Improving cosmological measurements from galaxy surveys by Angela Burden This thesis is submitted in partial fulfilment of the requirements for the award of the degree of Doctor of Philosophy of the University of Portsmouth. March 1, 2015 Abstract Reconstructing an estimate of linear Baryon Acoustic Oscillations (BAO) from an evolved galaxy field has become a standard technique in recent analyses. By par- tially removing non-linear damping caused by bulk motions, the real-space baryon acoustic peak in the correlation function is sharpened, and oscillations in the power spectrum are visible to smaller scales. In turn, these lead to stronger measurements of the BAO scale. Future surveys are being designed assuming that this improve- ment has been applied, and this technique is therefore of critical importance for future BAO measurements. A number of reconstruction techniques are available, but the most widely used is a simple algorithm that de-correlates large-scale and small-scale modes approximately removing the bulk-flow displacements by moving the overdensity field [1, 2]. The initial work presented in this thesis shows the practical development of a reconstruction algorithm which is extensively tested on the mock catalogues created for the two Baryon Oscillation Spectroscopic Survey (BOSS) Date Release 11 samples covering redshift ranges 0:43 < z < 0:7 and 0:15 < z < 0:43. The practical implementation of this algorithm is tested, looking at the effi- ciency of reconstruction as a function of the assumptions made for the bulk-flow scale, the shot noise level in a random catalogue used to quantify the mask and the method used to estimate the bulk-flow shifts. The reconstruction algorithm developed in Chapter 2 is applied to 5 differ- ent galaxy survey data sets. The algorithm was used directly to create the re- constructed catalogues used to extract the cosmological distance measurements published in [3, 4, 5, 6], the results and cosmological implications are presented. i The efficiency of reconstruction is also tested against external factors includ- ing galaxy density, volume and edge effects, and the impact for future surveys is considered. The results of this work are published here [7]. The measurement of linear redshift space distortions apparent in the observed distribution of matter provides information about the growth of structure and potentially provides a way of testing general relativity on large scales. The last chapter of the thesis presents a model of the reconstructed redshift space power spectrum in resummed Lagrangian perturbation theory [8] which is a new result. The goal of the work is to create a reconstruction algorithm that enhances the linear redshift space distortion signal measured from an evolved galaxy distribu- tion analogous to the improvement seen in the BAO signal post-reconstruction. ii Table of Contents Abstract i Declaration xiii Acknowledgements xiv 1 Introduction 1 1.1 The Standard Cosmological Model . .2 1.1.1 General Relativity . .3 1.1.2 The Cosmological Principle . .4 1.1.3 The Friedmann Equations . .6 1.1.4 The fluid description of components . .7 1.2 Inflation . 10 1.2.1 Initial perturbations from inflation . 12 1.3 Evolution of perturbations . 14 1.3.1 Newtonian perturbation theory . 16 1.3.2 Pre-recombination . 19 1.3.3 Recombination and decoupling . 22 1.3.4 Post-recombination . 24 1.4 Late time probes of cosmology . 33 1.4.1 Measuring the density field . 33 1.4.2 Redshift space distortions . 39 1.4.3 Weak Lensing . 44 1.4.4 Clusters of galaxies . 45 iii 1.4.5 Supernovae Type 1a . 47 1.5 Alternative cosmological models . 48 1.5.1 Problems with ΛCDM . 50 1.5.2 Modified Gravity . 51 1.5.3 Quintessence . 54 1.6 Summary and thesis outline . 54 2 The reconstruction method 56 2.1 Various algorithms . 57 2.2 The Lagrangian reconstruction method . 58 2.2.1 The observed galaxy displacement field in perturbation theory 59 2.3 Code development . 61 2.3.1 Application to CDM N-body boxed simulations . 62 2.3.2 Mock catalogues . 66 2.3.3 Application to survey mocks . 69 2.4 Measuring the BAO scale . 70 2.4.1 Measuring the power spectrum . 75 2.4.2 Modelling the power spectrum . 77 2.4.3 Fitting the BAO scale . 78 2.5 Change in effectiveness with method . 79 2.5.1 Smoothing length . 79 2.5.2 Number of randoms . 80 2.5.3 Finite difference method . 83 2.5.4 RSD removal . 92 2.6 Summary . 94 3 Application to galaxy surveys 96 3.1 The SDSS-III, BOSS CMASS sample . 99 3.1.1 Galaxy weights . 100 3.1.2 BOSS PTHalo mocks . 101 3.1.3 Reconstruction of CMASS, a comparison of algorithms . 101 3.1.4 Results, CMASS . 104 iv 3.2 The SDSS-III, BOSS LOWZ sample . 106 3.2.1 Reconstruction of the LOWZ sample . 107 3.2.2 Results, LOWZ mocks . 107 3.2.3 Results, LOWZ data . 109 3.3 The Red/Blue sample . 113 3.3.1 Improved weighting scheme . 114 3.3.2 Modified PTHalo mocks. 114 3.3.3 Reconstruction input parameters and results . 115 3.4 The MGS sample . 117 3.4.1 PICOLA mocks for MGS . 119 3.4.2 Reconstruction of the MGS sample . 119 3.4.3 Results, MGS sample . 120 3.5 Cosmological implications . 120 4 Reconstruction, survey properties 131 4.1 Change in effectiveness with survey density . 132 4.2 Change in effectiveness near edges . 138 4.3 Comparison to current predictions . 145 4.4 Summary and outlook . 149 5 Perturbation theory modelling of the power spectrum 152 5.1 SPT . 156 5.1.1 Non-linear Eulerian Perturbation Theory . 156 5.1.2 Solutions for an Einstein-de-Sitter Universe . 159 5.1.3 Arbitrary cosmologies, an approximation . 163 5.1.4 Non-linear power spectrum in SPT . 163 5.2 Lagrangian perturbation theory . 167 5.2.1 Linear LPT . 170 5.2.2 Second order LPT . 171 5.3 Resummed Lagrangian Perturbation Theory . 174 5.3.1 RLPT in real space . 175 5.3.2 RLPT redshift space . 187 v 5.4 Reconstructed Resummed LPT . 193 5.5 Reconstructed RLPT in redshift space . 196 6 Conclusions and future work 210 A Appendix 1 213 A.1 Hubble's Law . 213 A.2 The Zeldovich Approximation . 215 A.3 FKP weights . 216 A.4 Deriving the EFE . 219 A.4.1 The Friedmann Equation . 221 A.4.2 The Fluid Equation . 222 A.5 Wick's Theorem . 223 A.6 Enm Terms, LRT redshift space . 224 B Appendix 2 225 B.1 Histograms . 225 Bibliography 227 vi List of Tables 2.1 BAO scale errors recovered for different smoothing lengths from the LOWZ and CMASS mocks. 82 2.2 BAO scale errors recovered for different ratios randoms to mock data for the CMASS mocks. 85 2.3 CMASS, σα with/without RSDs removed during reconstruction . 92 h i 3.1 A description of the five SDSS galaxy samples that have been re- constructed with this algorithm. 98 3.2 Fitting results and BAO distance measurements made with data reconstructed using this algorithm. The consensus results refer to the average of P (k) and ξ(s) measurements. 121 3.3 Fitting results for the mock catalogues reconstructed using this al- gorithm. 123 3.4 Cosmological parameters constraints for a variations of the ΛCDM model for BAO + CMB data sets. These values are taken from [6]. The Planck only results are included (grey), to show how the parameter constraints for ΛCDM parameters are tightened with the additional BAO measurements. 129 4.1 BAO scale errors recovered for different survey densities from the LOWZ and CMASS mocks. 136 4.2 BAO scale errors recovered varying the percentage of the survey that lies along an edge. 145 vii List of Figures 1.1 Evolution of density parameter as a function of scale factor. 15 1.2 CMB angular power spectrum, model and data, fom [9]. 26 1.3 2D correlation function, BOSS DR11, from [10] . 35 1.4 Distance-redshift relation from BAO compared to Planck. 38 1.5 Constraints on γ and Ωm from combined probes, from [10]. 43 1.6 Constraints on ΩΛ and Ωm from SNe1a from [11]. 49 2.1 The reconstruction process shown as applied to a boxed N-body simulation. 64 2.2 Power spectrum showing RSD removal by reconstruction in an N- body simulation. 65 2.3 The number density of galaxies as a function of redshift for both the North galactic cap of CMASS and LOWZ data. 66 2.4 A slice through DR10 CMASS data showing the displacement vec- tors calculated by the reconstruction process. 71 2.5 1D profile of the smoothed CMASS DR10 galaxy density and ran- dom field. 72 2.6 A slice through DR10 CMASS with contours showing overdensity and displacement vectors from reconstruction. 73 2.7 A flow diagram outlining the steps of the reconstruction algorithm. 74 2.8 Average power spectra of CMASS and LOWZ mocks pre and post- reconstruction. 76 2.9 Average of 600 CMASS mock power spectra divided by the no- wiggle model pre and post-reconstruction. 78 viii 2.10 Recovered αpost and σα,post values as a function of smoothing h i h i scale for CMASS and LOWZ. 81 2.11 Number of randoms used vs efficiency of the reconstruction algorithm. 84 2.12 Lagrangian displacement field projected in 2D in the x,y plane f g from finite difference method and Fourier method. 88 2.13 Lagrangian displacement field projected in 2D in the x,z plane f g from finite difference method and Fourier method.