ xˆ kˆ dˆ. (2.96) · s ≈ · t ≈ ·

This is valid when the angle between xs and xt is small. A more complex model for redshift-space distortions (without this assumption) is beyond the scope of this thesis, but is a strong candidate for future work, which we discuss in Section 6.2. The parallel-plane approximation allows us to write the Fourier space expression for the line of sight peculiar velocity as

(k dˆ) v˜ (k) = iaHf · θ˜(k) (2.97) p − k2 µ = iaHf θ˜(k). (2.98) − k

Both the galaxy overdensity and peculiar velocity are additionally damped at non-linear scales. For the galaxy overdensity, this comes from the “fingers-of-God” effect: small- scale peculiar velocities around galaxy clusters enhance clustering along the line-of-sight. Peacock (1992) proposed a Gaussian damping function, which was later refined by Peacock & Dodds (1994):

δ˜s(k) δ˜s(k)D (k, µ, σ ) g → g g g 2 2 (kµσg) /2 = [bδ˜m(k) + fµ θ˜(k)]e− , (2.99) 2.3. Data Modiﬁcations 63

1 where σg is related to the pairwise velocity dispersion and has units of h− Mpc. The damping may also be modelled as a Lorentzian function in Fourier space (Ballinger et al., 1996), but it is generally accepted that there is very little diﬀerence between the two. We use the Gaussian form for convenience when it comes to the mathematical modelling. For the peculiar velocity, Koda et al. (2014) found that there was a strong damping contribution to the peculiar velocity auto-power spectrum when working with simulations. They empirically modelled this using a sinc function, such that the peculiar velocity becomes

v˜ (k) v˜ (k)D (k, σ ) p → p u u µ sin(kσ ) = iaHf θ˜(k) u , (2.100) − k kσu

1 where Koda et al. (2014) ﬁnd σu = 13.0 h− Mpc at z = 0. We use σu here to diﬀerentiate from the velocity dispersion term σv, which appears as an error in the logarithmic distance ratio auto-covariance (see Section 2.3.3).

The final modification we make is the introduction of the cross-correlation coefficient rg. This parameter allows for a more detailed galaxy bias relation, and is commonly used in RSD modelling (e.g. Dekel & Lahav, 1999; Burkey & Taylor, 2004; Blake et al., 2011b; Koda et al., 2014). It is defined such that it modifies the galaxy bias only for the galaxy- matter cross-power spectrum δ (k)δ (k ) = 1 r bP (k)δ3 (k k ), and we note that h g m 0 ∗i V g mm D − 0 setting rg = 1 recovers the standard linear bias model. Given that we are assuming δm = θ on linear scales (see Eq. 2.19), we apply the cross-correlation coefficient to the cross-power spectrum Pgθ in our RSD equations.

Given Eq. 1.62, we can calculate the basic anisotropic form of the power spectra from Eq. 2.99 and Eq. 2.100:

2 2 2 4 2 Pgg(k, µ) = b [Pmm(k) + 2rgβµ Pmθ(k) + β µ Pθθ(k)]Dg(k, µ, σg), (2.101) iaHfbµ P (k, µ) = [r P (k) + βµ2P (k)]D (k, µ, σ )D (k, σ ), (2.102) gv k g mθ θθ g g u u iaHfbµ P (k, µ) = − [r P (k) + βµ2P (k)]D (k, µ, σ )D (k, σ ), (2.103) vg k g mθ θθ g g u u aHfµ2 P (k, µ) = P (k)D2(k, σ ), (2.104) vv k θθ u u 64 Chapter 2. Theory such that

s s 0 1 3 δ˜ (k)δ˜ (k )∗ = P (k, µ)δ (k k0), (2.105) h g g i V gg D − s 0 1 3 δ˜ (k)˜v (k )∗ = P (k, µ)δ (k k0), (2.106) h g p i V gv D − 0 s 1 3 v˜ (k )δ˜ (k)∗ = P (k, µ)δ (k k0), (2.107) h p g i V vg D − 0 1 3 v˜ (k)˜v (k )∗ = P (k, µ)δ (k k0). (2.108) h p p i V vv D −

We note that the asymmetry between Eq. 2.102 and Eq. 2.103 arises from our deﬁnition of the direction of positive peculiar velocity, which is highlighted in Figure 2.2.

In the RSD modelling, we also account for all modifications discussed in the previous subsections: the modification to the amplitude of the fiducial power spectra by dividing fid through by σ8 (see Section 3.3.2), the transition from peculiar velocities to logarithmic distance ratios via the parameter κ (see Section 2.3.1), the damping from gridding the data (see Section 2.3.2), and the inclusion of error terms (see Section 2.3.3). However, we do not explicitly include them in the derivations below to avoid cluttering the mathematics.

Now that we have deﬁned the anisotropic power spectra, it is generally convenient to deal with them through the multipole expansion:

X∞ P (k, µ) = P`(k)L`(µ), (2.109) `=0 where P`(k) are the multipole power spectra and L`(µ) are the Legendre polynomials. Eq.

2.109 can then be evaluated for the required P`, which is done by multiplying each side by L`0 (µ) and taking advantage of the normalisation condition for Legendre polynomials:

Z 1 2 L`(x)L`0 (x)dx = δ``0 , (2.110) 1 2`0 + 1 − such that

Z 1 2` + 1 P`(k) = L`(µ)P (k, µ)dµ. (2.111) 1 2 − 2.3. Data Modiﬁcations 65

Galaxy Overdensity Auto-Covariance

Given the deﬁnition of the anisotropic power spectrum, we can write the galaxy overdensity auto-covariance as Z 1 ik r 3 C (x , x ) = P (k, µ)e · d k. (2.112) δgδg s t (2π)3 gg

Utilising Eq. 2.29 and 2.109: Z 1 X `0 3 C (x , x ) = P (k)L (kˆ dˆ)(2`0 + 1)i j 0 (kr)L 0 (kˆ rˆ)d k. (2.113) δgδg s t (2π)3 gg,` ` · ` ` · `,`0

This can then be expanded through the spherical harmonic addition theorem (Eq. 2.34):

1 Z X X 4π C (x , x ) = P (k) Y (kˆ)Y ∗ (dˆ) (2.114) δgδg s t (2π)3 gg,` (2` + 1) `m `m `,`0 m,m0 `0 4π ˆ 3 (2`0 + 1)i j`0 (kr) Y`∗0m0 (k)Y`0m0 (rˆ)d k. (2`0 + 1)

We now break up the integral into spherical coordinates d3k = k2 sin(θ)dφdθdk, noting that kˆ is a function of θ and φ, but dˆ and rˆ are not. This allows us to group the spherical harmonic functions into conﬁguration-space and Fourier-space pairs:

Z 2 1 ∞ X X 2 (4π) `0 C (x , x ) = k P (k) Y ∗ (dˆ)Y 0 0 (rˆ)i j 0 (kr) (2.115) δgδg s t (2π)3 gg,` (2` + 1) `m ` m ` 0 `,`0 m,m0 Z π Z 2π ˆ ˆ Y`m(k)Y`∗0m0 (k) sin(θ)dφdθdk. 0 0

The angular integral corresponds to the orthonormal condition of spherical harmonics (Eq.

2.33), producing the pair of delta functions δ`,`0 δm,m0 such that

Z 2 1 ∞ X X 2 (4π) ` C (x , x ) = k P (k) Y ∗ (dˆ)Y (rˆ)i j (kr)dk, (2.116) δgδg s t (2π)3 gg,` (2` + 1) `m `m ` 0 ` m which can be further reduced through the spherical harmonic addition theorem to Z 1 ∞ X C (x , x ) = k2P (k)L (rˆ dˆ)i`j (kr)dk δgδg s t 2π2 gg,` ` · ` 0 ` Z 1 ∞ X = k2P (k)L (cos γ)i`j (kr)dk (2.117) 2π2 gg,` ` ` 0 ` where γ is the angle between r and d as shown in Figure 2.1. 66 Chapter 2. Theory

The next step is to assess which values of ` are required for the expansion, and to determine the power spectrum multipole function at the required `. Given the form of the expansion (Eq. 2.109), the required values of ` are determined by the orders of µ that appear in the anisotropic power spectrum. For the galaxy-galaxy anisotropic power spectrum (Eq. 2.101), the orders of µ are 0, 2, 4. Recalling the deﬁnition of the power spectrum multipoles (Eq. 2.111):

Z 1 2` + 1 Pgg,`(k) = L`(µ)Pgg(k, µ)dµ 1 2 − Z 1 2` + 1 2 2 2 4 2 = L`(µ)b [Pmm(k) + 2rgβµ Pmθ(k) + β µ Pθθ(k)]Dg(k, µ, σg)dµ. 1 2 − (2.118)

It is now important to explain the general approach we take in evaluating the covariance equations. Given that the covariance must be calculated for every pair of positions to build the covariance matrix, we wish to avoid recalculating the covariance values wherever possible. Noting that the cosmological parameters we wish to vary (f, b, β) only modify the overall amplitude of the power spectra, the simplest approach is to calculate the covariance matrices that can be directly multiplied by these factors and sum them as necessary. In the case of the galaxy overdensity auto-covariance (given by Eq. 2.117 and 2.118), the complete covariance matrix can be expressed as

2 b 2 C = (C 0 + 2r βC 1 + β C 2 ), (2.119) δgδg 2π2 δgδg,β g δgδg,β δgδg,β where each of these covariance matrices will include the sum over ` of the power spectrum multipoles as well as the integrals over µ and k. The integral over µ can be evaluated analytically, whereas the integral over k is done numerically. We obtain the analytic expressions for the various covariances through Mathematica.

The covariances matrices for each order of β may then be expressed as the sum of integrand matrices for each value of `, which we label with K. For the β0 term:

Z 2 0 0 0 0 Cδgδg,β = k Pmm(k) Kδgδg,β ,`=0 + Kδgδg,β ,`=2 + Kδgδg,β ,`=4 dk. (2.120)

For a pair of positions (described by γ and r) the integrands have the following functional 2.3. Data Modiﬁcations 67 forms:

1 0 Kδgδg,β ,`=0 = √πErf(kσg)j0(kr), (2.121) 2kσg 5 k2σ2 2 2 0 g √ Kδgδg,β ,`=2 = 3 3 L2(cos γ) 6e− kσg + ( 3 + 2k σg ) πErf(kσg) j2(kr), (2.122) 8k σg − 9 k2σ2 2 2 K 0 = L (cos γ) 10e− g kσ (21 + 2k σ )+ (2.123) δgδg,β ,`=4 64k5σ5 4 − g g g 3(35 20k2σ2 + 4k4σ4)√πErf(kσ ) j (kr), − g g g 4 where Erf(x) is the error function. For the β1 term:

Z 2 1 1 1 1 Cδgδg,β = k Pmθ(k) Kδgδg,β ,`=0 + Kδgδg,β ,`=2 + Kδgδg,β ,`=4 dk, (2.124) where 1 k2σ2 1 g √ Kδgδg,β ,`=0 = 3 3 2e− kσg + πErf(kσg) j0(kr), (2.125) 2k σg − 5 k2σ2 2 2 K 1 = L (cos γ) 2e− g kσ (9 + 4k σ )+ (2.126) δgδg,β ,`=2 8k5σ5 2 g g g ( 9 + 2k2σ2)√πErf(kσ ) j (kr), − g g 2 9 k2σ2 2 2 4 4 K 0 = − L (cos γ) 2e− g kσ (525 + 170k σ + 32k σ ) (2.127) δgδg,β ,`=4 64k7σ7 4 g g g − g 3(175 60k2σ2 + 4k4σ4)√πErf(kσ ) j (kr). − g g g 4

Finally, the β2 term can be expressed as:

Z 2 2 2 2 2 Cδgδg,β = k Pθθ(k) Kδgδg,β ,`=0 + Kδgδg,β ,`=2 + Kδgδg,β ,`=4 dk, (2.128) 68 Chapter 2. Theory where 1 k2σ2 2 2 2 g √ Kδgδg,β ,`=0 = 5 5 2e− kσg(3 + 2k σg ) + 3 πErf(kσg) j0(kr), (2.129) 8k σg − 5 k2σ2 2 2 4 4 K 2 = L (cos γ) 2e− g kσ (45 + 24k σ + 8k σ )+ (2.130) δgδg,β ,`=2 32k7σ7 2 g g g g 3( 15 + 2k2σ2)√πErf(kσ ) j (kr), − g g 2 9 k2σ2 2 2 4 4 6 6 K 2 = − L (cos γ) 2e− g kσ (3675 + 1550k σ + 416k σ + 64k σ ) δgδg,β ,`=4 256k9σ9 4 g g g g − g 3(1225 300k2σ2 + 12k4σ4)√πErf(kσ ) j (kr). − g g g 4 (2.131)

Peculiar Velocity Auto-Covariance

The mathematics for the peculiar velocity auto-covariance is largely the same as what we used in the previous section. Elements of the covariance matrix have the same form as Eq. 2.117: Z 1 ∞ X C (x , x ) = k2P (k)L (cos γ)i`j (kr)dk, (2.132) vpvp s t 2π2 vv,` ` ` 0 ` where the multipole power spectra are given by

Z 1 2 2` + 1 aHfµ 2 Pvv,`(k) = L`(µ) Pθθ(k)Du(k, σu)dµ. (2.133) 1 2 k − For the velocity-velocity anisotropic power spectrum (Eq. 2.104), the orders of µ indicate that we require ` = 0, 2 for the multipole expansion.

Unlike the galaxy overdensity auto-covariance, there is only a single order of β, so we do not need to express the total covariance as a sum over orders of β, as in Eq. 2.119. Instead, we may jump straight to the expression in terms of integrand matrices K:

(aHf)2 Z C = P (k)D2(k, σ ) K + K dk. (2.134) vpvp 2π2 θθ u u vpvp,`=0 vpvp,`=2

For a pair of positions (described by γ and r) the integrands have the following functional 2.3. Data Modiﬁcations 69 forms:

1 K = j (kr) (2.135) vpvp,`=0 3 0 2 K = L (cos γ)j (kr). (2.136) vpvp,`=2 −3 2 2

We note that there is some similarity to the structure of the peculiar velocity auto- covariance without the parallel-plane approximation (Eq. 2.54), but the position dot- products are now represented by γ.

Cross-Covariance

Again, the mathematics for introducing RSD to the cross-covariance is largely the same as that used in the previous derivations. Elements of the covariance matrix have the same form as Eq. 2.117: Z 1 ∞ X C (x , x ) = C (x , x ) = k2P (k)L (cos γ)i`j (kr)dk. (2.137) δgvp δ v vpδg v δ 2π2 vg,` ` ` 0 `

Here, we have speciﬁed the equation in terms of the overdensity and velocity positions

(xδ, xv), as we did for the first derivation of the cross-covariance (see Eq. 2.64 and preceding text). By choosing to define r = x x , we account for the asymmetry δ − v of the cross-covariance, which appears as a sign difference when working in terms of fixed positions (xs, xt). Given there are outstanding factors of i, we note that the covariance expression above will only have the correct sign if calculated using the expression for Pvg,`. The multipole power spectra are given by

Z 1 2` + 1 iaHfbµ 2 Pvg,`(k) = L`(µ)− [rgPmθ(k) + βµ Pθθ(k)]Du(k, σu)Dg(k, µ, σg)dµ. 1 2 k − (2.138)

The orders of µ indicate that we require ` = 1, 3 for the multipole expansion. As with the galaxy overdensity auto-covariance, there are multiple orders of β, so we break up the covariance equation similarly to Eq. 2.139:

aHfb C = (r C 0 + βC 1 ). (2.139) vpδg 2π2 g vpδg,β vpδg,β

Again, we obtain the analytic expressions for the various covariances through Mathematica. The covariances matrices for each order of β may be expressed as the sum of integrand 70 Chapter 2. Theory matrices for each value of `, which we label with K. For the β0 term:

Z 0 0 0 Cvpδg,β = kPmθ(k)Du(k, σu) Kvpδg,β ,`=1 + Kvpδg,β ,`=3 dk. (2.140)

For a pair of positions (described by γ and r) the integrands have the following functional forms: 3 k2σ2/2 K 0 = L (cos γ) 2e− g kσ + (2.141) vpδg,β ,`=1 2k3σ3 1 − g g kσg √2πErf j1(kr), √2 7 k2σ2/2 2 2 K 0 = L (cos γ) 2e− g kσ (15 + 2k σ )+ (2.142) vpδg,β ,`=3 4k5σ5 3 g g g 2 2 kσg 3√2π( 5 + k σ )Erf j3(kr), − g √2 where Erf(x) is the error function. For the β1 term:

Z 1 1 1 Cvpδg,β = kPθθ(k)Du(k, σu) Kvpδg,β ,`=1 + Kvpδg,β ,`=3 dk, (2.143) where 3 k2σ2/2 2 2 K 1 = L (cos γ) 2e− g kσ (3 + k σ )+ (2.144) vpδg,β ,`=1 2k5σ5 1 − g g g kσg 3√2πErf j1(kr), √2 7 k2σ2/2 2 2 4 4 K 1 = L (cos γ) 2e− g kσ (75 + 16k σ + 2k σ )+ (2.145) vpδg,β ,`=3 4k7σ7 3 g g g g 2 2 kσg 3√2π( 25 + 3k σ )Erf j3(kr). − g √2

We have now constructed the appropriate covariance models, which account for the fact that we observe data in redshift space, rather than real space. 3 The Density-Velocity Cross-Correlation in the 6dF Galaxy Survey

3.1 Overview

In this study we apply the method developed in the previous chapter to the 6-degree Field Galaxy Survey (6dFGS). Our method is a novel way of analysing the relation between the peculiar velocity and galaxy overdensity fields: instead of modelling one field in terms of the other, we model the covariance between the two fields, which can be done analytically. This covariance can then be used in a maximum likelihood method to constrain our cosmological parameters of interest: the growth rate of structure f, and either the galaxy bias b, or the ratio of these β.

The aim of this chapter is to utilise the cross-covariance between peculiar velocities and galaxy overdensities to constrain the growth rate of structure to higher precision than can be obtained from peculiar velocities alone. We validate our approach by applying it to data from the GiggleZ N-body simulation, before analysing data from the 6dFGS, obtaining constraints for the growth rate of structure. This is the ﬁrst application of this method and this chapter serves to illustrate its eﬀect on cosmological constraints when used with current peculiar velocity and redshift data.

We begin by introducing the data in Section 3.2, and then discuss the theory and methodology in Section 3.3 (noting that most of the necessary theory has been developed in Chapter 2). Results for the simulation analysis are presented in Section 3.4 followed by the 6dFGS analysis in Section 3.5. We conclude with a summary of the study in Section 3.6.

71 72 Chapter 3. The Density-Velocity Cross-Correlation in 6dFGS

3.2 Data and Simulations

This analysis requires measurements of the galaxy overdensity field δg(x) and the radial peculiar velocity field vp(x). Both can be extracted from redshift surveys, although peculiar velocities also require an estimate of the true distance to the galaxy (see Section 1.2.5). These fields will be used to test our model of the covariance matrix of our data vector and infer constraints on several cosmological parameters (see Section 2.2 for a complete description of the model). In this section, we cover the observed and simulated data that will be used in this analysis.

3.2.1 6dFGS

We choose to work with data from the 6-degree Field Galaxy Survey (6dFGS; Jones et al., 2005, 2006, 2009) as it contains the largest single collection of peculiar velocity measurements currently available. 6dFGS is a redshift survey selected from the 2-Micron All-Sky Survey (2MASS) that covers the entire southern sky except for 10 degrees around the Galactic plane. It can be broken into two samples: the redshift sample, 6dFGSz, and the peculiar velocity sample, 6dFGSv, which is a subset of 6dFGSz. As of the final data release, 6dFGSz contains 125,071 extragalactic redshifts with a median redshift of z = 0.053. Our redshift sample for calculating the galaxy overdensity comes from the 6dFGS baryon acoustic oscillation analysis by Beutler et al. (2011), who selected galaxies with magnitude K 12.9 in regions of the sky that had a completeness ≤ value of 60% or greater, which yielded 75,117 galaxies. In 6dFGS, peculiar velocities were obtained using the Fundamental Plane method (for a general description of the Fundamental Plane, see Section 1.2.5; for the specific implementation, see Magoulas et al., 2012). We use the 6dFGSv sample as defined by 1 Springob et al. (2014), which required signal-to-noise ratios greater than 5A˚− and velocity dispersions greater than the resolution limit of the 6dF spectrograph (σ 112km s 1). 0 ≥ − This selection yielded 8,885 galaxies. Fig. 3.1 shows the sky and redshift distribution of both samples used in this work. Our analysis proceeds by gridding the data into cubic cells, as discussed in Section 2.3.2. Following Johnson et al. (2014), the logarithmic distance ratio sample is gridded 1 into cubes of length L = 10 h− Mpc. The redshift sample is gridded into cubes of length 1 L = 20 h− Mpc; we found that gridding at higher resolution did not affect our results so chose a larger gridding to reduce the dimension of our data vector. The gridding produced

Nδg = 1036 cells for 6dFGSz and Nη = 2977 cells for 6dFGSv. We now outline the steps 3.2. Data and Simulations 73

75◦ 6dFGSz 75◦ 6dFGSv 60◦ 60◦ 45◦ 45◦

30◦ 30◦

15◦ 15◦ 330◦ 300◦ 270◦ 240◦ 210◦ 180◦ 150◦ 120◦ 90◦ 60◦ 30◦ 330◦ 300◦ 270◦ 240◦ 210◦ 180◦ 150◦ 120◦ 90◦ 60◦ 30◦ 0◦ 0◦

15◦ 15◦ − − 30◦ 30◦ − − 45◦ 45◦ − − 60◦ 60◦ − 75◦ − 75◦ − −

0.012 0.018 0.024 0.030 0.036 0.042 0.048 0.054 Redshift

Figure 3.1 Distribution of the 6dFGSz and 6dFGSv samples, coloured by redshift. Both plots are equal-area Aitoff projections. involved in calculating the galaxy overdensity. When selecting our sample, it is important to consider galaxy bias. While galaxy bias evolves with redshift, we note that the small redshift range of 6dFGS means that bias (as a function of absolute magnitude) should be effectively independent of redshift. Con- sequently, we can keep the galaxy bias of our sample constant with redshift by selecting a volume-limited sample. This is where we select galaxies according to an appropriate absolute magnitude cut. Data from 6dFGS is apparent magnitude-limited, and we calculate the absolute magnitude using the k-correction from Mannucci et al. (2001). For simplicity, we limit our sample to the same volume as 6dFGSv, with a maximum redshift of z = 0.057; this defines the absolute magnitude cut: galaxies need to be brighter than M = 23.37 to exceed the 6dfGS apparent magnitude limit at all redshifts. We note that K − this calculation is informed by our fiducial choice for the Hubble constant (h = 0.6731), taken from the Planck 2015 cosmological parameters (see Table 3.1). After the cut, our redshift sample contains 20,796 galaxies.

The overdensity is related to the ratio between the number of galaxies in a cell Ncell, and the mean background number expected for that cell Nexp. Since the survey geometry and completeness affects the expected number of galaxies, we proceed by determining the selection function, assuming that the radial and angular components are separable. We obtain the radial selection function by taking a histogram of the sample and calculating the number density as a function of redshift n(z). We find a slight increase in number density with redshift, which can be explained by selection effects and our choice of k- correction, so we fit a second order polynomial to obtain a functional form for n(z). The number density values from the histogram, as well as our functional form for n(z) can be 74 Chapter 3. The Density-Velocity Cross-Correlation in 6dFGS

0.007 ] 3 0.006 Mpc) 1 −

h 0.005

0.004

) [galaxies0 per. ( 003 z n (

0.002

0.01 0.02 0.03 0.04 0.05 Redshift

Figure 3.2 The number density of the galaxy overdensity sample as a function of redshift n(z). The blue points show the values calculated from the histogram and the red line shows the functional form obtained by fitting a second order polynomial to the points. seen in Fig. 3.2. We use the angular selection function derived for 6dFGS by Jones et al. (2006). We calculate the number of galaxies expected in a given grid cell and normalise this to the total number of galaxies in our sample. From this, we calculate the overdensity using δ = Ncell 1, and obtain the shot noise for each cell assuming Poisson statistics as g Nexp p − σδg = 1/ Nexp. The logarithmic distance ratio measurement for each cell is the average of all logarithmic distance ratios in that cell, and we add the observational uncertainties in quadrature, converting this to an error in the mean. See Abate et al. (2008) for a discussion of the motivation behind this approach and Section 2.3.2 for a discussion on how it affects the modelling. Due to the distribution of galaxies in 6dFGS, there are cells that do not contain any galaxies. For example, a cell that covers the Galactic plane will be empty since the survey did not take any data in this region. For logarithmic distance ratios, a cell can only be empty if we have no measurement there: the cell has no information to contribute to our analysis and can be safely removed. The overdensity cells are more complicated, since an empty cell provides information about the overdensity field as long as it is within the survey volume. Thus, we keep empty overdensity cells if they fall within the 6dFGS survey volume and exclude them otherwise. Finally, since our sample is at very low redshift, we take the effective redshift for our 3.2. Data and Simulations 75 growth rate of structure measurement to be at z = 0.

3.2.2 Simulation

We use simulated peculiar velocities and halo positions from the GiggleZ simulation (Poole 1 3 et al., 2015) to validate our method. The volume of the simulation is 1 (h− Gpc) , and 3 9 1 it contains 2160 particles, each with a mass of 7.5 10 h− M . It was generated using × GADGET-2 (Springel, 2005), and halos and subhaloes were identified with the SUBFIND algorithm (Springel et al., 2001). The fiducial cosmology for GiggleZ is a spatially-flat ΛCDM fit to the Wilkinson Microwave Anisotropy Probe (WMAP) five-year data (with the addition of baryon acoustic oscillation and supernova data), and the cosmological parameters are listed in Table 3.1 in Section 3.3.2. Applying our method to simulated data allows us to check how well the approach recovers known input cosmological parameters, as well as the effects of introducing the cross-covariance in the absence of noise. We create a single approximate realisation of the 6dFGS survey from GiggleZ, constructing a hemisphere around an observer out to redshift z = 0.057 (the maximum redshift of the 6dFGSv sample), corresponding to a radius of 150 h 1 Mpc. For our galaxy overdensity sample, we use haloes with subhalo masses ∼ − 13 1 13.5 1 between 10 h− M and 10 h− M . This limits the variation in galaxy bias for our sample, as galaxy bias is related to halo mass (e.g. Seljak et al., 2005). Linear theory prescribes that all velocities should be fair tracers of the underlying matter distribution, so we use all velocities available in our chosen volume. We project the simulated velocities onto the line of sight to obtain the peculiar velocity, and then convert this to a logarithmic distance ratio, η, which is the observed quantity from the 6dFGS Fundamental Plane analysis (see Section 2.3.1). As with 6dFGS, we grid each sample to obtain a measurement of the galaxy overdensity and average logarithmic distance ratio in each cell. We find that gridding both samples 1 into cells of length L = 20 h− Mpc is sufficient for recovering the simulation cosmology and has the added advantage of improving the computation speed over a higher-resolution gridding (see Section 2.3.2). Since our samples from the simulation are not affected by a selection function, we can directly calculate the overdensity relative to the average number density for our sample. We note that this realisation is not a true mock of 6dFGS as it does not contain observational errors for the peculiar velocities. This is desirable since it allows us to perform a more accurate test of our approach. The number densities are also different: we estimate that our GiggleZ sample has a galaxy number density of n = 1.7 10 4 ( h 1 Mpc) 3 and g × − − − 76 Chapter 3. The Density-Velocity Cross-Correlation in 6dFGS logarithmic distance ratio number density of n = 1.0 10 2( h 1 Mpc) 3, whereas our η × − − − 6dFGSz sample has a number density of n = 3.6 10 3( h 1 Mpc) 3 and our 6dFGSv g × − − − sample has a number density of n = 1.5 10 3( h 1 Mpc) 3. Finally, for the purposes η × − − − of this test, we did not include the effects of redshift-space distortions in the simulations. This does not affect the ability of our GiggleZ sample validate our method, since we seek only to recover the input cosmological parameters. We utilise more sophisticated 6dFGS mock catalogues in the next chapter.

3.3 Theory and Methodology

Much of the underlying theory for our method has been covered in the previous chapter, but we now cover a few additional modiﬁcations speciﬁc to this analysis. Recalling the general form of the data vector and covariance outlined in Section 2.2.2 and substituting the logarithmic distance ratio in place of peculiar velocity, we have

    δ Cerr C g δgδg δ0 gη ∆ =   , C =   , (3.1) η C Cerr ηδ0 g ηη where the diagonal sub-matrices include errors (σ) as described in Section 2.3.3:

err 2 Cδgδg = Cδ0 gδg + σδgδg , (3.2) err 2 Cηη = Cηη0 + σηη, (3.3) where C0 indicates that the covariance has been corrected for the sample gridding (see Section 2.3.2). It is useful to note that the auto-covariances are constructed from a signal 2 term C0 and a noise term σ . The entries of the error matrices are given by

2 σδgδg (xs, xt) = σδg (xs)σδg (xt)δst, (3.4) 2 2 2 2 σηη(xs, xt) = σobs(xs)σobs(xt)δst + σsn(xs, xt)δst + κ σvδst, (3.5)

where for an observed value of δg or η at position xi, σδg (xi) is the shot noise contribution from the ﬁnite number of galaxies, σobs(xi) is the observed error in η from the Fundamental

Plane, σsn is the shot noise from averaging the value of η in each grid cell (see Eq. 2.90), κ is the quantity that allows for the conversion of modelled peculiar velocities to observed logarithmic distance ratios (discussed in Section 2.3.1), and σv is a stochastic velocity term to account for the breakdown of linear theory. 3.3. Theory and Methodology 77

The elements of each sub-matrix are given by

(bσ )2 Z P (k) C (x , x ) = 8 mm k2j (kr)Γ2 (k)dk, (3.6) δ0gδg s t 2 fid 2 0 δg 2π (σ8 ) (κaHfσ )2 Z P (k)1 C (x , x ) = 8 θθ (xˆ xˆ )[j (kr) + j (kr)] (3.7) ηη0 s t 2 fid 2 s t 0 2 2π (σ8 ) 3 · − (rˆ xˆ )(rˆ xˆ )j (kr) Γ2(k)dk, · s · t 2 η κaHfbσ2 Z P (k) C (x , x ) = 8 mθ k(xˆ rˆ)j (kr)Γ (k)Γ (k)dk, (3.8) δ0gη s t − 2 fid 2 t 1 δg η 2π (σ8 ) · κaHfbσ2 Z P (k) C (x , x ) = 8 θm k(xˆ rˆ)j (kr)Γ (k)Γ (k)dk. (3.9) ηδ0 g s t 2 fid 2 s 1 η δg 2π (σ8 ) ·

fid These largely have the forms developed in Section 2.2.3, with a few modifications: σ8 is the fiducial value of σ8 associated with the cosmology used to generate the power spectra (discussed in Section 2.2.4) and the Γ(k) functions are the gridding window functions (discussed in Section 2.3.2).

3.3.1 Eﬀect of Redshift-Space Distortions

The covariance functions above assume that the data is perfectly measured in real space, and ignore the effects of redshift-space distortions (RSD). We defer the full treatment of RSD to Chapter 4 because we wish to focus on the growth information in the velocity data and cross-correlation only. However, it is still important to account for the fact that the covariance will be amplified by the presence of RSD. In this analysis, we have absorbed redshift-space distortions into the galaxy bias such that we are effectively fitting for the redshift-space galaxy bias bs in terms of the real-space galaxy bias br:

2 1 b2 = b2 1 + β + β2 (3.10) s r 3 5 1 2 1 = f 2 + + , (3.11) 5 3β β2 where β f/b is the redshift-space distortion parameter. This relation was originally ≡ developed by Kaiser (1987), who showed that bs is boosted by the total contribution of redshift-space distortions to the power spectrum when integrating over all angles on the sky. This allows us to parametrize Eq. 3.6-3.9 in terms of f and β. This parametrization is convenient as traditional analyses that compare the peculiar velocity and galaxy overdensity ﬁelds (e.g. Pike & Hudson, 2005; Davis et al., 2011; Carrick et al., 2015) are able to constrain β, but not the galaxy bias (see Section 1.4 for an introduction to these 78 Chapter 3. The Density-Velocity Cross-Correlation in 6dFGS

Table 3.1 Cosmological parameters for the three cosmologies used in this analysis. The top section shows the 6 base parameters for standard ΛCDM: physical baryon density; physical dark matter density; reduced Hubble constant; scalar spectral index; scalar amplitude 1 1 (with pivot point k0 = 0.002 h Mpc− for GiggleZ and WMAP, and k0 = 0.05 h Mpc− for Planck); and reionization optical depth. The bottom section shows σ8, which is a derived parameter. GiggleZ Planck WMAP 2 Ωbh 0.02267 0.02222 0.02273 2 Ωch 0.1131 0.1197 0.1099 h 0.705 0.6731 0.719

ns 0.960 0.9655 0.963 A 2.445 10 9 2.195 10 9 2.41 10 9 s × − × − × − τ 0.084 0.078 0.087 ﬁd σ8 0.812 0.8417 0.7931 analyses). We will apply this parametrization to our 6dFGS analysis, but will continue to use f and br for the simulation data, since these are in real-space.

3.3.2 Generation of the Fiducial Power Spectra

We use two codes to generate the power spectra, CAMB and velMPTbreeze, which we discussed in Section 2.2.4. For our simulation tests, the power spectra are generated from the same cosmological parameters as used in GiggleZ, which allows us to check whether we recover the simulation input values. For the 6dFGS analysis, we use power spectra generated from the median values of the 6-parameter base ΛCDM model for the Planck 2015 TT-lowP data (Planck Collaboration et al. 2015c). To check how our results are aﬀected by this choice, we also use power spectra generated from the median values of the 6-parameter base ΛCDM model for the Wilkinson Microwave Anisotropy Probe (WMAP) ﬁve-year data (Komatsu et al. 2009). We list these parameters in Table 3.1.

3.3.3 Integration Bounds

In theory the integral over k in Eq. 3.6 - 3.9 must be evaluated from k = 0 to k = , ∞ which is not practical. As such, we must pick limits for the integral. The lower limit is dictated by the largest-scale mode that the ﬁelds are sensitive to. For the simulated data, 1 this scale is the side-length of the simulation box (Lbox = 1 h− Gpc), which corresponds 1 to kmin = 2π/Lbox = 0.006 h Mpc− . The 6dFGS data may be inﬂuenced by modes larger than the survey volume, so there is no explicit restriction on the value of the 3.3. Theory and Methodology 79

1 minimum wavenumber. We set kmin = 0.0025 h Mpc− , similar to values used in the peculiar velocity auto-covariance studies conducted by Johnson et al. (2014) and Macaulay et al. (2012). The upper limit is dictated by our ability to model the small, non-linear 1 scales. Johnson et al. (2014) found that setting kmax = 0.15 h Mpc− for the logarithmic distance ratio auto-covariance provided a good compromise between accurate recovery of fiducial parameters and constraining power (see figure 8 of that paper). We adopt this value for our logarithmic distance ratio auto-covariance and find that it is also suitable for the cross-covariance (for both simulation and 6dFGS data). For the galaxy overdensity auto-covariance, we find that there is a significant contri- 1 bution to the integral for k > 0.15 h Mpc− . We therefore add a second integral, which we refer to as the additional integral. This integral has a fixed galaxy bias and acts to increase the value of the covariance without needing to provide an advanced model of the bias on non-linear scales. The integral for the galaxy overdensity auto-covariance then becomes

2 Z kmax 2 Z kadd (bﬁtσ8) (baddσ8) Cδ0gδg (xs, xt) = f(k, xs, xt)dk + f(k, xs, xt)dk, (3.12) 2π kmin 2π kmax

1 where f(k) represents the integrand in Eq. 3.6. We choose kadd = 1.0 h Mpc− as this is 2 where the gridding window function Γδg (k) becomes close to zero. In Section 3.4.2 we show that the additional integral is required for the simulation data and determine the best-ﬁtting value for badd. We repeat this analysis for 6dFGS in Section 3.5.2 and show how the value of badd aﬀects the constraint on fσ8.

3.3.4 Evaluating the Likelihood Function

For our measured data vector and modelled covariance matrix, we wish to constrain the free parameters fσ8, bσ8 and σv; this can be achieved by evaluating the likelihood equation (Eq. 2.2) for a gridded parameter space (we implement a Markov chain Monte Carlo sampling method in the next chapter). We invert the covariance matrix by applying the 1 GNU Science Library Cholesky solver to the equation CΥ = ∆, which yields Υ = C− ∆. The exponent of the likelihood equation is obtained by multiplying this by 1 ∆T . We − 2 analyse our results using the publicly available ChainConsumer Python package (Hinton, 2016), which provides parameter constraints for gridded likelihood evaluations. Since this is the ﬁrst use of this approach to constrain cosmological parameters, we wish to examine how including the cross-covariance aﬀects our constraints compared to only using the auto-covariance pieces. To do this, we consider two tests: 80 Chapter 3. The Density-Velocity Cross-Correlation in 6dFGS

1. Assume the galaxy overdensity and logarithmic distance ratio ﬁelds are uncorrelated, C C 0 setting δ0 gη = ηδ0 g = .

C C 2. Include the cross-correlation, calculating δ0 gη and ηδ0 g .

We note that the first test is not a physically accurate model, as the two fields will be correlated. The test is still useful because fσ8 is only constrained by the logarithmic distance ratio auto-covariance, and bσ8 is only constrained by the galaxy overdensity auto- covariance, making the constraints from the first test the same as those that would come from working with each auto-covariance independently. This differs from the complete 2 model, which also constrains the product fbσ8. Consequently, comparing the constraints from these tests allows us to examine how the additional information provided by the cross-correlation affects the cosmological parameter constraints. Forecasts for cosmological surveys, which use the Fisher matrix formalism (discussed in more depth in Chapter 5), have shown that constraints on cosmological parameters improve when the cross-covariance is included (e.g. Howlett et al., 2016). While the constraint on the ratio of these parameters is improved due to the correlated sample variance of the fields, the individual parameters should also see improved constraints from needing to satisfy the extra relationship imposed by the cross-covariance. Abramo & Leonard (2013) demonstrated this by diagonalising the Fisher matrix for correlated data. Consequently, we expect to see improved constraints on the growth rate of structure when utilising the cross-covariance.

3.4 Simulation Results

3.4.1 Parameter Constraints

In this section we use data from an N-body simulation to validate our method and examine how the addition of the cross-covariance affects the analysis (discussed in Section 3.3.4). We constructed an approximate realisation of the 6-degree Field Galaxy Survey from GiggleZ as discussed in Section 3.2.2. Since the simulation data is in real space, all references to galaxy bias within this section refer to the real-space galaxy bias br. We expect to recover the standard model prediction of the growth rate of structure, 0.55 which is calculated from f(z = 0) = Ωm(z = 0) . We obtain the best-fitting value for the additional bias badd by fitting for this along with the standard bias bfit using only the galaxy overdensity data (see Section 3.4.2). We find bfit = 1.59 and badd = 1.61 for our GiggleZ galaxy overdensity sample. For the fiducial σ8 value of GiggleZ (listed 3.4. Simulation Results 81

0.48 0.48 8 8 0.40 0.40 fσ fσ 0.32 0.32

1.6 1.6

8 1.4 8 1.4 σ σ ﬁt ﬁt

b 1.2 b 1.2

1.0 1.0

30 45 60 75 0.32 0.40 0.48 1.0 1.2 1.4 1.6 30 45 60 75 0.32 0.40 0.48 1.0 1.2 1.4 1.6 σv fσ8 bﬁtσ8 σv fσ8 bﬁtσ8 (a) Cross-covariance excluded (b) Cross-covariance included

1 Figure 3.3 The posteriors of our free parameters, σv (km s− ), fσ8 and bfitσ8, with shaded 68% credible intervals, when fitting to the GiggleZ sample. The black dashed lines show the expected fσ8 and bfitσ8 values assuming the fiducial GiggleZ cosmology (Table 3.1) and an additional bias of badd = 1.60. Our analysis pipeline successfully recovers these values. in Table 3.1), we expect to recover fσ8 = 0.398 and bfitσ8 = 1.29. We do not set an expected value for σv, as this only serves as a nuisance parameter in this test. Applying 1 a gridding length of L = 20 h− Mpc for the galaxy overdensity and logarithmic distance ratio data is sufficient to recover the input parameters. We also test kmax values of 0.10 1 and 0.20 h Mpc− , and recover the input parameters at the 2σ level in both cases. We note that we present a thorough systematic analysis for our method in the next chapter.

C The posteriors for our free parameters when assuming no cross-covariance ( δ0 gη = C 0 ηδ0 g = ) are shown in Figure 3.3a, and the posteriors when including the cross-covariance are shown in Figure 3.3b. The maximum likelihood and median (with 68% credible intervals) are given in Table 3.2. We obtain a 7% measurement of fσ8 and a 5% measurement 2 of bﬁtσ8. Both tests recover the input parameters at the 1σ level with a reasonable χ /dof, validating our analysis pipeline.

The results of our GiggleZ test provide useful insight into our method. When setting the cross-covariance to zero, we would expect the bfitσ8-fσ8 contour to be uncorrelated, C C since δ0 gδg only constrains bfitσ8 and ηη0 only constrains fσ8. We calculated the Pearson correlation coefficient from our gridded likelihood result, finding ρ = 6.3 10 9, indicat- − × − ing an almost non-existent correlation. This is represented visually in Figure 3.3a, since 82 Chapter 3. The Density-Velocity Cross-Correlation in 6dFGS

Table 3.2 Maximum likelihood (ML) and median values (with 68% credible intervals) of our free parameters for the fit to the GiggleZ sample. We include the χ2 value and χ2/dof statistic for the maximum likelihood. No cross-covariance Cross-covariance ML Median ML Median 1 +5.8 σv (km s− ) 50.0 47.6 6.0 50.0 48.6 6.0 +0.027 − fσ8 0.390 0.399 0.026 0.390 0.395 0.026 +0− .077 +0.068 bfitσ8 1.3 1.303 0.075 1.275 1.279 0.066 − − χ2 2480.52 - 2484.09 - χ2/dof 0.99 - 0.99 - the contour forms an ellipse that is aligned with both axes. Consequently, if a correlation exists between the galaxy overdensity and logarithmic distance ratio fields, then including the cross-correlation should change the tilt of the ellipse. We see in Figure 3.3b that this is the case: the contour has become correlated and exhibits a positive slope. This highlights that the data are intrinsically correlated and that the cross-covariance has the power to constrain the relationship between bfitσ8 and fσ8. Here, we find a Pearson correlation coefficient of ρ = 0.48, indicating a moderate correlation, which is consistent with our visual interpretation. Multiple works have indicated that we expect to see an improvement in constraints when including cross-covariances (see Section 1.4.1). We find that this is the case for our correlated parameters, bfitσ8 and fσ8, although the improvement is greater for the galaxy bias. This may be in part because we do not include any observational uncertainty for the logarithmic distance ratio auto-covariance in order to test the recovery of the fiducial model more precisely. Coupling this with the fact that the number density for our logarithmic distance ratio sample (n = 1.0 10 2 ( h 1 Mpc) 3) is higher than that for η × − − − the galaxy overdensity sample (n = 1.7 10 4 ( h 1 Mpc) 3), we can also explain why g × − − − the fractional uncertainty is lower for fσ8 than for bσ8. If a parameter is already so well determined from having a high number density and low uncertainties, the cross-covariance may contribute less information. We expect this to change when working with the 6dFGS data.

3.4.2 Additional Bias

In this section, we show that the GiggleZ data requires the additional integral discussed in Section 3.3.3 and determine the best-ﬁtting value for badd for our galaxy overdensity 3.4. Simulation Results 83

2.0

1.6 8 σ ﬁt

b 1.2

0.8

0.6 1.2 1.8 baddσ8

Figure 3.4 68, 95 and 99% credible regions of bfitσ8 and baddσ8 when fitting the GiggleZ galaxy overdensity sample. The black dashed line represents the bfitσ8 we expect to recover given the mass range of our GiggleZ overdensity sample (see table 1 in Koda et al., 2014).

sample. We do this by evaluating the likelihood using only Cδgδg (φ):

1 1 δ T C φ 1δ δgδg = q exp g δ0 gδg ( )− g (3.13) L Nδ −2 (2π) g C0 (φ) | δgδg | where φ = (bﬁtσ8, baddσ8). We allow the baddσ8 parameter to vary over [0,2], where a value of 0 indicates that the additional integral is not required by the data when constructing the covariance. The results are given in Figure 3.4.

2 We find median values of bfitσ8 = 1.29 and baddσ8 = 1.306, with a χ /dof of 1.01. 2 badd = 0 is strongly disfavoured, with ∆χ = 241.27. We take this as sufficient evidence for including the additional integral in our analysis. The case where badd = bfit is within 1σ of our median, which might suggest that the covariance needs to be fitted beyond our 1 chosen kmax of 0.15 h Mpc− . However, we do not wish to include non-linear information in our likelihood analysis, so fix the value of badd rather than extending the value of kmax. Our fitted bias is consistent with the power spectrum fits to GiggleZ data by Koda et al.

(2014), which suggest a bias value of bσ8 = 1.2 for the mass range of our sample. 84 Chapter 3. The Density-Velocity Cross-Correlation in 6dFGS

Table 3.3 Maximum likelihood (ML) and median values (with 68% credible intervals) of our free parameters for the 6dFGS sample. We include the χ2 value and χ2/dof statistic for the maximum likelihood. No cross-covariance Cross-covariance ML Median ML Median 1 +63 +57 σv (km s− ) 175 174 74 225 210 63 − − +0.087 +0.067 fσ8 0.425 0.461 0.079 0.400 0.424 0.064 − − +0.087 +0.062 β 0.350 0.380 0.075 0.325 0.341 0.058 − − χ2 3835.34 - 3832.49 - χ2/dof 0.96 - 0.96 -

3.5 Data Results

3.5.1 Parameter Constraints

We now move to using the (fσ8, β, σv) parametrization discussed in Section 3.3.1. Our fiducial cosmology is the Planck 2015 base-set of parameters for ΛCDM, so we compare our result to the corresponding standard model prediction of fσ8 = 0.446. The observed data is in redshift space, so all references to the galaxy bias within this section refer to the redshift-space bias, bs. The best-fitting biases for our galaxy overdensity sample are bfitσ8 = 1.379 and baddσ8 = 1.50 (see Section 3.5.2), giving an expected β value of β = 0.364 from Eq. 3.11. We repeat the same test as performed for the simulated data, comparing the constraints with and without the cross-covariance. We do not expect the tilt of the fσ8-β contour to change appreciably, but do expect the uncertainty in both parameters to reduce. This is because the gradient of the fσ8-β contour is the galaxy bias, which we do not expect to be significantly affected by the inclusion or exclusion of the cross-covariance. C The posteriors for our free parameters when assuming no cross-covariance ( δ0 gη = C 0 ηδ0 g = ) are shown in Figure 3.5a, and the posteriors when including the cross-covariance are shown in Figure 3.5b. The maximum likelihood and median (with 68% credible intervals) are given in Table 3.3. +0.067 When including the cross-covariance, we measure fσ8 = 0.424 0.064 which is consis- − tent at the 1σ level with the prediction from ΛCDM using the Planck 2015 cosmological parameters. The fractional uncertainty in fσ8 is 18% when the cross-covariance is ignored +0.062 and drops to 15% when the cross-covariance is included. We also measure β = 0.341 0.058, − 3.5. Data Results 85

0.8 0.8

0.6 0.6 8 8

fσ 0.4 fσ 0.4

0.2 0.2 0.8 0.8

0.6 0.6

β 0.4 β 0.4

0.2 0.2

150 300 450 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 150 300 450 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 σv fσ8 β σv fσ8 β (a) Cross-covariance excluded (b) Cross-covariance included

1 Figure 3.5 The posteriors of our free parameters, σv (km s− ), fσ8 and β, with shaded 68% credible intervals, when ﬁtting to the 6dFGS sample. For comparison, the dashed lines show the fσ8 and β values when assuming the ﬁducial Planck 2015 cosmology (Table 3.1) and an additional bias of badd = 1.50.

+57 1 which is consistent at the 1σ level with our expectation, and σv = 210 63 km s− , which − 1 is consistent with the literature (σv usually takes a value between 100-300 km s− ). The χ2/dof value shows that the data is a good fit to our model. We see an improvement in the constraints on both fσ8 and β when including the cross-covariance, consistent with what we saw from the simulated data. The effect is visually evident in Figure 3.5, where the contour in β-fσ8 space shrinks considerably after adding the cross-covariance. We calculated the Pearson correlation coefficient between β and fσ8 from our gridded likelihood results, finding ρ = 0.97 when excluding the cross-covariance, and ρ = 0.96 when including it. Both cases exhibit strong correlation, which is consistent with the fact that

β is directly proportional to fσ8 by deﬁnition.

When the cross-covariance is introduced, the best-ﬁtting values of fσ8 and β both 2 decrease, and σv increases, while the χ statistic slightly decreases. σv only contributes to the diagonal elements of the logarithmic distance ratio auto-covariance, so any change in this parameter when introducing the cross-covariance will come from its coupling with fσ8. If including the cross-covariance lowers fσ8, then σv will increase to compensate for the change in the auto-covariance values along the diagonal. 86 Chapter 3. The Density-Velocity Cross-Correlation in 6dFGS

1.65

1.50 8 σ ﬁt

b 1.35

1.20

1.05

0.5 1.0 1.5 2.0 2.5 baddσ8

Figure 3.6 68, 95 and 99% credible regions of bﬁtσ8 and baddσ8 for the 6dFGS galaxy overdensity sample.

3.5.2 Additional Bias

In Section 3.3.3 we discussed the additional integral, which increases the value of the galaxy overdensity auto-covariance without directly ﬁtting for the galaxy bias on small scales where non-linear modelling is unreliable. To determine the best-ﬁtting bias for both integrals, we follow the same procedure as for the simulation data (see Section 3.4.2). The results are given in Figure 3.6.

2 We ﬁnd median values of bﬁtσ8 = 1.38 and baddσ8 = 1.50, with a χ /dof of 1.10. 2 badd = 0 is strongly disfavoured, with ∆χ = 3206.79. As with the simulation results, we

find that the case where badd = bfit is allowed by the data. Again, since we do not wish to include non-linear information, we fix badd at the median value and do not change kmax.

It is clear from Figure 3.6 that changing badd has little eﬀect on the value of bﬁt.

However, since both bfit and f appear in the cross-covariance, it is also important to examine how the value of badd affects our constraints on fσ8. We chose three different badd values (1.175, 1.675, 2.375) and ran the full likelihood analysis (using Eq. 2.2) for each. The fiducial power spectra for these runs were generated using the Planck 2015 cosmological parameters (see Section 3.3.2). We also investigated how the choice of cosmological parameters influences our results 3.5. Data Results 87

0.500

0.475

0.450

8 0.425 fσ 0.400

0.375

0.350

0.325

.675 .175 .675 .375 = 1 = 1 = 1 = 2 b add b add b add b add

Planck, Planck, Planck, WMAP5,

Figure 3.7 Maximum likelihood (solid bar) and 68% credible interval (shaded region) of fσ8 for the 6dFGS sample including the cross-covariance modelling. Results are shown for different values of the additional galaxy bias as well as using both the WMAP five-year and Planck 2015 fiducial cosmologies. by adding an additional likelihood run that used the WMAP five-year parameters to generate the fiducial power spectra. We display the 68% credible intervals around the maximum likelihood for fσ8 from the four runs in Figure 3.7.

When including the additional integral, we find that the value of badd does not significantly influence the constraints on fσ8. The maximum likelihood values are all consistent at the 1σ level, and the difference between the maximum likelihood value of fσ8 of the highest and lowest b is 0.04, which corresponds to around one-third of the statistical add ∼ uncertainty. Using the cosmological parameters from WMAP to construct the power spectra gives a lower fσ8 than using those from Planck for the same additional bias. However, we note that the difference is small, 0.02 between the maximum likelihood values. Given ∼ that the fractional uncertainties in our measurements of fσ8 are around 15%, the small 88 Chapter 3. The Density-Velocity Cross-Correlation in 6dFGS contribution of 3% from the choice of cosmological parameters is subdominant. However, ∼ analyses using data from future surveys will need to be aware that the constraint on fσ8 is influenced by the choice of cosmological parameter values used to generate the fiducial power spectra.

3.5.3 Direct Evidence of the Cross-Covariance

An interesting visualization of our results is to directly plot the analytic cross-covariance for the median values of our free parameters as a function of the separation of grid cells, which can then be compared to an estimate of the cross-covariance present in the 6dFGS data. We represent the estimated cross-covariance as

Σ = δ η δ η . (3.14) δgη h g i − h gih i

In principle, the average is performed for each pair of cells over many realisations of the data, which would produce a full covariance matrix that could be directly compared to our model. Given that we only have a single realisation of the 6dFGS sample, we instead perform the average for pairs with similar orientation within the survey, speciﬁcally their separation r, and their angle to the line of sight γ (see Figure 2.1). We split the range 1 < cos(γ) < 1 into four equal bins, as this prevents the signals from averaging out. The − 1 1 separation of each pair is assigned to bins of width 20 h− Mpc beginning at 20 h− Mpc 1 and ending at 240 h− Mpc. Performing the average for N pairs in each bin produces the covariance as a function of separation for each angular bin and we estimate the standard deviation of this quantity as

q 1 2 2 2 σδgη = δg η δgη . (3.15) √N h i − h i

We average the model covariance matrix for the median values of our free parameters using the same scheme. This can then be directly compared to our estimation from the data, which is shown in Figure 3.8. We note that the standard deviations will be underestimated since there will be an additional contribution from sample variance that we have not estimated. However, the agreement between our model and the estimated covariance of our data is reassuring. There is good visual evidence from the estimated amplitude that there is a non-zero cross- correlation between the peculiar velocity ﬁeld and galaxy overdensity ﬁeld on separations up to 50 h 1 Mpc, and that we are able to successfully model this. ∼ − 3.5. Data Results 89

0.02 0.02 1 < cos γ < 0.5 0.5 < cos γ < 0 − − − 0.01 0.01 i i

δη 0.00 δη 0.00 h h

0.01 0.01 − −

0 < cos γ < 0.5 0.5 < cos γ < 1 0.01 0.01 i i δη δη

h 0.00 h 0.00

0.01 0.01 − −

50 100 150 200 50 100 150 200 1 1 r (h− Mpc) r (h− Mpc)

Figure 3.8 Estimation of the 6dFGS cross-covariance δη , where the shaded contour rep- h i resents the covariance plus one standard deviation above and below. The black line shows the prediction from our 6dFGS model covariance for the median values obtained when including the cross-covariance.

This result is corroborated by Nusser (2017), who presented a direct measurement of the cross-correlation using peculiar velocities from the Cosmicﬂows-3 catalogue (Tully et al., 2016) and galaxy positions from the 2MASS Redshift Survey (Huchra et al., 2011). They also found evidence for a non-zero cross-correlation on scales up to 50 h 1 Mpc. ∼ −

3.5.4 Comparison to Existing Literature

Multi-tracer Approaches

Previous theoretical studies have shown that using multiple tracers of the underlying matter overdensity field can improve constraints on cosmological parameters (Seljak, 2009; McDonald & Seljak, 2009; Bernstein & Cai, 2011; Gil-Mar´ın et al., 2010; Abramo & Leonard, 2013, see Section 1.4.1 for a full review). In the context of this thesis, modelling the cross-covariance is an example of a multi-tracer analysis as we utilise the shared information between two tracers of the matter overdensity field. Blake et al. (2013) presented the first multi-tracer redshift-space distortion (RSD) analysis of observational data, using two galaxy samples with different biases from the Galaxy And Mass Assembly (GAMA) survey. They found a 10-20% improvement in their measurement of fσ8 when utilising the cross-covariance between their two galaxy tracers. We see behaviour that is consis- 90 Chapter 3. The Density-Velocity Cross-Correlation in 6dFGS tent with the expected outcomes from theoretical work and with the results presented by Blake et al. (2013): our constraints on fσ8 and β both improve when including the cross-covariance term. The fractional improvement on our measurement of fσ8 is 20%, and the fractional improvement on β is 26%.

Forecasts for the 6dFGS Cross-Covariance Analysis

Forecasts of parameter constraints have been performed for multi-tracer analyses that utilise galaxy overdensity and peculiar velocity data. Koda et al. (2014) and Howlett et al. (2016) presented forecasts for 6dFGS that can be compared to our results.

Koda et al. (2014) performed a Fisher matrix forecast for fσ8 and β given the properties of the peculiar velocity sample, 6dFGSv. Constraining the parameters out to 1 k = 0.1 h Mpc− , they found an expected fractional uncertainty of 25% on fσ8 when only using the peculiar velocity auto-covariance, falling to an expected fractional uncertainty of 15% when including the cross-covariance and galaxy overdensity auto-covariance. The complete covariance also gives an expected fractional uncertainty of 16% on β. Howlett et al. (2016) used the same modelling and formalism as Koda et al. (2014) but considered multiple extensions. For example, they modelled the full 6dFGSz and 6dFGSv samples. They found similar results to Koda et al. (2014), with the peculiar velocity auto-covariance providing an expected fractional uncertainty of 25.1% on fσ8, and expected fractional uncertainties of 11.2% on fσ8 and 12.3% on β when including the cross- covariance and galaxy overdensity auto-covariance. The tightening of their constraints relative to Koda et al. (2014) can be attributed to the increased number of galaxies from modelling the full 6dFGSz sample. Our analysis contains some differences in terms of modelling, sample and scales fitted, which we consider more closely in Chapter 5. However, we find that our results are in line with these forecasts. We obtain fractional uncertainties of 15% on fσ8 and 18% on β when using the cross-covariance, and 18% on fσ8 when only using the logarithmic distance ratio auto-covariance. The fractional uncertainties when including the cross-covariance are very similar to those from Koda et al. (2014) although larger than those from Howlett et al. (2016); this is likely due to their significantly larger galaxy overdensity sample. Our uncertainty from the logarithmic distance ratio auto-covariance is somewhat smaller than forecasted by both studies, in agreement with the results of Johnson et al. (2014) (see below), which we attribute to slight differences in assumptions. 3.5. Data Results 91

6dFGS Velocity and Redshift-Space Distortion Results

Johnson et al. (2014) constrained fσ8 using only the logarithmic distance ratio auto- +0.079 covariance. When using the 6dFGSv sample, they found fσ8 = 0.428 0.068, which has a − fractional uncertainty of 17%. While the fractional uncertainties are similar, our fσ8 value is higher when only using the logarithmic distance ratio auto-covariance. However, we find +0.067 a similar value of fσ8 = 0.424 0.064 once cross-covariance is included. This may be due − to subtle differences in modelling and approach between our two studies. Huterer et al. (2017) also presented an analysis of the peculiar velocity auto-covariance for 6dFGSv. We note that they applied their own Fundamental Plane model to the data. Their constraint +0.067 of fσ8 = 0.481 0.064 at an effective redshift of z=0.02 is consistent with our result and − that from Johnson et al. (2014). We can also compare our results to the RSD analysis for 6dFGS, performed by Beutler et al. (2012), who found fσ = 0.423 0.055 at an effective redshift of z = 0.067, and 8 β = 0.373 0.054, with fractional uncertainties of 13% and 14% respectively. Both of our constraints are consistent with this work at the 1σ level. We do not expect to perfectly recover β, as our use of a volume-limited sample will preference higher mass halos compared to the sample used by Beutler et al. (2012), which would increase our galaxy bias relative to theirs. Finally, we compare our result to measurements of the growth rate from 6dFGS by Achitouv et al. (2016). They present an RSD analysis of the galaxy-galaxy and galaxy- void correlation functions, utilising realistic mocks to estimate the uncertainties in their results. For the galaxy-galaxy correlation, their result of fσ = 0.42 0.06 is entirely 8 consistent with that from Beutler et al. (2012). They also find fσ = 0.39 0.11 when 8 analysing the void-galaxy correlation, which has a fractional uncertainty of 28%. Again, we are consistent at the 1σ level with both of these results. See Figure 3.9 for a visual comparison of the previous 6dFGS fσ8 measurements with the measurement from this work.

Density-Velocity Comparison Approaches

The practice of studying the relationship between the observed galaxy overdensity and peculiar velocity ﬁelds dates back to the 1990s and has revolved around using gravitational instability to link them (see Section 1.4.2 for a review). We compare our results to fσ8 constraints from several representative works: Pike & Hudson (2005), Davis et al. (2011) and Carrick et al. (2015). Pike & Hudson (2005) and Carrick et al. (2015) used similar methods where they utilised VELMOD (a velocity-velocity comparison approach developed 92 Chapter 3. The Density-Velocity Cross-Correlation in 6dFGS

0.55

0.50

0.45 8

fσ 0.40

0.35

0.30

This work

Davis et al. (2011)

Beutler et al. (2012) Carrick et al. (2015) Huterer et al. (2017) Johnson et al. (2014) Pike & Hudson (2005)

Achitouv et al.Achitouv (2017) - et gg al. (2017) - vg

Figure 3.9 Median values (solid bar) and 68% credible interval (shaded region) of fσ8 for this work. Other works from 6dFGS are shown in purple, and velocity-velocity comparisons are shown in pink. For Achitouv et al. (2016), gg corresponds the galaxy-galaxy result, and vg corresponds to the void-galaxy result. by Willick et al. 1997) with some extensions. Davis et al. (2011) expanded both the modelled peculiar velocity field and the observed field in terms of a set of basis functions and then compared the expansion coefficients between the fields. All three analyses used similar data sets, relying on redshifts collected from the 2-Micron All-Sky Survey (with various extensions), and peculiar velocities from the Spiral Field I-Band survey (again with extensions).

We do not compare β values, as the galaxy overdensity samples will be diﬀerent between

6dFGS and other surveys, which aﬀects the galaxy bias. We quote the normalised fσ8 results, which each survey produces by making an estimate of the galaxy bias for their sample. Pike & Hudson (2005) found fσ = 0.44 0.06, Davis et al. (2011) found fσ = 8 8 0.32 0.04, and Carrick et al. (2015) found fσ = 0.427 0.027. See Figure 3.9 for a 8 visual comparison of these results with the result from this work. 3.6. Summary 93

Our constraint on fσ8 when including the cross-covariance is consistent at the 1σ level with Pike & Hudson (2005) and Carrick et al. (2015), and only just so with Davis et al. (2011), with a similar fractional uncertainty to Pike & Hudson (2005). The most obvious diﬀerence between these approaches and ours is that we model the joint statistics, whereas these approaches make a model from one observation and compare it to the other. They also use more advanced modelling than our current approach: for example, Carrick et al. (2015) implemented a weighting scheme to model luminosity-dependent bias, whereas our linear bias approach is quite simplistic. However, an advantage of our approach is that the modelling can be easily extended by changing the power spectra in the covariance model, and there are multiple improvements that can be explored here (which are discussed in Section 6.2). Additionally, the covariance matrix framework makes propagation of errors straightforward, whereas this can be diﬃcult in the comparison approach.

3.6 Summary

We have presented the first joint statistical analysis of the observed galaxy overdensity and peculiar velocity fields in which the cosmological physics is modelled in the covariance, extending the work of Johnson et al. (2014). We also found evidence for a non-zero cross- covariance when testing against simulations, as well when we applied our method to data from the 6-degree Field Galaxy Survey. For logarithmic distance ratio and galaxy overdensity measurements drawn from 6dFGS, +0.067 we found the normalised growth rate of structure at redshift z = 0 to be fσ8 = 0.424 0.064, − which is consistent with the ΛCDM prediction for the Planck 2015 cosmological parameters. We also constrained the redshift-space distortion parameter for our sample, finding +0.062 β = 0.341 0.058. Our constraint on fσ8 improves on that from Johnson et al. (2014) who − quoted a fractional uncertainty of 17%, where we found a fractional uncertainty of 15%. This improvement is entirely consistent with the current theory of multi-tracer analyses: including the cross-covariance will improve constraints on the model’s cosmological parameters. Our results are also consistent with the redshift-space analyses of 6dFGS by Beutler et al. (2012) and Achitouv et al. (2016), as well as previous forecasts for 6dFGS from Koda et al. (2014) and Howlett et al. (2016). Finally, we also saw consistency with alternative methods of analysing the relationship between the galaxy overdensity field and the peculiar velocity field. This is the first maximum likelihood fit to the galaxy overdensity and logarithmic distance ratio auto- and cross-covariances, providing a new way of looking at multi-tracer approaches. Given the strong existing theoretical work in this area, it is hugely promising 94 Chapter 3. The Density-Velocity Cross-Correlation in 6dFGS to see concrete evidence of the statistical improvements that are predicted by Fisher matrix forecasting. Importantly, the flexibility offered by modelling the covariance acts as solid insurance that this method will continue to expand and improve, providing increasingly precise and accurate measurements of cosmological parameters as a function of scale and cosmological model. Refining this approach is particularly important, as several large peculiar velocity surveys will come online in the next few years. Taipan is forecast to obtain a fractional uncertainty on the growth rate of structure of at least 5% (da Cunha et al., 2017). We clearly still have much to gain from large-scale structure data, and taking advantage of the cross-correlation between different tracers is a strong step in improving our knowledge and understanding of the behaviour of our Universe. 4 Joint Growth Rate Measurements from Peculiar Velocities and Redshift-Space Distortions

4.1 Overview

In the previous chapter, we presented a new method for utilising the cross-correlation of galaxy overdensities and peculiar velocities to improve constraints on the growth rate of structure fσ8 and the redshift-space distortion parameter β. We clearly demonstrated that using the cross-covariance in addition to the galaxy overdensity and peculiar velocity auto-covariances provided better constraints than only using the auto-covariances, which corresponds to treating the two probes as independent. In this chapter, we continue this work by adding information from redshift-space distortions (RSD), which are the imprints of peculiar velocities on the galaxy overdensity field (see Section 1.2.4 for an introduction to RSD). In doing this, we construct a self-consistent framework for combining direct peculiar velocity and RSD measurements, which leads to improved growth rate of structure constraints. We begin by validating our improved model on a suite of mock catalogues (which have been significantly upgraded compared to those used in Chapter 3) for the 6-degree Field Galaxy Survey (6dFGS) before applying the model to the 6dFGS data. In addition to the improved model, we also conduct a rigorous investigation of the sensitivity of our fσ8 constraints to various model parameters and quantify this by estimating the systematic error for our final constraints. We begin by introducing the mock catalogues and the data sample (which has changed slightly compared to that from the previous chapter) in Section 4.2 and then discuss the updated theory and methodology in Section 4.3 (where most of the required theory has been developed in Chapter 2). Results from the mocks are presented in Section 4.4, followed by results from the 6dFGS data in Section 4.5. We conclude with a summary of

95 96 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD the study in Section 4.6.

4.2 Data and Simulations

As with Chapter 3, we apply our model to the 6-degree Field Galaxy Survey (6dFGS). We refer back to Section 3.2.1 for a summary of the survey and discuss the changes we have made to the sample for this analysis below.

4.2.1 6dFGS

Given that measuring the statistical imprint of RSD in the two-point clustering of galaxies requires a significant volume to reduce sample variance, we use a larger volume for our overdensity sample than in the previous analysis. We extend the maximum redshift from z = 0.057 (the same redshift as the peculiar velocity sample) to z = 0.1, giving a total of 70,467 galaxies. We keep the peculiar velocity sample, using all 8885 galaxies out to redshift z = 0.057. The first consequence of extending the redshift of the overdensity sample is that we are no longer able to generate a volume-limited sample like we did in Chapter 3. This is because the required absolute magnitude cut results in too few galaxies for our analysis, causing our measurement to be seriously limited by Poisson noise. In the previous analysis, we were able to use a absolute magnitude cut of M = 23.37, which gave us a sample − of 20,796 galaxies. At redshift z = 0.1, the necessary magnitude cut for a volume-limited sample is M = 24.53, which only gives 8539 galaxies. − Given that we’re no longer working with a volume-limited sample, the galaxy bias evolves with redshift, since higher luminosity galaxies are more biased. While our overdensity sample can still be treated as having a single effective bias, as it was in Beutler et al. (2012), there is a complication when it comes to the cross-correlation between overdensities and peculiar velocities. This is because the amplitude of the cross-correlation is proportional to the linear bias factor, but the value of the bias will be determined by the overdensities that directly influence the peculiar velocities, rather than the entire galaxy overdensity sample. Consequently, we naturally expect that the cross-correlation of peculiar velocities with a lower-bias overdensity sample will not be as strong as that with a higher-bias sample. Given that the peculiar velocity sample is limited to a lower redshift, we anticipate that the effective bias probed by the cross-correlation will be lower than that probed by the galaxy overdensity auto-correlation. We put forward an adjustment to the model to account for this in Section 4.3.3 and show evidence for the cross-correlation and 4.2. Data and Simulations 97 galaxy overdensity auto-correlation having different effective bias values in Section 4.4.3.

The second consequence is that the volume of our galaxy overdensity sample is now 1 signiﬁcantly larger. Gridding at the same length scale (Lδg = 20 h− Mpc) produces N = 5466 grid cells, resulting in a galaxy overdensity auto-covariance that has 5466 5466 δg × elements. Given that the time taken to solve our likelihood equation is largely dependent on inverting the covariance, we ﬁnd that using a covariance matrix of this size is infeasible. 1 To reduce it to a more manageable size, we increase the gridding scale to 30 h− Mpc, resulting in Nδg = 1633 grid cells. We note that this is still a larger number of cells than our previous analysis, which had Nδg = 1036 cells. We also increased the gridding scale for 1 1 the peculiar velocity sample from 10 h− Mpc to 20 h− Mpc, resulting in Nη = 908 grid cells (our previous analysis had Nη = 2977). This results in the total covariance having 1944 1944 elements, which is just under one quarter the size of the covariance in our × previous analysis. This reduction in size is advantageous as we require more likelihood evaluations for the Markov chain Monte Carlo method we implement in this analysis (see Section 4.3.4) compared to the gridded likelihood evaluation method we used previously (see Section 3.3.4).

Finally, the calculation of the expected number of galaxies in each cell according to the survey selection function Nexp (which is required to estimate the galaxy overdensity) has also been affected by the more complex number density function for this sample. In the previous analysis, the volume-limited sample had a fairly flat number density function n(z) and we fitted a functional form using a second-order polynomial, from which we calculated

Nexp (see Section 3.2.1). During this analysis, more sophisticated mocks of 6dFGS became available as part of the BAO reconstruction work by Carter et al. (2018). We discuss these mocks in more detail below, but highlight that we were able to use 600 random catalogues to determine the Nexp values for our analysis. The random catalogues were constructed to match the angular selection and redshift distribution of the simulations, and diﬀer from the mock data catalogues in that they do not contain large-scale structure. This means that there is less sample variance between the catalogues, producing a more reliable estimate of Nexp. When estimating Nexp for the 6dFGS data, the random catalogues were determined from the 6dFGS luminosity function (Jones et al., 2006), including the local survey completeness variations. Speciﬁcally, we gridded each random catalogue at the same resolution as the data and took Nexp to be the average number of galaxies in a given grid cell over all catalogues. The random catalogues also have many more galaxies than the mock data catalogues, so we normalised the calculated Nexp values such that the sum over all grid cells produces the same number of galaxies as the data. 98 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

4.2.2 Simulation

For this study, we used the random and data mock catalogues developed and used by Carter et al. (2018), which provide a more accurate modelling of the halo occupation of 6dFGS than the mock catalogue we created in Chapter 3. Each mock is drawn from a unique COmoving Lagrangian Acceleration (COLA; Tassev et al., 2013) simulation, where 1 3 each simulation has a box-length of 1.2 h− Gpc and contains (1728) particles with a mass 10 1 resolution of 2.8 10 h− M . The advantage in using the COLA method is that it is × faster than standard N-body techniques; the speed-gain comes from sacrificing accuracy on small scales, while retaining accuracy on large scales by exactly solving the linear perturbation theory equations. The fiducial cosmology used to generate the simulations is listed as the first column in Table 4.1. The mock catalogues were generated by taking a simulation snapshot at redshift z = 0.1 (close to the effective redshift of 6dFGS) and populating it using a halo occupation distribution (HOD) model. The HOD is informed by the number density function n(z) and projected correlation function wp(rp) of 6dFGS, and allocates both central and satellite galaxies to the N-body haloes. The random catalogues were populated by drawing Monte Carlo samples from the 6dFGS selection function, which accounts for both the angular and redshift distribution of galaxies in the survey. For these reasons, these mocks are a significant advancement on the mock catalogue we used in the previous analysis. We also improve upon our previous analysis by applying our method to ten mock catalogues, which helps us assess the reliability of our method. We take several additional steps to refine the ten mocks. We start by only using central galaxies for both our galaxy overdensity and logarithmic distance ratio sample. As with our previous analysis, our logarithmic distance ratio sample is obtained by taking the 8885 most massive centrals in the 6dFGSv redshift range z 0.057 and converting the given ≤ peculiar velocity into a logarithmic distance ratio (via Eq. 2.74). We introduce a new step to add mock measurement uncertainties to the logarithmic distance ratios: we draw a random offset η from the normal distribution (0, σ = 0.1), where σ represents offset N obs obs the typical level of uncertainty in the 6dFGSv logarithmic distance ratios, modify the true logarithmic distance ratio through ηmodified = ηtrue + ηoffset and set the observed uncertainty to σobs. This makes our mock logarithmic distance ratio sample more realistic in comparison to the mock we used in the previous analysis, where we did not specify mock measurement uncertainties. When calculating the galaxy overdensities, we use a slightly different method for calculating Nexp, which we estimate using the average n(z) function from the random catalogues associated with the mock catalogues. We did not use this 4.3. Theory and Methodology 99 method for the 6dFGS data because its angular selection function and number density function are not explicitly separable, hence the use of random catalogues determined from the luminosity function (see above). Finally, the data mock catalogues are gridded at the 1 1 same length-scales as the 6dFGS data: Lδg = 30 h− Mpc and Lη = 20 h− Mpc.

4.3 Theory and Methodology

The major advance in the modelling for this section is the use of the power spectrum multipoles to model the eﬀect of RSD on both the galaxy overdensity auto-covariance and the cross-covariance. As in Section 3.3, we start by stating the form of the data vector and covariance:     δ Cerr C g δgδg,RSD δ0 gη,RSD ∆ =   , C =   , (4.1) η C Cerr ηδ0 g,RSD ηη,RSD where the diagonal sub-matrices include various errors; these remain unchanged from their description in Eq. 3.2 - 3.5. The sub-matrices are now based on those developed in Section 2.3.4, where we used multipole expansion to compute the anisotropic power spectra. The sub-matrices (without the diagonal error components) are given by

2 Z 2 (bσ8) k X C0 = Pmm(k) K 0 + (4.2) δgδg,RSD 2π2 (σfid)2 δgδg,β ,` 8 `δδ X 1 Pmθ(k)2rgβ Kδgδg,β ,`+ `δδ 2 X 2 2 Pθθ(k)β Kδgδg,β ,` Γδg (k)dk, `δδ 2 Z 2 (κaHfσ8) Du(k, σu) X 2 C0 = Pθθ(k) Kv v ,` Γ (k)dk, (4.3) ηη,RSD 2π2 (σfid)2 p p η 8 `ηη 2 Z κaHfbσ8 kDu(k, σu) X C0 = − Pmθ(k) K 0 + (4.4) δgη,RSD 2π2 (σfid)2 δgvp,β ,` 8 `δη X 1 Pθθ(k)β Kδgvp,β ,` Γδg (k)Γη(k)dk, `δη 2 Z κaHfbσ8 kDu(k, σu) X C0 = Pmθ(k) K 0 + (4.5) ηδg,RSD 2π2 (σfid)2 vpδg,β ,` 8 `ηδ X 1 Pθθ(k)β Kvpδg,β ,` Γη(k)Γδg (k)dk, `ηδ 100 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

Table 4.1 Cosmological parameters for the four cosmologies used in this analysis. The top section shows the 6 base parameters for standard ΛCDM: physical baryon density; physical dark matter density; reduced Hubble constant; scalar spectral index; scalar amplitude 1 1 (with pivot point k0 = 0.002 h Mpc− for the WMAP cosmology and k0 = 0.05 h Mpc− for the mock and Planck cosmologies); and reionization optical depth. The bottom section shows σ8, which is a derived parameter. COLA Mocks Planck 2015 WMAP Planck 2018 2 Ωbh 0.02210 0.02222 0.02273 0.02212 2 Ωch 0.1166 0.1197 0.1099 0.1206 h 0.68 0.6731 0.719 0.6688

ns 0.96 0.9655 0.963 0.9626 A 2.215 10 9 2.195 10 9 2.41 10 9 2.092 10 9 s × − × − × − × − τ 0.09 0.078 0.087 0.0522 ﬁd σ8 0.82 0.8417 0.7931 0.8118

where K are the various integrand matrices for the appropriate multipole modes: `δδ =

(0, 2, 4), `ηη = (0, 2), `δη = `ηδ = (1, 3). The integrand matrices for the galaxy overdensity auto-covariance, logarithmic distance ratio auto-covariance and the cross-covariance are given in Section 2.3.4. We again have the same modifications that appeared in Section 3.3: fid σ8 is the fiducial value of σ8 associated with the cosmology used to generate the power spectra (discussed in Section 2.2.4); κ is the quantity that allows for the conversion of modelled peculiar velocities to observed logarithmic distance ratios (discussed in Section 2.3.1); and the Γ(k) functions are the gridding window functions (discussed in Section 2.3.2).

4.3.1 Generation of the Fiducial Power Spectra

Again, we use CAMB and velMPTbreeze to generate the fiducial power spectra (see Section 2.2.4). As with our previous analysis, we generate power spectra for the simulation tests from the cosmological parameters used to generate the COLA mocks and use the Planck 2015 parameter values (Planck Collaboration et al., 2015c) to generate power spectra when working with the 6dFGS data. To check how our 6dFGS results are affected by the underlying cosmological parameters (see Section 4.5.1), we also use power spectra generated from the median values of the 6-parameter base ΛCDM model for the Wilkinson Microwave Anisotropy Probe (WMAP) five-year data (Komatsu et al., 2009), and those from the Planck 2018 TT+lowE data (Planck Collaboration et al., 2018). The parameter values for all four cosmologies are listed in Table 4.1 4.3. Theory and Methodology 101

4.3.2 Integration Bounds

As in the previous chapter, we must specify the range we integrate over to get the model covariance (see Section 3.3.3). We use the same bounds for the mocks and the data: 1 1 kmin = 0.0025 h Mpc− and a fiducial value of kmax = 0.15 h Mpc− . We test how the value of kmax affects the constraints throughout Section 4.4 while working with the mocks, and how it contributes to the overall systematic error for our constraints from the data in Section 4.5.1. We also keep the additional integral (introduced in Section 3.3.3) as part 1 of our model, specifying kadd = 1.0 h Mpc− . We test whether the additional integral is justified while working with the mocks in Section 4.4.1.

4.3.3 Accounting for Diﬀerent Galaxy Bias Values

As discussed in Section 4.2, the bias of the galaxy overdensity sample will increase with redshift for the magnitude-limited sample considered in this chapter. It is important to account for this effect because the covariance model is a function of the effective bias over the redshift range. Given that our logarithmic distance ratio sample is located at preferentially lower redshifts than the galaxy overdensity sample, we expect that the cross- covariance will have a lower effective bias than the galaxy overdensity auto-covariance. We propose a simple modification that allows for the cross-covariance to have a lower effective bias value than the galaxy overdensity auto-covariance. Until now, the cross- covariance has used the same galaxy bias value as the galaxy overdensity auto-covariance, bσ8. We modify this using a scaling parameter αb, such that in the cross-covariance

bσ α bσ , (4.6) 8 → b 8 giving the overall transformation

C0 α C0 , (4.7) δgη,RSD → b δgη,RSD C0 α C0 , (4.8) ηδg,RSD → b ηδg,RSD

C while δ0 gδg,RSD remains unchanged.

Given that αb modifies the amplitude of the cross-covariance, we note that it is degen- 2 erate with the combined parameter fbσ8. In testing this model modification, we found that we could not freely vary αb along with fσ8 and bσ8 without entirely degrading the constraints on fσ8, even though αb only appears in the cross-covariance term. Conse- quently, we run the model on the mocks for different values of αb in Section 4.4.3 and 102 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD apply the value that best recovers the expected fσ8 to the data in Section 4.5. We discuss future avenues for estimating αb through the galaxy bias-luminosity relation in Section 6.2.

4.3.4 MCMC Sampling

In this analysis, we move from the grid-based likelihood evaluation used in Chapter 3 to a Markov chain Monte Carlo (MCMC) method. We use emcee (Foreman-Mackey et al., 2013), which is a Python implementation of the affine-invariant ensemble sampler for MCMC proposed by Goodman & Weare (2010). We used the OzSTAR supercomputing facility at Swinburne University of Technology, taking advantage of the fact that emcee parallelizes tasks through the mpi4py module. The MCMC chains in our analysis were run with 500 walkers taking 800 steps, which equates to 400,000 samples of our parameter space. We discard the first 150 steps as burn-in and confirm that the chains have converged using the Gelman-Rubin statistic (Gelman & Rubin, 1992). For m chains of length n, the statistic is calculated for each free parameter φ and is defined as

s Vˆ Rˆ = (4.9) W s 1 n 1 m + 1 = − W + B , (4.10) W n mn where Vˆ is the pooled variance, W is the average of each chain’s parameter variance:

m 1 X W = σ2 , (4.11) m φ,i i=1 and B is the variance of the mean parameter estimate between chains:

m n X B = (φ¯ φ¯)2. (4.12) m 1 m − − i=1 If the Gelman-Rubin statistic is close to one, then the chains have converged to the posterior distribution; we use the condition that Rˆ 1 must be less than 0.05 to satisfy − convergence. This convergence test is already implemented as part of the ChainConsumer analysis package (Hinton, 2016), which we use to analyse all of our emcee chains. 4.3. Theory and Methodology 103

4.3.5 Evaluating the Likelihood Function

The likelihood function (Eq. 2.2) is evaluated at each step for each walker. The covariance is effectively inverted by applying the Linear Algebra PACKage (LAPACK) Cholesky 1 solver to the equation CΥ = ∆, which yields Υ = C− ∆. We use the Python implementation of LAPACK available through the SciPy Linear Algebra package. The exponent of the likelihood equation is obtained by multiplying Υ by 1 ∆T . − 2 A major difference in the methodology of this analysis compared to that from the previous chapter is the introduction of the damping functions for RSD (see Eq. 2.99 and 2.100). These introduce two parameters (σg and σu) that cannot be varied as free parameters in the emcee runs. This is because they exist inside the integral over the wavenumber k and varying them would involve a recalculation of the entire covariance matrix, as opposed to just a rescaling of the covariance amplitude (see the justification that precedes Eq. 2.119). This has some consequence for how we choose to conduct our analysis, which we detail for both the mocks and data below:

6dFGS Mocks

The mock catalogues allow us to test various aspects of the model. We start with the model for the galaxy overdensity auto-covariance, checking whether baddσ8 is still justified as a free parameter, as well as how the values of kmax and σg affect the results. We then move on to testing our model of the logarithmic distance ratio auto-covariance, checking how the values of kmax and σu affect the results. Based on our findings from these tests, we

fix kmax, σg and σu and test our complete covariance model, including the cross-covariance in addition to the two auto-covariances. Given that we need to introduce αb as a fixed parameter in the cross-covariance model, we also test how its value affects the results. The

ﬁxed parameters values that lead to good recovery of fσ8 are chosen as the ﬁducial values for our model.

6dFGS Data

We then take the chosen fiducial values for kmax, σg, σu and αb from the various mock tests and run the galaxy overdensity auto-covariance, logarithmic distance ratio auto- covariance and complete covariance models with the 6dFGS data. Our primary result for this analysis is then the constraint on fσ8 and β that we obtain from the complete covariance model. We also estimate a systematic error for our constraints on fσ8 and β by varying our fixed parameters: kmax, σg, σu and αb. Finally, to understand whether 104 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD the underlying cosmological parameters affect our results, we run the complete covariance model using power spectra from two additional cosmologies, WMAP and Planck 2018 (see Section 4.3.1).

4.4 Simulation Results

As in Section 3.4, we wish to validate our method by testing whether our pipeline recovers the expected cosmology used to generate the 6dFGS mock catalogues (See Table 4.1). For the Ωm and σ8 values used to generate the mock catalogues, the expected growth rate of structure is fσ8 = 0.423. Given that the mocks are based on the galaxy distribution of the 6dFGS BAO sample (Beutler et al., 2011), we might expect a galaxy bias of around bσ 1.2, which would correspond to β 0.35. However, we have not directly estimated 8 ≈ ≈ the true galaxy bias for the mocks, nor is it quoted by Carter et al. (2018), so we leave it as a rough estimate and do not include it as a reference point in any of the plots for this section. We treat baddσ8 and σv as a nuisance parameters in this analysis and do not give expected values for them.

The nature of the covariance evaluation (discussed in Section 2.3.4) means that kmax,

σu and σg cannot be varied as free parameters in the analysis. Therefore, it is important to examine whether the choices we make for the values of these parameters affect the constraint on the growth rate. We start by evaluating the likelihood using only the galaxy overdensity and logarithmic distance ratio auto-covariances. Once optimal parameter values are established, we fix these and move on to evaluating the likelihood with the complete covariance, testing different values of αb.

4.4.1 Galaxy Overdensity Auto-Covariance

We begin by establishing our best estimates for the fixed parameters used in the galaxy overdensity auto-covariance model. The k-range we fit over is controlled by kmax, which 1 we take to be kmax = 0.15 h Mpc− based on our findings in Chapter 3. The damping 1 due to pairwise velocities is taken to be σg = 3.0 h− Mpc; this corresponds to a pairwise 1 velocity dispersion of 300km s− , which is a standard fiducial assumption (e.g. Peacock & Dodds, 1994; Blake et al., 2018). We take the galaxy cross-correlation coefficient to be rg = 1, which corresponds to the linear bias model. In Chapter 3 we introduced the additional bias term baddσ8 that allowed for the covariance to be boosted without directly fitting to non-linear scales. Keeping with that analysis, we do not model RSD in the galaxy overdensity auto-covariance model beyond kmax, making the non-linear covariance 4.4. Simulation Results 105

fσ8 β baddσ8 Mock 1 Mock 2 Mock 3 Mock 4 Mock 5 Mock 6 Mock 7 Mock 8 Mock 9 Mock 10

0.4 0.6 0.8 0.5 1.0 1.5 0.5 1.0 1.5

Figure 4.1 The median values and 68% credible intervals of fσ8, β and baddσ8 for ten 6dFGS mocks when using the galaxy overdensity auto-covariance. The expected value for fσ8 is shown by the dashed vertical line. We note that the best-fitting values of fσ8 are biased high, and discuss a possible reason for this in the text. We do not expect this bias to affect the fits to the data sample.

independent of the growth rate. Consequently, baddσ8 functions as a nuisance parameter in this analysis.

We start by evaluating the likelihood for the galaxy overdensity measurements from ten 6dFGS mocks. The marginalised constraints on the three free parameters for this model (fσ8, β, baddσ8) are given in Figure 4.1. We note that the credible intervals for each parameter have roughly consistent sizes across all ten mocks, which shows that the precision of our method is robust. It’s clear from the fσ8 panel that the method is biased high when recovering the growth rate; none of the fσ8 constraints have a median value that falls below the expected value of 0.423. This may be a consequence of the values used for the fixed parameters and we do not expect this to affect our fits to the 6dFGS data; we discuss both of these points later in this section. While we have not explicitly set an expected recovery value for β, we can see that the mocks show typically higher values than our estimated value of 0.35. This will partially be due to the higher values of the growth rate, but might also indicate that the mocks have a lower galaxy bias than expected. The relative positions of the β credible intervals between each mock are similar to those for fσ8, which indicates that the galaxy bias is consistent across the mocks. To get an appreciation for the degeneracies between the three parameters, we show the 106 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

1.6

1.2 β 0.8

0.4

1.50 8 σ 1.25 add b

1.00

0.25 0.50 0.75 1.00 0.4 0.8 1.2 1.6 1.00 1.25 1.50 fσ8 β baddσ8

Figure 4.2 The posteriors of our free parameters for Mock 1 when using the galaxy overdensity auto-covariance. The shaded area of each marginalised posterior represents the 68% credible interval. The dark shading on each contour indicates the 68% credible region and the lighter shading indicates the 95% credible region. The expected value for fσ8 is shown by the dashed vertical line. corner plot for Mock 1, which we take as a representative sample, in Figure 4.2. From this, we can see that we recover the expected growth rate of structure at the 2σ level despite the bias seen in Figure 4.1. The contours indicate that there is a slight degeneracy between

β and baddσ8, which is to be expected given that we saw similar behaviour between bfitσ8 and baddσ8 in the previous chapter (see Figure 3.6). There is minimal degeneracy between fσ8 and baddσ8, which is also consistent with our findings in the previous chapter. Finally, the tight slope of the β-fσ8 contour indicates that a single effective galaxy bias value is a reasonable model when fitting to linear scales.

Given that we see a biased recovery of the growth rate across the ten mocks, it’s impor- 4.4. Simulation Results 107 tant to further investigate our assumptions for the ﬁducial model. Our ﬁxed parameters

(kmax, σg) are viable candidates, as well as whether the model still requires the baddσ8 parameter. It’s also possible to vary the cross-correlation coeﬃcient rg, but lowering this parameter would only push the value of the growth rate up (see Eq. 4.2), so we choose not to vary it as part of the following checks.

The first thing we check is whether baddσ8 is still useful as a nuisance parameter. This is important as we have significantly changed the model for the galaxy overdensity auto- covariance by introducing RSD. The posteriors for the model with and without baddσ8 are shown in Figure 4.3, the corresponding median constraints (with 68% confidence intervals) are in Table 4.2 and maximum likelihood values (with the corresponding χ2 values) are in Table 4.3. We can see that excluding baddσ8 lowers the growth rate a small amount. The effect on β is greater because the additional power that was being contributed from the non-zero baddσ8 has been transferred to the linear scale bias instead, lowering β for a

ﬁxed growth rate. This can be seen in the change in slope of the β-fσ8 contour in Figure 4.3, where the larger bias value results in a shallower slope.

2 2 The diﬀerence in χ between including and excluding baddσ8 is ∆χ = 88.12 (lower than the ∆χ2 obtained in the previous chapter), and both models give reasonable χ2/dof values. Thus, we use the reduced Akaike information criterion (AICc; see Burnham & Anderson, 2004) to establish which model is preferred. It is deﬁned as

2k2 2k AICc 2k 2ln( ˆ) + − , (4.13) ≡ − L n k 1 − − where k is the number of free parameters, ˆ is the maximum likelihood and n is the L number of data points. The final term in the expression accounts for small sample sizes and vanishes for large n. We can see that this is a more advanced model comparison statistic than the χ2 since it penalises models with larger numbers of free parameters, which typically produce smaller χ2 values. The preferred model will have the lowest AICc value. One may also look at the difference in AICc values to assess whether multiple models are considered good fits. Defining ∆AICc = AICc AICc as the difference in i i − min AICc between model i and the preferred model, Burnham & Anderson (2004) suggest the rule of thumb that ∆AICci < 2 corresponds to substantial evidence for model i, and that

∆AICci > 10 suggests that model i is unlikely. We ﬁnd that the model where baddσ8 is included has the lowest AICc value, and the ∆AICc value for the baddσ8 excluded model is 69.4. So there is suﬃcient evidence for keeping baddσ8 in the model, which we do for the remainder of this work. 108 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

baddσ8 included baddσ8 excluded

1.6

1.2 β 0.8

0.4

1.50 8 σ 1.25 add b

1.00

0.25 0.50 0.75 1.00 0.4 0.8 1.2 1.6 1.00 1.25 1.50 fσ8 β baddσ8

Figure 4.3 The posteriors of fσ8, β and baddσ8 for the galaxy overdensity auto-covariance model ﬁt to Mock 1 when including or excluding baddσ8. The inner contour indicates the 68% credible region and the outer contour indicates the 95% credible region. The expected value for fσ8 is shown by the dashed vertical line. 4.4. Simulation Results 109

1 kmax = 0.10 hMpc− 1 kmax = 0.125 hMpc− 1 kmax = 0.15 hMpc−

2.0

1.5 β 1.0

0.5

1.50 8 σ 1.25 add b

1.00

0.3 0.6 0.9 1.2 0.5 1.0 1.5 2.0 1.00 1.25 1.50 fσ8 β baddσ8

Figure 4.4 The posteriors of fσ8, β and baddσ8 for the galaxy overdensity auto-covariance model ﬁt to Mock 1 for diﬀerent values of kmax. The inner contour indicates the 68% credible region and the outer contour indicates the 95% credible region. The expected value for fσ8 is shown by the dashed vertical line.

Next, we test how the value of kmax aﬀects the constraints. The posteriors for the model 1 1 at our best estimate of kmax = 0.15 h Mpc− as well as kmax = 0.10, 0.125 h Mpc− are shown in Figure 4.4, the corresponding median constraints (with 68% conﬁdence intervals) are in Table 4.2 and maximum likelihood values (with the corresponding χ2 values) are in Table 4.3. The systematic shifts in baddσ8 are entirely consistent with its function of boosting the overall covariance value. When we reduce the value of kmax, the non-linear covariance is generated by integrating over a larger k-range, so the multiplicative factor of baddσ8 is reduced in order to provide the same boost. The tighter constraints for baddσ8 when reducing the value of kmax are consistent with the fact that the parameter is being

ﬁtted for over a larger k-range. This is directly tied to the wider constraint on fσ8 and 110 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

1 σg = 3.0 h− Mpc 1 σg = 4.0 h− Mpc 1 σg = 5.0 h− Mpc

1.5

β 1.0

0.5

1.50 8 σ 1.25 add b

1.00

0.3 0.6 0.9 1.2 0.5 1.0 1.5 1.00 1.25 1.50 fσ8 β baddσ8

Figure 4.5 The posteriors of fσ8 and β for the galaxy overdensity auto-covariance model ﬁt to Mock 1 for diﬀerent values of σg. The inner contour indicates the 68% credible region and the outer contour indicates the 95% credible region. The expected value for fσ8 is shown by the dashed vertical line.

1 β for the kmax = 0.10 h Mpc− case. We interpret the degradation of the fσ8 constraint 1 in the kmax = 0.10 h Mpc− case as evidence that the k-range is too small to produce a 1 reliable fσ8 value. Consequently, we choose to keep kmax = 0.15 h Mpc− as the ﬁducial 2 value, given that it maximises the range over which fσ8 is ﬁt and has a slightly lower χ 1 than the kmax = 0.125 h Mpc− case.

Finally, we also test how the value of σg aﬀects the constraints. The posteriors for 1 1 the model at our best estimate of σg = 3.0 h− Mpc as well as σg = 4.0, 5.0 h− Mpc are shown in Figure 4.5, the corresponding median constraints (with 68% conﬁdence intervals) are in Table 4.2 and maximum likelihood values (with the corresponding χ2 values) are in

Table 4.3. We can see that there is a systematic shift in both fσ8 and β in proportion to 4.4. Simulation Results 111

Table 4.2 Median values (with 68% credible intervals) of fσ8, β and baddσ8 for various tests on Mock 1 when using the galaxy overdensity auto-covariance. Our ﬁducial model is 1 1 baddσ8 included, kmax = 0.15 h Mpc− , σg = 3.0 h− Mpc.

fσ8 β baddσ8 +0.21 Fiducial model 0.66 0.13 0.80 0.19 1.22 0.13 +0.13 +0− .16 baddσ8 excluded 0.62 0.12 0.64 0.15 – 1 −+0.09 −+0.26 kmax = 0.10 hMpc− 0.60 0.10 0.90 0.23 1.07 0.04 1 − −+0.21 +0.07 k = 0.125 hMpc− 0.63 0.13 0.75 1.09 max 0.18 0.06 1 +0− .21 − σg = 4.0 h− Mpc 0.70 0.13 0.85 0.19 1.23 0.13 1 +0.14 −+0.22 +0.13 σg = 5.0 h− Mpc 0.75 0.13 0.91 0.20 1.25 0.12 − − − the value of σg, while baddσ8 remains largely unaffected. This is consistent with the fact that σg controls the level of damping and that stronger damping will result in larger fσ8 values as the covariance compensates, similar to the trade-off between β and baddσ8. This effect has also been seen in other RSD studies (see figure 4 in Peacock et al., 2001). We can see that it is the shift in fσ8 that is causing the shift in β, rather than the galaxy bias, since the β-fσ8 contours appear to have very similar slopes. Given that lower values of 1 σg result in lower fσ8 values, we also attempted to run tests using σg = 2.0, 1.0 h− Mpc. Unfortunately, the numerical integration library we used to calculate the covariance matrix elements failed in both cases due to round-off errors. We anticipate that if we were able to construct the covariance for lower values of σg we would be able to recover the expected growth rate. The general systematic behaviour in this last test may give us some insight into why we don’t recover the expected growth rate for the mocks: it may be that a lower σg value would allow us to recover the growth rate. This would be consistent with the fact that we only use central galaxies for our mocks, given that centrals have lower pairwise velocity dispersion than satellites. This interpretation is supported by Blake et al. (2018), who saw a lower pairwise velocity dispersion in the centrals-only sample compared to the centrals+satellites sample when working with the same mocks (see figure 6 in that work). Given that the 6dFGS data contains both satellites and centrals we expect that this issue 1 will not impact the data analysis, and that σg = 3.0 h− Mpc will still provide a reasonable fit to the data. Consequently, we leave this as our fiducial value for the rest of the analysis. Finally, we show the marginalised constraints for each of the above tests relative to our 1 1 chosen fiducial model (baddσ8 included, kmax = 0.15 h Mpc− , σg = 3.0 h− Mpc) in Figure 4.6. We note that this is a visual representation of the median likelihood constraints from

Table 4.2. From this, we see that σg has the largest eﬀect on fσ8 of all three tests, and 112 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

Table 4.3 Maximum likelihood values of fσ8, β and baddσ8 for various tests on Mock 1 when using the galaxy overdensity auto-covariance. Our ﬁducial model is baddσ8 included, 1 1 2 2 kmax = 0.15 h Mpc− , σg = 3.0 h− Mpc. We also include the χ value and χ /dof statistic for the maximum likelihood values. 2 2 fσ8 β baddσ8 χ χ /dof Fiducial model 0.66 0.79 1.23 1612.44 0.99

baddσ8 excluded 0.61 0.62 – 1700.33 1.04 1 kmax = 0.10 hMpc− 0.70 0.87 1.06 1609.15 0.99 1 kmax = 0.125 hMpc− 0.62 0.72 1.09 1612.65 0.99 1 σg = 4.0 h− Mpc 0.69 0.83 1.23 1612.92 0.99 1 σg = 5.0 h− Mpc 0.75 0.91 1.25 1614.16 0.99

fσ8 β baddσ8

baddσ8 excluded 1 kmax = 0.10 hMpc− 1 kmax = 0.125 hMpc− 1 σg = 4.0 h− Mpc 1 σg = 5.0 h− Mpc

0.6 0.8 0.50 0.75 1.00 1.251.0 1.2 1.4

Figure 4.6 The median values and 68% credible intervals of fσ8, β and baddσ8 for various tests on Mock 1 when using the galaxy overdensity auto-covariance. The dashed lines and grey shaded regions represent the median value and 68% credible interval for our ﬁducial 1 1 model: baddσ8 included, kmax = 0.15 h Mpc− , σg = 3.0 h− Mpc. 4.4. Simulation Results 113

1 fσ8 σv (km s− ) Mock 1 Mock 2 Mock 3 Mock 4 Mock 5 Mock 6 Mock 7 Mock 8 Mock 9 Mock 10

0.2 0.4 0.6 100 200 300

Figure 4.7 The median values and 68% credible intervals of fσ8 and σv for ten 6dFGS mocks when using the logarithmic distance ratio auto-covariance. The expected value for fσ8 is shown by the dashed vertical line.

that the value of kmax has the largest effect on baddσ8. For fσ8, the various tests all return median values that are within 1σ of our fiducial case. We apply these model variations when fitting to the data to estimate a systematic error bar for our 6dFGS results in Section 4.5.1.

4.4.2 Logarithmic Distance Ratio Auto-Covariance

Again, we begin by establishing our best estimates for the fixed parameters used in the 1 logarithmic distance ratio auto-covariance model. We again take kmax = 0.15 h Mpc− as the boundary to our fitted k-range. The damping identified by Koda et al. (2014) is implemented as a sinc function and parametrized by σu (see Section 2.3.4). We take 1 σu = 13.0 h− Mpc as our best estimate, given that Koda et al. (2014) found this to be the best fit to their simulations. As in the previous section, we examine the constraints for all ten mocks when using our best estimates. The marginalised constraints for the free parameters of this model

(fσ8, σv) are given in Figure 4.7. As with the galaxy overdensity auto-covariance, the credible intervals for our free parameters are roughly consistent across all ten mocks. The method does not appear to be biased; the ten mocks are evenly distributed around the expected recovery value for fσ8, and the value is recovered at the 1σ level in six of the ten mocks. 114 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

240 ) 1 − 160 (km s v σ 80

0.2 0.4 0.6 0.8 80 160 240 1 fσ8 σv (km s− )

Figure 4.8 The posteriors of our free parameters for Mock 1 when using the logarithmic distance ratio auto-covariance. The shaded area of each marginalised posterior represents the 68% credible interval. The dark shading on each contour indicates the 68% credible region and the lighter shading indicates the 95% credible region. The expected value for fσ8 is shown by the dashed vertical line.

We again take Mock 1 as a representative sample and present the corner plot for this mock in Figure 4.8. There appears to be minimal degeneracy between fσ8 and σv, and the expected growth rate is well-recovered by the mock. In the previous chapter, we found +63 1 σv = 174 74 km s− for 6dFGS (see Section 3.5.1), and while consistent, the value we − +57 1 recover for Mock 1 is a little lower (σv = 84 49 km s− ). This could be caused by our − implementation of errors for the logarithmic distance ratio values measured from the mock.

This is because the total error is the quadrature sum of σv, which is a free parameter, and

σobs (see Section 2.3.3).

As in the previous section, we start by testing the effect of varying kmax. The posteriors 1 for the model at our best estimate of kmax = 0.15 h Mpc− as well as kmax = 0.10, 1 0.125 h Mpc− are shown in Figure 4.9 and corresponding constraints are in Table 4.4. We can see that changing the value of kmax has a negligible effect on the constraints from the logarithmic distance ratio auto-covariance. This is supported by the χ2 values, which only vary on the order of 0.1 between the three runs. Consequently, we choose to keep our best 1 estimate of kmax = 0.15 h Mpc− as our value for the fiducial model. 4.4. Simulation Results 115

1 kmax = 0.10 hMpc− 1 kmax = 0.125 hMpc− 1 kmax = 0.15 hMpc−

240 ) 1 − 160 (km s v σ

0.25 0.50 0.75 80 160 240 1 fσ8 σv (km s− )

Figure 4.9 The posteriors of fσ8 and σv for the logarithmic distance ratio auto-covariance model ﬁt to Mock 1 for diﬀerent values of kmax. The inner contour indicates the 68% credible region and the outer contour indicates the 95% credible region. The expected value for fσ8 is shown by the dashed vertical line.

We also test how changing the value of the damping parameter σu affects constraints. 1 The posteriors for the model at our best estimate of σu = 13.0 h− Mpc as well as σu = 1 11.0, 15.0, 17.0 h− Mpc are shown in Figure 4.10 and corresponding constraints are in Table 4.4. Again, the difference between our tested values is negligible, although we do see a slight trend in fσ8, where lower values of σu correspond to lower growth rates. This is consistent with the behaviour of the damping function, where lowering σu results in less 2 damping and a higher growth rate. Similarly to the kmax test, the χ values only vary on the order of 0.1. Given that our best estimate of σu provides a good fit, we choose to keep this as the value for our fiducial model. Finally, we show the marginalised constraints for each of the above tests relative to 1 1 our chosen fiducial model: kmax = 0.15 h Mpc− , σu = 13.0 h− Mpc in Figure 4.11. We note that this is a visual representation of the median likelihood constraints from Table

4.4. From this, we see that σu has the largest effect on fσ8, but it is still minimal. σv is unaffected by both the values of kmax and σu. For fσ8, the various tests all return median values that are well within 1σ of our fiducial case. We apply these model variations when estimating a systematic error contribution to our 6dFGS results in Section 4.5.1. 116 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

1 σu = 11.0 h− Mpc 1 σu = 13.0 h− Mpc 1 σu = 15.0 h− Mpc 1 σu = 17.0 h− Mpc

240 ) 1 − 160 (km s v σ 80

0.25 0.50 0.75 80 160 240 1 fσ8 σv (km s− )

Figure 4.10 The posteriors of fσ8 and σv for the logarithmic distance ratio auto-covariance model ﬁt to Mock 1 for diﬀerent values of σu. The inner contour indicates the 68% credible region and the outer contour indicates the 95% credible region. The expected value for fσ8 is shown by the dashed vertical line.

Table 4.4 Maximum likelihood (ML) and median values (with 68% credible intervals) of fσ8 and σv for various tests on Mock 1 when using the logarithmic distance ratio auto- 1 1 2 covariance. Our ﬁducial model is kmax = 0.15 h Mpc− , σu = 13.0 h− Mpc. The χ value and χ2/dof statistic are for the maximum likelihood values. 1 fσ8 σv (km s− ) ML Median ML Median χ2 χ2/dof +0.11 +57 Fiducial model 0.43 0.43 0.10 70 84 49 1054.78 0.91 1 +0− .12 +57− kmax = 0.10 hMpc− 0.44 0.44 0.11 75 85 50 1055.09 0.91 1 +0− .12 +57− kmax = 0.125 hMpc− 0.42 0.43 0.10 88 86 50 1055.00 0.91 1 −+0.11 −+56 σu = 11.0 h− Mpc 0.40 0.42 0.10 73 84 49 1054.67 0.91 1 +0− .12 +56− σu = 15.0 h− Mpc 0.44 0.45 0.11 86 86 50 1054.80 0.91 1 +0− .13 +56− σu = 17.0 h− Mpc 0.44 0.46 0.11 79 85 50 1054.86 0.91 − − 4.4. Simulation Results 117

1 fσ8 σv (km s− ) 1 kmax = 0.10 hMpc− 1 kmax = 0.125 hMpc− 1 σu = 11.0 h− Mpc 1 σu = 15.0 h− Mpc 1 σu = 17.0 h− Mpc

0.4 0.6 50 100 150

Figure 4.11 The median values and 68% credible intervals of fσ8 and σv for various tests on Mock 1 when using the logarithmic distance ratio auto-covariance. The dashed lines and grey shaded regions represent the median value and 68% credible interval for our 1 1 ﬁducial model: kmax = 0.15 h Mpc− , σu = 13.0 h− Mpc.

4.4.3 Complete Covariance

When running the full covariance model, we use the ﬁducial set-up that we established in the previous two sections:

b σ included as a free parameter, • add 8 k = 0.15 h Mpc 1 for both auto-covariances and the cross-covariance, • max − σ = 3.0 h 1 Mpc for the galaxy overdensity auto-covariance and cross-covariance, • g − σ = 13.0 h 1 Mpc for the logarithmic distance ratio auto-covariance and cross- • u − covariance.

In addition to these, we have also introduced a new parameter for the cross-covariance,

αb, which modifies the effective galaxy bias for the cross-covariance relative to that of the galaxy overdensity auto-covariance. Before proceeding, we establish that the effective bias of our sample is different for different redshift ranges. We do this by estimating the galaxy-galaxy power spectrum for two redshift ranges, using the 600 mock catalogues developed by Carter et al. (2018) and discussed in Section 4.2.2. The redshift ranges correspond to the limits of our logarithmic distance ratio sample (z < 0.057) and our galaxy overdensity sample (z < 0.1). We show the two estimated power spectra in Figure 4.12. 2 Nominally, we could estimate αb directly by taking the ratio of the two estimated power spectra (recalling that the amplitude of the galaxy power spectrum is proportional 118 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

z < 0.057 z < 0.10 ] 3

Mpc) 10 1 − h ) [( k ( gg P

2 1 10− 10− 1 k [h Mpc− ]

Figure 4.12 The mean and standard deviation of the galaxy-galaxy power spectrum from 600 mock catalogues, with uncertainties corresponding to those for one mock. The amplitude of the power spectrum for the z < 0.05 sample is clearly lower than that for the z < 0.10 sample, indicating that that the lower redshift sample has a lower eﬀective bias. 1 We note that the z < 0.10 points have been shifted to the right by ∆k = 0.003 h Mpc− for clarity. 4.4. Simulation Results 119

Table 4.5 Median values (with 68% credible intervals) of fσ8, σv, β and baddσ8 when running the complete covariance model with various values of αb on Mock 1. 1 Model fσ8 σv (km s− ) β baddσ8 +41 +0.050 αb = 0.95 0.301 0.038 204 43 0.328 0.048 1.25 0.13 +0.048 −+43 −+0.067 αb = 0.90 0.400 0.050 175 47 0.451 0.066 1.25 0.13 −+0.058 −+50 −+0.084 αb = 0.85 0.499 0.062 129 59 0.577 0.083 1.23 0.13 −+0.072 +55− +0− .11 αb = 0.80 0.560 0.074 97 54 0.66 0.10 1.22 0.13 −+0.082 −+55 −+0.12 αb = 0.75 0.577 0.078 87 50 0.68 0.11 1.22 0.13 −+0.083 −+55 −+0.12 αb = 0.70 0.586 0.080 82 48 0.70 0.11 1.22 0.13 − − − to b2). The ratio between each pair of points implies α = 0.94 0.03 when working with b the standard deviation values for a single mock. However, we still choose to run the model for different values of αb before selecting the fiducial value. We do this for two key reasons. Firstly, it is not trivial to estimate the redshift range that the logarithmic distance ratio sample (and hence the cross-covariance) is sensitive to. In the previous analysis, we showed 1 that the cross-correlation is non-zero up to separations of at least 50 h− Mpc, potentially more depending on the orientation of the galaxy overdensity-logarithmic distance ratio pair (see Figure 3.8 and surrounding text). This means that the effective bias for the cross-correlation is likely affected by overdensities beyond the boundary of the logarithmic distance ratio sample, a subtlety that the estimated value of αb from the power spectra ratio does not account for. Secondly, αb may be sensitive to additional effects beyond the difference in effective bias. For example, Eq. 4.7 and 4.8 show that αb reduces the amplitude of the cross-covariance relative to the two auto-covariances (although we note 0 1 that αb only reduces the amplitude of the β term and does not affect the β term). Additionally, it’s possible that the relative weight of different regions towards the signal- to-noise differs between the cross-covariance and the galaxy overdensity auto-covariance. Given these two factors, we determine that it is more appropriate to estimate the value of

αb by requiring recovery of the expected fσ8 value when working with the mocks. We start by running the full covariance model on our representative mock (Mock 1) with different values of αb. The lowest value we test is αb = 0.70; we consider lower values to be unphysical as they would translate to differences in the effective galaxy bias values that are implausible. The posteriors are shown in Figure 4.13, the corresponding median constraints (with 68% confidence intervals) are in Table 4.5 and maximum likelihood values (with the corresponding χ2 values) are in Table 4.6.

From Figure 4.13, we see that αb = 0.90 provides excellent recovery of the expected 2 growth rate. We note that the value producing the lowest χ is αb = 0.85, but given that 120 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

αb = 0.95 αb = 0.90 αb = 0.85 αb = 0.80 αb = 0.75 αb = 0.70

300 ) 1 − 200 (km s v

σ 100

1.2

0.9 β 0.6

0.3

1.75

1.50 8 σ 1.25 add b 1.00

0.2 0.4 0.6 0.8 100 200 300 0.3 0.6 0.9 1.2 1.00 1.25 1.50 1.75 1 fσ8 σv (km s− ) β baddσ8

Figure 4.13 The posteriors of fσ8, σv, β and baddσ8 for the complete covariance model ﬁt to Mock 1 for diﬀerent values of αb. The inner contour indicates the 68% credible region and the outer contour indicates the 95% credible region. The expected value for fσ8 is shown by the dashed vertical line. 4.4. Simulation Results 121

Table 4.6 Maximum likelihood values of fσ8, σv, β and baddσ8 when running the complete 2 covariance model with various values of αb on Mock 1. We also include the χ value and χ2/dof statistic for the maximum likelihood values. 1 2 2 Model fσ8 σv (km s− ) β baddσ8 χ χ /dof

αb = 0.95 0.301 205 0.328 1.25 2650.27 0.95

αb = 0.90 0.401 178 0.450 1.25 2643.87 0.95

αb = 0.85 0.501 141 0.573 1.23 2643.17 0.95

αb = 0.80 0.567 106 0.649 1.22 2650.87 0.95

αb = 0.75 0.571 90 0.670 1.22 2652.61 0.95

αb = 0.70 0.582 66 0.69 1.19 2652.93 0.95

2 the difference in χ between this and the αb = 0.90 fit is small, we choose αb = 0.90 as our fiducial value for the remainder of the analysis. This is because it gives a better recovery of the growth rate, which is our main criteria for assessing the tested values of αb. We also note that it is close (within 2σ) to our estimate of αb from the ratio of the power spectra for our two redshift ranges, which is reassuring. There is a clear systematic trend (although it is by no means linear) in that larger values of αb correspond to lower values of fσ8. Additionally, we also see that the posterior for fσ8 widens as αb becomes lower and that the posteriors appear to be converging. This behaviour ties back to the fact that αb reduces the amplitude of the cross-covariance model, causing it to contribute less to the overall covariance. The increase in the value of fσ8 and the widening of the posterior is consistent with the model favouring the auto-covariance information above the cross-covariance information, particularly since the galaxy overdensity auto-covariance model consistently returned a too-large growth rate. We note that the constraints from the low-αb cases are still better than the galaxy overdensity auto-covariance only case, since the logarithmic distance ratio auto-covariance is also con- tributing to the constraints. This interpretation also explains the behaviour of σv, which tends towards the logarithmic distance ratio auto-covariance constraint as αb is lowered (see Figure 4.8 for comparison). It’s clear that the cross-covariance has little influence on baddσ8, and consequently, the constraints are very close to those from the galaxy overdensity auto-covariance (see Figure 4.2).

We also attempted to run the model for αb = 1 and αb = 0.99, but found that the covariance matrix was not positive definite, so could not be inverted through the Cholesky solver we used (see Section 4.3.5). Covariance matrices are always positive semi-definite by construction, so we believe there could be a numerical stability issue that is causing the matrix to fail the positive definite requirement, possibly arising from the numerical 122 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

1 fσ8 σv (km s− ) β baddσ8 Mock 1 Mock 2 Mock 3 Mock 4 Mock 5 Mock 6 Mock 7 Mock 8 Mock 9 Mock 10

0.3 0.4 0.5 0.6 100 200 0.3 0.4 0.5 0.6 0.7 0.5 1.0 1.5

Figure 4.14 The median values and 68% credible intervals of fσ8, σv, β and baddσ8 for ten 6dFGS mocks when using the complete covariance with αb = 0.9. The expected value for fσ8 is shown by the dashed vertical line. integration that we perform when generating the covariance elements. We attempted to correct for this through an algorithm for finding the nearest positive semi-definite matrix that was developed by Higham (1988). We tried a Python implementation of the algorithm by Ahmed Fasih1, but found that it was too computationally expensive to embed within our emcee runs. Consequently, we do not present any results for αb = 1 or αb = 0.99. Finally, we show the median values and 68% credible intervals for all ten mocks when running the complete covariance model with αb = 0.90 in Figure 4.14. The recovery of fσ8 is excellent across all ten mocks, validating our choice of αb = 0.90 as the fiducial value for the remainder of the analysis. We note that β still appears to be a little higher than the anticipated value of 0.35, although it has been reduced compared to the results from only using the galaxy overdensity auto-covariance.

1https://gist.github.com/fasiha/fdb5cec2054e6f1c6ae35476045a0bbd 4.4. Simulation Results 123

4.4.4 Summary of Fiducial Model Results

We summarise the key constraints from running our fiducial model on our representative mock for the three covariance model cases in Table 4.7. It is clear that the uncertainty in the growth rate of structure has reduced significantly when using the complete covariance in comparison to using either of the auto-covariances alone: we see a 62% improvement in the uncertainty from the galaxy overdensity auto-covariance and a 53% improvement in the uncertainty from logarithmic distance ratio auto-covariance. For β the uncertainty improvement is 67% when going from the galaxy overdensity auto-covariance to the complete covariance. Given the sophistication of the mocks, we expect to see similar improvements when applying our fiducial model to the data in the next section.

While we believe that the reduction in uncertainty can be entirely attributed to the introduction of the cross-covariance, we note that tighter uncertainties can be a symptom of underlying tension in the model or data sets. It is important to consider this possibility here, since the galaxy overdensity auto-covariance model does not currently recover the expected growth rate for the mocks, which could result in tension when considered part of the complete covariance. However, given that the complete covariance model shows a reasonable χ2/dof (0.95), we do not believe that tension is impacting the uncertainty reduction in a signiﬁcant way. We also believe that this potential source of tension is only an issue for the mock catalogues and not for the data: the overestimation of the growth 1 rate for the galaxy overdensity auto-covariance is driven by our use of σg = 3.0 h− Mpc in the ﬁducial model, which may be too high for the mocks (since they only contain central galaxies). We believe this will be an appropriate value when working with the data (which contains satellites and centrals) and anticipate that the growth rate will no longer be overestimated.

Table 4.7 Median values (with 68% credible intervals) of fσ8, σv, β and baddσ8 for the galaxy overdensity auto-covariance, logarithmic distance ratio auto-covariance and complete covariance models for Mock 1. 1 Model fσ8 σv (km s− ) β baddσ8 Cerr +0.21 δgδg,RSD 0.66 0.13 – 0.80 0.19 1.22 0.13 err +0.11 +57 − Cηη,RSD 0.43 0.10 84 49 – – −+0.048 −+43 +0.067 C 0.400 0.050 175 47 0.451 0.066 1.25 0.13 − − − 124 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

4.5 Data Results

After comprehensive testing on the 6dFGS mock catalogues, we run the galaxy overdensity auto-covariance, logarithmic distance auto-covariance and complete covariance models on the 6dFGS dataset, ﬁxing the following parameters at the ﬁducial values determined from the mock analysis:

k = 0.15 h Mpc 1, • max − σ = 3.0 h 1 Mpc, • g − σ = 13.0 h 1 Mpc, • u − α = 0.9, • b where kmax determines the upper bound for the covariance integral (see Section 4.3.2), σg and σu are damping parameters in the redshift-space distortion (RSD) model (see Section

2.3.4), and αb allows the cross-covariance to have a different effective bias to the galaxy overdensity auto-covariance (see Section 4.3.3). Now that we are working with real data, we choose the fiducial cosmology to be that from Planck Collaboration et al. (2015c), which is summarised in Table 4.1. Given that we are working with a complete RSD model for our overdensity data, we directly compare our results to those from Beutler et al. (2012), who found fσ = 0.423 0.055 and β = 0.373 0.054 at an effective redshift of 8 zeff = 0.067. These are the most precise measurements of fσ8 and β available for 6dFGS, so serve as a useful point of comparison. We note that the logarithmic distance ratio +0.079 auto-covariance result from Johnson et al. (2014) of fσ8 = 0.428 0.068 may also be used − as a point of comparison. We present the posteriors of our free parameters in Figure 4.15, the corresponding median constraints (with 68% confidence intervals) in Table 4.8 and maximum likelihood values (with the corresponding χ2 values) in Table 4.9. 4.5. Data Results 125

Cηη Cδδ C

) 300 1 − 200 (km s v

σ 100

0.75

0.50 β

0.25

2.0 8

σ 1.6 add b 1.2

0.3 0.6 0.9 100 200 300 0.25 0.50 0.75 1.2 1.6 2.0 1 fσ8 σv (km s− ) β baddσ8

Figure 4.15 The posteriors of fσ8, β, baddσ8 and σv for 6dFGS when using the galaxy overdensity auto-covariance (Cδδ), the logarithmic distance ratio auto-covariance (Cηη), and the complete covariance (C). The results for fσ8 and β from Beutler et al. (2012) are indicated by the dashed lines. 126 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

Table 4.8 Median values (with 68% credible intervals) of fσ8, σv, β and baddσ8 for 6dFGS using the galaxy overdensity auto-covariance (Cδδ), the logarithmic distance ratio auto- covariance (Cηη), and the complete covariance (C). 1 Model fσ8 σv (km s− ) β baddσ8 +0.15 +0.13 +0.17 Cδδ 0.41 0.14 – 0.30 0.11 1.45 0.16 −+0.11 +54 − − Cηη 0.53 0.10 90 50 – – − −+44 +0.044 C 0.384 0.052 208 45 0.289 0.043 1.53 0.17 − −

Table 4.9 Maximum likelihood values of fσ8, σv, β and baddσ8 for 6dFGS using the galaxy overdensity auto-covariance (Cδδ), the logarithmic distance ratio auto-covariance (Cηη), and the complete covariance (C). We also include the χ2 and χ2/dof statistic for the maximum likelihood values 1 2 2 Model fσ8 σv (km s− ) β baddσ8 χ χ /dof

Cδδ 0.38 – 0.27 1.45 1774.45 1.09

Cηη 0.52 84 – – 847.69 0.94 C 0.380 208 0.286 1.52 2610.42 1.03

We find that our measurements of fσ8 and β for the three covariance analyses are self-consistent. This is a major achievement in that we have demonstrated that peculiar velocities and redshift-space distortions provide consistent measurements of the growth rate of structure for the same galaxy survey and modelling framework. Given that the two probes constrain the growth rate on different physical scales (peculiar velocities are sensitive to larger scales than RSD), the complete covariance analysis may be a promising way to test modified gravity models, which we discuss more in Section 6.2. We also note that our value of fσ8 is consistent at close to the 1σ level with the Planck 2015 + GR prediction of fσ8 = 0.446 at redshift z = 0 and at the 1σ level with the Planck 2018 +

GR prediction of fσ8 = 0.430 at redshift z = 0.

We calculate that the percentage uncertainties in fσ8 are 35% for the galaxy overdensity auto-covariance, 20% for the logarithmic distance ratio auto-covariance and 14% for the complete covariance. Most importantly, we see a 64% reduction in the uncertainty when going from the galaxy overdensity auto-covariance to the complete covariance, and a 50% reduction when going from the logarithmic distance ratio auto-covariance to the complete covariance. The improvement in going from the galaxy overdensity auto-covariance to the complete covariance is most notable in the fσ8-β contour of Figure 4.15, where the 2σ contour from the complete covariance sits well inside the 1σ contour from the auto-covariance.

We note that both the absolute uncertainties and percentage improvements for fσ8 are 4.5. Data Results 127 very close to those from the representative mock (see Table 4.7), although the mock did give slightly smaller uncertainties for the galaxy overdensity auto-covariance and complete covariance compared to what we found for the data. Conversely, the mock had much larger absolute uncertainties for β than we found for the data, although we ﬁnd that the reduction in uncertainty of 64% is similar to what we saw for the mock. We suspect that the larger uncertainty for the representative mock may be related to the unsuitable value for the pairwise velocity dispersion σg used in the model.

It’s also worth highlighting that the consistent fσ8-β slope between the galaxy overdensity auto-covariance and the complete covariance indicates that the two models prefer similar effective galaxy bias values. We note that this would not be the case without an appropriate value for αb, which allows the cross-covariance to be parametrized in terms of the galaxy overdensity sample’s effective bias. This supports our inclusion of this parameter and motivates further study in understanding how to model effective bias within the model covariance framework, which we discuss in Section 6.2.

4.5.1 Systematics

Given the increased precision of our constraint on the growth rate of structure, it is important to investigate how robust our result is to various systematics. This includes the

fixed parameters of our covariance model (kmax, σg, σu and αb), as well as the underlying cosmological parameters which inform the power spectrum models. For the fixed parameters, we’re able to estimate a systematic error contribution by varying the values of the fixed parameters and re-running the model. We assume that each systematic is independent, allowing us to vary a single parameter while holding the others fixed at their fiducial values. For each systematic s (k , σ , σ , α ), the ∈ max g u b systematic variance in parameter φ is

∂φ2 σ2 = (∆s)2, (4.14) s ∂s where we approximate the partial derivative using the central ﬁnite diﬀerence method:

∂φ φ(s + ∆s) φ(s ∆s) − − . (4.15) ∂s ≈ 2∆s

2 We note that σs are the diagonal elements of the full systematic covariance (e.g. eq. C4 in Zhang et al., 2017). We calculate the diagonal terms speciﬁcally because we are interested in the systematic error estimate for both fσ8 and β, rather than the variation of a pair of 128 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

Table 4.10 The minimum, maximum and step size used when calculating the systematic variance for each ﬁxed parameter. Systematic s Minimum Maximum ∆s 1 kmax [ h Mpc− ] 0.10 0.15 0.025 1 σg [ h− Mpc] 3.0 5.0 1.0 1 σu [ h− Mpc] 11.0 15.0 2.0

αb 0.875 0.925 0.025

Table 4.11 The systematic variance contributions to fσ8 and β for each ﬁxed parameter. 2 2 2 2 Parameter σkmax σσg σσu σαb fσ 2.87 10 6 8.07 10 6 1.18 10 6 3.67 10 3 8 × − × − × − × − β 1.09 10 5 8.04 10 7 1.17 10 6 2.34 10 3 × − × − × − × − parameters with a given systematic. The full systematic covariance is most useful when running an MCMC process with the statistical and systematic covariances estimated in advance (as in Zhang et al., 2017), which is not the case for our work. We then give the total systematic error as the sum in quadrature of each systematic:

s X 2 σsys = σi . (4.16) i=s

Where possible, we estimate the partial derivative using the fiducial point as the central point. We note that this is not possible for kmax and σg, where the fiducial value is the maximum and minimum tested value respectively. The minimum and maximum values and step size for each systematic are given in Table 4.10. We note that the size of the systematic variance will be affected by the step size. Consequently, we mostly use the same step sizes that we used when testing each fixed parameter throughout the simulation analysis in Section 4.4, which were chosen to encompass reasonable values for the fixed parameters. The only exception is in the case of αb, where we choose a smaller step size of ∆αb = 0.025, as the step size of ∆αb = 0.05 gives posteriors that recover the growth rate outside the 1σ level (see Figure 4.13). The systematic variance values are given in

Table 4.11 for fσ8 and β, noting that we don’t provide systematic variance estimates for our two nuisance parameters σv and baddσ8 since they are already marginalised over in the model ﬁts. It’s clear that αb is the dominant systematic for both fσ8 and β, with a systematic variance that is around three orders of magnitude larger than any of the other ﬁxed parameters. Consequently, it dominates the systematic error term, which we show in Table 4.12. 4.5. Data Results 129

Table 4.12 The total systematic error in fσ8 and β, shown with and without the contribution from αb. Parameter σsys (with αb) σsys (without αb)

fσ8 0.061 0.003 β 0.049 0.004

From this analysis, our final constraint (using the full covariance model) is fσ8 = +0.044 0.384 0.052(stat) 0.061(sys) for the growth rate of structure, and β = 0.289 0.043(stat) − 0.049(sys) for the redshift-space distortion parameter. Currently, the systematic error for each parameter is greater than the corresponding statistical uncertainty, which is driven by the behaviour of αb, specifically, its large degeneracy with fσ8. Given that our introduction of αb is a relatively simple method for accounting for the difference in effective bias across our samples, we believe that this systematic could be reduced or mitigated in future work, and suggest some avenues for this in Section 6.2.

Sensitivity to Cosmological Parameters

Our method is affected by the cosmological parameter values (those listed in Table 4.1) in two key ways. Firstly, the cosmological parameters alter the transformation of the observed coordinates (RA, dec, z) to Cartesian coordinates (x, y, z) in configuration space, which is required for our covariance model (see Section 2.3.2). Secondly, the cosmological parameters influence the shape of all three model power spectra Pmm, Pθθ and Pmθ. We note that the second point is more important, since the transformations from observed to Cartesian coordinates are independent of all the cosmological parameters to first order, 1 being at low redshift and with distances measured in h− Mpc units.

To test how sensitive our fσ8 constraint is to the choice of the cosmological parameter values, we use the values from the three CMB analyses listed in Table 4.1: the WMAP Year-5 results (Komatsu et al., 2009), Planck 2015 results (our ﬁducial model; Planck Collaboration et al., 2015c) and the Planck 2018 results (Planck Collaboration et al.,

2018). We show the matter power spectrum Pmm for each set of cosmological parameters in Figure 4.16. We note that the biggest difference occurs at large scales, consistent with the larger uncertainty from fewer large scale modes. However, we anticipate that this will have little effect on our overall constraint of fσ8 since this degenerate combination should account for any difference in the normalisation of the power spectrum amplitude (we note that each cosmological parameter set has a different fiducial value of σ8). 130 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

WMAP 5 Planck 2015 Planck 2018 ] 3 104 Mpc) 1 − h ) [( k ( mm P

2 1 10− 10− 1 k [h Mpc− ]

Figure 4.16 The matter power spectrum for each of the three sets of cosmological parameters used in our analysis.

Table 4.13 Median values (with 68% credible intervals) of fσ8, σv, β and baddσ8 for 6dFGS when using three sets of cosmological parameters, as listed in Table 4.1. 1 Model fσ8 σv (km s− ) β baddσ8 +0.049 +47 +0.043 +0.17 WMAP 5 0.385 0.050 186 50 0.295 0.042 1.50 0.16 − +44− +0− .044 − Planck 2015 0.384 0.052 208 45 0.289 0.043 1.53 0.17 +0.052 −+46 −+0.045 +0.17 Planck 2018 0.399 0.053 194 48 0.304 0.044 1.41 0.16 − − − −

We repeat our analysis for the complete covariance, including the data transformation, for the two additional cosmological parameter sets. We show the posteriors of our free parameters in Figure 4.17, the corresponding median constraints (with 68% conﬁdence intervals) in Table 4.13 and maximum likelihood values (with the corresponding χ2 values) in Table 4.14.

As anticipated, the choice of cosmological parameters has little eﬀect on fσ8. This conclusion is likely dependent on the caveat that each set of cosmological parameters has been determined from a CMB analysis and is an accurate reﬂection of our Universe’s true cosmology. Most interestingly, Planck 2015 and WMAP 5 give almost identical fσ8 constraints, which could be due to the overlap in their power spectra on intermediate 4.5. Data Results 131

Table 4.14 Maximum likelihood values of fσ8, σv, β and baddσ8 for 6dFGS when using three sets of cosmological parameters, as listed in Table 4.1. We also include the χ2 and χ2/dof statistic for the maximum likelihood values. 1 2 2 Model fσ8 σv (km s− ) β baddσ8 χ χ /dof WMAP 5 0.385 185 0.290 1.50 2611.63 1.03 Planck 2015 0.380 208 0.286 1.52 2610.42 1.03 Planck 2018 0.404 196 0.304 1.42 2611.63 1.03

WMAP 5 Planck 2015 Planck 2018

) 300 1 − 200 (km s v

σ 100

0.4

β 0.3

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2.0 8

σ 1.6 add b 1.2

0.2 0.3 0.4 0.5 100 200 300 0.2 0.3 0.4 1.2 1.6 2.0 1 fσ8 σv (km s− ) β baddσ8

Figure 4.17 The posteriors of fσ8, σv, β and baddσ8 for 6dFGS when using three sets of cosmological parameters, as listed in Table 4.1. The results for fσ8 and β from Beutler et al. (2012) are indicated by the dashed vertical lines. 132 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

1 scales (k = 0.003 to 0.007 h Mpc− ; see Figure 4.17). We note that it is possible to test a wider range of power spectrum models with our method. For example, Johnson et al. (2014) constrained the relative matter density Ωm and σ8 by constructing the power spectrum for diﬀerent pairs of these parameters and running the full covariance for each pair. However, the results from this test are suﬃcient to show that our constraint of fσ8 is robust to the choice of cosmological parameters, given that they have been determined from CMB analyses.

4.5.2 Comparison to Existing Literature

Previous Analysis

+0.065 +0.048 Our constraints from the previous chapter were fσ8 = 0.424 0.062 and β = 0.300 0.046, − − which we compare to the constraints from this analysis of fσ = 0.384 0.052(stat) 8 +0.044 0.061(sys) and β = 0.289 0.043(stat) 0.049(sys). For fσ8, the statistical uncertainty is − reduced by 18% and our new result is consistent at the 1σ level. For β, the statistical uncertainty is reduced by 7.4% and our new result is consistent at the 1σ level. We believe that the improvement in the statistical uncertainty comes from two sources: the improved covariance model, which now utilises the growth rate information present in RSD, and the larger galaxy overdensity sample used in this analysis. Our previous model neglected the information provided by RSD because it boosted the covariance by the average contribution of RSD over all angles on the sky. The improvement from the larger sample is due to two factors: there are more covariance entries because we’ve extended to higher redshift

(Nδg = 1633 cells compared to Nδg = 1036 cells) and a larger number of galaxies per cell at low redshift (which comes from no longer having the volume-limited sample), reducing the shot noise. We suspect that the overall improvement in the statistical uncertainty may also be slightly limited by the fact that we used a larger gridding scale, smoothing over more of the small-scale information. This might be mitigated in future by implementing better methods for eﬃciently inverting large covariance matrices, which we discuss in Sec- tion 6.2. We show a visual comparison of the complete covariance constraints in Figure 4.18.

Multi-Tracer Approaches

As in the previous chapter, we have demonstrated the benefits of using the shared information from multiple tracers, which supports the results of theoretical studies (see Section 1.4 for a review). Again, we compare the improvements we see in the statistical uncer- 4.5. Data Results 133 tainty to those seen by Blake et al. (2013), who presented the first multi-tracer approach applied to galaxy overdensity data from the Galaxy And Mass Assembly (GAMA) survey. In that study, they used two different galaxy overdensity samples with different galaxy bias values, comparing the power spectra of these samples to models, including the cross- power spectrum. Depending on the sample, they found a 10-20% improvement in their constraints of the growth rate when utilising the cross-power spectrum. In our analysis, we see significantly better improvements: 50% improvement compared to the logarithmic distance ratio only sample, and 64% improvement compared to the galaxy overdensity only sample. We believe that this is due, in part, to the strong independent constraints that the logarithmic distance ratio places on the growth rate. When coupled with the fact that the two samples (and their cross-correlation) constrain the growth rate in different ways, we believe that this explains our larger improvement on the growth rate constraint compared to the analysis from Blake et al. (2013), which only utilised RSD information.

Forecasts for the 6dFGS Cross-Covariance Analysis

We also compare our relative constraints to those forecasted for 6dFGS from two studies that use the Fisher matrix formalism: Koda et al. (2014) and Howlett et al. (2016). Both studies use the same RSD model as we do and forecast the relative constraints that can be obtained on fσ8 for various samples of 6dFGS. Koda et al. (2014) use 6dFGSv as the basis for both their galaxy overdensity and logarithmic distance ratio sample, and model the various covariances out to kmax = 1 0.1 h Mpc− . They forecast a 25% constraint on fσ8 for the logarithmic distance ratio auto-covariance and 15% when using the complete covariance. For β, they forecast a 16% constraint when using the complete covariance. We find a relative constraint on fσ8 of 20% for the logarithmic distance ratio auto-covariance and 14% for the complete covariance, and a relative constraint on β of 15% for the complete covariance. In this analysis, we find a constraint from the logarithmic distance ratio auto-covariance that is better than forecast, and note that we also saw this in both our previous analysis and that by Johnson et al. (2014). Again, this could be due to differences between the assumptions that went into the forecasting and our own analysis. We should expect to do better than Koda et al. (2014) because our galaxy overdensity sample goes to a higher redshift (z = 0.1 compared to z = 0.057), which we find to be the case, but only by a single percentage point. Howlett et al. (2016) used the complete 6dFGSv sample as the basis for their logarithmic distance ratio sample and close to the complete 6dFGSz sample as the basis for their galaxy overdensity sample (the upper limit on the redshift for their sample is z = 0.2). 134 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

1 Like Koda et al. (2014), they model the various covariances out to kmax = 0.1 h Mpc− .

They forecast a 25.1% constraint on fσ8 for the logarithmic distance ratio auto-covariance and 11.2% when using the complete covariance. For β, they forecast a 12.3% constraint when using the complete covariance. The similarity of their logarithmic distance ratio auto-covariance constraint to that from Koda et al. (2014) is consistent with the fact that they used very similar samples. The forecast constraints are better than our statistical uncertainties, which is unsurprising given they used a larger galaxy overdensity sample. As a final point, we note that both of these analyses assume that the galaxy overdensity auto-correlation and cross-correlation are responding to the same effective bias. At this stage, it is unclear what effect this might have on the precision of forecasting, since we found that the value of the relative effective bias (parametrized by αb in our study) has a significant effect on the posterior of fσ8.

6dFGS Redshift-Space Distortion and Velocity Results

One of the clear advantages of our approach is that it provides a new method to constrain the growth rate of structure. Consequently, it is informative to compare our results to those from other analyses of 6dFGS. Several RSD analyses have been performed using 6dFGS: Beutler et al. (2012) presented a traditional RSD analysis, Achitouv et al. (2016) looked at RSD in the void-galaxy cross-correlation, and Blake et al. (2018) presented a Fourier-space analysis using the RSD power spectrum multipoles. In addition to these, Johnson et al. (2014) and Huterer et al. (2017) both presented logarithmic distance ratio auto-covariance analyses using the 6dFGSv sample. Our galaxy overdensity auto-covariance analysis is most comparable to the results from Beutler et al. (2012). However, there are some minor differences that should be kept in mind. The galaxy overdensity sample used by Beutler et al. (2012) has a slightly lower magnitude cut (K 12.75 compared to K 12.9), and uses galaxies out to a higher ≤ ≤ redshift range (z . 0.2) than our sample, yielding 81,971 galaxies compared to the 70,467 galaxies used in our analysis. We also note that the Beutler et al. (2012) analysis employs the Feldman-Kaiser-Peacock (FKP; Feldman et al., 1994) weighting scheme to improve their statistical constraints, where we do not. By measuring the correlation function, they found fσ = 0.423 0.055 (13% relative uncertainty) and β = 0.373 0.054 (14% relative 8 uncertainty). These are significantly better (although still consistent at the 1σ level) +0.15 +0.13 than our galaxy overdensity auto-covariance results of fσ8 = 0.41 0.14 and β = 0.30 0.11. − − There are several factors that could explain this: Beutler et al. (2012) use a higher redshift sample, FKP weighting, and have access to smaller-scale information, which we lose by 4.5. Data Results 135 smoothing our model after gridding. We note that in terms of statistical uncertainties, +0.044 our complete covariance constraints of fσ8 = 0.384 0.052 and β = 0.289 0.043 are better − than those from Beutler et al. (2012), although this advantage is lost when considering the combined statistical and systematic uncertainty. As in this work, the analysis by Achitouv et al. (2016) also uses the Beutler et al. (2012) galaxy overdensity sample as a starting point. For their void-galaxy cross-correlation analysis, they take a volume-limited sample out to redshift z = 0.05, similar to the sample selection we made in our first analysis, and implement FKP weighting. They find fσ8 = 0.39 0.11 when fitting to the void-galaxy cross-correlation function, which is consistent with both our galaxy overdensity auto-covariance and complete covariance results at the 1σ level. Even with the lower redshift sample, this work provides a tighter constraint than our galaxy overdensity auto-covariance. We expect that the same factors that we highlighted when comparing to Beutler et al. (2012) are at play, especially the fact that the correlation fitting method may be accessing information on smaller scales. Recently, Blake et al. (2018) presented an RSD analysis that fits to the power spectrum multipoles rather than the correlation function, making it the first Fourier-space analysis of RSD for 6dFGS. We note that they use the same sample as us, the Beutler et al. (2012) sample out to redshift z = 0.1, and they implement FKP weighting. They find fσ = 0.38 0.12, which is consistent with both our galaxy overdensity auto-covariance 8 and complete covariance results at the 1σ level. In their analysis, they highlighted that their larger statistical uncertainty (relative to the standard correlation function analysis from Beutler et al., 2012) was likely due to the correlation function analysis accessing smaller scale information than was available in the multipoles analysis. This is consistent with the interpretation of our own results, and this coupled with the fact that Blake et al. (2018) also used FKP weighting could explain why our statistical uncertainty is slightly larger than theirs but more than double that from Beutler et al. (2012). Our method has been largely informed by that of Johnson et al. (2014), who effectively presented an logarithmic distance ratio auto-covariance analysis of 6dFGSv. They found +0.079 fσ8 = 0.428 0.068, which is consistent with our both our logarithmic distance ratio auto- − covariance and complete covariance results at the 1σ level. We note that their constraint +0.11 is better than our logarithmic distance ratio auto-covariance constraint of fσ8 = 0.53 0.10. − We suspect that this may be due to the fact that Johnson et al. (2014) used a gridding 1 1 scale of 10 h− Mpc where we used 20 h− Mpc. This would result in more covariance entries and potentially lower the statistical uncertainty. Finally, Huterer et al. (2017) performed a very similar analysis to Johnson et al. (2014) 136 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

+0.067 using 6dFGSv, but did not grid their sample. They found fσ8 = 0.481 0.064, which is again − consistent with our both our logarithmic distance ratio auto-covariance and complete covariance results at the 1σ level. We note that the lower absolute statistical uncertainty relative to Johnson et al. (2014) could be to do with the number of entries in the covariance matrix.

We show a visual comparison of our complete covariance constraint of fσ8 to these existing 6dFGS constraints in Figure 4.18.

Density-Velocity Comparison Approaches

While these approaches also take advantage of the shared information between the galaxy overdensity and peculiar velocity fields, they do so in a different way to our method. We refer the reader to Section 1.4.2 for more detail about these methods. We note that the most common method (which is employed by the following studies) is the velocity-velocity comparison, where gravitational instability theory is used to predict the peculiar velocity field directly from the galaxy overdensity field, and then the modelled velocity field is compared to the measured velocity field. The three studies we compare our results to are Pike & Hudson (2005), Davis et al. (2011) and Carrick et al. (2015), which are introduced briefly in Section 3.5.4. We note that all three use variations of the 2-Micron All-Sky Survey (2MASS) for the galaxy overdensity sample and variations of the Spiral Field I- Band survey for the velocity sample. Pike & Hudson (2005) found fσ = 0.44 0.06, Davis 8 et al. (2011) found fσ = 0.32 0.04 and Carrick et al. (2015) found fσ = 0.427 0.027. 8 8 Our complete covariance constraints are consistent with each of these at the 1σ level, both with and without the systematic error, and our statistical uncertainty is between that of Pike & Hudson (2005) and Davis et al. (2011). We show a visual comparison of our complete covariance constraint for fσ8 to these existing velocity-velocity constraints in Figure 4.18. While the consistency with these results is reassuring, we believe that additional work needs to be done to understand the difference in size of the absolute uncertainty between our approach and these more traditional velocity-velocity comparisons. We discuss potential opportunities for this in Section 6.2.

Cross-Correlation Only Analysis

Finally, we compare our constraint of fσ8 to that from Nusser (2017), who presented a ﬁt to the cross-correlation function for galaxy overdensities (from 2MASS) and peculiar velocities (from the Cosmicﬂows-3 catalogue). This is more similar to the analysis by 4.5. Data Results 137

Achitouv et al. (2016) than our analysis in that they model the cross-correlation as a function of separation, similar to how Achitouv et al. (2016) modelled the cross-correlation function between galaxies and voids. They found fσ = 0.40 0.08, which is consistent 8 with our complete covariance constraint at the 1σ level. The construction of this method means they only utilise the equivalent of our cross-covariance, rather than the complete covariance. This explains why we see tighter statistical uncertainties. We show a visual comparison of our complete covariance constraint for fσ8 to this constraint in Figure 4.18.

0.6

0.5 8 fσ

0.4

0.3

δδ ηη C

C C

Nusser (2017)

This Chapter - Previous Chapter

This Chapter - This Chapter - Blake et al. (2018) Davis et al. (2011)

Beutler et al. (2012) Carrick et al. (2015) Huterer et al. (2017) Johnson et al. (2014) Achitouv et al. (2017) Pike & Hudson (2005)

Figure 4.18 Median values (solid bar) and 68% credible interval (shaded region) of fσ8 for this chapter (shown in blue) and the previous chapter (shown in light green). Other works utilising 6dFGS are shown in purple, velocity-velocity comparisons are shown in red and the cross-correlation only analysis is shown in orange. 138 Chapter 4. Joint Growth Rate Measurements from Peculiar Velocities and RSD

4.6 Summary

We have presented a significant advancement on our previous analysis by including redshift- space distortions in our self-consistent model of the auto- and cross-covariance for the galaxy overdensity and peculiar velocity fields, thereby testing whether the same growth rate drives the amplitude of peculiar velocities and RSD. We have also advanced our previous analysis by testing our approach on a suite of realistic 6dFGS mock catalogues. We have performed a detailed analysis of how various model systematics affect our final growth rate constraint from 6dFGS and have provided a systematic error estimate in addition to our statistical uncertainty. Our constraints from the complete covariance model are fσ = 0.384 0.052(stat) 8 +0.044 0.061(sys) for the growth rate of structure and β = 0.289 0.043(stat) 0.049(sys) for the − redshift-space distortion parameter. We found that the statistical uncertainties were reduced by 64% when compared to the galaxy overdensity auto-covariance only constraint and 50% when compared to the logarithmic distance ratio auto-covariance only constraint.

Our current analysis provides an 18% improvement on the statistical uncertainty in fσ8 compared to our previous analysis. We believe this improvement is driven both by the improved model, which captures the information on the growth rate of structure encoded in the galaxy overdensity field through RSD, and by the use of a larger galaxy overdensity sample. The fact that our systematic uncertainties are larger than our statistical uncertainties is primarily driven by the degeneracy between the growth rate and the relative effective bias between the galaxy overdensity auto-covariance and cross-covariance, which we parametrized as αb. We anticipate that this could be mitigated by improving our underlying bias model to account for the fact that the cross-covariance is sensitive to a different effective bias than the galaxy overdensity auto-covariance. We found that our constraint is consistent with the ΛCDM prediction of fσ8 from the Planck 2015 cosmological parameters, as well as multiple analyses of galaxy overdensities and peculiar velocities from 6dFGS. This validates our method as a new approach for constraining fσ8 from large-scale structure and peculiar velocities. As with our previous analysis, we see obvious improvements in the statistical uncertainty when utilising the cross-covariance compared to either auto-covariance alone, or the naive constraint that one achieves by treating the two fields as independent. Once again, this supports the findings from the various theoretical studies on multi-tracer analyses, where accessing cross-correlations improves constraints. Our results also motivate additional improvements and refinements, as well as the application of this method to future large-scale structure and peculiar velocity surveys such as Taipan. 5 Forecasts for the Taipan Galaxy Survey

5.1 Overview

As highlighted in the previous chapter, we have developed a unique method that constrains the growth rate of structure by self-consistently modelling peculiar velocities, redshift- space distortions and their cross-correlation. Our success in applying the model to the 6-degree Field Galaxy Survey (6dFGS) naturally leads us to consider other surveys to which the model might be applied, with none so obvious as the Taipan Galaxy Survey. Designed as a successor to 6dFGS, Taipan stands to provide tight constraints on the growth rate of structure through its large peculiar velocity and galaxy redshift samples. Specifically, Taipan is forecast to constrain the growth rate to 4.5% after the first year of observations and 2.7% by the end of the survey (da Cunha et al., 2017). Given that these forecasts assume the use of all available information from the peculiar velocities and redshift-space distortions (including their cross-correlation), our method is well placed to provide these constraints once the data becomes available. These forecasts have been calculated using the Fisher matrix, a statistical tool that estimates the tightest possible constraints for a given set of observations. Consequently, it is important to ask how well our method performs relative to the Fisher matrix forecasts, and how this might affect the final constraints we are likely to achieve with Taipan. The aim of this chapter is to run our method on Taipan mocks and compare the results to the Fisher matrix forecast for an equivalent sample. Although we compared results and forecasts for 6dFGS in the previous chapters, we noted that the comparison was not always fair due to differences in the samples and assumptions; in this chapter, we aim to make the comparison as fair as possible. We begin with a description of Taipan and our mock sample in Section 5.2. We then introduce the theory of Fisher matrix forecasts and describe the construction of the Fisher

139 140 Chapter 5. Forecasts for the Taipan Galaxy Survey matrix and model covariance for our analysis in Section 5.3. We then present the results from the Fisher matrix forecast for our sample in Section 5.4, followed by the results from applying our covariance method to the mocks in Section 5.5. We analyse our results and compare them to the literature in Section 5.6, before concluding in Section 5.7.

5.2 The Taipan Galaxy Survey

The Taipan Galaxy Survey is a spectroscopic survey of the low-redshift universe that will be conducted on the UK Schmidt Telescope at Siding Spring Observatory. Scheduled to begin in March 2019, the survey will measure galaxy redshifts and peculiar velocities over a period of 4 years. The principal cosmological aims of the survey are precise constraints on the growth rate of structure fσ8 through measurements of redshift-space distortions and peculiar velocities, and the Hubble constant H0 through measurements of low-redshift baryon acoustic oscillations. Both constraints will enable scientists to test viable models of dark energy and modiﬁed gravity, which will lead to better understanding of the underlying cause of accelerating expansion. Taipan will be broken up into two phases:

Phase 1: Conducted during the ﬁrst year, Phase 1 aims to measure 300,000 galaxy • redshifts in the redshift range z . 0.2 and 33,000 peculiar velocities in the redshift range z . 0.1.

Final: By the end of the survey, the aim is to have measured 2 million galaxy • redshifts in the redshift range z . 0.4 and 50,000 peculiar velocities in the redshift range z . 0.1.

Both the galaxy redshift and peculiar velocity sample have been designed to extend 6dFGS, which is evident from the redshift distributions shown in Figure 5.1. The number density for the galaxy redshift sample highlights that Taipan is aiming to have a relatively constant number density out to higher redshifts than 6dFGS, which is achieved through a red colour selection; this design choice is a requirement of the baryon acoustic oscillation analysis, where the Final sample will enable a 1% measurement of H0. In terms of peculiar velocities, Taipan’s key contribution over 6dFGS is in the measurement of peculiar velocities beyond the redshift range of 6dFGSv, but we note that a signiﬁcant number of measurements will also added in the 6dFGSv redshift range (z 0.057). This ≤ improvement is largely enabled by the improved spectrograph, which will allow the survey 1 to measure velocity dispersions to 70km s− , compared to the 6dFGS velocity dispersion 5.2. The Taipan Galaxy Survey 141

Figure 5.1 The redshift distributions of the galaxy redshift sample (left) and peculiar velocity sample (right) for 6dFGS and both Taipan phases. The purple dotted line in the left-hand panel represents the number density of galaxies required to balance sample variance and shot noise for the baryon acoustic oscillation (BAO) measurement. (Credit: da Cunha et al., 2017).

1 limit of 112km s− . The inclusion of lower-dispersion galaxies signiﬁcantly increases the peculiar velocity sample size relative to 6dFGSv. It also allows peculiar velocities to be measured to higher redshifts, since the more accurate velocity dispersion measurements result in smaller scatter in the Fundamental Plane relation, leading to smaller distance uncertainties.

5.2.1 Taipan Mock Catalogues

For this study, we used mock catalogues of the Taipan galaxy redshift and peculiar velocity samples developed by Cullan Howlett for the Taipan Collaboration. We utilised eight mocks, each drawn from a single GADGET N-body simulation of 25603 particles in a 1 box with side-length of 1800 h− Mpc. The simulation was run with the following fiducial 2 cosmological parameters: the physical baryon density Ωbh = 0.02226, the physical dark 2 matter density Ωch = 0.1193, the reduced Hubble constant h = 0.6751, the scalar spectral index n = 0.9653, the scalar amplitude A = 2.130 10 9, the reionization optical depth s s × − τ = 0.063, and the derived value for the amplitude of matter fluctuations σ8 = 0.8150. The 1 observers for each mock are separated by 900 h− Mpc and the mocks have a maximum redshift z = 0.1; each mock is then a spherical volume of radius 300 h 1 Mpc, mean- ∼ − 1 ing there is at least 300 h− Mpc between each mock. While not perfectly independent, the separation is well beyond the scale at which peculiar velocities are sensitive to the overdensity field, so we treat the mocks as independent for the purposes of this analysis. 142 Chapter 5. Forecasts for the Taipan Galaxy Survey

The angular mask from 6dFGS was applied to each mock, such that the mocks cover the southern sky, with dec < 0 degrees and galactic latitude b > 10 degrees (which excludes | | the galactic plane).

The mock samples were generated by populating the N-body haloes with mock galaxies using subhalo abundance matching (SHAM; e.g. Conroy et al., 2006). This process involves assigning luminosities to haloes and subhaloes based on their properties; in this case, the halo with the largest circular velocity is assigned the highest luminosity, and so forth. Each halo and subhalo is assigned a single galaxy, with those assigned to haloes designated as central galaxies, and those assigned to subhaloes as satellite galaxies. Each mock galaxy in the peculiar velocity sample was assigned a value for the three Fundamental Plane parameters using the distribution described for 6dFGSv (Springob et al., 2014), which was then used to calculate the luminosity. The luminosities for the galaxy redshift sample were drawn from the Galaxy and Mass Assembly survey (GAMA) luminosity function (table 4 in Driver et al., 2012). We note that the listed J-band magnitudes for the luminosity function are AB magnitudes, which were converted to Vega magnitudes using table 1 in Blanton & Roweis (2007). J-band magnitude cuts were applied to both samples to match the proposed Taipan selection (see table 2 in da Cunha et al., 2017): JVega < 15.4 for the galaxy redshift sample, and JVega < 14.0 for the peculiar velocity sample. We converted the simulated peculiar velocities from the mock to logarithmic distance ratios using Eq. 2.74, and added mock measurement uncertainties by adding a random oﬀset drawn from the normal distribution (0, σ = 0.1), which is the same procedure we N obs used in the previous analysis (see Section 4.2.2). We note that we have kept the same observational uncertainty estimate as in the previous analysis; this corresponds to the conservative assumption that the logarithmic distance ratio uncertainties from Taipan will be similar to those from 6dFGSv. We subsampled the mocks to match the expected number densities of the Final phase samples. Finally, we gridded the mocks to obtain the overdensity and average logarithmic distance ratio in each cell, calculating the expected number of galaxies in each cell Nexp using random catalogues with the same angular and radial selection functions as the mocks. Given the increased redshift range for the 1 logarithmic distance ratio sample, we used a gridding scale of L = 30 h− Mpc for both this and the galaxy overdensity sample, which kept the covariance matrix for each mock at a reasonable size of 3450 3450 (for mock 1, N = 1636 and N = 1814, with similar ∼ × δ η numbers of cells in the other mocks). 5.3. Theory and Methodology 143

5.2.2 Sample for Analysis

We note that there are a number of key differences between the sample we use for our analysis and the specifications of the two Taipan phases. Although the mocks were designed using the Phase 1 specifications, we note that the number densities are slightly lower than expected for Taipan (particularly below redshift z = 0.02), which can be attributed to the mass-resolution limit of the simulation. We also note that the Taipan peculiar velocity sample selections listed by da Cunha et al. (2017) were made using an r-band cut, but the mocks were generated from a J-band cut due to the use of a J-band Fundamental Plane relation. While care was taken to match the number density of the mocks to the Phase 1 specification, the mock samples contain between between 40,000 and 45,000 peculiar velocities as opposed to the expected 33,000 for Phase 1. We also note that the redshift sample for the mocks can only be used out to redshift z = 0.1, rather than the maximum redshift for Phase 1 of z = 0.2. This is because we wish to treat the mock samples as independent; using a higher redshift would cause the mocks to have overlapping volumes. Consequently, the mock galaxy redshift samples only have between 145,000 and 155,000 galaxies out to redshift z = 0.1, as opposed to the approximately 300,000 expected out to redshift z = 0.2 for Phase 1. Given these factors, the mock samples we use don’t match the expected samples for Phase 1 or the Final phase. However, this is not a limitation for our analysis, since we are most interested in the comparison between Fisher matrix forecasts and the covariance method we’ve developed, rather than whether the Taipan mocks recover the forecasts in da Cunha et al. (2017). Consequently, having a well-understood, Taipan-like sample is the most important requirement for our analysis, which is what we’ve constructed from the simulation.

5.3 Theory and Methodology

5.3.1 Fisher Matrix Forecasts

The Fisher matrix method was designed to cheaply forecast the precision with which an experiment could constrain model parameters. Mathematically, the Fisher matrix is deﬁned as

∂2[ ln( )] Fij = − L , (5.1) ∂φi∂φj where is the likelihood function (defined in Eq. 2.2) and φ are the model parameters. L The Fisher matrix is defined as the expectation value of the likelihood second derivatives at 144 Chapter 5. Forecasts for the Taipan Galaxy Survey the maximum-likelihood point; this makes it an estimate of how quickly the likelihood falls off from the maximum-likelihood point, which is indicative of the function’s width. The Fisher matrix is an N N symmetric matrix for N model parameters. Once calculated, × one may invert the Fisher matrix to get the covariance matrix of the model parameters; the diagonal entries of the covariance matrix then give the expected variance in the model 2 parameters σφ. Importantly, the estimate of σφ from the Fisher matrix is the best you can possibly do for the specified observations; this is clearly expressed by the Cramér-Rao 2 bound which states σ 1/Fii. φi ≥ Introduced to astrophysics in the late 90s (Tegmark et al., 1997; Tegmark, 1997), the Fisher matrix forecast method has become a prominent part of cosmological survey design (e.g. Albrecht et al., 2009). For large-scale structure analyses, key work has shown that the Fisher matrix can be written in terms of modelled power spectra (Burkey & Taylor, 2004) or Gaussian density fields (McDonald & Seljak, 2009). In this study, we use the prescription featured in Koda et al. (2014) and the corresponding Fisher matrix forecast code1. Given the form of our likelihood function, one may evaluate Eq. 5.1 to find

T ∂∆ 1 ∂∆ 1 1 ∂C 1 ∂C Fij = C− + tr C− C− , (5.2) ∂φi ∂φj 2 ∂φi ∂φj where ∆ is the data vector, C is the covariance matrix, T represents the vector transpose operation, and tr represents the matrix trace operation. This relation shows the general case where both the data vector and covariance matrix may depend on the model parameters. For our model, the data vector contains the measured logarithmic distance ratios η and galaxy overdensities δg, and the model parameters are primarily the growth rate of structure fσ8 and the redshift-space distortion parameter β (the stochastic velocity σv and additional bias badd also feature as model parameters for sub-matrices of the covariance). Following work by McDonald & Seljak (2009) and Abramo & Leonard (2013), Koda et al. (2014) highlighted that the Fisher matrix can be written in terms of model power spectra by utilising the Fourier transform:

Z 3 3 " ˜ ˜ # 1 d xd k ˜ 1 ∂Σ ˜ 1 ∂Σ Fij = 3 tr Σ− Σ− . (5.3) 2 (2π) ∂φi ∂φj

In this expression, Σ˜ is a matrix of power spectra and errors that depend on the wavevector

1https://github.com/junkoda/vﬁsher 5.3. Theory and Methodology 145 k and position vector x respectively:

  2 Pgg(k) + σgg(x) Pgv(k) Σ(˜ k, x)   , (5.4) ≡ 2 Pvg(k) Pvv(k) + σvv(x) where the power spectra are the anisotropic power spectra (Eq. 2.101 to 2.104). The error 2 2 terms in the covariance, given by σgg(x) and σvv(x), take similar forms to those described in Section 2.3.3:

2 1 σgg = ng− , (5.5) 2 1 2 2 σvv = nv− (σv + σobs), (5.6) where the variance for the galaxy overdensity is given by the shot noise, and the variance for the peculiar velocities is the sum of the stochastic velocity and observational uncertainty from the distance indicator, multiplied by the shot noise. For the Fisher matrix, the observational uncertainty is modelled as

σ = H x , (5.7) obs 0| | where is the fractional uncertainty in the distance indicator and x is the distance to | | the galaxy. We note that the Fisher matrix model is in terms of peculiar velocities, rather than logarithmic distance ratios; this leads to the absence of the conversion factor κ from the Fisher matrix model (see Section 2.3.1 for the deﬁnition of κ). We note that this does not aﬀect our ability to compare the forecasts to our model covariance results, since both constrain the growth rate of structure.

To calculate the uncertainties in the cosmological parameters of interest, namely fσ8 and β, one must specify the model power spectra and uncertainties, invert the Fisher matrix and read oﬀ the forecast uncertainties in the model parameters.

5.3.2 Fisher Matrix Set-Up

We now specify the fiducial inputs for the Fisher matrix forecasts. The inputs are designed to match the specifications of our mocks, as well as the fiducial model choices we make for the covariance matrix model in the following section. We set the maximum redshifts for both samples as zδg = zv = 0.1, following the selections we described in Section

5.2.2. The power spectra are generated using the ﬁducial cosmology for the mocks; Pmm 146 Chapter 5. Forecasts for the Taipan Galaxy Survey

2 is generated using CAMB, and Pmθ and Pθθ are generated using Copter , a code that implements renormalized perturbation theory (Carlson et al., 2009). For the uncertainties, we use the average number density from all eight mock samples to calculate the shot noise 1 1 1 contributions ng− and nv− and we set σv = 300km s− . Given that the Fisher matrix is modelled in terms of peculiar velocities, we need to convert the logarithmic distance ratio error of σobs,η = 0.1 that we used for the mocks to a peculiar velocity error, speciﬁcally in terms of , which is the fractional uncertainty of the distance indicator (see Eq. 5.7). Recalling that the two quantities are related via κ (see Eq. 2.74), we calculate

σobs,η = κσobs,v = κH x 0| | 1 (1 + z) = H x . (5.8) ln(10) D(z)H(z) 0| |

At low redshifts, we can cancel the Hubble parameter and distance terms, and ignore the (1 + z) factor, leaving the approximation

σ , (5.9) obs,η ≈ ln(10) which gives the distance indicator fractional uncertainty value of = 0.23. We note that this is slightly higher than the anticipated distance indicator uncertainty expected for Taipan of 0.20, but this is consistent with our conservative choice for the mock logarithmic distance ratio uncertainty of σobs,η = 0.1. We set the damping parameters for the power spectrum models to be the same as the fiducial values in the previous chapter: σg = 1 1 3.0 h− Mpc and σu = 13.0 h− Mpc. We use a fiducial galaxy bias value of b = 1.12 and set the cross-correlation coefficient to rg = 1. Finally, we set the maximum wavenumber as 1 kmax = 0.15 h Mpc− , noting that the Fisher matrix model doesn’t include the additional bias factor badd that we introduced to the covariance model in Section 3.3.3.

Together, these settings ensure that the Fisher matrix forecast is for a hypothetical sample that closely matches the mock sample that we have constructed, while reﬂecting the model assumptions we’ve made and accounting for diﬀerences between our model and the Fisher matrix formalism developed by Koda et al. (2014).

2http://mwhite.berkeley.edu/Copter/ 5.4. Fisher Matrix Results 147

5.3.3 Covariance Matrix Model

We now cover the assumptions and settings for our covariance matrix model. Much of this is unchanged from the choices discussed in Section 4.3. Our covariance model is described by Eq. 4.2 to 4.5, which includes our prescription for redshift-space distortions. We generate the power spectra with CAMB and velMPTbreeze (see Section 2.2.4) using the fiducial cosmological parameters for the mock sample. We use the same wavenumber 1 1 bounds for the integrals as in Chapter 4: kmin = 0.0025 h Mpc− , kmax = 0.15 h Mpc− 1 and kadd = 1.0 h Mpc− (we include badd as a free parameter based on our analysis in the previous chapter). We include αb in the cross-covariance model, which parametrizes the difference between the effective galaxy bias for the galaxy overdensity and the cross- covariance (see Section 4.3.3). Given that we have significantly altered the sample, we test different values of αb for a single representative mock to establish the most appropriate value for our sample, then use this when running the complete covariance model on all eight mocks. We again perform our MCMC sampling using emcee, testing convergence with the Gelman-Rubin statistic (see Section 4.3.4), and use the same linear algebra library to solve the likelihood function as the previous analysis (see Section 4.3.5). As for fiducial values for model parameters, we use the same settings as the previous chapter for 1 1 the damping parameters (σg = 3.0 h− Mpc and σu = 13.0 h− Mpc) and set the cross- correlation coefficient to rg = 1.

5.4 Fisher Matrix Results

We begin by presenting the forecast uncertainties in the growth rate of structure fσ8 and redshift-space distortion parameter β for the three types of analyses we performed in the previous chapter: using the galaxy overdensity auto-covariance only, the peculiar velocity auto-covariance only, and the complete covariance. These are shown in Table 5.1. When going from the galaxy overdensity auto-covariance to the complete covariance, the

Table 5.1 Fisher matrix forecasts of the fractional uncertainties (in %) on fσ8 and β for our mock Taipan sample using the galaxy overdensity auto-covariance (Cδδ), the peculiar velocity auto-covariance (Cvv), and the complete covariance (C). Model fσ8 β

Cδδ 12.7 14.3

Cvv 15.0 – C 6.7 7.5 148 Chapter 5. Forecasts for the Taipan Galaxy Survey forecast anticipates a 47.2% reduction in the uncertainty on fσ8 and a 47.6% reduction in the uncertainty on β. When going from the peculiar velocity auto-covariance to the complete covariance, the reduction in uncertainty on fσ8 is 55.3%. These improvements are consistent with the results we’ve presented throughout this thesis. We can see how much inﬂuence the sample has on the forecasts by comparing the forecasts for our sample to those for the ideal Phase 1 sample from da Cunha et al. (2017). Figure 1 from da Cunha et al. (2017) shows approximate percentage uncertainties on the growth rate of structure for the Phase 1 sample; we note that they forecast 5% when ∼ using the galaxy overdensity sample, 10% when using the peculiar velocity sample, and ∼ 4.5% when using both samples and their cross-correlation. The fact that our mock galaxy redshift sample is limited to redshift z = 0.1 explains why they see a tighter constraint; the Phase 1 sample extends to redshift z = 0.2 and is expected to contain around double the number of galaxies as our sample. Interestingly, even though our sample contains more peculiar velocities than the expected Phase 1 sample, we see a larger fractional uncertainty in the growth rate. This is likely due to the fact that we use a more conservative fractional uncertainty estimate for the distance indicator than da Cunha et al. (2017) (23% compared to 20%), showing how important this factor is in achieving tight constraints with peculiar velocities. These diﬀerences explain why our complete covariance forecast of 6.7% is larger than the equivalent Taipan Phase 1 forecast of 4.5%.

5.5 Model Covariance Results

We now present the results of applying our model covariance method to our mock samples. Similarly to the forecasts, we present the results for the galaxy overdensity auto-covariance model, the logarithmic distance ratio auto-covariance model, and the complete covariance model. For each case, we present the marginalised constraints on the free parameters for all eight mocks, noting that the parameters have the same degeneracies seen in the previous chapter (see Figure 4.2 for the galaxy overdensity auto-covariance, Figure 4.8 for the logarithmic distance ratio auto-covariance, and Figure 4.13 for the complete covariance).

Throughout this section, we expect to recover the ΛCDM + GR prediction for fσ8, which is fσ8 = 0.429 for the ﬁducial cosmological parameters.

5.5.1 Galaxy Overdensity Auto-Covariance

We present the marginalised constraints for all eight mocks in Figure 5.2. Six out of eight mocks recover the ﬁducial growth rate of structure within the 68% credible interval and 5.5. Model Covariance Results 149

fσ8 β baddσ8 Mock 1 Mock 2 Mock 3 Mock 4 Mock 5 Mock 6 Mock 7 Mock 8

0.2 0.4 0.6 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Figure 5.2 The median values and 68% credible intervals of fσ8, β and baddσ8 for eight Taipan mocks when using the galaxy overdensity auto-covariance. The expected value for fσ8 is shown by the dashed vertical line. the size of the uncertainty for the growth rate is consistent across all mocks. There doesn’t appear to be any signiﬁcant bias in the recovery of fσ8, unlike what we saw when working with the 6dFGS mock catalogues in the previous chapter. This is likely because these mock samples contain satellites and centrals, meaning our chosen σg value is reasonable, which was not the case in the previous chapter. We also note that the relative positions of fσ8 and β across the mocks indicate that the mocks have consistent galaxy bias values. The ﬁts to all eight mocks had reasonable χ2/dof values (all between 1.00 and 1.01), indicating that the mock data was well predicted by the model. For ease of comparison to the forecasts, we calculate the average fσ8 and β values with average uncertainties: +0.115 +0.145 fσ8 = 0.453 0.113 (with a fractional uncertainty of 25.1%) and β = 0.485 0.131 (with a − − fractional uncertainty of 28.5%).

5.5.2 Logarithmic Distance Ratio Auto-Covariance

We present the marginalised constraints for all eight mocks in Figure 5.3. As with the galaxy overdensity auto-covariance fit, six out of the eight mocks recover the fiducial growth rate of structure within the 68% credible interval and we see consistently sized uncertainties for all eight growth rate constraints. The fits to all eight mocks had reasonable χ2/dof values (all between 0.95 and 1.02), indicating that the mock data was reasonably well predicted by the model. Again, we calculate the average fσ8 value with average +0.067 uncertainties: fσ8 = 0.408 0.062 (with a fractional uncertainty of 15.8%). − 150 Chapter 5. Forecasts for the Taipan Galaxy Survey

1 fσ8 σv (km s− ) Mock 1 Mock 2 Mock 3 Mock 4 Mock 5 Mock 6 Mock 7 Mock 8

0.3 0.4 0.5 0.6 50 100 150

Figure 5.3 The median values and 68% credible intervals of fσ8 and σv for eight Taipan mocks when using the logarithmic distance ratio auto-covariance. The expected value for fσ8 is shown by the dashed vertical line.

5.5.3 Complete Covariance

As in Chapter 4, we test different values of αb for a single representative mock to establish the value that gives the best recovery. Based on the results for the galaxy overdensity auto-covariance and logarithmic distance ratio auto-covariance models, we select Mock 7 as our representative mock. The parameter posteriors for various αb values are shown in Figure 5.4. We find that αb = 0.85 provides the best recovery of the fiducial growth 2 rate with a reasonable χ /dof of 0.97. We see similar behaviour to our αb test in the previous chapter (see Figure 4.13): the posterior on the growth rate widens and appears to converge as αb is lowered. Interestingly, the preferred αb value for the Taipan mock is lower than for the 6dFGS mocks used in the previous chapter. Given that our Taipan mocks contain peculiar velocities at higher redshifts, one might expect that these high- redshift peculiar velocities would be tracing the most massive structures, increasing the effective galaxy bias of the cross-correlation compared to the 6dFGS mocks and hence the value of αb. However, we believe that the value of αb could be affected by the uncertainty of the peculiar velocity tracers, and that the low-redshift velocities with small errors are more influential in determining the effective bias than the high-redshift velocities with large errors. It could also be affected by the Taipan mocks having a greater number density of galaxy redshifts towards redshift z = 0.1 than the 6dFGS mocks, since these will predominantly be high-luminosity galaxies. The presence of these galaxies will increase the effective galaxy bias of the galaxy overdensity sample and lower the value of αb. These additional influences on the behaviour of αb further motivate a more rigorous model for 5.5. Model Covariance Results 151

αb = 0.95 αb = 0.90 αb = 0.85 αb = 0.80 αb = 0.75 αb = 0.70

240 ) 1 − 160 (km s

v 80 σ

0.6 β 0.4

0.2 1.50

1.25 8 σ

add 1.00 b

0.75

0.30 0.45 0.60 80 160 240 0.2 0.4 0.6 0.75 1.00 1.25 1.50 1 fσ8 σv (km s− ) β baddσ8

Figure 5.4 The posteriors of fσ8, σv, β and baddσ8 for the complete covariance model ﬁt to Mock 7 for diﬀerent values of αb. The inner contour indicates the 68% credible region and the outer contour indicates the 95% credible region. The expected value for fσ8 is shown by the dashed vertical line. 152 Chapter 5. Forecasts for the Taipan Galaxy Survey

1 fσ8 σv (km s− ) β baddσ8 Mock 1 Mock 2 Mock 3 Mock 4 Mock 5 Mock 6 Mock 7 Mock 8

0.35 0.40 0.45 0.50 0.55 50 100 150 0.4 0.5 1.0 1.2 1.4

Figure 5.5 The median values and 68% credible intervals of fσ8, σv, β and baddσ8 for eight Taipan mocks when using the complete covariance with αb = 0.85. The expected value for fσ8 is shown by the dashed vertical line. the eﬀective galaxy bias of the cross-correlation, which we discuss in Section 6.2.

Given our choice to set αb = 0.85, we run the complete covariance model for all eight mocks and present the marginalised parameter constraints in Figure 5.5. The fiducial growth rate is recovered within the 68% credible interval for seven of the eight mocks, with no apparent bias. The fits to all eight mocks had reasonable χ2/dof values (all between 0.97 and 1.01), indicating that the each separate mock is a reasonable fit to the overall model. As with the previous two model fits, we calculate the average constraints across the mocks for ease of comparison with the forecasts: fσ = 0.422 0.044 (with 8 +0.054 a fractional uncertainty of 10.4%) and β = 0.448 0.052 (with a fractional uncertainty of − 11.7%).

5.6 Discussion of Results

5.6.1 Comparison of Fisher Matrix Forecasts to Model Covariance

We provide a summary of the results from both methods in Table 5.2. The largest discrepancy between the forecast and measured fractional uncertainties is for the galaxy overdensity auto-covariance analysis. The average fractional uncertainty from the mocks is 2.0 ∼ times the forecast uncertainty for both fσ8 and β. We believe this supports the conclusion we drew in the previous chapter that our galaxy overdensity auto-covariance model is not utilising all of the information available from redshift-space distortions. This is because the Fisher matrix provides constraints under the assumption that all available information is extracted from the data. We’ve established in the previous chapter that our method misses some information for two reasons. Firstly, the data are gridded, which results in a 5.6. Discussion of Results 153

Table 5.2 Results from the Fisher matrix and covariance model methods for our mock Taipan sample using the galaxy overdensity auto-covariance (Cδδ), the peculiar velocity (or logarithmic distance ratio) auto-covariance (Cvv or Cηη), and the complete covariance (C). For the Fisher matrix, we present the fractional uncertainties on the growth rate

σfσ8 and redshift-space distortion parameter σβ as percentages. For the covariance model, we present the average median values (with 68% credible intervals) for all eight mocks, in addition to the equivalent fractional uncertainties.

Method Model fσ8 σfσ8 (%) β σβ(%)

Fisher Matrix Cδδ – 12.7 – 14.3

Cvv – 15.0 – – C – 6.7 – 7.5 +0.115 +0.145 Covariance Model Cδδ 0.453 0.113 25.1 0.485 0.131 28.5 +0− .067 − Cηη 0.408 0.062 15.8 – – − +0.054 C 0.422 0.044 10.4 0.448 0.052 11.7 − loss of small-scale information when the covariance is multiplied by the gridding function 2 1 Γ (k, L) (see Eq. 2.85). For our gridding scale (L = 30 h− Mpc) and maximum fitted 1 2 wavenumber (kmax = 0.15 h Mpc− ), we note that Γ (kmax,L) = 0.15, indicating that the galaxy overdensity auto-covariance signal is strongly damped at this scale. Secondly, the data are not weighted with the Feldman-Kaiser-Peacock weighting scheme (Feldman et al., 1994), which is typically used to get optimal constraints from redshift-space distortions. We believe this loss of information explains the difference between the forecast constraints and those measured from the mocks, which further motivates various improvements to our method, discussed in Section 6.2. For the peculiar velocity auto-covariance analysis, we find only a small discrepancy between the forecast and the measured fractional uncertainties. The average fractional uncertainty from the mocks is only 1.05 times the forecast uncertainty for fσ8. This result speaks to the power of our method to utilise information from the peculiar velocity field. Because peculiar velocities trace larger cosmological scales than redshift-space distortions (discussed in Section 1.2.5), they are significantly less affected by the gridding required to generate the covariance, meaning our method recovers more of the available information. Finally, for the complete covariance, we find that the difference between the forecast and measured fractional uncertainties is between those for the two auto-covariance analyses. The average fractional uncertainty from the mocks is 1.55 times the forecast ∼ uncertainty for both fσ8 and β. Most encouragingly, the fact that the complete covariance result is closer to the Fisher matrix forecast than the galaxy overdensity auto-covariance result was shows that potential constraints using our method are not entirely limited by 154 Chapter 5. Forecasts for the Taipan Galaxy Survey the lower-performing galaxy overdensity auto-covariance component.

5.6.2 Comparison to Similar Analyses

While Fisher matrix forecasts are primarily used when designing cosmological surveys, several studies have compared forecasts to analysis methods to assess their accuracy. Wolz et al. (2012) presented a comparison of Fisher matrix forecasts to parameter constraints from MCMC posterior likelihood estimation. In particular, they worked with the specifications for future Dark Energy Task Force surveys of type Ia supernovae, baryon acoustic oscillations and weak lensing (Albrecht et al., 2006), running their MCMC estimation method on mock data that mimicked the proposed surveys. The study specifically focused on constraining the dark energy equation of state parameter and found that the Fisher matrix underestimated the fractional uncertainty of this parameter if the data likelihood was non-Gaussian or if there were significant parameter degeneracies that could not be broken by the data. They found that this was especially the case for geometric probes like type Ia supernovae and baryon acoustic oscillations, where the Fisher matrix underestimated the uncertainties in the equation of state by 30-70%. The underestimation for weak lensing was significantly lower, at the level of 5%. They reported that this was likely due to the fact that weak lensing probes both geometric and growth rate information, which broke parameter degeneracies. Font-Ribera et al. (2014) also presented comparisons between Fisher matrix forecasts and both existing and upcoming surveys. Most relevant to our work, they compared a Fisher matrix forecast for the Baryon Acoustic Oscillation Spectroscopic Survey Data Release 9 sample (BOSS-DR9) to the existing redshift-space distortion (RSD) analysis for that sample, presented by Reid et al. (2012). They forecast that an ideal RSD analysis would provide a 7% constraint on the growth rate of structure fσ8, and noted that Reid et al. (2012) found an 8% constraint. Both studies provide useful insight into our results. Our roughly 50% underestimation of the growth rate from the galaxy overdensity auto-covariance is consistent with the finding from Wolz et al. (2012) that the Fisher matrix may underestimate model parameters by anywhere between 30% and 70%. However, we believe the reasons for the underestimation are different in our case, in that it comes from loss of information on small scales rather than non-Gaussian likelihoods or parameter degeneracy. This is supported by the fact that Font-Ribera et al. (2014) found a much closer result between forecast and measurement than we did for our galaxy overdensity auto-covariance analysis. This confirms that traditional RSD analyses are capable of accessing the majority of the information 5.6. Discussion of Results 155 available and it is the loss of information in our method that is responsible for the underestimation. Like these two studies, our results support the use of the Fisher matrix in cosmological survey design, but such designs should account for the fact that the forecasts are idealistic and are unlikely to be realised when it comes to analysing collected data.

5.6.3 Comparison to 6dFGS Results

Given that our Taipan mocks extend the peculiar velocity and galaxy redshift samples of the 6-degree Field Galaxy Survey (6dFGS) used in the previous chapter, it’s informative to compare constraints. In the previous chapter, our constraint for the galaxy overdensity +0.15 auto-covariance model was fσ8 = 0.41 0.14, which corresponds to a fractional uncertainty − of 35%. In terms of comparing the reduction in absolute uncertainty, the average ∼ fσ8 constraint for our Taipan mocks had an absolute uncertainty of 0.114, which is a reduction of 21% compared to our 6dFGS result. For the logarithmic distance ratio auto- +0.11 covariance, our 6dFGS constraint was fσ8 = 0.53 0.10 (20% fractional uncertainty); the − average absolute uncertainty in our fσ8 constraint from Taipan mocks was 0.065, which is a reduction of 38%. Finally, for the complete covariance, our 6dFGS constraint was fσ = 0.384 0.052 (14% fractional uncertainty); the average absolute uncertainty in our 8 Taipan mocks was 0.044, which is a reduction of 15%. Given that the peculiar velocity sample for the Taipan mocks signiﬁcantly extends the 6dFGS sample, it is unsurprising that we see the greatest improvement in this analysis. From the galaxy overdensity auto- covariance analysis, we can see that increasing the number of observations does improve the constraints, since both samples have the same maximum redshift, but the Taipan mocks contain just over double the number of galaxies in the 6dFGS sample.

5.6.4 Implications for Taipan

Finally, we estimate realistic forecasts if our method were to be applied to the Taipan Phase 1 and Final samples. We do this by scaling the forecasts from da Cunha et al. (2017) by the diﬀerence we found between our Fisher matrix forecast and the constraints from the complete covariance model. For the complete covariance analysis, we found that the constraint from the model had uncertainties that were 1.55 times those from the equivalent forecasts. Consequently, we estimate that Taipan Phase 1 could provide a 7.0% constraint on the growth rate of structure and that Taipan Final could provide a 4.2% constraint (based on the original forecasts of 4.5% and 2.7% respectively). Even with the loss of information from gridding the galaxy overdensity sample, our results suggest that applying our method to the Final Taipan sample would still yield the survey’s overall 156 Chapter 5. Forecasts for the Taipan Galaxy Survey goal of at least a 5% constraint on the growth rate of structure. We note that this is 1 based on the same gridding scale (L = 30 h− Mpc) being applied to the data and that the constraints could improve or degrade depending on the scale used.

5.7 Summary

We have applied our covariance analysis to mocks of the Taipan Galaxy Survey and compared the resulting constraints to Fisher matrix forecasts for an equivalent sample. In doing so, we have assessed how well our method performs relative to the idealised constraints from the Fisher matrix. Out of the three possible applications of our method (galaxy overdensity auto-covariance only, logarithmic distance ratio auto-covariance only, and complete covariance), we found that the forecast for the galaxy overdensity auto-covariance method underestimated the mock constraints by the largest amount: the measured uncertainty was 2 times the forecast uncertainty. The smallest underestimation was for the logarithmic distance ratio auto-covariance, where the measured uncertainty was only 1.05 times the forecast uncertainty. Finally, the complete covariance showed an intermediate discrepancy, with the measured uncertainty being 1.55 times the forecast uncertainty. We attribute the discrepancy in the galaxy overdensity auto-covariance model case to the loss of small-scale information from gridding the galaxy overdensity, which is consistent with the results from Chapter 4. Given the original forecasts for Taipan presented by da Cunha et al. (2017), we calculated the expected fractional uncertainty on the growth rate of structure for the two phases of the Taipan Galaxy Survey by multiplying the forecasts by the discrepancy factor we found. If our method were applied to Taipan, we expect that the Phase 1 sample would yield a 7.0% constraint on the growth rate and that the Final sample would yield a 4.2% constraint. These constraints are highly competitive, especially since our analysis of the 6-degree Field Galaxy Survey (Taipan’s predecessor) yielded a 14% constraint of the growth rate. Such tight constraints of the growth rate of structure in the low-redshift universe hold great promise for testing viable models of accelerating expansion, particularly modiﬁed gravity models. 6 Conclusions and Future Work

6.1 Conclusions

Over the course of this thesis, we have developed, tested and applied a new method that self-consistently utilises the cosmological information available from direct peculiar velocities and redshift-space distortions, including the information from their cross- correlation. This is a major achievement in the field of large-scale structure cosmology, as our method is the first to model the galaxy overdensity auto-correlation, peculiar velocity auto-correlation and the cross-correlation within the same framework, enabling simulta- neous constraints on the growth rate of structure fσ8 and the redshift-space distortion parameter β through a maximum likelihood method. By applying our method to data from the 6-degree Field Galaxy Survey (6dFGS), we have demonstrated the key benefits of our method, confirming that the cross-correlation adds additional and unique information, beyond what can be extracted from peculiar velocities and redshift-space distortions alone. We have also shown that our method can be applied to the upcoming Taipan Galaxy Sur- vey, and will be key in realising one of the main cosmological goals for the survey: a less than 5% constraint on the growth rate of structure. We now summarise the key findings from each chapter. In Chapter 2, we presented all of the necessary theory that we have developed as part of our method. Building on various works that have separately analysed the peculiar velocity and galaxy overdensity fields, we constructed a complete model covariance for these fields, including their cross-covariance. The model neatly fits into a maximum likelihood analysis, allowing us to simultaneously constrain fσ8 and β. We began by modelling the galaxy overdensity auto-correlation in real space, before constructing a full redshift-space model, which takes advantage of the growth rate information encoded in redshift-space distortions. Throughout, we took care to enfold observational effects into the model, making

157 158 Chapter 6. Conclusions and Future Work it physically accurate. Consequently, our method involves very little direct modiﬁcation of the measured data from large-scale structure surveys, which we see as a major advantage.

In Chapter 3, we applied our method to a subset of data from 6dFGS after testing our model on simulations. Given this was the first application of our model, we chose not to model the effect of redshift-space distortions, but focussed on constraining the growth rate primarily through the peculiar velocity measurements. To understand the effect of introducing the cross-covariance model, we constrained the parameters with and without +0.087 +0.087 it. When excluding the cross-covariance, we found fσ8 = 0.461 0.079 and β = 0.380 0.075. − − +0.067 +0.062 When including the cross-covariance, we found fσ8 = 0.424 0.064 and β = 0.341 0.058, − − showing that our constraints on both parameters significantly improved when utilising the cross-covariance. Our results are consistent with the findings from theoretical multiple- tracer studies, as well as forecast improvements from Fisher matrix studies of large-scale surveys. Our constraints are also consistent with existing results from peculiar velocity and redshift-space distortion analyses of 6dFGS. As a part of the work in this chapter, we also presented direct evidence for the existence of the cross-correlation signal by measuring it as a function of pair separation directly from the data. We showed visual evidence for 1 a non-zero cross-correlation on scales up to 50 h− Mpc.

In Chapter 4, we advanced the analysis of the previous chapter by including redshift- space distortions in our model of the galaxy overdensity field, allowing us to access information on the growth rate of structure from both auto-covariances as well as the cross- covariance. We also improved the analysis by increasing the redshift range of our galaxy overdensity sample for 6dFGS, and by performing thorough tests of our method using sophisticated mock catalogues. In addition to the statistical uncertainty from the maximum likelihood approach, we also quantified the systematic uncertainty by analysing how various model parameter choices impacted our constraints on the growth rate of structure and redshift-space distortion parameter. Using the complete covariance model, we +0.044 found fσ8 = 0.384 0.052(stat) 0.061(sys) and β = 0.289 0.043(stat) 0.049(sys). We − found that adding the cross-covariance reduced the statistical uncertainty in the growth rate by 64% compared to only using the galaxy overdensity auto-covariance, and by 50% compared to only using the logarithmic distance ratio auto-covariance. Overall, utilising the growth rate of structure information from redshift-space distortions and increasing the sample resulted in an 18% reduction in the absolute uncertainty in the growth rate relative to our result from Chapter 3. We found that the systematic uncertainty was dominated by the parameter we used to model the difference in the effective galaxy bias of the cross-correlation and the galaxy overdensity auto-correlation, αb, and believe this could 6.2. Future Work 159 be significantly reduced by improving our model of galaxy bias in the cross-correlation, which we discuss below. In Chapter 5, we looked to the future of low-redshift galaxy surveys by applying our method to mocks of the Taipan Galaxy Survey, which will begin observations this year. As part of this analysis, we took the opportunity to run a Fisher matrix forecast that matched our mock sample and model parameter choices, enabling a fair comparison between forecast and actual performance. We found that the Fisher matrix gave overly optimistic constraints on fσ8 and β to varying degrees. The difference in results from the two methods was strongest for the galaxy overdensity auto-covariance only analysis, where our average uncertainty in the model parameters from the mocks was 2 times that from the forecasts. We believe this is from the loss of small-scale information from the gridding process required in our method, but believe this loss could be lessened in the future, which we discuss in the next section. On the other hand, we found that the logarithmic distance ratio auto-covariance only analysis gave a growth rate constraint that was only 1.05 times the forecast constraint. This highlighted how effective our method is in extracting the available information on the growth rate of structure from peculiar velocities. The complete covariance performed between the two; we found that the constraints from our method were 1.55 times those from the Fisher matrix forecast. Given that forecasts for Taipan expect a 4.5% constraint on the growth rate in the first year, and a 2.7% constraint by the end of the survey, we used the 1.55 factor to convert these to more realistic constraints of 7.0% and 4.2% respectively. We note that these are significantly better than what we achieved in our analysis of 6dFGS, which speaks to the constraining power of Taipan. We conclude that applying our method to Taipan could deliver one of the survey’s key cosmological goals: a less than 5% constraint on the growth rate of structure. In summary, we believe the work we have presented in this thesis demonstrates a powerful way to access cosmological information in the low-redshift universe, taking advantage of the cross-correlation between redshift-space distortions and peculiar velocities to provide tight constraints on the growth rate of structure.

6.2 Future Work

Throughout the thesis, we’ve identiﬁed several potential improvements to our method, which we now discuss. In addition to this, we also highlight new and interesting research opportunities that follow on directly from this thesis. 160 Chapter 6. Conclusions and Future Work

6.2.1 Model and Method Improvements

In Chapter 4 we introduced αb to parametrize the difference in the effective galaxy bias of the galaxy overdensity auto-correlation and cross-correlation. While we found that this parameter was sufficient to recover the growth rate of structure in the mocks, it was the dominant source of systematic uncertainty in our final constraints. This can be linked to the fact that αb represents more than just the difference between effective bias values; it directly influences the amplitude of the cross-covariance, such that a lower value of αb may signify that the cross-correlation between peculiar velocities and galaxy overdensities is weaker than anticipated by our model. We discovered this behaviour when fitting our model with different values of αb, noting that the posteriors on fσ8 appeared to converge for increasingly small values of αb (see Figure 4.13 and Figure 5.4). We also saw that the

αb value was lower for the Taipan mocks than the 6dFGS mocks, which implied that it may depend on how the model weights peculiar velocities as tracers of the overdensity field according to their observed uncertainties, as well as the redshift distribution of the galaxy overdensity sample. Noting that the difference in effective bias arises from being unable to use a volume- limited sample, we can turn to work on the bias-luminosity relation for inspiration. For example, Beutler et al. (2013) characterised the bias-luminosity relation for 6dFGS; this could be used to modify the cross-correlation model to account for the expected galaxy bias when considering how a given peculiar velocity responds to a particular galaxy overdensity. An alternative would be to modify the data directly such that the galaxy overdensity sample had a constant galaxy bias over the whole volume, as implemented by Carrick et al. (2015). We feel that the first option is more consistent with the general approach of our method, where we have tried to capture as much of the physics as possible in the model, as opposed to manipulating the data to fit a simpler model. It may also be possible to refine the method for estimating αb from the measured power spectra from two redshift ranges, as presented in Figure 4.12. However, given the uncertainty in the redshift range over which peculiar velocities are sensitive to the galaxy overdensity field, we do not believe this is the optimal approach. We believe significant further research is required to implement any of these solutions for our method, which includes validation with simulations before application to data. Throughout this thesis (and explicitly in Section 2.3.4), we have assumed that the parallel-plane approximation was sufficient when modelling redshift-space distortions. This approximation has been shown to break down for galaxy pairs with wide opening angles, which is common for large-area surveys at low-redshift, like 6dFGS and Taipan. We note 6.2. Future Work 161 that redshift-space distortions may be modelled without this limit, as shown by Szalay et al. (1998) and Szapudi (2004), and that such modelling was included in the original 6dFGS redshift-space distortion analysis by Beutler et al. (2012). To our knowledge, the cross-correlation model has not been derived without assuming the parallel-plane approximation, which would need to be done before it could be adopted self-consistently into our model.

Finally, we touch on perhaps the largest limitation of our method, which is the computational effort required to solve our likelihood equation. The time taken to solve the likelihood (specifically through inverting the covariance model) is directly linked to the size of the covariance matrix, which is informed both by the volume of the survey and by how finely we grid the sample. Recalling that the likelihood must be solved for each step in an MCMC process, our method becomes increasingly computationally expensive for larger samples and smaller grid scales, both of which are desirable. While we have been able to achieve acceptable inversion speed by utilising Cholesky solvers from efficient linear algebra packages, we often needed to make sacrifices such as increasing the gridding scale. Unfortunately, this lowered the amount of small-scale information we could extract from redshift-space distortions, giving a wider posterior for the growth rate of structure than is typical from traditional redshift-space distortion analyses.

We note that considerable work has been done in improving the computational effi- ciency of Wiener filtering, which also involves the inversion of sizeable matrices. Specif- 1 1 1 ically, one must evaluate (C− + N − )− , where C is the covariance matrix, and N is the noise matrix. This is a challenging problem if there is no single basis in which the covariance and noise matrix are both sparse, but can be solved using the messenger field method, where an additional field mediates between the basis where C is sparse and the basis where N is sparse (e.g. Elsner & Wandelt, 2013; Alsing et al., 2016). We note that we do not perform exactly the same inversion, but there are undoubtedly parallels between the two problems. Given this information, it would also be worth investigating whether there is a basis where our covariance matrix model is sparse, and whether it is computationally efficient to convert to this basis, perform the covariance inversion and convert back. Such an investigation was well beyond the scope of this thesis, but we highlight that it represents a possible way to address the largest computational bottleneck in our analysis and deserves significant consideration in future work that wishes to apply the method we have developed here. 162 Chapter 6. Conclusions and Future Work

6.2.2 Research Opportunities

Given that Johnson et al. (2014) provided scale-dependent constraints on the growth rate of structure for peculiar velocities using a similar model formalism, we believe implementing a similar capability for the complete covariance is a natural extension of the work presented in this thesis. Based on the improvements we saw in the growth rate when including the cross-covariance, we expect similar improvements in constraints of the growth rate in scale-dependent bins. Tighter constraints would enable better tests of modified gravity models, such as those presented by Johnson et al. (2016). Our method is also nicely set up to look for signatures left behind by non-Gaussian perturbations present during the inflationary period of the Universe. Such perturbations are a feature of alternative inflation models and they imprint a scale-dependent bias in the galaxy distribution. Consequently, any deviation from Gaussianity in the early universe modifies galaxy clustering on very large scales; the strength of the deviation is parametrized by fNL. While large scales are difficult to measure in low-redshift surveys (making it difficult to constrain fNL), adding peculiar velocities and their cross-correlation with the galaxy distribution may tighten such constraints. For example, Howlett et al. (2016) used Fisher matrix forecasts to show that the cross-correlation helps break degeneracies between fNL and β, which improves constraints on fNL by up to 40%. This has already been leveraged by Ma et al. (2013), who constrained fNL in the local universe using a velocity-velocity comparison analysis (such as those described in Section 1.4.2), but we note that it would also be possible to modify our method to constrain fNL. One of the major benefits of our method is that any model where the power spectrum is proportional to the growth rate of structure could be substituted. We could potentially use this feature to constrain the growth rate under the assumption of a specific modified gravity model (such as f(R) gravity), rather than under the assumption of general relativity. This could be done by using power spectra for modified gravity models, such as those produced by MGCAMB (Hojjati et al., 2011). Such research would provide interesting insights into how growth rate of structure constraints respond to the assumed cosmological and gravitational model. Finally, we believe that Taipan provides a useful opportunity to perform our analysis in tandem with a more traditional velocity-velocity comparison analysis, such as the analysis presented by Carrick et al. (2015). While both methods are accessing similar information from the cross-correlation between peculiar velocities and galaxy overdensities, velocity- velocity comparison studies report much smaller errors than our method, even when using smaller samples. Consequently, we believe a fair comparison of these methods, using 6.2. Future Work 163 the same data, would provide significant insight into the differences in assumptions or procedures that might be causing this discrepancy. We also believe that both methods are likely to benefit from such a comparison; the methods approach a similar challenge from very different perspectives, meaning that each could learn a significant amount from the other. Both the potential improvements and applications of the work presented in this thesis highlight the promising future that lies ahead for combined studies of large-scale structure and peculiar velocities in the low-redshift universe. This thesis has shown that the complementary nature of these two cosmological probes results in statistical improvements that are highly desirable in the age of precision cosmology. We believe that the application of our method to future low-redshift surveys will be critical in testing viable explanations of accelerating expansion, especially when combined with powerful constraints from the high-redshift universe.

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