Higher-order statistics of weak gravitational lensing Susan Pyne Thesis submitted for the degree of Doctor of Philosophy Supervisors: Examiners: Dr Benjamin Joachimi Prof. Antony Lewis Prof. Hiranya V. Peiris Dr Andreu Font-Ribera Department of Physics and Astronomy University College London July 2020 I, Susan Pyne, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. This thesis is dedicated to my mother who sadly did not live to see its completion, but is always in my thoughts. Yet what are all such gaieties to me Whose thoughts are full of indices and surds? x2 + 7x + 53 11 = 3 Lewis Carroll: Four Riddles ABSTRACT The next decade will see the advent of unprecedentedly large cosmological surveys, optimised to provide data about the gravitational lensing of galaxies. This opens up the possibility of exploring methods and statistics which are out of reach of current surveys. In this spirit, this thesis focuses on exploiting three-point and higher-order weak lensing statistics. First we consider the deflection of three-point correlation functions by weak lensing, a small, subtle signal which is not accessible with current surveys. We derive a general expression for the lensing deflection but show that its detection must await even larger and deeper surveys. We next consider the information content of the weak lensing bispectrum. We confirm that using the bispectrum as well the power spectrum can help to reduce statistical errors on cosmological parameters. Moreover we show that the bispectrum can help mitigate two major systematic uncertainties, the intrinsic alignment of galaxies and redshift errors. We find that these affect the power spectrum and bispectrum differently, and that using the bispectrum can facilitate self- calibration. This is a promising finding which could be extended to other systematics. Future surveys will probe small, non-linear scales so in a Bayesian weak lensing analysis it may not be valid to approximate the likelihood as Gaussian. We discuss theoretical alternatives in Fourier space and show that the real space weak lensing likelihood is also theoretically non-Gaussian. In practice if a Gaussian likelihood is assumed then the covariance matrix should be calculated at a fixed point in parameter space. Working to the accuracy required by future surveys, it is important to choose this point optimally. We develop an emulator for the weak lensing power spectrum covariance and demonstrate an iterative method to determine this fixed cosmology in a principled way. 9 IMPACT STATEMENT This thesis is part of a body of work currently being undertaken across the cosmology community to prepare for the next generation of galaxy surveys, including Euclid, the Dark Energy Spectroscopic Instrument and the Rubin Observatory Legacy Survey of Space and Time. The science goals of these surveys focus on some profound mysteries about the Universe, in particular the nature and properties of dark matter and dark energy. To this end the surveys will look deeper into the Universe, and hence farther back in time, than current surveys do, gathering an unprecedented wealth of highly accurate and detailed data about tens of billions of galaxies. This exquisitely accurate data will transform weak gravitational lensing, the subject of this thesis, into one of a handful of truly precision cosmological probes. These next-generation surveys are financed by public investment of hundreds of millions of dollars from countries across the globe; their data is an extremely valuable resource which cosmologists have an obligation to exploit fully. This thesis focuses on ways to get more out of the new data. We may be able to detect signals which cannot be measured in current surveys. Equally it may become practicable to obtain new information from higher-order statistics which at present are not viable. These statistics do not require additional data, just more complex analysis of data which will be collected anyway for conventional analysis. Even more fruitfully, as this thesis shows, higher-order statistics provide a promising new way of handling systematic uncertainties in weak lensing, again without the need for extra data. This is important because controlling systematics will become more pressing as abundant data reduces statistical errors. Finally, new insights can be gained by taking into account fine details which were previously inaccessible or by relaxing simplifying assumptions. This thesis considers examples of all these possibilities for extracting extra value from future weak lensing surveys and paves the way for three-point statistics to become a standard tool for analysing weak lensing data. Similar methods may also be feasible for other cosmological probes. 10 ACKNOWLEDGEMENTS First I would like to thank my supervisor, Benjamin Joachimi, for taking on a rather ‘non-standard’ student in the first place, but also for his patient support and scientific guidance, for making time for regular meetings despite his busy schedule, for helping me to do things I didn’t know I could, for the breadth and depth of his knowledge of weak lensing, and for reliably being right. I couldn’t wish for a better supervisor. I also acknowledge use of his comprehensive cosmological code. In many parts of my work I used his code for two-point statistics as a basis for my three-point code. Next I owe a debt of gratitude to my second supervisor, Hiranya Peiris, an outstanding role model and inspiration for any female cosmologist, who was ready give me time when I needed a listener, and who always provided wise insights and constructive advice. It has been a real privilege to learn from her. I am particularly grateful to Fabian Kohlinger¨ and Alessio Spurio Mancini for helpful advice about using their KiDS likelihood for the work in Chapter 4, and to Stephen Stopyra for helping me to get to grips with Mathematica. My thanks also to John Deacon and especially Edd Edmondson for their expertise in sorting out my IT problems, even when they were my own fault, and to Anita Maguire for efficiently dealing with administrative matters and keeping us all happy. Thank you also to all the UCL colleagues who over the course of my PhD have given me advice, tips, bits of code, or just been friendly. I must also thank my family. Even in her final illness my mother was gently amused by the idea of me doing a PhD. This thesis is dedicated to her. My daughter Ros has been another good listener and source of cheerful, intelligent support, reversing the normal mother-daughter relationship. Last but certainly not least I thank my husband Geoff who encouraged me to embark on a PhD in the first place, and supported me wholeheartedly throughout. He has helped me enormously by acting as a sounding board for ideas and initiating many stimulating scientific discussions from the perspective of an experimentalist. I’m grateful also for his practical skills: he has often solved technical and IT problems, and gave me one of my best ever birthday presents - a PC which he designed for fast calculations with big datasets. I couldn’t have done this without him. 11 THESIS ROADMAP The first chapter of this thesis is a three-part overview of necessary background to the rest of the work. The first part outlines fundamental topics in modern cosmology, including some aspects of its historical development. The second introduces the main cosmological probes which are referred to in subsequent chapters, in particular weak gravitational lensing. The third part summarises methods and techniques which are used later in the thesis. In Chapter 2 we discuss the weak lensing deflection of three-point correlation functions. This is a potential source of information which is not accessible through current surveys. This chapter is based on Pyne, S., Joachimi, B. and Peiris, H.V. (2017), ’Weak lensing deflection of three-point correlation functions’, Journal of Cosmology and Astroparticle Physics 2017(12), 043. Chapter 3 investigates the information content of the weak lensing bispectrum. This is again in three parts. The first is a theoretical account of the composition of the weak lensing bispectrum covariance matrix. In the second part we investigate the information content of the weak lensing bispectrum using Fisher matrix analysis. Finally we explore whether the bispectrum can help to mitigate two major sources of systematic uncertainty in weak lensing, intrinsic alignments and redshift errors. Chapter 4 covers several slightly eclectic topics related to likelihoods and covariance matrices used in Bayesian estimation of cosmological parameters. First we look theoretically at the Gaussianity of power spectrum likelihoods and review alternative non-Gaussian likelihoods. We also investigate a hierarchical model of weak lensing two-point correlation function likelihoods. Finally we discuss the parameter-dependence of the power spectrum covariance matrix and the cosmology at which the covariance matrix should be calculated. The final chapter draws together conclusions from the whole thesis and discusses the potential impact of the work for next-generation surveys. 12 CONTENTS Table of Contents 13 List of Figures 17 List of Tables 21 1 Introduction 23 1.1 Cosmology . 23 1.1.1 Historical perspective . 23 1.1.2 General relativity and the Friedmann equations . 23 1.1.3 Cosmological redshift . 29 1.1.4 Components of the Universe . 30 1.1.5 Evolution of the Universe . 32 1.1.6 Structure formation and perturbation theory . 33 1.1.7 The matter power spectrum . 38 1.2 Cosmological probes . 40 1.2.1 Cosmic microwave background . 40 1.2.2 Baryon acoustic oscillations . 42 1.2.3 Weak gravitational lensing . 48 1.3 Methods and techniques . 58 1.3.1 Clustering statistics . 58 1.3.2 Halo model .