University of Southern California Dissertations and Theses

Cosmological study with galaxy clusters detected by the Sunyaev-Zel'dovich effect

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Cosmological study with galaxy clusters detected by the Sunyaev-Zel'dovich effect

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Content COSMOLOGICAL STUDY WITH GALAXY CLUSTERS DETECTED BY THE SUNYAEV-ZEL’DOVICH EFFECT by Suet-Ying Mak A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS & ASTRONOMY) August 2013 Copyright 2013 Suet-Ying Mak Dedication To my family ii Acknowledgments I want to express my sincere gratitude to my thesis advisor Prof. Elena Pierpaoli for her continuous teaching, guidance, and patience throughout my Ph.D study. I especially appreciatethefactthatshegives creditaboutmyworkandhelpmetogetopportunitiesto give presentations. In particular, sheputin alot of effort in helpingmeto work within the PlanckCollaboration, whichisagoodlearningexperienceandpreparemefortheacademic career. I would not have accomplished the thesis work without her guidance. I want to thank Stephen Osborne for the wonderful collaboration works on the bulk flow measurements. He has provided me a very good start in writing computer codes on CMB simulations. I thank Fabian Schmidt for his enlightened teaching and discussions on our work of modified gravity. He taught me a lot on, not only on aspects of modified gravity, but also various aspects of cosmology. I am impressed by his breath of knowledge andishonoredtoworkwithhim. IthankLorisColumboforhisusefuldiscussionsonCMB analysis and it was a good experience of having a joint colloquium with him at USC on the Planck latest results. And I also want to thank Tsz Yan Lam for his warm hospitality during my short visit at IPMU. ThroughouttheyearsatUSC,Imetalotofgreatcolleagues,whoentertainandbrighten my Ph.D life. Many thanks go to Ming Chak Ho who is an extremely good friend and helps me a lot in setting up things living in Los Angeles, Hsiao-Husan Lin who endures iii my frequent visits to her office and has nice chat while I am at school, and Kok-Wee Song and Nicolo’ Macellari who studied with me during our first year, and Jennifer Shitanishi, Moncho Rey Raposo, and Andrey Egorov who work together in the cosmology group and have useful discussions. Most importantly, I would not have started and finished the Ph.D work without the tremendous and selfless support of my families. My work is dedicated to them. The research is supported by the various USC fellowships, including the Provost’s fellowship and John Staffer fellowship. iv Table of Contents Dedication ii Acknowledgments iii List of Figures ix List of Tables xiii Abstract xiv Chapter 1: Introduction 1 1.1 Galaxy Clusters as Cosmological Probes . . . . . . . . . . . . . . . . . . . . 2 1.2 The Sunyaev-Zel’dovich Effect. . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Thermal Sunyaev-Zel’dovich Effect . . . . . . . . . . . . . . . . . . . 6 1.2.2 Kinetic Sunyaev-Zel’dovich Effect . . . . . . . . . . . . . . . . . . . . 8 1.3 Modeling SZ cluster survey . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.1 Cluster count . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.2 Limiting mass: selection function . . . . . . . . . . . . . . . . . . . . 11 1.4 Current Status on SZ clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Objective of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Chapter 2: The search of Cosmic Bulk Flow with the Planck satellite 17 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Theoretical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.1 Expected Bulk Flow Velocity from ΛCDM model . . . . . . . . . . . 21 2.3.2 Models of Cosmic Bulk Flow . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.1 Assumptions and Definitions . . . . . . . . . . . . . . . . . . . . . . 24 2.4.2 Outline of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.3 Optical depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 v 2.5 Cluster Catalogs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5.1 eRosita All-Sky Survey . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5.2 The Planck Catalog . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5.3 ROSAT All-Sky cluster survey . . . . . . . . . . . . . . . . . . . . . 29 2.6 Sky Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.7 Filtering: Reconstruction of the Kinetic SZ signal . . . . . . . . . . . . . . 32 2.7.1 Matched filter (MF) . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.7.2 Unbiased Kinetic SZ filter (UF) . . . . . . . . . . . . . . . . . . . . . 34 2.8 Analysis Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.8.1 Weighted Least Square Fitting of Dipole . . . . . . . . . . . . . . . . 34 2.8.2 Translating from μK to km/s . . . . . . . . . . . . . . . . . . . . . . 35 2.9 Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.9.1 CMB and instrument noise . . . . . . . . . . . . . . . . . . . . . . . 37 2.9.2 Thermal SZ signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.9.3 Uncertainty in optical depth and random cluster peculiar velocity . 40 2.9.4 Extragalactic Point Sources . . . . . . . . . . . . . . . . . . . . . . . 40 2.10 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.10.1 Precision and detection threshold of Bulk Flow Measurement . . . . 43 2.10.2 Recovering the Bulk Flow Direction . . . . . . . . . . . . . . . . . . 47 2.10.3 Effect of the systematic error components on the amplitude of the recovered bulk flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.10.4 Performance of Filters . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.11 Latest measurements of Bulk Flow on Planck CMB data . . . . . . . . . . . 50 2.12 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Chapter 3: Modified Gravity 56 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2 Theoretical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2.1 f(R) gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2.2 Cluster abundance inf(R) . . . . . . . . . . . . . . . . . . . . . . . 59 3.2.3 Halo clustering inf(R) . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.2 Non-Gaussian likelihood . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.1 Number counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.2 Power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4.3 Combined constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.4.4 Uncertainties in scatter of mass observable relations . . . . . . . . . 72 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 vi Chapter 4: Probing Inflation: Primordial Non-Gaussianity 83 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2 Primordial Non-Gaussanity . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2.1 Mass Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2.2 Halo Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.4.1 Cluster counts and Power Spectrum . . . . . . . . . . . . . . . . . . 92 4.4.2 Combined constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5.1 Uncertainties in scatter of mass observable relations . . . . . . . . . 96 4.5.2 Comparison to previous work . . . . . . . . . . . . . . . . . . . . . . 97 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Chapter 5: Weighing Neutrino Mass 101 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 Impact of neutrino masses on growth of the large scale structures . . . . . . 103 5.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.3.2 Notes on CMB Fisher matrix . . . . . . . . . . . . . . . . . . . . . . 104 5.4 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.4.1 Cluster number count and power spectrum . . . . . . . . . . . . . . 105 5.4.2 Cluster probes + CMB . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.4.3 Self-calibration and uncertainty of nuisance parameters . . . . . . . 107 5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.5.1 Parameter Degeneracies . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.5.2 Survey sensitivities to neutrino mass constraints . . . . . . . . . . . 110 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Chapter 6: Conclusions 114 Appendix A: Derivation of Filter Kernel 117 A.1 Data Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 A.2 Filtered Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 A.3 Unbiased Matched Filter for kSZ signal . . . . . . . . . . . . . . . . . . . . 118 A.4 Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Appendix B: Modeling SZ signal of individual clusters 122 vii AppendixC:CosmologicalConstraintsfromSZclustersurveyusingFisher matrix (FM) Analysis 125 C.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 C.2 Cosmological parameter estimation . . . . . . . . . . . . . . . . . . . . . . . 126 C.2.1 FM of Cluster Number counts . . . . . . . . . . . . . . . . . . . . . 126 C.2.2 FM of Cluster Power spectrum . . . . . . . . . . . . . . . . . . . . . 127 C.2.3 Fiducial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 C.3 CMB Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 C.4 Calibration parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Bibliography 132 viii List of Figures 1.1 ConstraintsonthedarkenergydensityΩ X andequation ofstatew 0 inaflat universe from the combination of all cosmological datasets. The combined constraints are w 0 =−0.991±0.045 and Ω X =0.740±0.012. Credit: [188]. 6 1.2 Spectral distortion of the CMB spectrum due to SZ effect. Credit: [25]. . . 7 1.3 Left: Mass limit of cluster surveys. Right: The redshift distribution of clusters in the Planck (black), ACTPol (blue), SPT (green), and SPTPol (magenta) survey in the fiducial ΛCDM cosmology. . . . . . . . . . . . . . . 12 2.1 The dipole direction in Galactic coordinates of current bulk flow measure- ments. (a) CMB b =48.26±0.03 ◦ , l =263.99±0.14 ◦ [74]; (b) Local Group b = 30± 3 ◦ , l = 276± 3 ◦ [90]; (c) KAKE b = 34 ◦ , l = 267 ◦ ; (d) [192] b = 8±6 ◦ , l = 287±9 ◦ ; (e) [98] b = −28±27 ◦ , l = 220±27 ◦ ; (f) [72] b =0±11 ◦ , l =263±13 ◦ ; (f) [53] b =6±6 ◦ , l =282±11 ◦ . The size of the colored region is proportional to the amplitude of the measured bulk flow. The color code represents the convergence depth of the bulk flow, in units of Mpc h −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Thetheoretical bulkflowvelocity inΛCDMcosmology, usingselection func- tions forthethreeclustersurveys, smoothedover atophatwindowfunction W(kR) with the comoving sphere centered on the observer with radius R. Also plotted is the amplitude of the expected bulk flow with uniform selec- tion function. The shaded region is the uncertainty of the expected bulk flow from sample variance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Left: Theminimummassasafunctionofredshiftforaclustertobeincluded in the RASS, Planck, and EASS catalogs. The cluster mass that we use in thecatalogs ismax[10 14 M ⊙ ,M lim (z)]. FortheRASSsample, wefitEq.(2.7) tothearchived catalog andobtaintheM lim valuesshowninthisplot. Right: The differential cluster redshift distribution per square degree for the three surveys above the limiting mass. The ΛCDM cosmology is assumed and σ 8 = 0.8 is used for the number counts of the Planck and EASS clusters. The observed number count of the RASSclusters are overlaid with the solid lines for comparison with the fitted number counts (dash dot line). . . . . . 28 ix 2.4 Left: Multi-frequency matched filter signed to detect the kSZ signal. Right: Unbiased multi-frequency matched filters designed to remove the tSZ com- ponents . Both filters are constructed using the Planck beam profiles as the cluster radial profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5 The systematic error contributions to the bulk flow measurement using the RASS (top), Planck (middle), and EASS (bottom) cluster samples: instru- ment noise and CMB (cross), thermal SZ (asterisk), uncertainty in the kinetic SZ signal due to scatter in both the optical depth and cluster pecu- liar velocity (diamond), and the contamination from radio point sources (triangle for our simulation and solid line for PSM simulation). The data points are the average values from 100 realizations and the error bars are the standard deviations of the one-sided distribution. . . . . . . . . . . . . 38 2.6 Recovered bulk flow velocity for clusters z ≤ 0.5 ( Top: RASS, Middle: Planck,Bottom: EASS)fromsimulatedmapswithinputbulkflowvelocities logarithmically spaced between 100 and 10,000 km/s, filtered with the MF (black cross) and UF (blue diamond). The solid line represents perfect recovery. Thedash lines quotethe 95% confidenceinterval of theinputbulk flowvalues. AllresultsarebasedonmapscontainingkineticandthermalSZ signals, but simulated with different systematic components. Plots labelled Noise andCMBare frommapscontaining instrumentnoise andCMB; plots labelled thermal SZ are from results containing thermal SZ emission and hence estimate the level of thermal SZ bias. . . . . . . . . . . . . . . . . . 44 2.7 Recovered bulk flow velocity for three inputbulk flows v =500 (black), 1000 (yellow), and 5000 (cyan) km/s as a function of redshift for the Planck cluster sample. The recovered bulk flows are measured from simulations that contain all systematic errors we consider. . . . . . . . . . . . . . . . . . 45 2.8 The 95% upper limit to the error in the angle measurements for all the three cluster surveys. The l = 30 ◦ , b = 280 ◦ data points correspond to the direction of the input bulk flow that the simulations in Fig. 2.6 are based on. 48 3.1 The fractional deviation of the number density between f(R) and ΛCDM models due to all effects of f(R) (dotted lines), Δn ln effect only (dashed lines), anddynamical mass effect only(dot-dashed lines)evaluated at f R0 = 10 −4 (top), f R0 = 5×10 −5 (middle), and f R0 = 2×10 −5 (bottom) respec- tively. Colors of the lines refer to different surveys which follow the scheme in Fig. 1.3. For the largest field value, the effect on the mass function dominates the enhancement of the cluster abundance at low z, while the dynamical mass effect dominates at z& 0.3. The Planck and SPTPol sur- vey have the lowest mass threshold at z <0.2 and z > 0.2 respectively and hence are most sensitive to the f(R) effects for small field values. . . . . . . 77 x 3.2 Mass limit of cluster surveys in ΛCDM (solid) and f(R) gravity (dashed) with f R0 =10 −4 (left) and f R0 =3×10 −5 (right). The mass limits in f(R) are reduced due to the effect on dynamical mass measurements (Sec. 3.2.1). 78 3.3 Relative deviations in the f(R) halo power spectrum from ΛCDM, i.e. ΔP h /P h for the Planck survey, with|f R0 | = 10 −5 . Upper left. Total devia- tion. Upper right. Deviation due to P L (k) only. Lower left. Deviation due to halo bias b L only. For this value of f R0 , the dynamical mass effect on the power spectrum is negligible and therefore we do not show it here. The redshift and scale dependence in the relative deviations from other cluster surveys are similar to the ones shown here. . . . . . . . . . . . . . . . . . . 79 3.4 Fully marginalized 68% confidence level (CL) constraints on f R0 from the number count of clusters only (using Planck CMB priors), as a function of the step size Δf R0 , for the surveys considered in this paper. The red dotted line indicates σ f R0 = Δf R0 . For a given survey, the intersection of this line with the predicted constraints yields the final expected constraint (Sec. 3.3.2). Solid (dashed) lines represent the case when dynamical mass is (is not) considered. The sharp upturn at Δf R0 . 5×10 −5 is due to the chameleon mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.5 SameasFig.3.4, butfromthepowerspectrumofclustersonly(usingPlanck CMB priors). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.6 Fully marginalized constraints on f R0 from the power spectrum of clusters only, as a function of maximum cluster redshift z max . Δf R0 =7×10 −6 was used for all values shown here. . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.7 Combined dN/dz + P(k) 68% CL marginalized constraints on f R0 as a function of the step size Δf R0 for the different surveys. As in Fig. 3.4, the red dotted line shows the identity σ f R0 =Δf R0 . . . . . . . . . . . . . . . . 81 3.8 Joint constraints on the Compton wavelength λ C0 (in Mpc; see Eq. (3.11)) and (counterclockwise from top left) Ω Λ , σ 8 , w o , and w a . All curves denote 68%confidencelevel, andarefornumbercountsonly(blue),powerspectrum only (cyan), and combination of the two (green). The results are shown for the Planck survey with Δf R0 =5×10 −6 . . . . . . . . . . . . . . . . . . . . 82 4.1 The deviation from the gaussian number count when f NL =100. . . . . . . 90 4.2 Relative deviations of the non-gaussian power spectrum from the gaussian power spectrum, i.e. ΔP/P G for the Planck survey and f NL =100. . . . . 90 4.3 Fully marginalized constraints on f NL from the power spectrum of clusters only, as a function of minimum wavenumber k min (upper) and maximum cluster redshift z max . (lower). . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.4 Jointconstraintsonthef NL and(counterclockwisefromtopleft)σ 8 ,w o ,and w a . All curves denote 68% confidence level, and are for number counts only (blue), power spectrum only (cyan), and combination of the two (green). The Planck CMB power spectrum priors are assumed. . . . . . . . . . . . 96 xi 5.1 Joint constraints ontheM ν andw 0 . All curves denote68% confidencelevel, and are for number counts only (blue), power spectrum only (cyan), and combination of the two (green), Planck CMB (yellow), and Planck CMB LE (dotted yellow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.2 Fully marginalized constraints on M ν from the power spectrum of clusters + Planck CMB prior, as a function of maximum wavenumber k max (left) and minimum cluster mass M min (right). . . . . . . . . . . . . . . . . . . . 111 C.1 The dependence on redshift (left) and wavenumber (right) of the effective volume (Eq. (C.11)) for a single mass bin and each survey: Planck (black), SPT (green), SPTPol (magenta), and ACTPol (blue). The effective volume is a weak function of wavenumber k but strongly depends on the redshift. . 129 xii List of Tables 1.1 Properties of SZ cluster survey . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1 Characteristics of the Planck channels . . . . . . . . . . . . . . . . . . . . . 25 2.2 Parameters of cluster catalogs . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1 Marginalized constraints (68% confidence level) from the two cluster probes dN/dz and P(k), as well as the combination of both for the four SZ sur- veys. The results are combined with forecasted constraints from the Planck CMB. We also indicate the step size Δf R0 used for each survey and probe. The results reported as approximate values refer to the case in which the constraints do not match any of the explicitly evaluated step size. . . . . . . 70 3.2 Relativeimprovementinconstraintsonf R0 ,i.e. σ no prior f R0 /σ weak f R0 ,wheninclud- ing weak priors on the mass-observable relation (see text). In each case, Δf R0 is that given in Tab. 3.1 for the corresponding survey/probe. . . . . 74 4.1 Marginalized 1σ errors on f NL . The labels in the second column means the following. standard: The standard setup as indicated in Sec. 4.2.1 and Sec. 4.2.2; less conservative: The conservative mass limit of the cluster survey is considered (see Sec. 4.4.2); photo-z: Redshift binning correspond to the error of photometric redshift, i.e. Δz = 0.1; one mass bin: No mass slicing in the Fisher matrix, i.e. only one mass bin. . . . . . . . . . . . . . 91 4.2 Same as Tab. 3.2, but for f NL constraints. . . . . . . . . . . . . . . . . . . . 97 5.1 Marginalized 1σ errors on M ν (in units of eV). . . . . . . . . . . . . . . . . 107 5.2 SameasTab.3.2,butforM ν constraints. AfiducialM ν =0.3eVisassumed. The best case is obtained by the Planck survey, σ Mν =0.08 eV (same for + CMB or + CMB LE). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.1 Summary of the best parameter constraints (dN/dz + P(k) + CMB). . . . 116 xiii Abstract Inthiswork,wepresentvariousstudiestoforecastthepowerofthegalaxyclustersdetected bytheSunyaev-Zel’dovich (SZ)effectinconstrainingcosmological models. TheSZeffectis regarded as one of the new and promising technique to identify and study cluster physics. With the latest data being released in recent years from the SZ telescopes, it is essential to exploretheirpotentials inprovidingcosmological information andinvestigate theirrelative strengths with respect to galaxy cluster data from X–ray and optical, as well as other cosmological probes such as Cosmic Microwave Background (CMB). One of the topics regard resolving the debate on the existence of an anomalous large scale bulk flow as measured from the kinetic SZ signal of galaxy clusters in the WMAP CMB data. We predict that if such measurement is done with the latest CMB data from the Planck satellite, the sensitivity will be improved by a factor of > 5 and thus be able to provide an independent view of its existence. As it turns out, the Planck data, when using the technique developed in this work, find that the observed bulk flow amplitude is consistent with those expected from the ΛCDM, which is in clear contradiction to the previous claim of a significant bulk flow detection in the WMAP data. We also forecast on the capability of the ongoing and future cluster surveys identified through thermal SZ (tSZ) in constraining three extended models to the ΛCDM model: modified gravity f(R) model, primordial non-Gaussianity of density perturbation, and the xiv presence of massive neutrinos. We do so by employing their effects on the cluster number count and power spectrum and using Fisher Matrix analysis to estimate the errors on the model parameters. We find that SZ cluster surveys can provide vital complementary information to those expected from non-cluster probes. Our results therefore give the confidence for pursuing these extended cosmological models with SZ clusters. xv Chapter 1 Introduction One of the most popular theoretical aspects of nowadays cosmology is the behavior of gravity on large scales. Galaxy clusters are the biggest gravitationally bound objects in ourUniverseandseveraloftheirglobalpropertiessuchasabundanceandclusteringonlarge scales can be predicted accurately with theoretical models (e.g., [180, 169]). Astronomers believethatthegalaxiesthatweobservetodaygrewgravitationallyoutofsmallfluctuations in the density of the universe and they are pulled together by gravity to form larger halos (cluster of galaxies). These objects are multi-component systems that consist of dark matter and baryons. They are very massive with typical mass of 10 14 M ⊙ and size of a few Mpc across. They emit radiation almost across the entire electromagnetic spectrum: X–ray from thermal bremsstrahlung emission of ionized hot gas; optical and infrared from stellar components such as galaxies and intra-cluster light; millimeter wavelength from the inverse Compton scattering within clusters; radio from synchrotron emission of relativistic electrons. Therefore, galaxy clusters are extensively used to probe the evolution of the large scale structure formation and as a tool to test the standard cosmological models. In fact, early works of cluster-based cosmology were focused on the X–ray and optical band (see review in Sec. 1.1), since large samples of clusters were initially identified in these wavelengths: ∼10 4 clusters in optical [e.g. 89, 179] and∼10 3 clusters in X–ray [e.g. 137]. These traditional methods are being complemented by new approaches, in particu- lar the observations in the microwave regime through the Sunyaev-Zel’dovich (SZ) effect. 1 Measurements of this effect provide distinctly different information about cluster prop- erties than X–ray data, while combining X–ray and SZ data leads to new insights into cluster science. The SZ effect is independent of redshift, therefore it provides a unique way to probe the large scale structure of the universe as traced by galaxy clusters. Future observations with SZ effect are promising and expect to find hundreds to thousands of new galaxy clusters. Therefore I dedicate this work to investigate potentials of SZ clusters in extracting cosmological information. I first give an overview of the probes used to do cosmology with X–ray and optical detected clusters, and their potentials in extending to SZ clusters, in Sec. 1.1. Theprincipleof SZ effect is explained in Sec. 1.2. In Sec. 1.3 I will presentthemodelinconstructingSZclustersurveys. ThecurrentstatusofSZobservations is given in Sec. 1.4 and the outline of this thesis is given in Sec. 1.5. 1.1 Galaxy Clusters as Cosmological Probes In this section, I first give a brief review on what we have learnt in recent years from different types of measurements, mainly X–ray and optical, of galaxy clusters. On the general picture, these measurements agree with other probes such as Cosmic Microwave Background Radiation (CMB) and type 1A supernovae (SN) that our universe can be described by the Lambda Cold Dark Matter (ΛCDM) model: dark energy (73%), with sub-dominant dark matter (23%), and a small minority of baryonic matter (4.6%) [92]. The main strength of cluster data is that it is very sensitive to the dynamics and evolution of the growth of structure of the universe. This helps to break parameter degeneracies because it provides complementary information to geometric measurements such as CMB. Specifically, cluster data are very powerful and unique in the following aspects: Local abundance The number of clusters can be predicted with theoretical models, e.g. ΛCDM model. Comparison between the theoretical prediction and the observations 2 can give constraints on the cosmological parameters. Various measurements of cluster counts in X-ray [186, 136, 135, 15, 4, 188, 113] and optical [155] provide constraints of the amplitude of the matter power spectrum on the scale of 8 Mpc/h (roughly the scale of cluster size, σ 8 ), matter density (Ω m ), dark energy (Ω Λ and w), particle physics and neutrinoproperties. Theseresultsalsosuggestthatclusterdataalonearevery competitive with the CMB constraints from WMAP and further tighten the precision when combined with other datasets such as Baryon Acoustic Oscillation (BAO) and SN (Fig. 1.1). I defer the detail of the formalism to calculate the cluster abundance from cluster sur- veysinSec.1.3.1, whichisusefulfortheapplicationsofthisprobetoconstraincosmological parameters from SZ measurements as discussed in Chapter. 3, Chapter. 4, and Chapter. 5. Furthermore, in Chapter. 2 we will construct simulations of SZ emission to study cluster peculiar velocities using the theoretical prediction of cluster abundance. Gas mass fraction The mass fraction of hot gas, f gas (z) ≡ Ω b (z)/Ω m (z), where Ω b is the baryon density, inside massive cluster is recognized as a distance estimator since f gas (z)∝d(z) 3/2 . Thereforethis quantity as measured from massive clusters can beserved asstandardcalibration sourcetotesttheexpansionhistory oftheuniverse. Theadvantage off gas isthat,relativetoclustercount,asmallsampleofclusterswouldbeenoughtodeliver competitive constraints. Constraints on cosmological parameters obtained from f gas in X- ray [3, 113] agrees with those of cluster count, thus offering a good consistent check on the result from cluster probes. Measurement of the gas fraction requires imaging data, i.e. resolved cluster profiles, to obtain the cluster mass. X–ray data for low redshift clusters is well suited to this purpose, far surpassing SZ data since currently only a few SZ clusters are resolved. Since future SZ surveys are aimed at measuring the statistical quantities of clusters but not the resolved profiles, we are not going to study the constraints from gas mass fraction. 3 Velocity field Peculiar velocity field is generated from primordial perturbations and is related to density field through the continuity equation, so it is a direct probe of matter density [16, 36, 35, 17] and the gravitational potential [43]. The peculiar velocity field was initially measuredfromsurveys ofgalaxy (objects that clustersarecomposed of), however, these measurements have low signal to noise (S/N) and are limited to low redshifts due to large uncertainties in distance determinations. Latest measurements using the kinematic Sunyaev Zel’dovich (kSZ) effect [177] of clusters (Sec. 1.2) have offered an alternative way to probe the velocity field since the signal is independent of distance. We will investigate in this work the usefulness of this technique in Chapter. 2. Clustering The Fourier transform of the spatial distribution of galaxy clusters, i.e. the power spectrum, is a useful statistics in extracting cosmological information. This is given by the two point correlation function hδ(k)δ(k ′ )i of the matter density field. The matter power spectrum P m (k) is defined by hδ(k)δ(k ′ )i = (2π) 3 P m (k)δ 3 ( ~ k− ~ k ′ ), where δ 3 () is the Dirac delta function. The theoretical treatment to derive P m (k) can be found in [110]. In practice, one can compute it, for any given set of cosmological parameters, using public available code CAMB [102] which will be employed in this work. Given that structures form only at the peaks of density fluctuations, the power spectrum measured from large scale structures (galaxies or clusters) only probe a portion of the underlying matter distribution. As a result the cluster power spectrum do not necessarily trace the matter power spectrum. We can account for this difference by modifyingthe cluster power spectrum with the linear bias function b L (M): P cluster (k) =b 2 L (M)P m (k) (1.1) The bias function can be calibrated through N-body simulations [e.g. 181] and we will return to its analytical form when we encountered it. 4 Theclusterpowerspectrumisusefulincosmology becauseitsshape,amplitude, aswell asitsevolutionhistorydependsoncosmology. Constraintsfromtheclusterpowerspectrum from X-ray and optical measurements are relatively weak (e.g. 0.07 < Ω m < 0.38 [189]). Alternatively, the baryon acoustic oscillation signature in the power spectrum of clusters can also probethe comic distance which depends on cosmology. However, [49, 8, 66] found that the signal is weak (S/N <2) and thus can not be used to constrain cosmology. Nevertheless, the cluster power spectrum is found to be useful in constraining models of inflation and modified gravity. The measurements from SZ detected clusters are partic- ularly suitable for these investigations. This is because the corresponding effects of these models on structure formation are more prominent for very massive objects and located at larger redshift which are probed efficiently by SZ cluster surveys. Power spectrum mea- surements have not been made from SZ cluster survey yet. As such we will forecast of its usefulness to constrain cosmological parameters, like we do with number count. We defer the details of its application to Fisher matrix analysis in Appendix. C.2.2. 1.2 The Sunyaev-Zel’dovich Effect The SZ effect [178] is a secondary CMB anisotropy caused by the scattering of CMB photonsbyhighenergyelectrons, suchasthosecomposingtheintra-clustermedium(ICM). A comprehensive review of the science of the SZ effect can be found in [13]. Of particular interest in the microwave observations is the fractional change in temperature of the CMB photons caused by SZ effect which can be categorized as two components: the kinetic (kSZ) and thermal (tSZ) Sunyaev-Zeldovich effect, as describe below. The SZ effect has became a new technique to identify galaxy clusters in recent years because it has a unique spectral feature that helps to distinguish itself from other signal on the CMB sky. Onekey issue of using cluster as cosmological probe is to relate the observable (in this case the SZ 5 Figure 1.1: Constraints on the dark energy density Ω X and equation of state w 0 in a flat universe from the combination of all cosmological datasets. The combined constraints are w 0 =−0.991±0.045 and Ω X =0.740±0.012. Credit: [188]. signal) and the total mass of cluster, since this is the quantity most readily predicted from theoreticalmodels. Observations[e.g.146]andnumericalsimulations[e.g.121]suggestthat the amplitude of the SZ effect is an excellent mass proxies. As a result, the SZ selected clusters are considered to be one of the most powerful means to constrain cosmology. 1.2.1 Thermal Sunyaev-Zel’dovich Effect WhenCMBphotons passthroughthecenter ofa massivecluster, about1% ofthephotons would interact with an energetic ICM electron. The resulting inverse Compton scattering preferentially boosts the energy of the CMB photon and cause a small distortion in the CMB blackbody spectrum. Such spectral distortion, when expressed as a temperature change ΔT at dimensionless frequency x =hν f /k B T CMB , is given by ΔT tSZ T CMB =f(x)y =f(x) Z r cluster n e k B T e m e c 2 σ T dl (1.2) 6 Figure 1.2: Spectral distortion of the CMB spectrum due to SZ effect. Credit: [25]. where T CMB =2.73 Kelvin, y is the Compton parameter, σ T is the cross section of the Thomson scattering, n e is the electron number density, T e is the electron temperature, k B is theBoltzmann constant, m e c 2 isthe electron restmassenergy, ν f is thefrequency of the CMBphotons, andf(x)= x e x +1 e x −1 −4 (1+δ SZ (x,T e ))isthefrequencydependenceof the SZ effect, δ SZ (x,T e ) is the relativistic correction to the frequency dependence. As shown in Fig. 1.2, the tSZ effect has a unique spectral signature: the minimum and maximum temperature changes occur at ν = 143 GHz and ν = 353 GHz respectively with a null at ν =217 GHz. For typical clusters, T e ∼10 keV gives ΔT tSZ ∼100μK. It is worth noting that ΔT tSZ /T CMB is independent of redshift, as shown in Eq. (1.2). This unique feature makes the thermal SZ effect a potentially powerful tool to identify clusters at high redshift. The quantity that is most relevant for finding clusters with SZ surveys (see Sec. 1.3) is the integrated tSZ signal over the solid angle of the cluster: the Y parameter: Y = Z ΔT tSZ dΩ cluster ∝ N e hT e i D 2 A ∝ MhT e i D 2 A (1.3) 7 where N e is the total number of electrons in the cluster, hT e i is the mean electron tem- perature, D A is the angular diameter distance, and M is the mass of the cluster. The parameter Y, which is the integrated pressure of the cluster, provides a clean measure of the total thermal energy of the cluster. Unlike the Compton parameter y, this integrated parameter has a redshiftdependence from the angular diameter distance D A (z). However, D A (z) is fairly flat at high redshift and that cluster of a given mass will be denser and therefore hotter at high redshift. Therefore, clusters detected with the SZ effect, above some mass threshold, would have little redshift dependence. 1.2.2 Kinetic Sunyaev-Zel’dovich Effect If the cluster is moving with respect to the CMB rest frame, there will be an additional temperaturedistortiontotheCMBphotonsduetotheDopplereffectoftheclustervelocity on the scattered CMB photons. In the non-relativistic limit, the spectral signature of the kSZ effect is a pure temperature change with magnitude: ΔT T CMB kSZ =− v p c τ (1.4) where v p is the line-of-sight peculiar velocity of the cluster, τ = σ T R n e dl is the optical depth of the cluster. For typical galaxy clusters with τ ∼ 10 −3 and v ∼ 500 km/s, we expect a CMB temperature change of ΔT kSZ ∼ 5μK at the location of a galaxy cluster. To date, the kSZ effect has not been measured in individual galaxy clusters. Nevertheless, if many galaxy clusters are moving in the same direction, their velocity field creates a dipolar pattern in the CMB radiation at large scale and leads to a net dipole moment C 1,ksz =T 2 CMB hτi 2 V 2 bulk /c 2 [82], wherehτi is themean optical depthof thecluster sample. We will investigate the detectability of this signal in detail in Chapter. 2. 8 In individual clusters, the tSZ effect is typically larger than the kSZ effect: ΔT kSZ ΔT tSZ ∼0.1f(x) v p 300 km/s T e 5keV −1 (1.5) Nevertheless, the dipole component of the kSZ effect may dominate over the statistical dipole component of the tSZ effect if a bulk flow is present. Furthermore, the different frequency dependence of the two effects allows us to extract the kSZ signal. 1.3 Modeling SZ cluster survey The first set of clusters detected by large scale observations of the SZ effect, such as the South Pole Telescope (SPT [24, 184]), the Atacama Cosmology Telescope (ACT [94, 114], and the Planck satellite [146, 143], are released recently with promise of more to come. As mentioned, the advantage of the SZ surveys, as compared to X-ray and optical surveys, is thattheSZsignalisindependentofredshift. Thereforethesesurveyswouldprovide∼1000 new clusters at intermediate to high redshifts upon completion which would happen in the next 2–3 years. In this work, we will investigate to what extent these new SZ cluster surveys are expected to test cosmological models. To do so, one needs a theoretical model that describes the cluster count and spatial distribution expected from a survey, given cosmological assumption and specification of the observations. 1.3.1 Cluster count The number of galaxy clusters observed in a survey per mass and redshift bin can be predicted by: dN dMdz (M,z) =ΔΩ dn dM (M,z) dV dzdΩ (z) (1.6) 9 where ΔΩ is the solid angle in a given direction, dV/dzdΩ = D 2 A c/(H(z)(1 +z)) is the physical volume per unit solid angle per unit redshift [132] which depends on cosmology. The halo mass function, dn dM (M,z), is the number density of galaxy clusters per mass bin at a given redshift z, which is given by dn dM (M,z)= ρ m M dlnσ −1 dM f(σ) (1.7) f(σ) is called the multiplicative function which describes the fraction of volume collapsed to form clusters. Analytical expression is first derived by [147] (hereafter PS), and later calibrated in many N-body simulations with progressive improvement in accuracy. We have adopted different forms of the halo mass function: the Jenkins mass function [75] inChapter.2,theShethandTormenmassfunction[169]andtheTinkermassfunction[180] in Chapter. 3, Chapter. 4, and Chapter. 5: f Jenkins (σ −1 )=0.315exp(− lnσ −1 +0.64 3.88 ) (1.8) νf Sheth&Tormen (ν)=A st r 2 π 0.75ν 2 [1+(aν 2 ) −0.3 ]exp[−0.75ν 2 /2] f Tinker (σ)=A t 1+ σ b −a exp − c σ 2 The parameters A st , A t , a, b, c are fitting parameters given in the respective references. ρ m is the matter density at present time (z = 0), ν = δ c /σ(M,z) is the normalized peak height [77, 9], δ c =1.686 is the critical collapse threshold in which objects can formed [48]. σ is the variance of the linear density field in a sphere of radius R, which is related to the mass by M =4πρ m (z =0)R 3 /3 and is given by σ(M) 2 = 1 2π 2 Z ∞ 0 k 2 dkP m (k)|W(kR)| 2 (1.9) 10 whereW(x)=3(sinx−xcosx)/x 3 is thefilter function of aspherical top hat. Thisshould not be confused with the previous mentioned parameter σ 8 , in which σ 8 measures the normalization of the matter power spectrum on the scale of 8 Mpc/h. Byintegratingoverthemassandredshiftintervalofinterest,weobtainthetotalnumber of clusters N expected in a survey with sky coverage f sky , redshift range [z i ,z f ] and mass range [M lim ,M max ]: N = Z f sky dΩ Z z f z i dz Z Mmax M lim dM dN dM dz(M,z) (1.10) The adaptation of the number count in the Fisher matrix analysis can be found in Appendix. C.2.1. 1.3.2 Limiting mass: selection function In Eq. (1.10), the limiting mass M lim (z) (usually depends on redshift) is the minimum mass that a cluster can be detected in a flux limited survey. Clusters that have M <M lim are too faint to be detected and would not make it into the survey catalog. Hence a well understanding of the selection function, i.e, limiting mass as a function of redshift, of a given survey is crucial in constructing a realistic catalog. In the following, we will describe thespecificationofthefourSZsurveysthatareconsideredthroughoutthiswork. Whilewe trytoobtainasrealisticsurveyspecificationsaspossible, thelack ofpreviouslargesamples of SZ clusters necessarily makes these quantities somewhat uncertain. In particular, the relation between cluster mass and SZ signal is still imperfectly known (e.g. [5, 150, 122, 138]). The properties are summarized in Tab. 1.1. The final mass limits as a function of redshift and the resulting expected number of clusters for each survey is shown in Fig. 1.3. The Planck Catalog The European Space Agency’s Planck mission is the third generation space mission to 11 Table 1.1: Properties of SZ cluster survey Survey Area (sq. deg) No. of clusters Planck 30000 1000 SPT 2500 500 SPTPol 625 1000 ACTPol 4000 500 Figure 1.3: Left: Mass limit of cluster surveys. Right: The redshift distribution of clusters in the Planck (black), ACTPol (blue), SPT (green), and SPTPol (magenta) survey in the fiducial ΛCDM cosmology. measure the anisotropy of the CMB. It is imaging the whole sky with an unprecedented combination of sensitivity (ΔT/T ∼ 2×10 −6 per beam at 100 - 217 GHz), angular reso- lution (5 ′ at 217 GHz), and frequency coverage (30−857 GHz). A blind multi-frequency search of the SZ clusters in the all-sky maps yields a sample of 189 clusters with S/N >6 in the early release [146] and later 1227 clusters in the latest data release [141]. We expect such sample will expand in the future when the validation process of cluster can- didates is completed, and it will produce a cluster sample with median redshift ∼ 0.3 (see Fig. 1.3, upper left panel). In this work, we model the selection function for this Planck samplebyassumingahighlevel ofcompleteness (about90%) whichcorrespondsto 12 Y 200,ρc ≥2×10 −3 arcmin 2 [119], where Y 200,ρc is the integrated comptonization parameter within r 200,ρc , the radius enclosing a mean density of 200 times the critical density. This choice is justified since the expected yield is consistent with the Planck catalog. For an SZ survey, its flux limit can be translated into a limiting mass by using simulation-calibrated scaling relations [167]: M lim,200ρc (z) 10 15 M ⊙ = " D A (z) Mpc/h 70 2 E(z) −2/3 Y 200,ρc 2.5×10 −4 # 0.533 . (1.11) Inordertomitigatetheeffectofoverestimationofunresolvedclustersatlowredshift,we further restrict M lim,200ρc to beat least 10 14 M ⊙ at all z. With all these criteria, our model find∼1000clusters,whichisconsistentwith[141]. WhilewekeepY 200,ρc =2×10 −3 arcmin 2 as our reference minimum value for presentation of the main results, we will also discuss predictions for a lower mass threshold, correspondingtoY 200,ρc =10 −3 arcmin 2 . With such threshold, the completeness of the S/N >5 sample is reduced to about 70% and the total number of clusters is 2700. SPT and SPTPol The South Pole Telescope is a 10m diameter telescope operating at the South Pole that aims to conduct large area mm and sub mm wavelength surveys of the CMB since 2007. It is currently observing the sky with a sensitivity of 18μK/arcmin 2 at 148 GHz, 218 GHz, and 277 GHz. This survey covers Ω≈ 2500 square degrees of the southern sky (between 20h ≥ RA ≥ 7h, −65 ◦ ≤ δ ≤ −30 ◦ ) with a projected survey size and cluster mass limit well matched to the Stage III survey specification of the Dark Energy Task Force [184]. For the mass limits, we employ the calibrated selection function of the survey by [184]. This is based on simulations and used to provide a realistic measure of the SPT detection 13 significance and mass. Disregarding the scatter in the fitting parameters for this relation, we use here: M lim,200¯ ρ (z) 5×10 14 M ⊙ h −1 = " p ξ 2 −3 6.01 ! 1+z 1.6 −1.6 # 1/1.31 (1.12) whereξ isthedetectionsignificance. FortheSPTsurvey,wetakeclustersdetectedatξ >5 which ensure a 90% purity level. Currently, the SPT team is setting a low redshift cut at z cut = 0.3 in their released cluster sample, due to difficulties in reliably distinguishing low-redshiftclustersfromCMBfluctuationsinsinglefrequencyobservations. Nevertheless, withupcomingmulti-frequencyobservations, alower cutz cut =0.15willlikely beattained. We therefore apply this cut in our work. With this, the SPT survey is expected to detect ∼500 clusters. In addition to this, we also consider the upcoming SPT polarization survey (hereafter SPTPol)whichwill haveanincreased sensitivity of4.5μK/arcmin 2 at150 GHzfora3year survey and sky coverage of 625 square degrees. We scaled the mass limits by a factor of 3.01/5.95 inEq. (1.12) to match withthe expected mass limits of SPTPol clusters (Benson 2011, private communication). We again use z cut = 0.15, resulting in a total expected number of∼1000 clusters. While these are the limits we use for our main results, we also discuss outcomes that consider a lower mass limit, corresponding to ξ =4.5 (80% purity). With this mass limit, SPT would find 800 clusters and SPTPol about 1400 clusters. ACTPol The Atacama Cosmology Telescope is a 6m diameter telescope located at the Atacama desert. It has been observing a portion of the southern sky since 2008 consisting of two strips of the sky, each 4 degrees wide in declination and 360 degrees around in right ascen- sion, one strip is centered at δ =−5 ◦ , and the other is centered at δ =−55 ◦ [167]. With a sensitivity of ≈ 35μK/arcmin 2 , only about 100 clusters are expected to be detected. 14 Instead, we turn to the newly developing dual-frequency (150 GHz and 220 GHz) polar- ization sensitive receiver (hereafter ACTPol [124]) to be deployed on ACT in 2013. One of the three ACTPol observing seasons will have a wide survey covering 4000deg 2 to a target sensitivity of 20μK/arcmin 2 in temperature at 150 GHz. With the wide field, they aim to find ∼ 600 clusters in the ACTPol survey. The survey is 90% complete above a limiting mass of M lim,200¯ ρ = 5×10 14 M ⊙ h −1 (Sehgal 2011, private communication), and we therefore assume this as our redshift-independent mass limit for ACTPol. As in SPT, the ACT team also put a low redshift cut in their parameter determination works and we likewise take z cut = 0.15 for ACTPol, resulting in a total expected number of∼ 500 clus- ters. We also present in the discussion section the results corresponding to a lower mass limit, M lim,200¯ ρ =4×10 14 M ⊙ h −1 , which would result in a catalog of about 1000 clusters. 1.4 Current Status on SZ clusters Preliminary results have been reported from cluster counts measured from the latest SZ survey: SPTsurvey(18clusters[11]and100clusters[153]clusters),ACT(21clusters[62]), and Planck (189 clusters [146] and 1227 clusters [143]). These works, when combined with CMB information from the WMAP data, yield Ω m = 0.25−0.29, σ 8 = 0.77−0.80, all consistent with each other to within 1σ. The tSZ angular power spectrum as proposed by [91], which is a sensitive probe of the σ 8 parameter since the amplitude C l ∝ σ 8 8 , is recently measured by [172, 152] at small scales ∼ 5 arcmin and [140] at large scales 0.17 ◦ −3.0 ◦ . TheyfoundΩ m =0.28−0.33 andσ 8 =0.74−0.81. Inparticular,[172]present comprehensive constraints on the base ΛCDM model as well as parameters describing extension to this standard model. Theseinitial resultsarebasedonthepreliminarilyphaseofthosesurveys, withpromise of expansion in sample size and refined analysis to come in the near future. The data are 15 expected to produce interesting constraints once the surveys are completed. Furthermore, wecanapplytotheselargesampleswithnewtypesofmeasurements, suchasclusterpower spectrum and peculiar velocity field, to give consistency check on current constraints. 1.5 Objective of this work Inthiswork,weexploretheprospectsofmakingthesenewmeasurementsandthemeritsof thefutureSZsurveysinprovidingconstraintstotheextendedΛCDMmodel. InChapter.2, we study the possibility of measuring peculiar motions of distant galaxy clusters using the kinetic SZ effect with the Planck satellite. We then forecast the constraints, using the Fisher matrix analysis as described in Appendix. C, on various types of physics: modified gravitymodelwhichisconsideredasthealternativetodarkenergy(Chapter.3);primordial non-Gaussianitycontributiontothestatisticaldistributionsofthedensityfluctuationwhich helps to reveal physics of inflation and the very early universe (Chapter. 4); masses of neutrinos that have direct influences on the growth of cosmic structures (Chapter. 5). We summarize our results in Chapter. 6. 16 Chapter 2 The search of Cosmic Bulk Flow with the Planck satellite 2.1 Introduction Peculiar velocities, along with inhomogeneities, can be used to constrain cosmology. Pecu- liar velocities are induced by inhomogeneities in the underlying mass distribution. This is different from the Hubble flow, which makes objects always move away from us due to the expansion of the universe, that it instead pulls objects together since it is induced by the gravitational force of the matter field. Large scales structures such as galaxies and clusters probethematterdistribution,thereforetheyaregoodtracersofthevelocityfields. Onlarge scale, we can consider the average velocities of objects within a volume of the space. We referthislargescalepeculiarvelocityasthebulkflow. Thestandardinflationarymodelpre- dicts that the rms bulk velocity within a sphere of radiusR decreases linearly with comov- ing distance in the ΛCDM universe with V rms (R >50−100 Mpc h −1 )≈250( 100 Mpc h −1 R ) km/s [83]. Galaxy cluster peculiar velocity surveys at scales R ≤ 60 Mpc h −1 generally agree with theoretical predictions of the cluster bulk velocities. However, recent measure- ments at larger scales (R ≥ 100 Mpc h −1 ) indicate that the bulk flow velocity is signifi- cantly largerthantheΛCDMpredictionwithstatistical significanceupto3σ [82, 53]. This ”dark flow”, in analogy of the other dark imponderables like dark energy and dark matter, therefore presents a challenge to the ΛCDM model. 17 Many attempts were carried out to estimate these statistical measures of the peculiar bulk flow but the direction in which the dark flow move relative to us and the convergence depthwereofteninconsistentamongvariousmeasurements. Themeasuredbulkflowdirec- tions on the sky are illustrated in Fig. 2.1. The main reason is the problematic nature of thedistanceindicators usedtodeterminethepeculiar velocities. Randomerrors, including intrinsicscatterinthedistanceindicatorandmeasurementerrors,andnontrivialerrorsdue tononuniformsamplingofthegalaxies thatserveasvelocity tracers werealmostinevitable in any measurements. [80] have proposed a method aimed at determining the largest scale bulk flows from galaxy clusterpeculiarvelocities measuredusingthekineticSZeffect. Ifmanygalaxy clus- ters aremoving with acoherent motion withrespectto theCMBrestframe, the kinematic part of the SZ signal acquires a dipole moment. Since the kSZ signal is proportional to line of sight velocity, such a measurement directly probes the bulk flow, free of distance measurementerrors. Severalauthorsattempted tomeasurethebulkflowbymeasuringthe kSZeffect in WMAP data. [82] (hereafter KAKE, andlater [81]) firstutilized this method, claiming a large-scale flow with v >600 km/s out to∼575 Mpc h −1 , without sign of con- vergence to the ΛCDM predicted value. However, by repeating the same method [85] did notdetectastatistically significantbulkflow.[129](hereafterOMCP),usedfiltersdesigned to enhance the signal to noise of the kinetic signal and found no significant velocity dipole in the WMAP 7 year data. More specifically, they found a 95% bulk flow upper limits of the order of 4600 km/s in the direction of the KAKE. They also demonstrated that CMB and instrument noise dominate the uncertainties. 18 Figure2.1: ThedipoledirectioninGalacticcoordinatesofcurrentbulkflowmeasurements. (a) CMB b = 48.26±0.03 ◦ , l = 263.99±0.14 ◦ [74]; (b) Local Group b = 30±3 ◦ , l = 276±3 ◦ [90]; (c) KAKE b = 34 ◦ , l = 267 ◦ ; (d) [192] b = 8±6 ◦ , l = 287±9 ◦ ; (e) [98] b = −28±27 ◦ , l = 220±27 ◦ ; (f) [72] b = 0±11 ◦ , l = 263±13 ◦ ; (f) [53] b = 6±6 ◦ , l = 282± 11 ◦ . The size of the colored region is proportional to the amplitude of the measured bulk flow. The color code represents the convergence depth of the bulk flow, in units of Mpc h −1 . 2.2 Objective In this work, we apply the scheme of OMCP to study the capability of Planck data and future cluster surveys to measure bulk flows. The use of Planck maps is expected to produce improved results with respect to the WMAP case because of reduced instrument noise, wider frequency coverage (ensuring better foregrounds’ subtraction) and increased spatial resolution of this mission. In addition, Planck will also produce the first all-sky 19 SZ survey with a median redshift of z = 0.3 (Planck Blue Book) and will ensure a better extraction of the kSZ signal for any cluster sample considered. Weinvestigate towhatextenttheuseof Planck maps, incombination withdataforthe ROSAT clusters which was used in previous works, improve on bulk flow determination from WMAP. We also assess the expected performances of bulk flow measurements for upcomingall–sky cluster catalogs, such as the ones derived from Planck and eRosita satel- lites. Such samples are more abundant and extend to higher redshifts than the ROSAT. Ourgoalsare: (i)todeterminethesensitivityoftheclustervelocitydipolemeasurement with the Planck specifications and assess the nature of the uncertainty; (ii) to study which cluster survey can best constrain the bulk flow; (iii) to study the performanceof the filters used in OMCP with the Planck setup. This chapter is organized as follows. The bulk flow velocity expected from the ΛCDM model is calculated in Sec. 2.3. In Sec. 2.4, we briefly describe the procedure we use to measure the bulk flow velocity. In Sec. 2.5 and Sec. 2.6 we give details of the SZ and X-ray cluster catalogs we use and describe the procedure we adopt to generate simulated SZ maps. In Sec. 2.7, we present the two filters we use to reconstruct the kSZ signal from the CMB maps. In Sec. 2.8, we describe the analysis pipeline we use to measure and calibrate the cluster dipole. In Sec. 2.9, we describe the systematic effects that may contaminate our results. The results are presented in Sec. 2.10, followed by the latest measurements from the Planck data when applying the current method in Sec. 2.11. Our conclusions are in Sec. 2.12. Throughout this chapter, we assume a ΛCDM cosmological model with Ω m =0.3, Ω Λ =0.7, h=0.72, w =−1, σ 8 =0.8. 20 2.3 Theoretical Aspects 2.3.1 Expected Bulk Flow Velocity from ΛCDM model In the linear theory of structure formation the peculiar velocity field of galaxy clusters is related to the matter overdensity through the continuity equation: ~ v k =if(z) δ k k ˆ k (2.1) where f(z)≡ a D 1 dD 1 da =H(z) dD 1 (z) dz , and D 1 is the growth factor [95, 26]:. D 1 (z)=(1+z) −1 5Ω m (z) 2 [Ω m z 4/7 −Ω Λ (z)+(1+Ω m z/2)(1+Ω Λ (z)/70)] (2.2) At scales much larger than the size of a cluster, correlated peculiar velocities within a given region wouldlead to acoherent motion, or bulkflow, of theobjects insidetheregion. Assuming the distribution of clusters is isotropic, the bulk flow velocity within a region of size R is given by: σ 2 v (z)=f 2 (z) Z dk P m (k) 2π 2 |W(kR)| 2 (2.3) where W(kR) is the Fourier transform of the top-hat window function, R = R z 0 c/H(z ′ )dz ′ is the comoving radius and P m (k) is the present-day linear matter power spectrum. The cosmology dependence is embedded in the matter power spectrum P m (k) and its time evolution. Inparticularthermsvelocity, σ v , isproportionaltotheamplitudeofthematter power spectrum σ 8 which is known with an uncertainty <10% (e.g. [97]). We incorporate the effect of the different redshift distributions of the clusters in the catalogs we use by weighting the bulk flow velocity by a selection function φ(z). The 21 selection function takes into account the fact that the survey is not complete at all of the redshifts we use. Then, σ 2 v (z)= Z dk P m (k) 2π 2 |W(kR)| 2 Z z 0 φ(z ′ )f(z ′ ) dV dz ′ dz ′ 2 (2.4) where V(z) is the comoving volume, φ(z) is the comoving number density or selection function. We calculate φ(z) from the halo mass function (Sec. 1.3.1) and cluster mass limits for each survey (Sec. 2.5), φ(z)= ¯ n(z)= Z ∞ M min (z) dM dn(M,z) dM (2.5) where M min (z) is the limiting mass of object in the cluster survey at redshift z. φ(z) is normalized such that R z 0 φ(z ′ )(dV(z ′ )/dz ′ dΩ)dz ′ =1. Fig. 2.2 shows the rms bulk flow velocity predicted by Eq. (2.4) for the three cluster samples considered in this chapter. The effect of the selection function is small in which the velocities among the three cluster samples are no different than 1% at all redshifts. For comparison, the rms bulk flow computed with no selection function is also plotted. The sample variance of the velocity distribution is the largest source of uncertainty. For a Gaussian density field the amplitude of the bulk flow has a Maxwellian distribution with the probability of having a velocity lying between V and V +dV given by P(V)dV ∝ V 2 exp(−1.5V 2 /σ 2 v )dV. The95% confidencelimits onthemeasuredbulkflow ofamplitude V are then V/3<V <1.6V. This is the shaded region in the plot. In addition to large scale correlated cluster velocities, we also include the peculiar velocity on cluster scale (R∼8 Mpc h −1 ) to each cluster in our simulations. 22 Figure2.2: ThetheoreticalbulkflowvelocityinΛCDMcosmology,usingselectionfunctions for the three cluster surveys, smoothed over a top hat window function W(kR) with the comovingspherecenteredontheobserverwithradiusR.Alsoplottedistheamplitudeofthe expected bulk flow with uniform selection function. The shaded region is the uncertainty of the expected bulk flow from sample variance. 2.3.2 Models of Cosmic Bulk Flow Several theories havebeensuggested toexplain thehighbulkflowvelocities atlarge scales. One explanation is that pre-inflationary fluctuations in scalar fields on superhorizon scales gives a titled universe ([182], [84], [120]). In this picture, matter slides from one side of our Hubble volume to the other, producing an intrinsic CMB dipole anisotropy as seen in the matter restframe. Thisinhomogeneity generates a bulkflow with correlation length of order the horizon size. Alternatively, [194] have showed that a strengthened gravitational attraction at late times can speed up structure formation and increase peculiar velocities. Using N-body simulations, they found an enhancement in large scale, R > 100 Mpc h −1 , bulk flow velocities of up to∼ 40% relative to the ΛCDM cosmology. A similar approach using modified gravity is discussed in [1] and [87]. 23 2.4 Methodology 2.4.1 Assumptions and Definitions In this section we define the different physical processes contributing to the sky signal and describe the Planck instrumentation. Theobserved dipolesignal at clusters’ locations has, in principle, several contributions: a 1m =a CMB 1m +a noise 1m +a tSZ 1m +a kSZ 1m +a point 1m +a Gal 1m (2.6) where a i 1m are the dipole coefficients of the CMB, instrument noise, thermal SZ, kinetic SZ, extragalactic radio and infrared point sources, and galactic signal. An appropriate temperature mask can be used to suppress the galactic components and fit for the cluster velocitydipoleoutsidethegalacticregion. Wethereforedonotconsidertheeffectofgalactic foregrounds in this work and apply a cut at |b|≤ 7 ◦ to the simulated Planck maps. The contribution from infrared point sources is not well characterized at the present time and no model is available to accurately describe their abundances within galaxy clusters. We therefore do not consider the infrared point source signal in this work, and only simulate the first 5 terms in Eq. (2.6). Our goal is to separate a kSZ 1m from the other components and calculate its amplitude in Planck maps using different cluster samples. The detector noise levels and beam sizes we usearesummarizedinTab.2.1. AlthoughCMBobservationsbyPlanck cover9frequencies from 30 GHz to 857 GHz, we only consider the 6 channels between 44 GHz and 353 GHz because of large potential foreground contamination outside this range. 2.4.2 Outline of the Method The steps we take to estimate the cluster kinetic SZ dipole velocity are: 24 Table 2.1. Characteristics of the Planck channels Planck Channel 1 2 3 4 5 6 Center frequency ν 0 (GHz) 44 70 100 143 217 353 Resolution Δθ (FWHM) 26. ′ 8 13. ′ 1 9. ′ 2 7. ′ 1 5. ′ 0 5. ′ 0 Noise level σ N (mK) 3.02 5.40 3.2 2.50 3.64 10.94 Note. — Characteristics of the Planck LFI-receivers (column 2- 3) and HFI-bolometers (column 4-7): center frequency ν 0 , angular resolution Δθ in FWHM, and instrument noise variance σ N per pixel per observation (thermodynamic temperature units) [99]. We scaled the σ N corresponding to 3. ′ 4 (Healpix n side =1024). 1. Wesimulateclustercatalogsusinghalomodelandclusterselfsimilarscalingrelations to relate the observed SZ signal to the properties of cluster. We do so for three different cluster samples from the ROSAT All-Sky Survey (hereafter RASS), Planck, and eRosita All-Sky Survey (hereafter EASS). 2. Full-sky maps of the simulated kinetic and thermal SZ are created in the 6 Planck frequency channels together with realizations of the CMB. These maps are con- volved with Gaussian beams and detector noise is added using the properties given in Tab. 2.1. 3. The maps are filtered to enhance the kinetic SZ signal while suppressing the CMB and instrument noise terms and removing the thermal SZ signal. 4. The filtered maps are used to calculate the dipole moment of the reconstructed kSZ signal. Thedipole is calculated at the cluster positions usingdifferent redshift shells. 25 5. Temperature dipoles (in units of Kelvin) are converted into velocity dipoles (in units of km/s) by a calibration matrix M, which is calculated from kSZ realizations with pre-definedbulkflowamplitudes,i.e. a V =Ma T , wherea T anda V arethemonopole and dipole coefficients in temperature and velocity units respectively. We iterate the above pipeline to study the properties of the recovered bulk flows by (a) using different filters; (b) varying the amplitude of the input bulk flow velocity; (c) using different cluster catalogs; (d) inputing different systematic components into the maps to determine their importance. 2.4.3 Optical depth The calibration of conversion from dipole amplitude to bulk flow velocity, through the calibration matrixM(Sec. 2.8.2), requiresclusterpositionandoptical depth. Inthiswork, we assume optical depth is measured in the observed samples with negligible measurement errors and consider the scatter in the Y −M relation for Planck and EASS clusters and L−T relation for RASS clusters when producing our simulations (Appendix. B). When dealing with clusters from observations, we have to derive the optical depth of the cluster sample to reconstruct the kinetic SZ signal. We briefly outline below how we can recover optical depth from SZ and X-ray clusters respectively. For SZ observation, the optical depth of each cluster is directly obtained from their SZ flux, or equivalently the integrated compton Y-parameter, by τ = Y(m e c 2 )/(k B T e ) if the electron temperature T e is known. The temperature can be obtained from X-ray measurements through the L−T scaling relations. For X-ray observations, RASS and EASS, we can calculate the optical depth from the electron density by τ =σ T R n e dl. We followOMCPtodeterminetheclusterelectrondensity(seeequation8inOMCP)usingthe Bremsstrahlungemissionluminosity [58] andtherelation T X =(2.76±0.08)L 0.33±0.01 X [193] 26 to calculate the electron temperature of the cluster. For the RASS clusters, propagating the error in observed L X to optical depth gives uncertainty of∼15%. The error in L X for the EASSclusters is likely to besmaller since the background noise for eRosita is expected to be lower. The uncertainty in the optical depth for RASS is therefore an upper limit for EASS clusters. 2.5 Cluster Catalogs MeasuringthebulkflowfromthekSZdipoleiseasiertobedoneonall skyclustercatalogs. This would allow us to better determine the bulk flow direction on the whole sky. Here we use the only all sky SZ survey, i.e. the Planck survey, that we introduced in Sec. 1.3.2. In addition, we also employ representative all sky surveys in the X–ray: the combined ROSAT All-Sky Survey and eRosita All-Sky Survey. Each survey has its own advantages: the RASS sample already exists and has been used for previous bulk flow measurements (e.g. KAKE and OMCP ); the Planck cluster sample contain about three–five times as many clusters, extend to greater distances and, being an SZ survey, is subject to different selection effects than RASS; the EASS cluster sample will be larger than both the ROSAT and Planck samples and will extend to greater redshift z ∼ 1. All three surveys provide isotropicsamplesoutsideregionsatlow galactic latitudes. All surveysareflux-limited, and so at any redshift z only objects of mass M > M lim (F lim ) are included in the catalog. A summary of the catalog properties is listed in Tab. 2.2. 2.5.1 eRosita All-Sky Survey The eRosita mission 1 is expected to be launched in 2012 and perform the first all-sky X-ray imaging survey in the X-ray energy range up to 5 keV with a limiting flux of 1 mission definition document at http://www.mpe.mpg.de/erosita/MDD-6.pdf 27 Figure 2.3: Left: The minimum mass as a function of redshift for a cluster to be included in the RASS, Planck, and EASS catalogs. The cluster mass that we use in the catalogs is max[10 14 M ⊙ ,M lim (z)]. For the RASS sample, we fit Eq. (2.7) to the archived catalog and obtain the M lim values shown in this plot. Right: The differential cluster redshift distribution per square degree for the three surveys above the limiting mass. The ΛCDM cosmology is assumed and σ 8 =0.8 is used for the number counts of the Planck and EASS clusters. The observed number count of the RASS clusters are overlaid with the solid lines for comparison with the fitted number counts (dash dot line). F lim,0.5−5 keV = 1.6× 10 −13 erg s −1 cm −2 . The EASS is expected to yield a catalog of a few tens of thousands of clusters out to redshift z ≈ 1.3. Multi-band optical surveys are planned to provide photometric and spectroscopic redshifts for the EASS clusters with masses above 3.5×10 14 M ⊙ h −1 [21]. The second phase of the eRosita survey (the wide survey)will detect moreclusters by usinglonger exposuresthanthe firstphase(theall sky survey). However, itwillonlycoverabouthalfoftheskyresultinginanon-isotropiccluster sample and so we only use clusters expected from all sky survey. We apply a galactic cut at|b|≤ 7 ◦ giving a sky fraction 0.727. The limiting mass of cluster to be included in the sample at redshift z is given by [52]: M lim,200 (z) 10 15 M ⊙ h −1 = 1 E(z) 4πd 2 L (z) F lim,0.5−2keV /c b 1.097×10 45 ergs −1 1/1.554 (2.7) 28 where E(z)≡ H(z)/H 0 is the normalized Hubble parameter, d L (z) is the luminosity dis- tanceandc b isthebandcorrection factor whichconverts thebolometricfluxtotheeRosita energy0.5−2keV. Weestimatec b byassumingaRaymond-Smith[151]plasmamodelwith metalliticity of 0.4Z ⊙ , a cluster temperature of 4 keV, and a Galactic absorption column density of n H = 10 21 cm −2 . For consistency, we use the virial mass definition throughout this work and we convert M 200 to M v (and write M v as M hereafter) using the conversion fitting formula by [68]. To create mock catalogs, we assign a mass and redshift to each cluster in the EASS sample and simulate the properties of the clusters using X-ray scaling relations (described in Appendix. B). 2.5.2 The Planck Catalog The detailed description is introduced in Sec. 1.3.2. In this work, we take the optimistic mass limit, i.e. Y 200,ρ,c = 10 −3 arcmin 2 which would yield about 2700 clusters. As with EASS, we do not simulate clusters with galactic latitude|b|≤7 ◦ to minimize the Galactic signal. 2.5.3 ROSAT All-Sky cluster survey The RASS sample, consisting of clusters from the REFLEX [14], BCS [45], eBCS [44], CIZA[47, 88], andMACS[46]catalogs comprisesatotal of 827 clusters. All of theclusters have spectroscopic redshifts with z <0.5. We simulate the SZ signal of the clusters in this catalog from the observed X-ray properties via scaling relations, while the redshifts and positions on the sky are taken from the catalog. Fig. 2.3 (left) shows the limiting mass for clusters to be included in the RASS catalog. We fit Eq. (2.7) to the 827 RASS clusters and find an effective flux limit of F lim = 1.2× 29 Table 2.2. Parameters of cluster catalogs Catalog RASS Planck EASS units f 0.5−2keV Y 200 f 0.5−2keV ergs −1 cm −2 arcmin 2 ergs −1 cm −2 Flux limit 1.2×10 −12 10 −3 1.6×10 −14 z ∗ median 0.091 0.246 0.211 hN cl i 827 2700 33000 hτi (7.1±2.2)×10 −3 (8.5±1.7)×10 −3 (6.1±1.5)×10 −3 Note. — ∗ The median redshift is computed from the simulated cluster sample with a cut-off redshift z=0.5 for all three cluster samples. 10 −12 ergs −1 cm −2 . In Fig. 2.3 (right) we show the cluster number counts dN/dz/dΩ, for our cluster catalogs. 2.6 Sky Simulations We create full-sky SZ maps using a semi-analytical approach. We follow the method presented in in [40] and [190] that used the halo model and cluster gas properties. The simulations include SZ emission, primary CMB anisotropy and instrument noise, while diffuse Galactic foregrounds are excluded. Clusters are simulated with properties expected from the surveys described in Sec. 2.5 with number densities given by the halo mass function n(z,M). Each simulated cluster is then assigned a mass and redshift and the gas properties (Appendix. B) are then derived. We then assign N clusters to a bin (ΔM,Δz) from a random Poisson distribution with an average value given by equation Eq. (1.10). The mass range is taken to be 10 14 −10 16 M ⊙ withabinsizeoflog(ΔM)=0.1. Theminimummassofclusteratredshiftzthatcanenter 30 the simulated catalog is given by M min = max[10 14 M ⊙ ,M lim ], where M lim is the limiting mass of the corresponding survey from Eq. (2.7) and Eq. (1.11). The redshift range z is 0−0.3 with a bin size of Δz = 0.02. Having obtained the mass and redshift distributions of the clusters, we distribute the clusters over the whole sky. For simplicity, we ignore the spatial correlation among clusters and assign a random position to each cluster. In the nonrelativistic limit the distortion to the CMB temperature of the kinetic SZ effectisgivenbyEq.(1.4)butwiththepeculiarvelocityreplacedby~ v = ~ V bulk +~ v peculiar . We give an overall bulk velocity ~ V bulk to the whole cluster sample which is a free parameter of the simulation. Besides a bulk motion, we also give each cluster a random peculiar velocity ~ v peculiar drawn from a Gaussian distribution with variance given by Eq. (2.3) with R =8 Mpc h −1 . Maps of CMB anisotropies are generated from the angular power spectrum using the CMBfast code [168] for a flat ΛCDM cosmology. The CMB and SZ maps are combined and then smoothed with a Gaussian beam with sizes given in Tab. 2.1. Finally, we add the instrument noise and briefly describe here how the noise maps are generated. The instrument noise of Planck’s receivers consist of two components: a power law component in frequency f (also known as the 1/f noise) and the white noise component. The 1/f part of the noise spectrum arises from zero point drifts of the detector gain on large time scales. Calculating the full Planck noise covariance is computationally unfeasible and so we restrict our analysis to inhomogeneous white noise. We use the noise variance per pixel σ N per observation from [99] as listed in Tab. 2.1. Inhomogeneous noise is obtained by scaling the noise variance σ N by 1/ √ n obs , where n obs denotes the number of observations per pixel. Maps of n obs (number of hits of each pixel) are readily available via the Planck Sky Model. As we shall see later in Sec. 2.10, the Planck noise level is so low that the instrumentnoise is notthe dominantsource of uncertainty to our bulkflow measurements. Therefore the choice of noise simulations have small effect of the final results. 31 Synthesized maps are generated in the HEALPix pixelization scheme [57], with resolu- tion of Nside=1024. 2.7 Filtering: Reconstruction of the Kinetic SZ signal The kinetic SZ signal from a cluster is embedded in the CMB and instrument noise, and tSZ emission is also present at the same location. We therefore filter the maps to increase the signal-to-noise of our cluster dipole measurement. We use two types of linear multifrequency filters to reconstruct the cluster kinetic SZ signal. Thefilters are constructed with the aim of minimizing the CMB and noise variance in the map. We leave the derivation of the filter shapes to the Appendix.A. Thefirstfilter is a matched filter (hereafter MF; [61, 64]) that is optimized to detect the kSZ signal. The second filter (hereafter UF) is subject to the additional constraint of removing the tSZ signal at the cluster location. The UF was first proposed by [63] for use on flat patches of the sky and later [159] extended the scheme for use on full sky maps. Both filters use all of the frequency channels and take into account the statistical correlation of the CMB and instrument noise between different frequencies. The filter kernels are: matched filter: Φ MF l = C l −1 γ B l (2.8) unbiased filter : Φ UF l = C −1 l Δ (αB l −βF l ) (2.9) where α = P lmax l=0 F T l C −1 l F l , β = P lmax l=0 B T l C −1 l F l , γ = P lmax l=0 B T l C −1 l B l , and Δ = αγ− β 2 . WeassumetheclustersarepointsourcesinthePlanck mapsandsoB l arethespherical harmonic coefficients of the Planck beam function, F l =f(ν)B l where f(ν) gives the tSZ 32 Figure 2.4: Left: Multi-frequency matched filter signed to detect the kSZ signal. Right: Unbiased multi-frequency matched filters designed to remove the tSZ components . Both filters are constructed using the Planck beam profiles as the cluster radial profiles. frequency dependence,C l =C CMB l +C noise l is a matrix at every multipole giving the sum of the cross power spectra of the CMB and instrument noise between frequency channels. The filters are normalized such that the filtered field gives the amplitude of the kinetic SZ signal at the central pixel of each cluster. Given the spatial resolution of Planck we compute the filter kernels out to multipolel max =3000. Fig. 2.4 shows the MF and UF for the 6 Planck frequency channels we use, using the Planck beam FWHM and pixel noise variance. 2.7.1 Matched filter (MF) The MF is less complicated than the UF since most of the features are only seen at the cluster scales (1000≤ l ≤ 2500) while the smallest and largest scales are suppressed. As with the UF, the channels ν = 100, ν = 143, ν = 217 GHz are given the largest weights. The matched filter is more efficient than the UF at removing the CMB and instrument noise components because it forces the scales at which the noise and CMB dominant to 33 disappear. However, since the MF is not designed to remove the thermal SZ signal, the amplitudes of the filter kernels are all positive. 2.7.2 Unbiased Kinetic SZ filter (UF) TheUF is constructed to give an unbiased estimate of the kinetic SZ signal at theclusters’ locations. At large and intermediate scales (l≤1000) the channels belowν =100 GHz are subtracted from those above. The CMB fluctuations which dominate the signal at these scales are therefore suppressed. At cluster scales (1000≤l≤2500) all of the filter kernels have positive amplitudes and the 100, 143, 217 and 353 GHz channels are given the largest weight. At 100 GHz and 143 GHz the tSZ signals are negative, approximately zero at 217 GHz and positive at 353 GHz. The channels are weighted so that the tSZ signal is zero. Although the tSZ signal is smaller at 100 GHz than the other three channels, the pixel noise level is lower (σ N ≈ 6μK) and so the 100 GHz channel also has substantial weight. Atsmallerscales(l≥2500)thesignalisdominatedbyinstrumentnoiseandsoall channels are suppressed by the filter. 2.8 Analysis Pipeline 2.8.1 Weighted Least Square Fitting of Dipole We fit the real spherical harmonic coefficients of the monopole, a 0 , and dipole terms, a 1x , a 1y and a 1z , to the filtered maps using a weighted least square fit which is based on the Healpix IDL procedure remove dipole [57]: a T =(X T WX) −1 X T Wu (2.10) 34 where a T is a vector of best fit monopole and dipole coefficients, u is the filtered map (e.g.Eq. (A.5)), W is a matrix with diagonal elements equal to the weight given to each pixel of the map, and X is a matrix giving the contribution of the fitting function to each pixel. We give the central cluster pixel of the map a weight W i = 1/σ 2 N,i , where σ N,i is the i–th pixel noise variance, all other pixels are given zero weight. The matrix X T WX is a mixing matrix that couples different spherical harmonic modes together. The effect of the mask W is that a least square fitting process can then be used on the masked map to determine the dipole from clusters alone. To account for the mode coupling, we therefore simultaneously fit for modes l =0−3. Thenoise variances are calculated from 100 filtered CMB and noise realizations. We also tried weighing each pixel by an estimate of thesignal to noise, i.e. W i =τ i /σ 2 N,i , but we find a larger systematic bias due to tSZ contamination in the recovered bulk flow than when usingthe former weighting scheme. We therefore use a weighting scheme that involves only the pixel noise. We fit the dipole only at central cluster pixels because our filters are optimized to reconstruct the source amplitude if the source were centered at that pixel. 2.8.2 Translating from μK to km/s We construct a 4× 4 matrix a V = Ma T to convert the dipole from temperature units (μK) to velocity units (km/s), in which the matrix elements having units of km/s/K. This involves the use of simulated kinetic SZ maps that can account for the attenuation of the optical depth by the beam convolution and the filtering process. In principle one can perform this unitconversion in observations, e.g. Planck temperature sky maps, using calibrated SZ simulations based on estimation of the optical depth of each clusters. This is the case in OMCP that they used the simulated kSZ maps of the RASS clusters to calibrate WMAP data. On the other hand, in this work we assume the optical depth, hence the kSZ signal, of each cluster is known and calculate the matrix M from our 35 sky simulations. Note that the matrix elements depend implicitly on the average optical depth of the cluster sample, therefore it is specific to individual cluster experiment and we construct it separately for the three cluster surveys. We now describe how we create this matrix. We generate four different sets of kinetic SZ signal only realizations with a given bulk flow amplitude: one with a monopole velocity and the other three with a dipole velocities in the x, y, z directions. We choose an amplitude of of 10,000 km/s, which is large enough such that the recovered signal is not confused by uncertainty of optical depth and random peculiar velocity. Since the kinetic SZ signal of each cluster is proportional to the optical depth, the elements of the matrixM depend implicitly on the average optical depth of the cluster sample. Each set of maps are passed through the pipeline to obtain the a 1m . The four a 1m from each set of realizations are the elements of each row of M −1 . By repeating this procedurefor thefourkinetic SZsignal only realizations, we constructthe16 elements of the calibration matrix. We check that the off-diagonal elements ofM are at most 1% of the diagonal ones for all of the cluster samples considered. Weperformthisexercise20timesandassigntheaveragevaluefromthese20realizations tothematrix elements. For each of theRASS,Planck, andEASSsample, wecalculate one averageMandapplyittothefilteredtemperaturemaps. Duetotheintrinsicscatterofthe Y −M relation introduced in the simulations, the average optical depth of any simulated cluster sample might deviate from the expected value from the characteristics associated with that catalog, this may lead to calibration error (Sec. 2.9.3). 2.9 Error estimation The error budget to the dipole coefficients contains several terms: 36 σ 2 a V =σ 2 CMB+noise +σ 2 kSZ +σ 2 tSZ +σ 2 PS (2.11) where σ CMB+noise is the residual from the CMB and instrument noise, σ 2 kSZ is the error in the kinetic SZ signal due to the intrinsic scatter of the Y −M relation and the random component of the galaxy cluster peculiar velocity that is not part of the bulk flow, σ 2 tSZ is the thermal SZ residual, and σ 2 PS is the contamination from radio or infrared point sources associated with clusters. In principle, the last three may be correlated, while we are assuming here that correlations are negligible. The errors on the three dipole coefficients a 1m are correlated and the uncertainty in the best fit value can be described by a covariance matrix N. We compute the covariance matrixofeachtypeoferrorinEq.(2.11)bypassingsimulationsofthecomponentsthrough our pipeline and performing the dipole fit on them. The scatter σ in dipole coefficients then provides an estimate of the noise correlations between the dipole directions, i.e. N= a V a T V . Then, χ 2 =(a V,rec −a V,in ) T N −1 tot (a V,rec −a V,in ) (2.12) wherea V,rec is thebestfitdipolecoefficients of therecovered velocity anda V,in is theinput velocity dipole coefficients. The confidence level of a given a V can be calculated from the χ 2 probability distribution with 3 degrees of freedom. Fig. 2.5 shows the contribution from each systematic effect to the dipole velocity. We discuss these results in the following subsections. 2.9.1 CMB and instrument noise TheCMBsignalisnotcompletelyremovedbyourfiltersbecausethereisCMBemissionon clusterscales withthesamefrequencydependenceasthekSZsignal. Weevaluatetheerror 37 Figure 2.5: The systematic error contributions to the bulk flow measurement using the RASS (top), Planck (middle), and EASS (bottom) cluster samples: instrument noise and CMB (cross), thermal SZ (asterisk), uncertainty in the kinetic SZ signal due to scatter in both the optical depth and cluster peculiar velocity (diamond), and the contamination from radio point sources (triangle for our simulation and solid line for PSM simulation). The data points are the average values from 100 realizations and the error bars are the standard deviations of the one-sided distribution. from the CMB and instrument noise by measuring the bulk flow of 300 filtered CMB plus noise maps. As a further check on this procedure we also calculate the dipole on a single CMB plus noise realization and repeat the procedure with 100 realizations of the cluster sample, each having a different spatial distribution. The results from the two procedures are similar with errors that are both gaussian with similar variance. 38 As expected the UF errors are larger than the MF ones by about 30% for the three cluster samples. We findthat the error is largest for the RASSsample and smallest for the EASS sample, since the EASS sample contains more clusters. 2.9.2 Thermal SZ signal We expect the thermal SZ signal to be entirely removed when using the unbiased matched filter (UF). However, maps filtered with the MF suffer from a systematic bias by the unfiltered thermal SZ signal that contaminates the kinetic SZ signal. In addition to the effect of filtering, the intrinsic scatter of the cluster y-parameter would also introduce furtherincomplete removal of thethermal signal andresultinginscatters aroundthemean recovered velocities. To evaluate the level of thermal SZ bias and uncertainty, we generate simulated maps containing only tSZ signal. We find that in thermal SZ signal only maps the average systematic bias (±1σ uncer- tainty as represented by the error bars in Fig. 2.5) to the cluster dipole are v = 313±48 km/s for RASS,v =77±33 km/s for Planck, and v =19±8 km/s for EASS when filtered bytheUF,andv =885±66km/sforRASS,v =171±77km/sforPlanck, andv =47±20 km/s for EASS when filtered by the MF. The thermal bias is most serious for MF filtered RASS clusters while the 1σ uncertainty is the largest for MF filtered Planck clusters. This isduetothefactthattheaverage optical depthisthelargestfor Planck clustersandhence the largest intrinsic scatter (and uncertainty). We find a significant monopole: v a 0 = 730 (RASS), 1440 (Planck), and 870 (EASS) km/s when filtered by the UF, and v a 0 = 3500 (RASS), 3890 (Planck), 2180 (EASS) km/s when filtered by the MF. The UF is more effective at removing the thermal SZ signal than the MF, by a factor of ∼ 5. Though the monopole velocities are large compared to the thermal dipole, they have no effect on the bulk flow measurement. 39 2.9.3 Uncertainty in optical depth and random cluster peculiar velocity Wefindthattheclusterrandompeculiarvelocity notpartofthebulkflowgivesanegligible dipole with an average magnitude of v < 50 km/s for all three cluster samples and is independent of the type of filter used. The error due to uncertainty in the optical depth dependsontheamplitudeofthepeculiarvelocity sinceσ kSZ ∝vσ τ . Wefindthatσ τ ≈15% for the Planck cluster sample and σ τ < 15% (OMCP) for the RASS and EASS clusters. The uncertainty for Planck clusters is relatively larger because it contains more abundant massiveclustersandhencepropagatelarger errorstotheoptical depthforthesamescatter in the cluster Y-parameter. As mentioned earlier, this uncertainty introduces a calibration error when translating from μK to km/s because of the difference of the average optical depth and the calibration matrixM. However, since we choose a large calibration velocity of 10,000 km/s, the calibration error is only about 5% and negligible when compared to the error due to CMB and noise. 2.9.4 Extragalactic Point Sources At small scales extragalactic point sources are significant contaminants of CMB maps. They give a Poisson noise contribution to the measured angular power spectrum and a non-Gaussian signature in themaps [33]. Radio pointsources have a falling spectrumwith α∼0.7 whereS∝ν −α , whereasinfraredpointsources have arisingspectrumwithtypical spectral indices α∼3. Thus infrared sources tend to bethe more significant contaminants at higher frequencies where the SZ effect is large. Recent measurements such as the SPT detected 47 IR sources at S/N> 4.5 [187]. We note that an appropriate approach would be to correlate the infrared sources with galaxy clusters and therefore we attempted to estimate the level of infrared point source contamination to the cluster kSZ dipole by 40 utilizing the infrared point source simulations by [166] available online 2 . These maps are full-sky simulations of the microwave sky matched to the most recent astrophysical observations intended for testing the data reduction pipeline for the ACT experiment. In order to follow our analysis pipeline, we select only the four frequency maps 90, 148, 219, 350 GHz that are overlapping with the Planck frequencies. We do not apply any flux cut and thus bright infrared sources are also included. We note that an appropriate approach wouldbetocorrelate theinfraredsourceswithgalaxy clusters. However thiswouldrequire knowledgeoftheirspatialcorrelationfunctionofIRpointsourcescaused,e.g. bylensing,in whichcurrentlythereisnoreliablemodeltoaccountforit. Asanalternative, weassumean uncorrelated spatial distribution between infrared signal in the simulation and the galaxy clusters, and we fit the dipole for simulated cluster catalogs with properties expected from the three survey but with positions randomly selected on the full sky. We repeat this for 100 times each for the Planck and EASS survey and do once for the RASS survey since their positions are known. The method we use here will give a crude estimate of how IR point sources may contaminate our results. We find that the mean infrared source dipoles are <25 km/s for all the three survey and filtered with both filters. Since we also include bright infrared sources in the simulations, the obtained small values suggest it would not affect our bulk flow measurement. Radio sources are an important contaminant of SZ measurements as they fill in the SZ temperature decrement if the sources are within the cluster being observed, leading to an underestimation of the SZ signal. We expect a negligible contribution since the cluster signal is strongest at frequencies where the radio point source signal is small. 2 http://lambda.gsfc.nasa.gov/toolbox/tb_cmbsim_ov.cfm 41 Toestimatethelevel ofradiocontamination totheclusterkSZdipolewesimulatemaps of the radio point source emission. To do this we use the radio luminosity function at 1.4 GHz of [105] extrapolated to higher frequencies using, dn(logP ν ) dlogP ν = Z dn(logP 1.4 ) dlogP 1.4 f(α)dα (2.13) where P ν is the radio luminosity in units of WHz −1 at frequency ν, α is the spectral index, and f(α) is the probability distribution of the spectral indices at 1.4 GHz, which is taken to be a Gaussian distribution with mean ¯ α=0.51 and rms σ α =0.54. f(α) is taken to be f(α+0.5) for ν >90 GHz to account for possible steepening of α at high frequencies. The RLF at 1.4 GHz is given by the fitting formula: log dn dlogP 1.4 =u− s b 2 + logP 1.4 −x w 2 −1.5logP 1.4 (2.14) where u = 37.97, b = 2.40, x = 25.80, and w = 0.78 [105]. Here n is the radio source number density within r 200 . We integrate Eq. (2.13) to obtain the expected number of radio sources, N radio , in a cluster and assign fluxes to the N radio sources using the RLF. SincebrightsourceswillbemaskedinthePlanck mapswealsomasksourceswithS >S lim ν , whereS lim ν istheupperlimitof theradiofluxatfrequencyν given by[108,Table2]leaving a radio map with sources having S <500mJy. We find no significant cluster dipole from radio point sources in the RASS, Planck, or EASS cluster samples. In the redshift shell z =0−0.5 we find signals of v <10 km/s for EASS and v ∼ 20 km/s for both Planck and RASS clusters when filtered with either the UF or MF. We can further verify the result for the RASS sample by using more realistic radio point source simulations that considers information from sources observed by NVSS, and simulations from the Planck Sky Model (PSM [39]). As before, we mask bright radio 42 sources in the PSM simulation. The result from the PSM simulation is shown in Fig. 2.5. The radio dipoles from the PSM are consistently larger than our radio simulations at all redshift shells, but are of a similar order of magnitude with v∼50 km/s. 2.10 Results We generate sets of simulations containing CMB, noise, thermal SZ signal, and kinetic SZ signal with 68 bulk flow velocities logarithmically spaced between 100 and 10,000 km/s. In this section we describe how well we recover the input bulk flow. 2.10.1 Precision and detection threshold of Bulk Flow Measurement 43 Figure 2.6: Recovered bulk flow velocity for clusters z ≤ 0.5 ( Top: RASS, Middle: Planck, Bottom: EASS) from simulated maps with inputbulkflow velocities logarithmically spaced between 100 and 10,000 km/s, filtered with the MF (black cross) and UF (blue diamond). The solid line represents perfect recovery. The dash lines quote the 95% confidence interval of the input bulk flow values. All results are based on maps containing kinetic and thermal SZ signals, butsimulatedwithdifferentsystematiccomponents. Plots labelledNoiseandCMBarefrommapscontaining instrument noise and CMB; plots labelled thermal SZ are from results containing thermal SZ emission and hence estimate the level of thermal SZ bias. 44 Figure 2.7: Recovered bulk flow velocity for three input bulk flows v =500 (black), 1000 (yellow), and5000 (cyan)km/sasafunctionofredshiftforthe Planck clustersample. The recovered bulk flows are measured from simulations that contain all systematic errors we consider. Fig. 2.6 shows the recovered bulk flow values V recovered versus the input values V input for all clusters with redshift z ≤ 0.5. The dashed lines in the figures are the 95% upper limitstotherecoveredbulkflowvelocities andarecomputedbygenerating10,000Gaussian distributeddipolecoefficients ateach of thex, y, andzdirections havingmeanequal to the input bulk flow velocity and thermal bias, and variance equal to the noise levels described in Sec. 2.9, i.e. a sim 1m =(a in 1m +a bias 1m )±a noise 1m . Theresulting magnitudeof the dipolevelocity follows the χ 2 distribution, the magnitude and angle error (Sec. 2.10.2) at 95% are then computed. We determine the precision of the velocity measurement by computing the deviation of the values V recovered from their corresponding V input with a parameter defined to quantify the deviations as follows: σ logV = s P N i=0 (logV recovered,i −logV input,i ) 2 N−2 (2.15) 45 where the summation runs over the range of V input being considered and N = 68. This parameter allows us to compare the level of dispersion among the results from the various experiments and surveys. We provide the values for each cluster sample in Tab. 6.1. Taking a representative velocity of V input =500 km/s we find that the recovered veloc- ities are overestimated by 120% with a scatter of 16% when evaluated using the RASS clusters using the MF (V recovered = 1085±82 km/s), but perfectly recovered with only a scatter of 5% when evaluated on the EASS clusters using the UF (V recovered = 500±24). The large overestimation in MF filtered RASS clusters is due to tSZ contaminations which has the largest effect for the RASS clusters. We find that the recovered velocity has the largest uncertainty for MF filtered Planck clusters, with a scatter of 25%, and is consis- tent with the error analysis that the uncertainty due to intrinsic scatter in Y-parameter is largest for Planck clusters. With the sensitivities we find, measurements using either the Planck or EASS cluster sample will be able to detect a bulk flow as large as that claimed by KAKE. If the bulk flow is consistent with the ΛCDM prediction of v = 30 km/s for redshift shell extending to z = 0.5, our analysis pipeline would recover a 95% upper limit to the velocity of, when the UF is used, v =470 km/s for RASS cluster, v =160 km/s for Planck cluster sample and v =60 km/s for EASS cluster sample. Fig. 2.7 shows the recovered signal in different redshift shells at three input bulk flow velocities: V input =500, 1000, and5000km/s,estimatedfrom100realizations. Forillustra- tion purpose, only the result from the Planck cluster sample is shown. At these velocities, the recovered velocities are shown to be consistent with the input values within 1σ at red- shifts z = 0.1. At lower redshifts, the recovered velocities are noise dominated. At this redshift limit, the Planck cluster catalog will contain approximately 400 clusters. If a bulk flow of amplitude ≈ 500 km/s is present our results suggest that a sample of about 400 clusters would allow Planck observations to constrain the bulk flow. 46 2.10.2 Recovering the Bulk Flow Direction The direction of any bulk flow is not predicted by the ΛCDM model. We can estimate the error in the direction of a detected dipole for each of the three surveys. For V input = 500 km/s towards l = 280 ◦ and b = 30 ◦ the average values of the dipole components, (hV x i, hV y i, hV z i) from 100 simulations, are listed in Tab. 6.1. The direction is recovered with deviation < 20% for the Planck and EASS cluster samples (except forhV x i of the Planck sample filtered with the MF), but is unconstrained for RASS clusters. We find that the error is larger in the V x and V y directions due to the galactic cut. We also calculate the angle errors for a range of input velocities for four different input directions: the three perpendicular directions x (l = 0 ◦ , b = 0 ◦ ), y (l = 90 ◦ , b = 0 ◦ ) and z (l = 0 ◦ , b = 90 ◦ ), as well as the reference V input used above. The 95% upper limit to the angle errors, Δα 95 , is shown in Fig. 2.8. For maps filtered with the MF (UF) the 95% confidence limit on the dipole direction for a 500 km/s input bulk flow is Δα 500 95 =62 ◦ (34 ◦ ) for RASS clusters, 34 ◦ (14 ◦ ) for Planck clusters, and 9 ◦ (4 ◦ ) for EASS clusters. The errors are smaller than the discrepancies in the observed bulk flow directions. Therefore Planck should be able to better constrain the region where the bulk flow points to. 2.10.3 Effect of the systematic error components on the amplitude of the recovered bulk flow The dominant systematic error is tSZ emission, and the dominant statistical error is CMB and instrument noise and uncertainty in the Y-parameter. They are velocity-independent and dominates for v < 1000 km/s. At large velocities, the calibration error due to the scatter in optical depth dominates and gives a <10% error in the velocity. 47 Figure 2.8: The 95% upper limit to the error in the angle measurements for all the three cluster surveys. The l =30 ◦ , b =280 ◦ data points correspond to the direction of the input bulk flow that the simulations in Fig. 2.6 are based on. 48 Using inverse variance weights in the dipole fitting (as opposed to weighting by optical depth) is found to minimize the tSZ contribution. The recovered bulk flows are generally overestimated by≈20% when W =τ/σ 2 noise is used to weight pixels in the dipole fit. 2.10.4 Performance of Filters The overall performance of both the MF and UF agree with expectations that maps fil- tered by the MF have higher kSZ signal to noise ratios while maps filtered by the UF have smaller thermal SZ bias. The UF is slightly a better filter in the Planck data since the dominant uncertainty of the measurement comes from the thermal SZ bias, and the recovered velocities are more accurately determined. Nevertheless, the thermal SZ bias is not completely removed by the UF. This is because we assume a β model for the cluster profile, but assume a point source model to construct the filters. We could assume a β model to construct the filters but we do not consider this here. We now compare the filter performance in the WMAP channels which are studied in OMCP (see their Figure 11) with the Planck channels in this work (Fig. 2.6), with the use of theRASScluster samplewhich isusedin bothanalysis. InOMCP theMF was foundto bemore sensitive than the UF, butwe findthe opposite. Thevelocity detected at 95% CL with the MF (UF) is reduced from∼5000 km/s (∼ 10,000 km/s) in the WMAP channels to ∼ 1000 km/s (∼ 1000 km/s) in the Planck channels. Considering that the amplitude of observed bulk flow velocities are a few hundreds of km/s at scales r < 300 Mpc h −1 (Fig. 2.1), the filters in the WMAP channels do not have the sensitivity to measure bulk flow velocities at these scales. The differences between the WMAP and the Planck filters is expected: the error due to instrument noise is largely reduced because the noise levels of Planck are much lower than WMAP; the wider frequency coverage in the SZ sensitive regime also allows filters in the Planck channels to suppress the thermal SZ bias. 49 2.11 LatestmeasurementsofBulkFlowon Planck CMBdata Recently, I have the opportunity to apply the current scheme to measure the bulk flow on the Planck data taken during the first 15 months of observations in [145], as part of the study of constraining cluster peculiar velocities. For the sake of completeness, I will present the essence of the results (see Section 4.2.2 in [145]) which adopt the procedures presented in this work, but refers readers to the reference for more details. In general, we use the unbiased matched filter developed in Sec. 2.7 and the analysis pipeline as outlined in Sec. 2.8 to fit for the kSZ dipole on the Planck CMB map. A few modificationsanddifferencesfromthisworkwereemployedtosuitforPlanck specifications: 1. Data: For the frequency maps used, we present results only on the four lowest HFI channels, i.e. 100, 143, 217, and 353 GHz, and drop the two highest LFI channels, 44 and 70 GHz. We verified that extending the analysis to 44 and 70 GHz like we do here gives consistent results and does not significantly improve the constraints. Furthermore, instead of using the three cluster surveys mentioned in Sec. 2.5, we use the Meta Catalogue of X–ray detected Clusters of galaxies (MCXC [137]) which is the extension to the RASS sample. It contains 1750 clusters with a mean redshift of z∼0.18 and the spatial distribution is shown in Figure 2 in [145]. 2. Filter: Instead of the point source model, we adopt the universal pressure profile from [6] with a characteristics scale of θ 500 = 8 ′ convolved with the beam profile. We have verified that this choice of the cluster profile would allow us to construct the optimal filter such that the map is matched filtered. This also results in a reduced amount of noise coming from thermal SZ bias and instrument noise+CMB components. As mentioned, we construct the filters using only the four lowest HFI channels. 50 3. Errorestimation: Wedonotignorethecontamination fromtheGalactic components to the bulk flow measurement and make use of the PSM simulations to assess its contribution. These simulations contain our current best knowledge of the diffuse Galactic component. Since Planck is most sensitive at frequencies above 100 GHz where the dust emission dominates, we only consider this component. The main results are reported in Figure 9 and Table 2 in [145]. Briefly, the Planck data give dipole coefficient amplitudes consistent with those expected from the ΛCDM scenario, once one has taken into account the contamination from Galactic foregrounds and other signals. The apparent bulk flow measured in the redshift range 0≤z≤1 is 614 km/s. The dominant source of systematic error is the diffuse Galactic component which provides a non-negligible contribution to the dipole signal of 529 km/s, as measured in the PSM simulations. The dominant statistical error is the CMB and instrument noise which amountsto140−290km/s(dependsondifferentdirectionsonthesky). Whenwetakeinto account the bias induced by the Galactic foreground (by subtracting from the measured dipole as a vector), the fraction of the observed bulk flow, 350 km/s, is within 95% of the error on bulk flow induced by tSZ, CMB and instrument noise (893 km/s). This scenario is also seen in the measuredbulk flows within spheresof comoving radiusr≤2400 Mpc/h. Finally, we notice that the direction of the observed dipole is quite different from both theCMBdipoleandtheresultofKAKE.Italignsbetterwiththedirectionofthecollection of MCXC clusters in the map, which happens to be in a low instrument noise area of the sky,asonewouldexpectfromanoise-inducedmeasurement. Nevertheless, theupperlimits to the bulk flow with this approach are about a factor of five better than what was found using WMAP data (OMCP), thus showing the capability of Planck data to perform this measurement. Our result therefore is in clear contradiction with the claim in KAKE 51 2.12 Conclusion and Discussion We investigated Planck performances in determining bulk flow velocities using the kinetic Sunyaev-Zeldovich effect. We characterize the sensitivity of the bulk flow velocity mea- surement using simulated Planck data combined with three representative all sky galaxy cluster surveys: the archived ROSAT All-Sky Survey, the Planck cluster catalog and the eRosita All-Sky Survey. We employ two different types of filters, a matched filter and an unbiased kinetic SZ filter, to maximize the cluster signal to noise ratio. The main results are (see also Tab. 6.1): 1. The use of simulated Planck sky maps instead of WMAP ones in combination with the RASS catalog reduces the velocity that can be detected at 95% CL, from∼5000 km/s(∼10,000 km/s)to∼1000km/s(≈ 1000km/s)whenfilteredbytheMF(UF). 2. Usingallclusterswithz≤0.5wefindthatabulkflowof500km/swouldbemeasured in a strongly biased way with the RASS sample with both filters. The same velocity is measured with 0–5% bias with EASS and Planck clusters, with uncertainties of 25–75 km/s. Thesenumbersshouldbecomparedwiththermsvelocity outtoz =0.5 in ΛCDM: 30 km/s (95% upper limit: 48 km/s). 3. If the bulk flow is consistent with ΛCDM prediction of v = 30 km/s at z = 0.5, our analysis pipeline would obtain a 95% upperlimit to the recovered velocity (when the UF is used) of v = 470 km/s for RASS cluster, v = 160 km/s for Planck cluster sample and v = 60 km/s for EASS cluster sample. This allows us to measure the departure from ΛCDM if the measured bulk flow is in excess of these estimates. 4. Planck can also constrain a recovered bulk flow direction. For V input = 500 km/s, the 95% upper limit to the angle errors (when the UF is used) are Δα 95 ≈ 30 ◦ for the RASS clusters, Δα 95 ≈15 ◦ for the Planck clusters, and Δα 95 ≈5 ◦ for the EASS 52 clusters. These uncertainties are lower than the discrepancies observed in bulk flow directions atdifferent optical depthuptonow (Fig. 2.1). Theerrorsinthedirections contained within the galactic plane are larger than in the perpendicular one due to the galactic cut at low latitudes. 5. TheerrorinthekineticSZdipoleisdominatedbytheunfilteredCMBandinstrument noise and intrinsic scatter of the Y-parameter, with a systematic thermal SZ bias. The contamination from extragalactic radio sources is negligible and can be safely ignored. The noise level is lowest for the EASS sample since it contains the most clusters. 6. TheUFismoresensitiveandeffectivethantheMFinsuppressingthebiasinducedby the thermal SZ in the velocity reconstruction. This is in contrast to the performance of the two filters on the WMAP maps, where the MF was much more sensitive than the UF. Differences in performances are to beexpected, as the two experiments have differentcharacteristics. AsPlanckhasmuchlower noise, thetSZcontamination now plays a more major role in setting the bias and errors. Planck increased frequency coverageandresolution,however,enablesthefilters(UFinparticular)tomitigatethe effects of the thermal SZ signal, especially when a large cluster sample is considered. 7. The current scheme is applied to the Planck data and MCXC clusters to search for bulk flow signal in the 0≤ z ≤ 1 universe. The Planck data give dipole coefficient amplitudesconsistentwiththoseexpectedfromtheΛCDMscenario, whichisinclear contradiction to the claim in KAKE. We have not considered here the effect of infrared point sources, correlations in the signal between CMBand clusters, large–scale (non bulk flow) peculiar velocities, and error determination in extracting optical depth from the survey in hand. Some of these could 53 present further improvements to this initial study, which demonstrate the superior poten- tials Planck has in determining the bulk flow. The work presented in this chapter is based on the publication: Mak, D. S.Y., Pierpaoli, E., & Osborne, S. 2011, ApJ, 736, 116 And the method presented here is recently applied to the latest Planck CMB data to constrain the peculiar velocities of galaxy clusters, and the publication can found via: Planck Collaboration 2013, submitted to A & A, arXiv:1303.5090 54 Table 2.3. Summary of the results noise components RASS Planck EASS MF UF MF UF MF UF V 500 rec noise + CMB + thermal SZ bias 1085±82 720±94 530±126 516±73 507±33 500±24 hvxi (75 km/s) 129±94 131±138 108±133 88±68 80±40 75±29 hvyi (-426 km/s) −75±105 −434±133 −432±123 −431±73 −427±37 −428±26 hvzi (250 km/s) 1065±82 526±66 240±91 255±49 256±28 243±19 Δα 500 95 62 ◦ 34 ◦ 34 ◦ 14 ◦ 9 ◦ 4 ◦ σ logV,bin1 0.71 0.40 0.12 0.09 0.09 0.09 σ logV,bin2 0.28 0.14 0.09 0.04 0.02 0.01 σ logV,bin3 0.08 0.05 0.04 0.02 0.01 0.01 σ logV,bin4 0.02 0.03 0.03 0.02 0.01 0.01 σ logV,whole 0.42 0.24 0.08 0.06 0.05 0.05 V 500 rec (hVreci) only thermal SZ bias 1100±60 724±82 530±115 515±63 511±30 504±14 hvxi (75 km/s) 125±73 142±98 82±109 82±60 104±30 101±15 hvyi (-426 km/s) −71±75 −445±110 −437±112 −436±63 −427±34 −428±14 hvzi (250 km/s) 1082±63 534±46 253±86 250±46 255±29 244±11 Δα 500 95 62 ◦ 33 ◦ 32 ◦ 14 ◦ 9 ◦ 3 ◦ σ logV,bin1 0.68 0.40 0.12 0.10 0.07 0.04 σ logV,bin2 0.27 0.15 0.06 0.03 0.03 0.02 σ logV,bin3 0.09 0.06 0.04 0.02 0.02 0.01 σ logV,bin4 0.03 0.03 0.03 0.02 0.01 0.01 σ logV,whole 0.41 0.24 0.08 0.06 0.05 0.03 Note. — This table lists the main results of the bulk flow measurements for clusters within z ≤0.5 in the three cluster surveys considered and for maps containing different components of uncertainties. The first five quantities are based on 100 realizations with input velocity of 500 km/s at l = 280 ◦ and b = 30 ◦ . V 500 rec is the average value of the recovered velocity, hvxi , hvxi , hvxi are the average of the individual directions. For ideal recovery, they should have values (75, -426, 250) km/s. Δα 500 95 is the 95% upper limit to the angle error. The last five quantities σ logV,bini are the deviation parameters as defined in Eq. (2.15), where bin i refers to the binned velocity range considered in the calculation: bin 1= 100–500 km/s, bin 2= 500–1000 km/s, bin 3= 1000–5000 km/s, bin 4= 5000–10000 km/s. The whole range is for velocities 100–10000 km/s. 55 Chapter 3 Modified Gravity 3.1 Introduction Theobservedcosmiclate-time acceleration isrevealing usfundamentalaboutastrophysics. Themostacceptedexplanationforthisisthepresenceofdarkenergy. Withinthecontextof the standard ΛCDM model, dark energy is associated with small, non-zero vacuum energy equivalent to the cosmological constant term in the Einstein’s equations which drives the cosmic acceleration. Alternative suggests that the theory of gravity needs to be modified forextremelylargedistancescales, orthatwehappentoliveinaverylargeregionofunder- density. Thisscenariooffersthemotivationtoexploretheoriesthatparametrizedepartures from General Relativity (GR; e.g. [71]), and an answer to the question of whether dark energy is needed at all. Therelevantforceintheformationofstructureintheuniverseonlargescalesisgravity, the abundance and clustering properties of galaxy clusters are therefore very sensitive and have a large impact from modifications to the behavior of gravity. In recent years, clusters have received considerable interest as a probeof gravity [115, 42]. Modifications to gravity generically change the growth of large-scale structure (e.g., [73, 32]), and clusters at the high-mass tail of the mass function are especially sensitive to changes in the growth rate. Here, we focus on a class of modified gravity model called f(R) models (see [176] for a review), usingthe functional form proposed in [71]. This model involves an extra degree of freedom, or a new force, that enhances gravitational forces on scales relevant for structure formation, and produces the acceleration without a true cosmological constant. In the 56 parameter range of interest here, it is indistinguishable from ΛCDM through geometric probes (CMB, Supernovae, H 0 , BAO measurements). However, gravitational forces are modified on smaller scales. Furthermore, the model includes the chameleon screening mechanism which restores General Relativity in high-density environments. Thus, this model is able to satisfy all current constraints on gravity. Structure formation in this modified gravity model is now understood on all cosmological scales. The linear regime of structure formation in this f(R) model has been studied in [71]. The non-linear structure formation was investigated using dedicated N-body simulations in [130, 131, 162, 196]. This allows for fully self-consistent constraints and forecasts to be made for this model. Theconstraintsonthismodelhavebeenrecentlymadein[163]whousedasampleofX– rayclusters,andin[107]whousedanopticalSloanclustersample. Aconsistencytestofthe GRplussmoothdarkenergyframeworkusingclusterswasdonein[149]. Inthischapter,we exploreto whatextent, usingtheFisher Matrix approach, theSZcluster surveysdescribed in Sec. 1.3 are expected to constrain f(R) models through cluster number counts and clustering. Cluster surveys are sensitive to both the geometry (expansion history) of the Universe and the growth of structure. The geometry affects the volume covered by a given solid angle and redshiftinterval, while the growth affects the number density of clusters at a given mass within this volume. For the range of f(R) models considered in this work, chosen to be compatible with solar system constraints, the change in volume is negligible while the one in abundance and clustering of clusters is significant. The chapter is organized as follows. We begin by presenting the parametrization of modified gravity effects on the cluster abundance and clustering in Sec. 3.2. In Sec. 3.3 we specify the methodology and the Fisher matrix analysis adopted here. The forecasted constraints are presented in Sec. 3.4. We discuss our results in Sec. 3.5 and conclude in Sec. 3.6. 57 3.2 Theoretical modeling 3.2.1 f(R) gravity In the f(R) model (see [125, 176] and references therein), the Einstein-Hilbert action is augmented with a general function of the scalar curvature R [27, 126, 20], S G = Z d 4 x √ −g R+f(R) 16πG . (3.1) Here and throughout c= ~ =1. This theory is equivalent to a scalar-tensor theory (if the function f is nontrivial). The additional field given by f R ≡df/dR mediates an attractive force whose physical range is given by the Compton wavelength λ C = a −1 (3df R /dR) 1/2 . On scales smaller thanλ C , gravitational forces are increased by 4/3, enhancing thegrowth of structure. This additional attractive force implies that matter cluster together more strongly than they do in GR. Therefore more and larger clusters would form, the empty regions would become emptier, and the particles would move faster overall. These effects can in principle be observed via the abundance and clustering properties of rare massive dark matter halos. A further important property of such models is the non-linear chameleon effect which shutsdowntheenhancedforcesinregionswithdeepgravitational potential wellscompared with the background field value, |Ψ| > |f R ( ¯ R)| [86, 71]. This mechanism is necessary in order to pass Solar System tests which rule out the presence of a scalar field locally. Thus, Solar System tests constrain the amplitude of the background field to be less than typical cosmological potential wells today (∼10 −5 ). In this work, we will choose the functional form introduced by Hu & Sawicki [71]: f(R)=−2Λ R R+μ 2 , (3.2) 58 with two free parameters, Λ, μ 2 . Note that as R → 0, f(R)→ 0, and hence this model does not contain a cosmological constant. Nevertheless, as R≫μ 2 , the function f(R) can be approximated as f(R)=−2Λ−f R0 ¯ R 0 R , (3.3) with f R0 =−2Λμ 2 / ¯ R 2 0 replacing μ as the second parameter of the model. Here we define ¯ R 0 = ¯ R(z = 0), so that f R0 = f R ( ¯ R 0 ), where overbars denote the quantities of the background spacetime. Note that f R0 < 0 implies f R < 0 always, as required for stable cosmological evolution. If|f R0 |≪ 1, the curvature scales set by Λ =O(R 0 ) and μ 2 differ widely and hence the R≫μ 2 approximation is valid today and for all times in the past. The background expansion history thus mimics ΛCDM with Λ as a true cosmological constant to order f R0 . Therefore in the limit |f R0 | ≪ 10 −2 , the f(R) model and ΛCDM areessentially indistinguishablewithgeometric tests. Thelineargrowth rateis identical to that of ΛCDM on scales larger than λ C , and becomes strongly scale-dependent on smaller scales [71]. Note that we have chosen a model whose expansion history is close to ΛCDM by construction. In general, there is sufficient freedom in the free function f to emulate any givenexpansionhistory[175]. Hence,belowwewillalsoallowtheexpansionhistorytovary, parametrized by effective dark energy parameters w 0 and w a . Further, while we choose a specificfunctionalformforf(R)here,itisstraightforwardtomapconstraintsontodifferent functional forms (see [54] for details). In the following, for notational simplicity f R0 will always refer to the absolute value of the field amplitude today. 3.2.2 Cluster abundance in f(R) Studying structure formation in f(R) gravity beyond linear theory is complicated by the non-linear field equation for the scalar field f R , the non-linearity being responsible for the 59 chameleon mechanism. The field equation needs to be solved simultaneously with the evo- lution of the matter density. This has been done in the self-consistent N-body simulations of [130]. The abundance of dark matter halos (mass function) and their clustering (halo bias) in the f(R) simulations was studied in [162]. Since these simulations are very time-consuming, they cannot be used to exhaust the cosmological parameter space. Instead, we use a simple model developed in [162] based on spherical collapse and the peak-background split in order to predict the cluster abundance and their linear bias. Inordertodescribetheeffect off(R)gravity on thehalo massfunction, weemploy the Sheth-Tormenprescriptionforthecomoving numberdensity ofhalos, as given inEq. (1.9). In [169], the cluster mass definition is the virial mass which corresponds to the mass enclosed at the virial radiusr v , at which the average density is Δ v times the mean density. We transform the virial mass to the desired overdensity criterion Δ = 500/Ω m assum- ing a Navarro-Frenk-White [123] density profile[69], and assuming the mass-concentration relation of [18] (note that the rescaling depends very weakly on the assumed halo concen- tration for the values of Δ used here). We thus obtain the mass function of halos in the ST prescription, n (ST) , from n (ST) v . Theeffects off(R)modifiedgravity enter in two ways inthis prescription: first, weuse the linear power spectrum for the f(R) model in Eq. (1.9). Second, we assume modified spherical collapse parameters which were obtained by rescaling the gravitational constant by 4/3 during the collapse calculation as well as the corresponding linear growth extrap- olation to obtain δ c . This corresponds to the case where the collapsing region is always smaller than the Compton wavelength of the field. [162] showed that this case always underestimates thef(R)effects on the mass function and bias, and hence serves as conser- vative model. For our fiducial cosmology at z = 0, we obtain GR collapse parameters of δ c = 1.675, Δ v = 363, while the modified parameters are given by δ c = 1.693, Δ v = 292. 60 The Sheth-Tormen prescription itself does not provide a very accurate prediction for the abundance of clusters in ΛCDM in the entire redshift range relevant for SZ surveys. Since more precise parametrizations are available, we only use the ST prescription to predict the relative enhancement of the cluster abundance in f(R). Specifically, after rescaling to our adopted mass definition, we take the ratio of the two and multiply it by the ΛCDM mass function from Tinker [180], dN dz ≡n(M,z) =n (T) ΛCDM (M,z) n (ST) f(R) (M,z) n (ST) ΛCDM (M,z) , (3.4) where we use the parameters given in their Appendix B. Note that for small field values and at high masses, the predicted f(R) mass function in fact becomes smaller than that for ΛCDM. Since this effect is not seen in the simulations, we conservatively set the mass function ratio to 1 whenever it is predicted to be less than 1. Fig. 3.1 shows the number of clusters as a function of redshift expected from the four surveysconsideredinthiswork(seeSec.1.3), andtherelativedeviationsofthef(R)model from the ΛCDM model for different values of f R0 (dashed lines). The f(R) modifications aremostprominentatlowredshiftsz.0.4, sincethechangesinthelinearpowerspectrum are restricted to progressively smaller scales towards higher redshifts. Further, for f R0 < 5× 10 −5 , we see the strongest effects for surveys with the lowest mass thresholds, in particular Planck (for z < 0.15) and SPTPol. This is a consequence of the chameleon mechanism which suppresses the mass function enhancement above progressively lower masses as f R0 decreases. There are negligible differences between f(R) and ΛCDM for f R0 <3×10 −5 athighhalomasses. Hence, themassthresholdofagivensurveydetermines what field values can be probed by number counts. Further, we have to take into account the effect of modified gravity on the mass- observable relation. The SZ effect is a dynamical mass measure, as the decrement Y is 61 proportional to the velocity dispersion(pressure)of electrons. Inmodified gravity, dynam- ical mass estimates are generally different from the actual mass due to the presence of the additional gravitational force which enters the virial equation. As shown in [160], the dynamical mass is related to the true mass via M dyn =g 3/5 M, (3.5) where g is a weighted integral of the force modification over the object which describes the effect on the virial equation. In principle, g should be weighted by the SZ emissivity and observational windowfunction. However in theinterest of simplicity, andsince weare only interested in an approximate forecast, we simply weight the modified forces by the matter density ρ NFW (r) of the halo, assuming an NFW profile [123]. Further, we assume the host halo is spherically symmetric. We then have g= R rv 0 drr 2 ρ NFW (r)g(r)rdΨ N /dr R rv 0 drr 2 ρ NFW (r)rdΨ N /dr , (3.6) whereΨ N istheNewtonian potential ofthehalo, foundbysolving (see[160] foran explicit expression) ∇ 2 Ψ N =4πGρ NFW , (3.7) and g(r) is the force modification. In order to calculate the force modification, we have to solve the chameleon field equation for an NFW halo [160]. This calculation is compu- tationally expensive, so we instead use a simple model which describes the exact results reasonably well [160]; in fact it underpredicts the exact result for the force modification, and thus is a conservative estimate. Specifically, g(r)≈1+ 1 3 M(<r)−M(<r scr ) M(<r) (3.8) 62 Here, r scr is the outermost radius at which the condition |Ψ N | ≥ 3| ¯ f R |/2 is met. In the large-field limit this condition is never met, so that r scr = 0 and g(r) = 4/3 throughout. Eq.(3.5)thenyieldsM dyn /M =(4/3) 3/5 ≈1.22. Forsufficientlysmallfields,thechameleon mechanism becomes active so that g(r)→ 0 for r < r scr , thus modeling the screening of the modified force. In this case, M dyn will interpolate between M and 1.22M. We show in Fig. 3.2 the mass threshold of the four SZ cluster surveys in ΛCDM (solid) and the f(R) dynamical mass effect to these thresholds (dashed). Fig. 3.1 also shows the dynamicalmasseffectontheobservedclusterabundance(dash-dottedlines). Notethatthe dynamical mass effect is not simply additive to the mass function enhancement, since the latter dependson mass as well. Due tothe steepness of thehalo mass function at thehigh- mass end, the fact that M SZ = M dyn is larger than the true mass M significantly boosts the abundance of detected clusters above the mass threshold. The two effects of enhanced growth and increased M dyn both contribute to increase the observed cluster abundance. For z . 0.4, the mass function enhancement provides a significant contribution to the overall change in number counts, while at higher redshifts the increase in dynamical mass is the dominant effect. 3.2.3 Halo clustering in f(R) In addition to the halo abundance, f(R) modified gravity also affects the clustering of halos. This effect comes from two sources: first, the matter power spectrum is enhanced onsmallscales bytheincreasedgravitational forces. Second, thelinearbiasb L (M)ofhalos at a given mass M is reduced, since at a fixed mass halos are less rare inf(R) than in GR. 63 The power spectrum of clusters of mass M is modeled as in Eq. (1.1). For the linear bias b L (M), we adopt an analogous prescription as for the mass function Eq. (3.4), b L (M,z)=b (T) L,ΛCDM (M,z) b (ST) L,f(R) (M,z) b (ST) L,ΛCDM (M,z) , (3.9) whereb (T) L,ΛCDM denotesthebiasfittingformulafrom[181],andb (ST) L isthepeak-background split bias derived from the Sheth-Tormen mass function, b L (M)=1+ aν 2 −1 δ c + 2p δ c [1+(aν 2 ) p ] , (3.10) where ν,a,p are defined after Eq. (1.9). Note that ν is given in terms of the virial mass M v , and thus for a given mass and redshift ν differs in f(R) due to both the modified spherical collapse parameters and the different linear power spectrum. For the matter power spectrum in Eq. (1.1), we use the linear theory power spectrum for f(R) and ΛCDM. As shown in [131], this describes the non-linear power spectrum at z = 0 measured in f(R) N-body simulations up to scales k ∼ 0.2h/Mpc. In order to minimize the impact of non-linearities on the power spectrum and its covariance, we limit our Fisher matrix to modes with k less than 0.1h/Mpc. Note that including smaller scales will further improve the constraints; however, a more sophisticated model including non- linear and/or scale-dependent bias, and the non-linear matter power spectrum would be necessary in this case. Thus, the effect on the cluster power spectrum is due to three combined effects: enhancement of the linear power spectrum ΔP L (k), halo bias Δb L (M,z), and the dynam- ical mass effect M dyn . Fig. 3.3 shows the relative deviation ΔP h /P h of the cluster power spectrum in f(R) with respect to ΛCDM for the Planck survey as a function of redshift and wavenumber k. Plots for the other surveys investigated here show similar z− and 64 k−dependences, though the amplitude of each effect dependson thesurvey. Here, we have assumed one mass bin M >M lim (z) and f R0 = 10 −5 . Similar to dN/dz, we plot the total effect (upper left), and separately the effect due to ΔP L (k) (upper right), and Δb L (M,z) (lower panel). For this field value, the dynamical mass effect is irrelevant since the clus- ters detectable by Planck are chameleon-screened. The departure from ΛCDM is mainly driven by Δb L (M,z) which shows a strong redshift dependence, and only mildly affected by ΔP L (k) which is k-dependent and only relevant on small scales. Given that the power spectrumis shot-noise dominated at all scales for the cluster samples considered, theeffect on the linear halo bias in fact is the most important contribution to the f(R) constraints from the cluster power spectrum. 3.3 Analysis 3.3.1 Methodology We use the standard Fisher matrix analysis (Appendix. C) to investigate the constraining power of the SPT, Planck SPTPol and ACTPol surveys (Sec. 1.3) on the f(R) grav- ity model. The modified gravity effects on the mass function, halo bias, matter power spectrum, and mass-observable relation are considered. The self-calibration approach is adopted to account for the uncertainties in the mass-observable scaling relation (Sec. C.4). TheCMB prior (Sec. C.3) is also considered in addition to the cluster probes. We take the ΛCDM model as the fiducial model (Sec. C.2.3). The f(R) modification is parametrized using the field amplitude f R0 at z = 0, or the Compton wavelength λ C0 at z = 0 (see Sec. 3.3.2). Our fiducial value is f R0 = λ C0 = 0. As we shall see in Sec. 3.4, the field amplitudeparameter f R0 shows degeneracies with some of the cosmological parameters, so that the CMB prior also helps in further constraining f R0 . 65 3.3.2 Non-Gaussian likelihood AninherentassumptionintheFishermatrixapproachisthatthelikelihoodcanbeapprox- imated as Gaussian around its maximum; in other words, that one can do a reasonably accurate Taylor expansion of lnL in all parameters. Unfortunately, this is not the case for the parameter f R0 , as the derivatives of the likelihood with respect to f R0 diverge at the fiducial value f R0 =0 (see Fig. 5 in [163]). Thus, we choose the Compton wavelength λ C0 as aparameter instead off R0 , wherefor thef(R)model andfiducialcosmology considered here, λ C0 ≈32.53 r |f R0 | 10 −4 Mpc. (3.11) With this choice, lnL becomes analytic at the fiducial value λ C0 = 0. Specifically, we calculate the derivatives numerically as dlnL dλ C0 = lnL(λ C0 )−lnL(0) λ C0 , (3.12) where λ C0 is the Compton wavelength evaluated at the chosen step size f R0 = Δf R0 throughEq.(3.11), andLdenotesthelikelihoodfromeitherdN/dz orP(k). Unfortunately, the likelihood is still strongly non-Gaussian in the direction of λ C0 , and the constraints dependonthestepsizeΔf R0 chosentoevaluatetheFishermatrixelementsinEq.(C.1). In principle, one would have to evaluate the full likelihood with a MCMC approach, and then perform a marginalization to obtain proper forecasted constraints. Here, we opt instead for a simpler approach. We evaluate the Fisher matrix for a range of step sizes Δf R0 , and then quote the constraints for which σ(f R0 )=Δf R0 is satisfied. One can easily show that this gives the correct answer in the ideal case where the likelihood is Gaussian in all other parameters. Note that while we always use λ C0 as parameter in the Fisher matrix, we 66 will quote constraints in terms of f R0 in order to facilitate comparison with the literature, using λ C0 only to show parameter degeneracies. 3.4 Results We begin by discussing constraints from number counts (Sec. 3.4.1) and power spectrum (Sec. 3.4.2) separately, before moving on to combined constraints (Sec. 3.4.3) and the impact of external priors on the nuisance parameters (Sec. 3.4.4). 3.4.1 Number counts As discussed in Sec. 3.3.2, the Fisher constraints depend on the value of Δf R0 adopted to evaluate the numerical derivatives in the Fisher matrix. Fig. 3.4 shows the pro- jected constraints for the different surveys as a function of Δf R0 . The sharp upturn at Δf R0 ∼ 3×10 −5 (SPT and ACTPol), Δf R0 ∼ 2×10 −5 (SPTPol) and Δf R0 ∼ 9×10 −6 (Planck) signals the transition to the chameleon-screened regime, where the mass function enhancement becomes negligible [162]. The shape of this transition depends on the mass limits of the different surveys, as more massive halos are screened for larger values of f R0 . The figure clearly shows that, with number counts alone, constraints cannot be tighter than σ f R0 ∼ 10 −5 . Nevertheless, this still constitutes an order of magnitude in improve- ment over current constraints. It should also be noted that the use of the dynamical mass inthecalculationsleadstoasignificantimprovementinconstraintsinthelarge-fieldregime where the chameleon mechanism is not active. The precise constraints obtained at the intersection σ f R0 =Δf R0 are listed in Tab. 3.1, along with the step size used for each survey. The relative constraining power of the differentsurveyscaneasily beinterpretedbylookingatΔN/N showninFig. 3.1. Thebest survey toconstrainf(R)withnumbercounts is Planck, which showsprominentdeviations 67 in ΔN/N at low redshift, and yields a 68% CL constraint of σ f R0 = 2×10 −5 . Although SPTPol shows significant differences in number counts out to large redshifts, the relatively small survey volume compared to Planck limits the performance in constraining f(R) to σ f R0 ≃3×10 −5 . Itisinterestingtonoticethatwhiletheiroverallperformanceissimilar,the constraints leverage on clusters in almost disjoint redshift ranges. Therefore these surveys provide complementary information on f(R) constraints from number counts, making the overall resultlesssusceptibletospecificissuesrelatedtoeitherloworhighredshiftclusters. An investigation of whether the combination of both cluster samples yields a significant improvement on the expected f(R) constraints would be worthwhile, but is beyond the scope of this paper. The other two surveys also present results highly competitive with current constraints, and not very different from SPTPol (σ f R0 = 3× 10 −5 ). A better investigation with a proper likelihood would be necessary in order to make more precise statements. 3.4.2 Power spectrum Fig. 3.5 shows the constraints from the clustering of clusters alone as a function of step size. The constraints generally worsen as the step size decreases to very small values. This is because the likelihood around the fiducial model (ΛCDM) scales as λ C0 a , where a>1, and hencethe derivatives go to zero as the step size decreases. However, constraints do not worsen dramatically as the step size crosses the chameleon threshold, because the modificationtothehalobiasinf(R)persistsevenifthehalosarechameleonscreened[162]. Furthermore, the deviations in the matter power spectrum on small scales also persist for field values f R0 < 10 −5 . As expected, the use of the dynamical mass does not affect the constraints for small field values where the entire cluster sample is chameleon screened. The constraints from power spectrum only are summarized in the second column of Tab. 3.1. For Planck (as well as marginally for SPTPol) the constraint on f R0 from the 68 cluster power spectrum is tighter than that from the abundance only. This is mainly because the power spectrum retains sensitivity to f(R) effects even when the halos are chameleon screened. For ACTPol and SPT the power spectrum yields slightly less con- straining power than number counts, as the disadvantage of not having all-sky coverage is not compensated by the relatively low mass threshold. In order to investigate what cluster redshift range contributes to the f R0 constraints, we theconstraints (for Δf R0 =7×10 −6 fixed)as functionof themaximum cluster redshift considered in Fig. 3.6. For surveys with mass limits which decrease with redshift, i.e. SPT and SPTPol, constraints improve up to z max = 1, while for Planck all the information is derived from clusters below z ≈ 0.3, and for ACT the constraining power comes from clusters below z ≈ 0.5. It is especially interesting to compare results from ACTPol and SPT, which detect a comparable number of clusters overall but with a different redshift distribution. Fig. 1.3 shows that ACTPol has a significantly higher number of clusters than SPT out to z ≈ 0.5, and a lower mass limit out to z ≈ 0.3. Yet the constraints from the cluster power spectrum are worse for ACTPol than SPT, due to the contribution from z > 0.5 clusters for SPT (Fig. 3.6). How well each survey can realize their potential constraining power clearly depends on the precise M lim (z) achieved in the final cluster sample. 3.4.3 Combined constraints Fig. 3.7 shows constraints on f R0 when combining both number counts and clustering, as a function of the step size Δf R0 . The dependence on Δf R0 is similar to the case of power spectrum-only and number counts-only constraints at small and large step size respectively. Combining the two probes helps to break degeneracies and better constrain the nuisance parameters. As a result, the constraints on f R0 show improvements with respect to those derived from power spectrum or number counts alone (third column in 69 Table 3.1: Marginalized constraints (68% confidence level) from the two cluster probes dN/dz and P(k), as well as the combination of both for the four SZ surveys. The results are combined with forecasted constraints from the Planck CMB. We also indicate the step size Δf R0 used for each survey and probe. The results reported as approximate values refer to the case in which the constraints do not match any of the explicitly evaluated step size. Parameter dN/dz P(k) dN/dz + P(k) Planck ΔfR0 2×10 −5 7×10 −6 5×10 −6 fR0 2.04×10 −5 7.58×10 −6 5.13×10 −6 ΩMh 2 1.10×10 −3 1.09×10 −3 1.07×10 −3 ΩΛ 0.17 9.84×10 −2 3.18×10 −2 σ8 6.31×10 −3 6.26×10 −3 6.15×10 −3 Ω b h 2 1.31×10 −4 1.30×10 −4 1.29×10 −4 ns 3.33×10 −3 3.30×10 −3 3.27×10 −3 w0 0.64 0.36 0.11 wa 2.34 13.10 1.11 ACTPol ΔfR0 3×10 −5 3×10 −4 2×10 −5 fR0 ∼ 3×10 −5 3.23×10 −4 ∼ 3×10 −5 ΩMh 2 1.10×10 −3 1.10×10 −3 1.09×10 −3 ΩΛ 0.17 0.17 0.13 σ8 6.31×10 −3 6.32×10 −3 6.27×10 −3 Ω b h 2 1.31×10 −4 1.31×10 −4 1.31×10 −4 ns 3.33×10 −3 3.33×10 −3 3.30×10 −3 w0 0.65 0.64 0.48 wa 2.18 11.50 1.45 SPT ΔfR0 3×10 −5 5×10 −5 2×10 −5 fR0 ∼ 3×10 −5 6.60×10 −5 ∼ 3×10 −5 ΩMh 2 1.10×10 −3 1.09×10 −3 1.09×10 −3 ΩΛ 0.17 8.82×10 −2 6.47×10 −2 σ8 6.31×10 −3 6.28×10 −3 6.26×10 −3 Ω b h 2 1.31×10 −4 1.31×10 −4 1.31×10 −4 ns 3.33×10 −3 3.30×10 −3 3.29×10 −3 w0 0.65 0.31 0.23 wa 2.28 5.11 0.94 SPTPol ΔfR0 2×10 −5 2×10 −5 2×10 −5 fR0 ∼ 3×10 −5 ∼ 3×10 −5 2.04×10 −5 ΩMh 2 1.10×10 −3 1.09×10 −3 1.09×10 −3 ΩΛ 0.17 6.06×10 −2 3.63×10 −2 σ8 6.32×10 −3 6.26×10 −3 6.24×10 −3 Ω b h 2 1.31×10 −4 1.31×10 −4 1.30×10 −4 ns 3.33×10 −3 3.29×10 −3 3.29×10 −3 w0 0.62 0.22 0.14 wa 2.16 2.82 0.88 70 Tab.3.1). WhilePlanck reachesaconstraintofσ R0 ≈5×10 −6 ,ACTPol, SPTandSPTPol achieve 2−3×10 −5 . Among the four surveys, the Planck survey thus yields the tightest constraintsregardlessofwhichclusterprobeisbeingused. TherelativemeritofthePlanck survey is due to its large area, which allows to detect massive clusters on the whole sky, and its ability to detect low redshift clusters. Fig. 3.1 shows that in the small-field regime (f R0 ∼ 10 −5 ), the low redshift clusters drive the constraints for Planck while low mass clusters do so for SPTPol. Uptonow,wepresentedresultswithconservativemasslimits,i.e. clustersareexpected to be detected with S/N ≥ 5 for all the surveys. We also examined improvements in the constraint σ f R0 when using more optimistic mass limits for each survey, according to what is outlined in section Sec. 1.3. For all surveys, the constraints from number counts only are hardly affected, since they are mainly set by the chameleon threshold. In each case, the larger cluster sample does improve the power spectrum constraints. However, only for Planck does this yield a significant improvement in the combined constraints (by a factor of 1.5 to 3×10 −6 ), while for ACTPol, SPT, and SPTPol, the improvement in combined constraints is marginal. Fig. 3.8 illustrates the most important degeneracies of λ C0 with standard cosmological parameters for the Planck survey. Here, we show λ C0 instead of f R0 for purposes of pre- sentation. The most prominent degeneracies are with the amount and equation of state of dark energy (Ω Λ , w o and w a ). Clearly, the combination of both observables yields a signif- icant reduction in degeneracies in all cases. The degeneracy with dark energy parameters also explains why the combined constraints on f R0 are slightly better for SPT than for ACTPol, even though the constraints from number counts and clustering separately are very similar for the two surveys. By probing higher redshifts more effectively, SPT is able to better break degeneracies with dark energy parameters. 71 Constraintsonmodifiedgravity showlittlewiththepowerspectrumnormalization (see Fig. 3.8). This is due to the fact that the high number of clusters detected allows for good characterization oftheshapeofthemassfunctionbeyonditsoverall normalization. Similar but somewhat weaker degeneracies are present for the other surveys. 3.4.4 Uncertainties in scatter of mass observable relations Throughout this work, we have assumed a functional form for the scaling relations and then allowed the data to calibrate the parameters that characterize it. In practice, we can expect some external constraints on these parameters by using detailed studies of individual clusters or combining different information from optical, lensing, X-ray and SZ measurements. This procedure is possible thanks to the large number of clusters that are expected to be detected in these surveys. Current strategies for deriving constraints from cluster surveys, however, rely on the calibration of scaling relations as obtained by a small subset of well studied clusters. In general, allowing more freedom to the scaling relation parameters may avoid biases induced by incorrect scaling relations but can also result in a degradation of the final result. In order to investigate the degradation of σ f R0 due to this self-calibration, we repeat the forecasts assuming different priors on the four “nuisance” parameters. The result is summarized in Tab. 3.2 for the number counts, clustering, and combined, and for the four surveys. Here,the“weakprior”caseassumespriorsonthenuisanceparametersofΔB M,0 =0.05 and Δα =1, as well as Δσ M,0 =0.1 and Δβ =1, as suggested by comparison between X- rayandlensingclustermassmeasurement(e.g.,theXMM-Newtonmeasurementspresented in [195]). The combined constraints on f R0 are smaller than those for the default, no prior case, by about25% for Planck 80% for ACTPol, and 50% for SPTandSPTPol. Themost prominent improvements are seen in number counts only constraints (e.g. a factor 3.8 for SPT). 72 The “strong prior” case assumes that all four nuisance parameters are fixed at their fiducialvalues. Thisassumption,whichisanywaynotrealistic,wouldleadtoimprovements of about one order of magnitude with respect to the self-calibration results. This result suggests that although self-calibration does not in general lead to major degradations in the constraints, good prior information on normalization and scatter in the mass-observable relation can improve constraints considerably in particular for the ACTPol and SPT/SPTPol surveys. On the other hand, it is important to keep in mind that self-calibration relies on a specific parametrization of the mass-observable relation and its scatter, and external mea- surements are important to validate these assumptions. As a worst-case scenario, we also consideredthecaseofasinglemassbinforeachsurvey, i.e. neglecting allmassinformation on individual clusters. The fully marginalized, combined constraints on f R0 (without any priors on bias and scatter) worsen by approximately a factor of two for Planck and ACT- Pol. On the other hand, both SPT and SPTPol constraints degrade by only∼10%, since these surveys have a large lever arm in redshift. While these constraints are considerably worse than when using mass bins, the Planck SPT, and SPTPol constraints with a single mass bin still improve over current upper limits. 3.5 Discussion It is worth comparing our forecasted constraints on f R0 with those obtained in [163] ([107] obtainedsimilarconstraints). Bycombining49ChandraX-rayclustersandusinggeometric constraints from CMB, supernovae, H 0 , and BAO, they found an upper limit of f R0 < 1.4×10 −4 (95% CL), including only the statistical error. Our forecasted constraints are tighter by a factor of ∼ 3−4 (ACTPol, SPT, SPTPol) and ∼ 15 (Planck), respectively. The main reasons for the tighter constraints are: the significantly larger cluster samples 73 Table 3.2: Relative improvement in constraints on f R0 , i.e. σ no prior f R0 /σ weak f R0 , whenincluding weak priors on the mass-observable relation (see text). In each case, Δf R0 is that given in Tab. 3.1 for the corresponding survey/probe. Survey f R0 (10 −6 ) λ C0 (Mpc/h) dN/dz P(k) dN/dz dN/dz P(k) dN/dz +P(k) +P(k) weak Planck 1.00 1.05 1.26 1.02 1.02 1.12 ACTPol 2.14 1.82 1.91 1.01 1.18 1.38 SPT 3.80 1.12 1.48 1.03 1.01 1.21 SPTPol 1.29 1.0 1.45 1.13 1.00 1.21 yielded by these surveys, the use of the dynamical mass (which improves number count constraints), and the inclusion of the clustering of clusters as an observable. As shown in Sec. 3.4, the latter in fact provides the dominant constraining power for these surveys in the small field limit. Furthermore,theconstraintsin[163]aredominatedbythesystematicuncertaintyinthe clustermassscale,andincludingthissystematicincreasestheupperlimittof R0 .3×10 −4 . The constraints presented here are marginalized over the cluster mass scale, and hence already include this systematic. Indeed, the combination of power spectrum and number counts is essential in order to realize self-calibration without loosing constraining power. One interesting finding of our study is that the chameleon screening mechanism, a necessary ingredient in this modified gravity model in order to satisfy Solar System con- straints, has a qualitative impact on the constraints. In particular, the number counts by themselves cannot push constraints below f R0 ∼ 10 −5 due to this effect, while they yield the tightest constraints for larger field values. Similarly, the importance of the dynamical mass effect is controlled by the chameleon threshold. This is expected to hold for other modified gravity scenarios as well, as long as the respective screening mechanism depends 74 mainly on the host halo mass (or potential well) of the cluster. On the other hand, screen- ing mechanisms that mainly dependon the average interior density, such as the Vainshtein mechanism employed in braneworldandgalileon models, will show aqualitatively different behavior [161, 160] (see [31] for a study of the related symmetron mechanism). For such models, theutility ofnumbercountswill notbelimited tocertain parameter ranges. Thus, taking into account the screening mechanism is crucial for obtaining realistic constraints on any viable modified gravity model, both for forecasts and when using actual data. All of the surveys considered here reach the limit set by the chameleon mechanism on the constraints from number counts. The Planck survey achieves the tightest constraints both due to its large volume, which reduces the sample variance especially in the cluster power spectrum, and due to its ability to detect clusters at z < 0.15. For example, if we limit the Planck cluster sample to z≥0.15, the combined constraints in f R0 degrade by a factor ofthreeto∼2×10 −5 . Wethusexpectthatsignificantimprovements inconstraining power are achievable for ground-based SZ surveys if the minimum cluster redshift can be reduced. Several improvements upon our treatment here are possible. First, our model for the f(R) effects on mass function and bias of halos is conservative. In order to investigate this, we repeated the forecast using the standard as opposed to modified spherical col- lapse parameters in the model prediction [162]. In case of the Planck survey, the fully marginalized, combined constraint is tightened by a factor of 5−6, constraining f R0 to less than 10 −6 . This prescription overestimates the f(R) effects in the small field regime (f R0 . 10 −5 ) and thus leads to overly optimistic constraints. Nevertheless, the improve- ment in constraints signals that it is worth developing a more accurate model for the f(R) effects on halomass functionand bias(e.g., along thelines of [103]). Given theimportance of the cluster power spectrum in the constraints, an accurate model for the modified halo 75 bias will be crucial. Furthermore, a model for the cluster power spectrum on mildly non- linear scales would also lead to tighter constraints by allowing k max to be increased above the value of 0.1h/Mpc adopted here. 3.6 Conclusions ThelargeclustersamplesexpectedfromcurrentandupcomingSZsurveyscanbeexploited to place tight constraints on modifications to gravity. We have shown that the Planck cluster sample will allow for morethan oneorder of magnitude improvement in constraints on thefield parameter f R0 over currentobservational constraints, even whenmarginalizing over the expansion history (parametrized by w 0 ,w a ) and bias and scatter in the mass- observable relation. Similarly, SPT, SPTPol and ACTPol should provide improvements of about a factor 3–4. Using number counts only, the Planck cluster catalog should be able to reduce errors to σ f R0 = 2×10 −5 in the near future. The inclusion of the cluster power spectrum as a probe greatly improves results especially in the small field limit. The best constraint we obtain is for Planck (combined constraints, σ f R0 = 5×10 −6 ) and is mainly driven by the power spectrum. These constraints push into the regime not ruled out by Solar System tests [71]. Even with self-calibration, a good understanding of the cluster selection function will be necessary to realize this potential however. On the theoretical side, a better description of the modified gravity effects on halo mass function and bias should allow for further improvements. In addition, the use of a proper likelihood function wouldconstituteanimportantvalidationoftheresultsobtainedherewiththeFishermatrix approximation. The work presented in this chapter is based on the publication: Mak, D. S.Y., Pierpaoli, E., Schmidt, F., & Macellari, N. 2012, Phys. Rev. D, 85, 123513 76 Figure3.1: Thefractional deviation of thenumberdensity betweenf(R)andΛCDMmod- els due to all effects of f(R) (dotted lines), Δn ln effect only (dashed lines), and dynamical masseffect only(dot-dashedlines)evaluated atf R0 =10 −4 (top), f R0 =5×10 −5 (middle), and f R0 = 2×10 −5 (bottom) respectively. Colors of the lines refer to different surveys which follow thescheme in Fig. 1.3. For thelargest field value, theeffect on themass func- tion dominates the enhancement of the cluster abundance at low z, while the dynamical mass effect dominates at z & 0.3. The Planck and SPTPol survey have the lowest mass threshold at z < 0.2 and z > 0.2 respectively and hence are most sensitive to the f(R) effects for small field values. 77 Figure 3.2: Mass limit of cluster surveys in ΛCDM (solid) and f(R) gravity (dashed) with f R0 =10 −4 (left) and f R0 =3×10 −5 (right). The mass limits in f(R) are reduced due to the effect on dynamical mass measurements (Sec. 3.2.1). 78 Figure3.3: Relative deviationsinthef(R)halopowerspectrumfromΛCDM,i.e. ΔP h /P h for the Planck survey, with |f R0 | = 10 −5 . Upper left. Total deviation. Upper right. Deviation duetoP L (k) only. Lower left. Deviation dueto halo biasb L only. For this value of f R0 , the dynamical mass effect on the power spectrum is negligible and therefore we do not show it here. The redshift and scale dependence in the relative deviations from other cluster surveys are similar to the ones shown here. 79 Figure 3.4: Fully marginalized 68% confidence level (CL) constraints on f R0 from the number count of clusters only (using Planck CMB priors), as a function of the step size Δf R0 , for thesurveys considered in this paper. Thered dotted lineindicates σ f R0 =Δf R0 . Foragivensurvey,theintersectionofthislinewiththepredictedconstraintsyieldsthefinal expected constraint (Sec. 3.3.2). Solid (dashed) lines represent the case when dynamical mass is (is not) considered. The sharp upturnat Δf R0 .5×10 −5 is due to the chameleon mechanism. Figure 3.5: Same as Fig. 3.4, but from the power spectrum of clusters only (using Planck CMB priors). 80 Figure3.6: Fullymarginalized constraintsonf R0 fromthepowerspectrumofclustersonly, as a function of maximum cluster redshift z max . Δf R0 = 7×10 −6 was used for all values shown here. Figure3.7: CombineddN/dz +P(k)68%CLmarginalizedconstraintsonf R0 asafunction of the step size Δf R0 for the different surveys. As in Fig. 3.4, the red dotted line shows the identity σ f R0 =Δf R0 . 81 Figure 3.8: Joint constraints on the Compton wavelength λ C0 (in Mpc; see Eq. (3.11)) and (counterclockwise from top left) Ω Λ , σ 8 , w o , andw a . All curves denote 68% confidence level, andarefornumbercountsonly(blue), powerspectrumonly(cyan), andcombination of the two (green). The results are shown for the Planck survey with Δf R0 =5×10 −6 . 82 Chapter 4 Probing Inflation: Primordial Non-Gaussianity 4.1 Introduction Cosmological inflation has emerged as the most popular scenario of the early universe and itpredictsanearscaleinvariantpowerspectrumandclose–to Gaussiandistributionforthe primordial curvature inhomogeneities that seeds large scale structure (LSS). Various infla- tionary models produce different levels of departures from Gaussianity. For example, the slow–roll and single field inflation model produces tiny amount of departures from Gaus- sianity, while other models predict sizable amount of primordial Non-Gaussianity (NG) that could be observed with current experiments such as CMB and galaxy clusters [e.g. 10, 29]. Therefore, any detection of primordial NG would open a new and extremely informative window on the physics of inflation and the very early Universe. Current measurements of the CMB [e.g. 92, 144] and LSS [e.g. 173] found that the distribution of primordial fluctuations is consistent with Gaussianity, however, that bound is still several orders of magnitude away from testing primordial NG at the level predicted by slow–roll inflation. Galaxy clusters are in principle the sensitive and powerful tool for this purpose because they trace the rare, high mass tail of density perturbations. As a result, the changes in the shape and evolution of the mass function of dark matter halos are most sensitive to departures from Gaussianity. The effect of NG on the mass function 83 has been investigated in e.g. [117, 118, 59, 79, 111, 165] and was validated by large- scale cosmological simulations with non-Gaussian initial conditions [59, 60, 38, 41]. More recentlyNGeffectsonthelargescaleclusteringofcollapsedhaloeswerestudiedby[38,116, 183, 96]. They found that the linear biasing parameter acquires a scale dependence, which modifies the power spectrumof the the distribution of cosmic structures most prominently at large scales. This unique signature serves as a powerful way to constrain and forecast the nature of NG assumption. For instance, predictions for the cluster abundance and clustering of galaxy clusters expected from various surveys with primordial NG conditions were presented in [52, 154]. In this chapter, we aim to forecast the capability of ongoing and future SZ cluster surveys in constraining primordial NG, using the Fisher matrix approach. Several ini- tial studies have explored such possibility with future galaxy cluster surveys in different wavebands, e.g. X-ray [158, 139] and optical [51, 37, 127], and they all showed that the constraints from galaxy clusters are quite strong. With the use of SZ cluster survey, we would complete thespectrumof cosmological applications withgalaxy clusters. Thesesur- veys will detect clusters at high redshift to test non-Gaussianity in the regimes where its effects are pronounced, that is, the high mass tail of the mass function and the large scale power spectrum of the cluster distribution. This makes the SZ clusters more favorable since X–ray and optical clusters are not as efficient in detecting high redshift clusters. This chapter is organized as follows. We begin by presenting the parametrization of primordial non-Gaussianity effects on the halo abundance and clustering in Sec. 4.2. In Sec. 4.3 we describe the analysis adopted in this work and specify details of the Fisher formalism. The forecasted constraints are presented in Sec. 4.4. We discuss our results in Sec. 4.5 and conclude in Sec. 4.6. 84 4.2 Primordial Non-Gaussanity Early universe models predict deviation from Gaussian initial conditions. Primordial non- Gaussianity induced by inflationary models can be conveniently parametrized by a non- linear coupling parameter f NL and the Bardeen’s gauge invariant potential Φ. This can be written as the sum of a linear Gaussian term and a non-linear second order term that encapsulates the deviation from Gaussianity [157, 55, 185, 93] Φ=Φ G +f NL (Φ 2 G − Φ 2 G ) (4.1) The parameter f NL determines the amplitude of the non-Gaussianity. In this work, we adopt the large scale structureconvention for definingthe fundamental parameter f NL . As noted by [60], the primordial value of Φ has to be linearly extrapolated at z = 0 in the LSS convention, so that f NL =g(z)/g(0)f CMB NL ≈1.3f CMB NL for the ΛCDM model. We note that the factor 1.3 is approximate since the growth factor g(z) is cosmological dependent. This means that any constraints gathered from CMB data should be increased by 30% in order to comply with the convention adopted here. We define here several terms which appear in subsequent text. The relation between the power spectrummatter density fluctuation extrapolated at z =0, P( ~ k), andthe power spectrum of the Newtonian potential, P Φ ( ~ k) is P( ~ k)T 2 (k) = 2T(k)k 2 3H 2 0 Ω m,0 2 P Φ ( ~ k)=M 2 R (k)P Φ ( ~ k) (4.2) The primordial matter power spectrum is scale free, i.e. P( ~ k) = Ak n , and therefore the potential power spectrum can be written as P Φ ( ~ k)= 9AH 4 0 Ω 2 m,0 4 k n−4 ≡Bk n−4 . In the case of non-Gaussianity, the random field of the potential Φ cannot bedescribed by the power spectrum P Φ = Bk n−4 alone. Higher order moments, in particular the 85 bispectrum B Φ ( ~ k 1 , ~ k 2 , ~ k 3 ), are required. The bispectrum is defined on the basis of the Fourier transform of the three point correlation function D Φ( ~ k 1 )Φ( ~ k 2 )Φ( ~ k 3 ) E as follows D Φ( ~ k 1 )Φ( ~ k 2 )Φ( ~ k 3 ) E ≡(2π) 3 δ D ( ~ k 1 + ~ k 2 + ~ k 3 )B Φ ( ~ k 1 , ~ k 2 , ~ k 3 ) (4.3) The shape of the non-Gaussian bispectrum is related to the fundamental physics of the early universeandtheevolution of the inflation field. A wideclass of inflationary scenarios lead to non-Gaussianity of the local type in which the bispectrum of the Bardeen’s poten- tial is maximized for squeezed configurations. Example of this scenario is the curvaton model which involves an additional contribution to the curvature perturbations by a light field [109]. The parameter f NL of the local type is a constant in space and time with f NL ≪1, and is expected to be of the same order of the slow-roll parameters [50]. In such case, the bispectrum has a simple form [28]: B Φ ( ~ k 1 , ~ k 2 , ~ k 3 )=2f NL B 2 (k n−4 1 k n−4 2 +k n−4 1 k n−4 3 +k n−4 2 k n−4 3 ) (4.4) 4.2.1 Mass Function Non-gaussianity lead to a modified mass function with respect to the gaussian case and are usually expressed as perturbations of the gaussian mass function. There are several prescriptions of the corrections in mass functions of collapsed objects (e.g. [117], [106]). In this work, we adopt the approach of [106] (hereafter LMVJ), in which the probability distribution for the smoothed dark matter density field is approximated using the Edge- worth expansion truncated at the the first few orders. The LMVJ approach was shown to give reasonable agreement with full numerical simulations of structure formation [60], pro- vided that the linear overdensity threshold for collapse is corrected for ellipsoidal density perturbations, i.e. Δ c →Δ c √ q where q≈0.75. 86 In this prescription, the non-Gaussian mass function n NG (M,z) can be written as a function of a Gaussian one, n G , multiplied by a non Gaussian correction factor R(M,z), n NG (M,z) = R(M,z)n G (M,z) (4.5) whereR(M,z)≡n G,PS /n NG,PS , andn G,PS andn NG,PS aretheGaussianandnon-Gaussian mass function respectively computed according to the Press and Schechter formula [147]. In this work, we adopt the formula from [180] for the Gaussian mass function n G (M,z) whichisbasedontheΛCDMcosmology. Followingtheirwork,wealsoconsidertheredshift dependence of the mass function using the simple scaling of the z = 0 fitting parameters with (1+z) α . We disregard the cosmological dependence of mass function (and halo bias) here since there is a lack of analytical expression that formulate the systematic errors induced by varying the cosmological parameters. We notice that the accuracy of the mass functionishigh(upto5%errorinsimulation)evenforslightvariationincosmology, except with the dark energy parameters [180]. More studies are required to determine quantita- tively the effects of variation of dark energy densities and equation of state parameters to the mass function. Using LMVJ approach, the correction factor R is R(M,z)=1+ 1 6 σ 2 M δ c S 3 ( δ 4 c σ 4 M −2 δ 2 c σ 2 M −1)+ dS 3 dlnσ M ( δ 2 c σ 2 M −1) (4.6) where δ c ≡ Δ c /D(z) is the critical density for collapse, D(z) is the linear growth factor, σ M is the rms of primordial density fluctuations on the scale corresponding to mass M, S 3 ≡ f NL μ 3 (M)/σ 4 M is the normalized skewness, μ 3 is the third order moment. For the local non-Gaussianity, μ 3 can be computed as 87 μ 3 (M)= f NL (2π 2 ) 3 R ∞ 0 dk 1 k 1 M R (k 1 )P(k 1 ) R ∞ 0 dk 2 k 2 M R (k 2 )P(k 2 ) R 1 −1 dμM R (k 12 ) h 1+2 P(k 12 ) P(k 2 ) i (4.7) wherek 2 12 =k 2 1 +k 2 2 +2μk 1 k 2 . This integral is computational intensive. In order to reduce the workload of this calculation, we instead use the fitting formula of S 3 by [30] which is shown to have sub-percent accuracy: S 3 = 3.15×10 −4 f NL σ 0.838 R (4.8) We verified the accuracy of this fitting formula by directly comparing this with the numer- ical table publicly available online 1 , and we found that they agree with each other to percent level. 4.2.2 Halo Bias Thehalo bias acquires an extra scale dependencedueto primordial non-Gaussianity of the local type [116], i.e. b NG (M,z,k) =b G (M,z)+Δb(M,z,k) (4.9) where Δb(M,z,k) =[b G (M,z)−1]δ c Γ R (k) (4.10) The term Γ R (k) encapsulates the dependence on the scale and mass. For the local non- Gaussianity it can be written as 1 http://icc.ub.edu/ ˜ liciaverde/nongaussian.html 88 Γ R (k)= 2f NL 8π 2 M R (k)σ 2 R Z ∞ 0 dk 1 k 2 1 M R (k 1 )P(k 1 ) Z 1 −1 dμM R (k 12 ) P(k 12 ) P(k) +2 (4.11) wherek 2 12 =k 2 1 +k 2 +2μk 1 k. ThefunctionΓ R (k) is flatat small scales butscales as k −2 at large scales (k≥0.01 Mpc/h), so that a substantial deviation in the halo bias is expected atthosescales. Weadopttheformulafrom[181] fortheGaussianbiasb G . Unlikethemass function, itwas shownthatthehalobiasfromthis fittingformuladonotshowevidencefor significant redshift evolution [181]. Therefore we do not consider the redshift dependence here. 4.3 Analysis Following theanalysis in Chapter. 3for modifiedgravity work, we performa Fisher matrix analysis to forecast the capability of the four SZ cluster surveys: SPT, Planck, SPTPol and ACTPol surveys, in constraining the deviations from Gaussian distribution of primor- dial density perturbations. We use the constraining power of the cluster number counts and clustering properties to forecast limits on the f NL parameter. The primordial non- Gaussianity effects on the mass function and halo bias are considered, and are illustrated in Fig. 4.1 and Fig. 4.2. Unlike the work in modified gravity, the NG effects vary on different length scales, i.e. the k values, in the cluster power spectrum. This is because primordial NG of the local type modifies the shape of the power spectrum of galaxy clusters by introducing a scale- dependentbiasonverylargescales. Thereforethechoiceoflargestscalethatcanbeprobed by a cluster survey would significantly affect the constraints on f NL . In practice, this is taken into account by introducing the window function of the survey in the calculation of 89 Figure 4.1: The deviation from the gaussian number count when f NL =100. Figure 4.2: Relative deviations of the non-gaussian power spectrum from the gaussian power spectrum, i.e. ΔP/P G for the Planck survey and f NL =100. P(k). However, for simplicity, we approximate the influence of the window function by considering a cut off in the choice of k min . We use k min = 10 −3 for the all-sky Planck survey, and k min =10 −2 for the partial sky surveys. 90 Table 4.1: Marginalized 1σ errors on f NL . The labels in the second column means the following. standard: The standard setup as indicated in Sec. 4.2.1 and Sec. 4.2.2; less conservative: Theconservativemasslimitoftheclustersurveyisconsidered(seeSec.4.4.2); photo-z: Redshift binning correspond to the error of photometric redshift, i.e. Δz = 0.1; one mass bin: No mass slicing in the Fisher matrix, i.e. only one mass bin. Probes Planck ACTPol SPT SPTPol dN/dz standard 987 1473 2319 1463 less conservative 497 1069 1810 1230 photo-z 1540 1588 2355 1462 one mass bin ∞ ∞ ∞ ∞ P(k) standard 7 24 21 15 less conservative 4 16 17 13 photo-z 8 24 20 15 one mass bin 160 272 267 124 dN/dz + P(k) standard 7 24 20 15 less conservative 4 16 16 13 photo-z 7 24 20 15 one mass bin 156 269 255 119 Ontheotherhand,weadopttheself-calibrationforthemass-observablescalingrelation like we do in the work of modified gravity. We will study the effect of assuming different priors on the four nuisance parameters on the f NL constraints in Sec. 4.5.1. As a final note, we note that the CMB power spectrum does not add constraints on f NL , although it contains information about the primordial non-Gaussianity as we men- tioned in introduction, and thereforewe compute the CMB Fisher matrix for the Gaussian perturbation. 91 4.4 Results 4.4.1 Cluster counts and Power Spectrum The first four rows of Tab. 4.1 summarize the marginalized f NL constraints from dN/dz. Underthestandardsetup: assumingwehavespectroscopicredshiftsΔz =0.05andbinning in cluster mass of ΔlogM = 0.3, the f NL constraints are ∼ 10 3 . The relative constrain- ing power of the different surveys can easily be interpreted by looking at ΔN shown in Figure 4.1. The Planck survey, which has the largest ΔN at z < 0.3, gives the tightest constraints of σ f NL = 987 at the 68% CL. The other three surveys are less stringent and the constraints are >80% worse than Planck. We study how the constraints change if we ignore the redshifts and consider combining information on all redshifts. With only one redshift bin, none of the surveys is able to constrainf NL . Thisimpliesredshiftinformationoneachclustersisessentialinconstraining f NL using number counts. So far we have assumed the optimistic scenario in which the cluster redshifts are spectroscopic. We relax this choice by using the photometric redshifts (i.e. Δz =0.1),thisworsenstheconstraints,particularlyforPlanck inwhichσ f NL increased by 59% . The constraints from power spectrum only are summarized in the row 5-9 in Tab. 4.1. Overall the constraints are almost two orders of magnitude better than that from number counts only. This is because the scale dependence of the halo bias is very sensitive to non- gaussianity (e.g. [164, 116, 38]). The best constraint come from the all-sky Planck survey, with σ f NL = 7. This result is largely due to its large sky coverage (f sky ≈ 0.7). It was shown in previous studies [e.g. 106, 60] that the deviation of the halo bias, and hence the power spectrum, between non-gaussian and gaussian case is most prominent at large scale. We illustrate this inFig. 4.2 withtheΔP/P in differentredshiftandwavelength bins. The 92 survey that probes longer wavelength modes, as in the case of Planck with k min = 10 −3 , therefore has the highest sensitivity to f NL . We here quantify the sensitivity of the non-gaussianity constraints to the adopted k min value in the analysis of the power spectrum. We address this by computing σ f NL as a function of k min , as shown in Fig. 4.3. As expected, the constraints significantly improve when we consider smaller k values. This is partly because more information is included in the Fisher matrix and, more importantly, the effect of NG on halo bias is most prominent at the largest scales. It is interesting to note that if we limit k min to 10 −2 Mpc/h also for the Planck survey, then the derived f NL constraint would be similar to, and even slightly worse than the one derived from the SPTPol survey. It is worthwhile to note that the effective volume V eff also impacts the f NL constraints by its z and k dependence on the Fisher matrix, as shown in Fig. C.1. This is particularly usefulinunderstandingtherelativemeritsofthethreepartialskysurveysthathavesimilar survey area and k-range. Indeed the relative constraining power of these four surveys is reflected in the redshift distributions of the effective volume: while SPTPol has the largest V eff at z > 0.7 (upper panel of Fig. C.1), its error on f NL is similar and is 40% smaller than ACTPol and SPT. To further understand which redshift range contributes most to the f NL constraints, we show in Fig. 4.3 the f NL constraints as a function of the maximum cluster redshifts. If we limit the cluster samples to z≤0.6, the ACTPol survey gives tighter constraint than SPTPol and SPT. As we extend to larger maximum redshift, SPTPol survey then gives tighter constraint than SPT and ACTPol survey as its V eff (z) is decreasing at a smaller rate. This result shows that the constraints leverage on clusters in disjoint redshift ranges and these surveys provide complementary information on f NL constraints from power spectrum. 93 Figure4.3: Fullymarginalizedconstraintsonf NL fromthepowerspectrumofclustersonly, as a function of minimum wavenumber k min (upper) and maximum cluster redshift z max . (lower). 4.4.2 Combined constraints The constraints on f NL when combining both number counts and power spectrum are summarized in rows 10-13 of Tab. 4.1. While combining the two probes helps to break degeneracies and improves constraints on cosmological parameters, the constraint on f NL does not improve. The constraining power is mainly driven by the power spectrum. We note that the best constraint estimated here (σ f NL = 7 from Planck cluster survey) is a factor of 2 (when accounting the difference in f LSS NL and f CMB NL ) better than the current upper limits as measured by CMB experiments (e.g. 32± 21 [92]) and a factor of 14 better than large scale structure probes (e.g. Δf NL =±96 [173]). The most recent Planck CMB result finds that f NL = 2.7±5.8 [144], which is very similar to the one probed by clusters here. Next generation all-sky X-ray surveys [158, 139] would achieve σ f NL ≈ 10. Note that constraints from CMB are based on different physics than large scale structures and suffered from different systematics. The SZ clusters surveys therefore provide a useful complement to the f NL information we can derive from the CMB maps. 94 Fig.5.1illustratesthemostimportantdegeneraciesoff NL withcosmologicalparameters (σ 8 , w 0 , w a ) for the Planck survey. These plots clearly demonstrate the complementarity that the number counts and clustering have to constrain f NL and standard cosmological parameters,particularlytheredshiftdependencepartofequationofstateofdarkenergyw a . It is obvious that the f NL is almost non-degenerate with other cosmological parameters. Only mild degeneracies are seen in the joint constraint with w 0 and w a . Therefore the constraints on other parameters have little effect on f NL . It is important to keep in mind that these results are based on a conservative mass limits, i.e. clusters are expected to be detected with S/N ≥5 for all surveys. We examine improvements in the f NL constraints when using more optimistic mass limits for each survey according to what is outlined in Sec. 1.3. Using number counts only, the constraint from Planck is improved by 50%, while constraints from other surveys are only slightly improved (by 16− 27%). However, these constraints are worse than those from power spectrum. Using power spectrum only, the constraints are improved only marginally for SPTPol and SPT, but largely by 30−40% for Planck and ACTPol. It is interesting to notice that ACTPol is now slightly better in constraining f NL than SPT while it was the opposite case when using the conservative mass limits. This is mainly due to the larger increase in detected clusters for the ACTPol survey in the optimistic mass limits. As a worse case scenario, we also consider the case when there is no information on the cluster mass, i.e. one mass bin. For all surveys, the combined constraints are significantly worsened by a factor of > 10. Yet, the SPTPol constraint with single mass bins is still comparable to current upper limits. 95 Figure 4.4: Joint constraints on the f NL and (counterclockwise from top left) σ 8 , w o , and w a . All curves denote 68% confidence level, and are for number counts only (blue), power spectrumonly (cyan), and combination of thetwo (green). The Planck CMB power spectrum priors are assumed. 4.5 Discussion 4.5.1 Uncertainties in scatter of mass observable relations So far, we assumed no prior on the nuisance parameters in the mass-observable relations: B M,0 , σ M,0 , α, β. Following Sec. 3.4.4, we repeat the forecasts on different priors on these four nuisance parameters to better understand the importance of self-calibration of systematic uncertainties on the non-Gaussianity constraints. 96 Table 4.2: Same as Tab. 3.2, but for f NL constraints. Probes prior Planck ACTPol SPT SPTPol dN/dz weak 1.63 1.54 2.13 2.22 strong 7.68 3.48 5.41 4.39 P(k) weak 1.00 1.01 1.01 1.01 strong 1.02 1.04 1.01 1.02 dN/dz+P(k) weak 1.01 1.02 1.01 1.01 strong 1.66 3.46 3.15 2.80 The results are summarized in Tab. 4.2. We find that the f NL constraints only slightly improves when using number counts only and has negligible improvement in the joint constraints when considering the weak prior. In the case of strong prior, the constraints are significantly tightened by a factor of 4−8 from numbercount only, butnegligibly from power spectrum only. This results in appreciable improvements in the joint constraints for the partial sky surveys but not for Planck, as measured from the cluster. As the f NL joint constraint is essentially driven by the power spectrum, we conclude that it is largely insensitive to priors on nuisance parameters. This is consistent with the scenario shown in [158] as they calculated the ratio of non-Gaussian to Gaussian at a wide range of values of the mass bias B M0 and scatter σ M0 . They showed that the ratio on number counts change by less than 0.1%, and is negligible on the effective bias except at the very large scales (40% at k≈ 10 −3 Mpc −1 ). This justifies the weak dependence of the f NL constraints on prior since the choice of prior affects more the prediction for the cluster counts on f NL , thus show smaller dependence than that from clustering measurements. 4.5.2 Comparison to previous work Forecast of the local-type non-gaussianity was previously done with galaxy cluster probes in other wavelengths, e.g. future X-ray surveys ([158] and [139]) and optical surveys ([51], 97 [37], [127]). This study is the first attempt in exploring the f NL constraints from the SZ cluster survey and our results are in broad agreement with these previous studies. A detailed comparison between our work and these studies is not straightforward because of the different survey specifications, prescriptions for the mass function and halo bias, and the type of cluster probes used. The more interesting comparison can be made with the results from [158] and [139] who putconstraints with cluster number counts and clustering and similar subsamples. In terms of cluster sample ever considered by other authors, the subsample from eRosita [139] of the most massive 1000 clusters (M 500,crit ≥ 2.2×10 14 h −1 M ⊙ at z ≥ 1, ”magnificent 1000”) is the most similar to the Planck cluster sample. This is because the limiting mass of the Planck catalog behaves in a more similar way to the X-ray catalog due to its large beams that significantly smoothes the signal, especially when the angu- lar size of the cluster is small. We forecast a tighter constraint (σ f NL = 7) than theirs (σ f NL = 26), when both assuming photometric redshift information and adding thePlanck CMB prior. One should keep in mind that the non-gaussianity effect is more prominent at high redshift, sothe ”magnificent 1000” which contains cluster at z≥1 shouldin principle be more sensitive to f NL constraint. The discrepancies could probably be due to a number of differences in the analysis: (a) they considered different types and number of nuisance parameters that used to model the uncertainties in the scaling relations, such as L X −M, T X −M. This is a conservative approach but at the same time degrade their constraints considerably. (b) the mass–redshift distribution of the X–ray sample are different from our SZ samples. The SZ samples tend to have higher fraction of massive clusters, i.e. the regime where the NG effects are more prominent. This makes the SZ clusters more sensitive to the f NL constraints. In terms of cluster probes, we use the same ones in [158], except that we additionally consider slicing the P(k) in mass bins. This makes our constraint comparable to theirs 98 even if our samples have much lower statistics (σ f NL =7 for≈1000 clusters in the Planck survey vs σ f NL =11 for∼10 6 clusters in the WFTX Wide survey). On the other hand, if we do not consider mass bins in P(k) then our constraint degrade to σ f NL =156, which is anorderofmagnitudeworse. Theimprovementmainlycomesfromthemassdependenceof the halo bias which in turnprovides extra information of the shapeof thepower spectrum. This suggests the slicing in mass bins compensates for the poor statistics. 4.6 Conclusions In this work, we exploited the large cluster samples expected from current and upcoming SZ surveys to place constraints on primordial non-Gaussianity of the local type. Making use of the cluster number counts and power spectrum, and taking into account the self- calibration of mass-observable scaling relations, we employed the Fisher matrix analysis to forecast the sensitivities of various SZ surveys in constraining the f NL parameter. The main results are presented in Sec. 4.4. We find that the induced scale-dependence of halo bias (through a term that is proportional to k −2 ) by local type non-Gaussianity provides a very effective way to put strong constraints on f NL . This makes the power spectrumamorepowerfulprobethannumbercounts, byatleasttwoordersofmagnitudes. As a result, all–sky surveys such as Planck, which probe better the most NG sensitive regimeat largescales, aremorefavorable to measuref NL . Thebestconstraint weobtain is fromPlanck (combinedconstraints, σ f NL =7)andismainlydrivenbythepowerspectrum. The partial–sky surveys SPT, SPTPol, and ACTPol, however, are a factor of > 2 less constraining than Planck, with σ f NL =13−24. The best constraint we obtained, which is based on conservative assumptions on uncertainties in mass-observable relations and mass thresholds, is a factor 2 better than that measured from WMAP CMB and is comparable to that from Planck CMB (f NL = 2.7±5.8). The SZ cluster surveys therefore provide a 99 useful complement to the f NL information we can derive from CMB maps. However one should note that these results are based on different physics and suffering from different systematics than the probes considered in this work. We also show that f NL have little degeneracy with other cosmological parameters and it is only mildly degenerate with w 0 and w a . Thus the constraints on other parameters have little effect on f NL . We investigate the sensitivity of our results to various aspects of survey specification in Sec. 4.5. We find that the errors on f NL are mainly driven by the power spectrum and therefore they are insensitive to priors on nuisance parameters. Only when we have perfect knowledge on the uncertainties of these parameter we can improve the constraint by a factor of 1.7-3.5, with Planck being the least benefited. This, however, requires very good understanding of the mass calibration which may be possible with multi–frequency followup measurements. In addition, the cluster selection function plays a less significant role in which σ f NL is marginally improved when we consider more optimistic mass limits. Ourresultsareinbroadagreement withpreviousstudiesthatexplorestheconstraining power of f NL from X–ray and optical cluster surveys. The values of σ f NL we obtained are comparable to the ones quoted in these other works even if the SZ samples contain at least two orders of magnitude less clusters. This is because, unlike previous authors, we considered mass slicing in power spectrum when calculating the Fisher matrix and this greatly improves the constraints. This suggests the slicing in mass bins compensates the poorstatistics oftheSZclustersamples. WerealizethattheSZsurveysweconsideredhere are very promising and their cluster catalogues would be available in the very near future, since most of them are operating. Therefore, our f NL forecasts can readily be referenced. The work presented in this chapter is based on the publication: Mak, D. S.Y. & Pierpaoli, E. 2012, Phys. Rev. D, 86, 123520 100 Chapter 5 Weighing Neutrino Mass 5.1 Introduction Measuring masses of neutrinos is one the major goals of particle physics and cosmology. While atmospheric and solar neutrino oscillation experiments are sensitive to neutrino flavor, mixing angle, and the mass difference among different species, cosmological data are instead more sensitive to the absolute mass scale M ν = P m ν . In fact, the most stringent upper bound of the total neutrino mass is coming from CMB and large scale structures since massive neutrinos leave detectable imprints throughout the history of the universe. Most recently, [142] obtained M ν < 0.23 eV at 95% C.L. by combining CMB data and BAO from Sloan Digital Sky Survey (SDSS) In this chapter, we explore the prospects of employing SZ cluster survey in constrain- ing neutrino mass. Galaxy clusters are in principle a powerful tool for probing neutrino properties. Neutrino becomes nonrelativistic after the epoch of decoupling if its mass scale is smaller thanO(0.1) eV. The relativistic behavior of neutrinos, as opposed to cold dark matter, causes the suppression of matter perturbations on small scales with respect to the case in which neutrinos are massless and all dark matter is cold. The presence of massive neutrinos affects the growth rate of perturbations in the linear regime, and, as a consequence,theshapeofthematterpowerspectrumandclusterabundance. Currentmea- surements from X-ray cluster surveys obtained a tight upper limit of M ν < 0.33 eV [188] by combining measurements of Chandra X-ray observations of galaxy clusters, CMB from WMAP 5yeardata, BAOandtype1Asupernova(bothfromHSTKeyProject). Similarly, 101 measurement from galaxy power spectrum from the SDSS-III BAO survey + CMB + SN found M ν <0.34 eV [197]. Subsequently, several works were dedicated to discuss the prospects of utilizing large future surveys of large scale structures (galaxy or galaxy clusters) in different wavelengths (e.g. [191, 148, 76, 23, 22, 171, 170, 19]). These works showed that constraints of neutrino mass depend on assumptions of the underlying cosmology (e.g. inclusion of dark energy or flatness), cluster physics, and the use of external priors (e.g. CMB lensing extraction). Here we revisit the analysis to forecast the constraint of the total neutrino mass, in the frameworkofflatΛCDMuniverseand,likepastconstraints,thestandardscenariowithonly three neutrino species. We use cluster abundance and power spectrum as the observables that will be obtained from the four SZ cluster surveys described in Sec. 1.3. With respect to previous works [e.g 171, 170] which also employ SZ cluster surveys, we provide a more realistic survey specifications to characterize the cluster detection and include the self- calibration to characterize the uncertainties of the mass-observable relations. We also discuss the degeneracy of the neutrino mass with dark energy, which is lacking in previous studies, and compare the strength of cluster probes with CMB on the constraining power of neutrino mass. The chapter is organized as follows. In Sec. 5.2 we discuss the effects of M ν on the large scale structures. In Sec. 5.3 we present the methodology and specify details on the Fisher matrix formalism. Themainresults arepresentedSec. 5.4anddiscussedinSec. 5.5. Finally, a conclusion is presented in Sec. 5.6. 102 5.2 Impact of neutrino masses on growth of the large scale structures The presence of massive neutrinos mildly affects expansion history but significantly impacts the growth of structure through free-streaming. Fluctuations on comoving scales that enter the horizon when neutrinos are still relativistic may be reduced in amplitude because neutrinos would tend to leave the perturbation. This effect, that is neutrino mass dependent, typically occurs on scales below the free–streaming scale: k fs = 1.5 p Ω M h 2 /(1+z)(M ν /eV)Mpc −1 . Thus, the growth of any structures that have lengthscalesmallerthan∼1/k fs willbelessefficient. Asmallerneutrinomassincreasesthe free-streamingscale,butalsoreducestheneutrinofractionwithrespecttothetotalamount of dark matter, mitigating the overall suppression. As a result of these dependences mea- surements of the large scale structures such as cluster number counts and power spectrum can be used to place constraints on neutrino masses. Thelate-time evolution of perturbationsinaΛCDM cosmology with massive neutrinos can be accurately described by the product of a scale dependent growth function and a time dependent transfer function. For example, [100] derived a reasonable approximation to the analytical expression of the transfer function for small scales. In this work, we employ the transfer function determined numerically from CAMB [102] which provides precise estimate on the matter power spectrum and include non-linear effects at large–k limit in which the analytical expressions fail to give an accurate estimate. 103 5.3 Analysis 5.3.1 Methodology Our analysis closely follows the treatment in the modified gravity and primordial non- Gaussianity work. Unlike the previous two works that the cluster abundance and clus- tering are being modified to model the according theory, we here simply use the results of numerical simulations from [180] for the cluster mass function n(M,z), and [181] for the halo bias in the ΛCDM cosmology. In this work, we adopt the fiducial values of the cosmological models described in Sec. C.2.3, except that we additionally add M ν to the parameter list. Unless otherwise specified, we take M ν =0.3 eV as the fiducial value. 5.3.2 Notes on CMB Fisher matrix [133] noted that the CMB power spectra likelihood function for the neutrino mass differs from the gaussian case due to strong parameter degeneracies, particularly for models with manyparameters. TheseauthorssuggesttheuseofCMBlensingextraction informationin order to sharpenthe likelihood and make it better approximated by Gaussian. The Planck CMBlensingextraction(LE)isconsideredtobeaverypromisingwaytoconstrainneutrino mass (e.g. [100]). The CMB anisotropies obey Gaussian statistics in the absence of weak lensing, and therefore they are fully described by the temperature and polarization power spectrum. Weak lensing, however, introduces non-gaussianity in both the temperature and polarization anisotropies [168, 12]. Therefore, extracting the lensing information from CMB(e.g. usingquadraticestimators [67, 70, 65, 128])wouldprovidethelensingpotential and delensed CMB anisotropies, and hence extra information to the Fisher matrix. In the following, we refer to the Fisher matrix results obtained from CMB lensing extraction as the CMB LE. As shown in [101], CMB LE is useful in providing strong neutrino mass constraints and potentially breaking of the major neutrino mass degeneracies with other 104 parameters [133]. While very promising, the exploitation of higher order statistics may suffer from subtle ways from the effect of galactic and extragalactic contaminants. For this reason, wealso consider constraints coming from theCMB power spectrumonly (with lensing) when combining probes with cluster’ s ones. WenotethatthelatestPlanck resultswerereleasedduringthepreparationofthiswork. TheyderiveatightupperlimitofM ν ≤0.93eVwhenusingCMBdataaloneandM ν ≤0.23 eVwhenfurthercombinedwithBAOdata. Nevertheless, theselimitsuseinformationfrom polarization of the WMAP data and not from the Planck data itself (the Planck CMB polarization data will be employed in the next data release). Therefore instead of using these numbers as priors on M ν constraints, we derive our Planck CMB prior, like we do for the previous two chapters, that takes into account the Planck polarization information whichisbelievedtobebetterthanthatfromWMAP.Thusthispriorshouldbeconsidered as the self-contained and improved one than the current constraint in [142]. 5.4 Results 5.4.1 Cluster number count and power spectrum Tab. 5.1 summarizes the neutrino mass constraints from the Fisher matrix analysis for Planck CMB (with and without LE), cluster number counts, and power spectrum for the four cluster surveys. Constraints of M ν from cluster number counts alone are better than powerspectrumones, however, each ofthemis veryweak whenconsideredseparately, with σ Mν >4 eV. When combining information from both probes, the constraints are improved significantly by a factor of 4−8. The best case is obtained from the Planck cluster survey with σ Mν =0.94 eV, whereas the constraints from other surveys are a factor of two worse. 105 5.4.2 Cluster probes + CMB Adding the Planck CMB priors breaks degeneracies (see Sec. 5.5.1) and improves the constraints (number count or power spectrum alone) further by a factor of > 4 (without LE) and > 5 (with LE). When including all the information but LE, i.e. count + power spectrum + CMB, we find the best constraint comes from the Planck and ACTPol cluster survey with σ Mν = 0.23 eV. This is 80% better than that obtained from Planck CMB alone (σ Mν = 0.41 eV). Including CMB priors also shrinks the difference in σ Mν among different surveys in which it is now σ Mν = 0.23− 0.30 eV. Similar results are obtained when we add the CMB LE and the best constraint is σ Mν = 0.17 eV. This suggests that the improvements in σ Mν are mainly driven by CMB information. We note that a perfect cleaning of all the astrophysical foregrounds is assumed when computing the CMB Fisher matrix in this work. Foreground contamination dominates at small angular scales (e.g. l ≥ 1000) and would introduce extra non-gaussianity and spoil the lensing extraction process [34]. Nevertheless, [101] found that the effect of no foreground subtraction in Planck CMB (with and without LE) only degrades the M ν constraint marginally (by 9%). Therefore,ourresultsthatinvolveCMBinformationcanbeconsideredtoberobustagainst foreground contamination. We repeat the analysis with a fiducial M ν = 0.1 eV instead to investigate the effect on the constraint with less massive neutrinos. The results are very close (within 15%) to those for M ν = 0.3 eV when using cluster probes only, and are almost the identical when CMB priors are added. 106 Table 5.1: Marginalized 1σ errors on M ν (in units of eV). no prior + CMB prior + CMB LE prior CMB CMB LE Survey NC P(k) Comb NC P(k) Comb NC P(k) Comb fiducial Mν = 0.3 eV 0.41 0.21 Planck 4.06 7.83 0.94 0.29 0.29 0.23 0.20 0.20 0.17 ACTPol 6.04 11.73 3.33 0.29 0.29 0.23 0.20 0.20 0.17 SPT 12.44 12.45 2.12 0.33 0.31 0.30 0.21 0.20 0.20 SPTPol 12.59 7.81 1.79 0.32 0.30 0.28 0.21 0.20 0.19 fiducial Mν = 0.1 eV 0.52 0.19 Planck 5.09 19.98 0.77 0.37 0.42 0.23 0.18 0.19 0.15 ACTPol 17.97 48.63 2.83 0.43 0.38 0.26 0.19 0.18 0.16 SPT 6.56 31.39 1.78 0.43 0.39 0.28 0.19 0.18 0.17 SPTPol 12.59 7.81 1.79 0.32 0.30 0.28 0.21 0.20 0.19 5.4.3 Self-calibration and uncertainty of nuisance parameters To estimate the effect of self-calibration of systematic uncertainties on the neutrino mass constraints, we repeat the forecasts with different priors on the four nuisance parameters as summarized in Tab. 5.2. We first discuss the results when applying a ”weak” prior. In the case of cluster count + power spectrum, the 1σ error reduces marginally for SPT and SPTPol, but significantly (by a factor of two) for Planck and ACTPol. This results in σ Mν =0.48 eV for the Planck cluster survey which is competitive with the CMB only constraint. In the case of adding the CMB (with and without LE) priors, the 1σ errors generally reduce by a factor of two and resulted in, for the best case as obtained by the Planck survey, σ Mν =0.08 eV, which corresponds to a S/N ≈4 detection for M ν ≥0.3 eV Similar results are obtained when applying a ”strong prior”, i.e. assuming a perfect knowledge of cluster true masses. The constraints are improved significantly by 66−236% in the case of cluster count + power spectrum, anda factor 2−3 when theCMB priors are further added. The best constraint is, again with the Planck cluster survey, σ Mν = 0.07 eV which is a relative marginal improvement with respect to the weak prior case. While it 107 is unrealistic to have perfect knowledge on the mass observable relations, one can achieve similar scenario by restricting the analysis to a relatively small subset of clusters for which follow upobservations are available. Thiswould ensurea sample with well calibrated mass proxies. For example, it has been shown in [188] that the ability to constrain dark energy parameters from a small sample of≈ 50 well calibrated X-ray clusters is comparable to a larger sample of≈10000 optical clusters (e.g. SDSS [155]). Unlike other parameter constraints (e.g. non-Gaussianity with galaxy clusters [112]), the results of the weak prior are sufficiently close to the those from the strong prior. The prospectof achieving theweak prior conditions is promising, e.g. clusters detected in weak lensing measurements or a subsample of objects having extensive multi-wavelength follow- up. Therefore the cluster probes are good enough to provide interesting M ν constraint even without perfect knowledge of the scaling relations. As a final remark, we would like to compare our count + CMB result with [170] which similarly presented M ν constraints assuming perfect knowledge of cluster mass and used Planck cluster count+CMB. Ourresult(σ Mν =0.17 eV)isafactor of2.8 worsethanthat obtained in [170]. We note that the discrepancy is due to the different assumption on the total numbercounts: ≈6000 in [170] and≈1000 in this work for the Planck survey if a5σ survey detection limit is assumed. Our estimate is based on the conservative assumption thatensureshighlevelofcompleteness(90%)andrealisticmasslimitsthatvaryatdifferent redshifts, while [170] assumed a constant and lower mass threshold. 5.5 Discussion 5.5.1 Parameter Degeneracies The dark energy equation of state w 0 and M ν is one of the major parameter degenera- cies.Fig.5.1showsthe1σ constraintsonM ν andw 0 computedfromclusternumbercounts, 108 Table 5.2: Same as Tab. 3.2, but for M ν constraints. A fiducial M ν = 0.3 eV is assumed. The best case is obtained by the Planck survey, σ Mν = 0.08 eV (same for + CMB or + CMB LE). Probes prior Planck ACTPol SPT SPTPol dN/dz+P(k) weak 1.97 2.43 1.18 1.06 strong 2.34 3.36 1.67 1.66 dN/dz+P(k) + CMB weak 2.76 2.16 2.12 2.53 strong 3.25 2.53 2.33 2.88 dN/dz+P(k) + CMB LE weak 2.25 1.67 1.64 1.95 strong 2.63 1.92 1.77 2.19 power spectrum, combination of the two, with and without LE of the Planck CMB. The contour for number count shows a clear diagonal alignment, and the degeneracy direction can beunderstood as follows: an increase in neutrino mass suppresses the growth of struc- ture formation, this can be compensated by a larger rate of accelerated expansion (i.e. more negative w). The constraints from power spectrum is less degenerate but show dif- ferent degeneracy directions. As a result, combining information from both probes greatly improve the constraints. To see the effect of w 0 on M ν constraint, we derive σ Mν again by marginalizing over w 0 and w a . We find that, as expected, only the constraints from number count are affected (improve by a factor of >2), while those from power spectrum arebarelyaffected. Furthermore, onlymodestimprovements areobtainedwhencombining number count and power spectrum in this case. The degeneracy between curvature Ω K and neutrino mass M ν is also known to be significant and impact on both M ν and the number of neutrino species N eff , which could affect the constraints coming from CMB [e.g 134, 174]. However we note that the cluster probes used in this work are related to the growth of structures which are not sensitive to Ω K . Thus we expect that including Ω K in the Fisher matrix analysis would not impact our results. It is out of the scope of this work to study in depth the impacts of including 109 Figure 5.1: Joint constraints on the M ν and w 0 . All curves denote 68% confidence level, and are for number counts only (blue), power spectrum only (cyan), and combination of the two (green), Planck CMB (yellow), and Planck CMB LE (dotted yellow). an extended set of parameters (e.g. Ω K , N eff ). Nevertheless it would be potentially interesting to study their effects for growth of structures and we leave it for future works. 5.5.2 Survey sensitivities to neutrino mass constraints In order to better understand what aspects of SZ surveys would improve the constraints on neutrino mass, we repeat the Fisher matrix calculation that includes different range of wavenumber k and cluster mass M. One of the major dependences is the maximum k values (k max ) as it determines the smallestscalesthatcanbeprobedbyasurvey. Theeffectofmassiveneutrinoisparticularly prominent at small scales (large k values), in which free streaming of neutrinos prevent structure formation. In this work, we use the same k max = 0.1 h/Mpc in the power spectrumFisher matrix for all surveysconsidered. We donot attempt to increase thek max beyond this value to avoid the non-linear effects at smaller scales. Furthermore, this scale is the limit that can be reached by SZ cluster surveys. Instead we study the dependence 110 Figure 5.2: Fully marginalized constraints on M ν from the power spectrum of clusters + Planck CMB prior, as a function of maximum wavenumber k max (left) and minimum cluster mass M min (right). whensmallscalemodesarelost, asshowninFig.5.2(left). Theeffect oflosingsmallscales information begins at k ≈ 0.06 h/Mpc, which corresponds to the free streaming scale k fs at z = 2, with M ν = 0.3 eV. The prospect of using smaller scale modes in constraining neutrino mass would be coming from galaxy surveys which can probe down to k ≈ 0.5 Mpc/h. A number of studies forecasted the neutrino mass constraints from future galaxy surveys (e.g. [156] (BOSS) and [23] (EUCLID-like)) and found that the improvement in σ Mν beyond k =0.1 Mpc/h is marginal (see Fig. 6 in [156]). Theotherrelevantdependenceisthelimitingclustermassthatdifferentiatesthevarious SZsurveys. WeshowinFig.5.2(right)thefullymarginalizedσ Mν fromthepowerspectrum + Planck CMB as a function of minimum cluster mass used in the calculation. It is clear that a deeper survey that can probe down to lower mass would improve the constraints. It is also interesting to note that the SPTPol survey, despite of its small sky coverage, can perform better than the ACTPol and SPT survey because it can detect clusters down to ≈ 2× 10 14 M ⊙ . Therefore a deep survey can compensate for the sky coverage when constraining neutrino mass. 111 5.6 Conclusion In this work, we explored the possibility of using future and upcoming SZ cluster surveys to constrain neutrino masses. We employ the Fisher matrix analysis to forecast the sensi- tivities of various SZ surveys in constraining the total neutrino mass M ν in the context of flat ΛCDM cosmology. We do so by making use of the cluster number counts and power spectrum, and taking into account the self-calibration of mass-observable scaling relations. In general, we findthat theM ν constraints from cluster number count and power spec- trum is weak if they are considered separately, duemainly to strong parameter degeneracy between M ν and w 0 , especially in the case of cluster number count. However, such degen- eracycanbebrokenifthetwoprobesarecombined, whichhelpstoimprovetheconstraints considerably. For example, a sample of ≈ 1000 clusters obtained from the Planck cluster survey gives σ Mν = 0.94 eV (0.43 eV with weak prior). The constraints can be further improved whencombined with CMBpriors. Thebestconstraint is obtained for the Planck and ACTPol survey, with σ Mν = 0.23 eV (CMB) and σ Mν = 0.17 eV (CMB LE). This is ≈80(25)% improvement with respect to the CMB (CMB LE) only constraint. The use of CMB lensing extraction can better help the cluster only constraints because it determines the neutrino’s free streaming effect on the matter power spectrum and break some of the parameter degeneracies. While we find that M ν constraint is mainly driven by CMB and the addition of cluster probes, i.e. number count + power spectrum, to CMB only helps marginally, the use of clusters is still beneficial if we have good control of cluster systematics. For example, when applying a weak prior on the mass-observable relation, the 1σ error on M ν as obtained from cluster count + power spectrum goes down to 0.48 eV and is competitive with CMB only constraint. If we further combine with CMB priors, σ Mν reduces to 0.07 eV, which correspondstoa≈4σ detection forM ν ≥0.3eV. Theprospectofachieving theweak prior 112 conditionsispromising,e.g. clustersdetectedinweaklensingmeasurementsorasubsample of objects having extensive multi-wavelength follow-up. Therefore, cluster measurements are useful, as an independent probe of the M ν with respect to the CMB, in tightening the current bound on M ν . We find that a deeper cluster survey that detects smaller mass clusters, e.g. down to 2×10 14 M ⊙ like SPTPol, improves neutrino mass constraints. This is because of the effect offreestreamingofmassiveneutrinosthatpreventsstructureformationtohappenatsmall scales. Likewise, the availability of the small scale modes, i.e. the maximum k values that can be probed by a cluster survey, also helps the constraints. We show that the modes at k≥0.06 h/Mpc are important as they help decreasing σ Mν significantly. The work presented in this chapter is based on the publication: Mak, D. S.Y. & Pierpaoli, E. 2013, arXiv:1303.2081 113 Chapter 6 Conclusions The use of galaxy clusters detected by the Sunyaev-Zel’dovich effect to do cosmology is a new and exciting area of research in recent years. The unique spectral features and that the signal is independentof distance make the SZ effect a very promising tool to search for massive clusters at high redshifts. Large catalogs of clusters selected from observations of theSZeffectarecurrentlyunderconstructionfrombothgroundandspacebasedtelescopes. In this work we have explored the prospects of using these upcoming SZ cluster surveys to place constraints on various aspects of cosmology. In Chapter. 2 we use the kinetic SZ effect to measure the peculiar motions of galaxy clusters, in particular the large scale bulk flow. Recent measurement of bulk flow using the kSZ signal in WMAP data has contradicted the prediction from the ΛCDM model. However, the existence of this anomalous bulk flow is still controversial since the measure- ment is limited by the WMAP sensitivity. In this work, we test this finding against the newer data fromthe Planck mission whichhas widerfrequency coverage andbetter spatial resolution than WMAP, and help narrow down uncertainties in the measurements. We predict that the sensitivity of bulk flow measurement when using the Planck CMB maps improves by a factor of > 5, depending on the cluster sample used and filtering scheme. As an initial study, this demonstrates the superior potentials Planck has in determining the bulk flow. Most recently, we are able to apply this method on the Planck data [145] and we find that the bulk flow is consistent with the ΛCDM expectations and is in clear contradiction to thepreviousclaim of thelarge scale bulkflow. Thisresultalso constitutes 114 an unprecedented piece of evidence for the local homogeneity of the Universe in the large scale. InChapter.3,Chapter.4,andChapter.5,weinvestigate quantitatively thepotentialof futureSZclustersurveysinconstrainingthephysicsoftheforcelaw,theearlyuniverse,and theenergycontentrespectively. Specificallyweemploytheimprintsofvariouscosmological models have on the cluster abundance and clustering properties to probe the error of the model parameters. A summary of the best parameter constraints from our Fisher matrix analysis is tabulated in Tab. 6.1. In general, we find that the SZ cluster surveys are able to give comparable constraints on general cosmological parameters of the ΛCDM model to current measurements from non-cluster probes. Similarly, this is true for constraints on the extension of the ΛCDM model that modifies the gaussianity of density fluctuation distribution or includes non-zero neutrino masses. And in the case of modified gravity f(R) model, SZ clusters can improve on the current measurements from X-ray by at least an order of magnitude. The strength of SZ surveys comes from their abilities in detecting massive and high redshift clusters. These make them the very sensitive probes of these extended ΛCDM models as those effects on the structureformation are more prominentat extreme distances and masses. We find that the constraining power is much stronger for power spectrumof clusters , except for the constraints of neutrinomasses, than the cluster abundance. It alsooften provides complementary information with cluster abundancethat helps breaking parameter degeneracies when both probes are combined. Thus this work demonstrates that cluster power spectrum is a new promising tool in the upcoming SZ surveys to do cosmology. ThefirstPlanck datareleasejustcameoutduringthepreparationofthisthesis. Aquick comparison show that the properties of the predicted Planck cluster survey in this work are consistent with the first release of Planck catalog [144]. Furthermore, the constraints derived from this work also agree to the cluster science from the Planck collaboration [e.g. 115 Table 6.1: Summary of the best parameter constraints (dN/dz + P(k) + CMB) Parameter 1σ constraints Survey General ΛCDM cosmology Ω M h 2 7.41×10 −4 Planck Ω Λ 8.88×10 −2 SPTPol σ 8 3.80×10 −3 Planck Ω b h 2 1.25×10 −4 Planck n s 2.20×10 −3 Planck w 0 0.34 SPTPol w a 0.97 SPTPol Modified gravity f(R) model f R0 4.8×10 −6 Planck primordial non-Gaussianity f NL 7 Planck Massive neutrinos M ν (w/CMB LE) 0.17 Planck 143, 140]. Aspointedoutin[143], thecosmological parameterdetermination usingclusters is currently limited by the knowledge of mass observable relations. Further investigations towardsthisdirectionarerequiredinexploitingthefullpowerofgalaxyclusters,andwould alleviate the tension as seen between the CMB and cluster constraints. 116 Appendix A Derivation of Filter Kernel A.1 Data Model In real space, the observation field can be described as s ν (θ)=y(0)(f ν −V)B ν (θ)+n ν (θ) (A.1) where y(0) is the y parameter at the cluster center (defined in Eq. (B.7)), f ν = x(e x + 1)/(e x − 1)− 4, and x = (hν)/(k B T CMB ) is the frequency response of the tSZ signal, V =v r m e c/k B T e gives the signal in kSZ effect, B ν (θ) is the convolved cluster profile given by: B ν (θ) = Z dΩ ′ p(θ ′ )b ν (θ−θ ′ ) (A.2) = ∞ X l=0 B l0,ν Y 0 l (cos(θ)) (A.3) where B l0,ν = p (4π)/(2l+1)b l0,ν p l0,ν , p(θ) and b ν (θ) are the cluster spatial profile and beam function respectively, and n ν is the noise consisting of instrumental noise and CMB . The noise map and the cross power spectrum C of each of the two components satisfy n lm,ν 1 n l ′ m ′ ,ν 2 =C l,ν 1 ,ν 2 δ ll ′δ mm ′ (A.4) 117 A.2 Filtered Field Let Φ ν and u ν be the filter and filtered field respectively at frequency ν, then u ν (θ)= Z dθ 2 s ν (θ)Φ ν (θ)= X lm u lm,ν Y m l (β) (A.5) with u lm,ν = q 4π 2l+1 s lm,ν Φ l0,ν The total signal in all frequency channels is u(θ) = P ν u ν (θ). A useful quantity to consider is the variance of the filtered map: σ 2 u = (u(θ)−hu(θ)i) 2 (A.6) We introduce vector notation for the set of frequency dependent quantities, e.g. ~ B l = {B l,ν 1 ,B l,ν 2 ,·B l,ν N ,} Since hu(θ)i = D P l h y c ( ~ F l −V ~ B l )+ ~ n l i ~ Φ l E , where we have used the notation ~ F l = {f ν B l0,ν } ν , then we have σ 2 u = *" X l (y c ~ F l −y c V ~ B l ) ~ Φ l + ~ n l ~ Φ l −(y c ~ F l −y c V ~ B l ) ~ Φ l # 2 + = *" X l ~ n l ~ Φ l # 2 + Therefore σ 2 u = P l ~ Φ T l C l ~ Φ l . A.3 Unbiased Matched Filter for kSZ signal In order to derive the optimal filter for the kSZ signal, we want to minimizeσ 2 u subject the the following constraints: 118 1. The filter is an unbiased estimator of KSZ signal at the source location, such that X l ~ B l ~ Φ T l =1 (A.7) 2. The filter should remove the TSZ signal at the source location, i.e. X l ~ F l ~ Φ T l =0 (A.8) The functional variation of σ 2 u with respect to ~ Φ l subject to constraints Eq. (A.7) and Eq. (A.8) can be obtained using Lagrangian multipliers. The Lagrange function is defined as: L= X l ~ Φ T l C l ~ Φ l +λ 1 (1− X l ~ B l ~ Φ T l )+λ 2 X l ~ F l ~ Φ T l (A.9) Minimizing the Lagrange function with respect to the filter function ~ Φ l , Δ ~ Φ T l L= X l h C l ~ Φ l −λ 1 ~ B l +λ 2 ~ F l i =0 ⇒C l ~ Φ l =λ 1 ~ B l −λ 2 ~ F l ⇒ ~ Φ l =C −1 l (λ 1 ~ B l −λ 2 ~ F l ) (A.10) Thejob now is to findtheconstants λ 1 andλ 2 . Usingconstraint Eq. (A.7) and Eq. (A.10), X l ~ B l ~ Φ T l ~ Φ l = X l h λ 1 ~ Φ l ~ B T l C −1 l ~ B l −λ 2 ~ Φ l ~ B T l C −1 l ~ F l i ⇒1=λ 1 X l ~ B T l C −1 l ~ B l −λ 2 X l ~ B T l C −1 l ~ F l ⇒1=λ 1 γ−λ 2 β (A.11) 119 Similarly, using constraint Eq. (A.8) and Eq. (A.10), X l ~ F l ~ Φ T l ~ Φ l = X l h λ 1 ~ Φ l ~ F T l C −1 l ~ B l −λ 2 ~ Φ l ~ F T l C −1 l ~ F l i ⇒1=λ 1 X l ~ F T l C −1 l ~ B l −λ 2 X l ~ F T l C −1 l ~ F l ⇒1=λ 1 β−λ 2 α (A.12) where α = P l ~ F T l C −1 l ~ F l , β = P l ~ F T l C −1 l ~ B l , and γ = P l ~ B T l C −1 l ~ B l Solving Eq. (A.11) and Eq. (A.12), we obtain λ 1 = β αγ−β 2 , and λ 2 = α αγ−β 2 . Thus we get the filter kernal: ~ Φ UF l = C −1 l αγ−β 2 (α ~ B l −β ~ F l ) (A.13) It is easy to verify that this filter kernel satisfies the two constraints. The filter can be interpretated in this way: 1. Δ≡αγ−β 2 is the normalization such that P ~ B T l ~ Φ l =1 2. The term α Δ ~ B l comes from the kSZ constraint, ensuring that the filter gives the kSZ signal at the source location. 3. The term− β Δ ~ F l comes from the TSZ constraint. Its purpose is to suppress the tSZ signal such that tSZ signal vanishes at the source location. 4. Thefilteredfieldsateachchannelareweightedbytheinverseofthecovariancematrix C −1 l and then combined to form the final filtered signal. 120 A.4 Matched Filter The derivation of the matched filter is similar to that for the unbiased matched filter, except that constraint Eq. (A.7) is used. Therefore we have the Lagrange function: L= X l ~ Φ T l C l ~ Φ l +λ(1− X l ~ B l ~ Φ T l ) ⇒Δ ~ Φ T l L= X l h C l ~ Φ l −λ ~ B l i =0 ⇒ ~ Φ l =C −1 l (λ ~ B l ) (A.14) Using constraint Eq. (A.7) and solving for λ, we find λ = 1/γ. Thus we get the filter kernel: ~ Φ MF l = C −1 l γ ~ B l (A.15) 121 Appendix B Modeling SZ signal of individual clusters The SZ signal of galaxy clusters is characterized by the Compton y parameter, which in turn depends on the gas properties of individual clusters. The next step is to model the electron density n e (r), temperature T e (r) and the integrated Compton Y parameter. Assuming that clusters are self-similar, virialized and isothermal, we obtain the mass- temperature relation from the virial theorem [78]: k B T e =β −1 T (1+z) Ω m Δ V (z) Ω m (z) 1/3 M cl 10 15 M ⊙ h −1 2/3 keV (B.1) where β T = 0.75 is the normalization constant under the assumption of hydrostatic equi- librium and isothermality, Δ V is the mean overdensity of a virialised spherewhich we have calculated using the fitting formula from [136], M cl is the mass of the cluster within the virial radius r vir , M cl =4π/3 Δ V ρ crit r 3 vir . Solving for r vir gives, r vir = 9.5103 1+z Ω m Δ V (z) Ω m (z) −1/3 M cl 10 15 M ⊙ h −1 1/3 h −1 Mpc (B.2) The assumption that clusters are isothermal is in good agreement with XMM-Newton observations of outer cluster regions which find that the cluster temperature profiles are isothermal within±10% up to≈r vir /2 [7]. 122 Theelectrondensitywithinthevirialradiusdescribedbyaβ model([190]andreference herein), n e (r)=n e0 1+ r 2 r 2 c −3/2β (B.3) wherer c isthecore radius.[56] obtain arelationship between thecoreradiusandthevirial radius: r c (z) =0.14(1+z) 1/5 r vir (B.4) We choose β = 2/3 to match [190] which describes the X-ray surface brightness profile of observed clusters. The central electron density n e0 is normalized by R r vir 0 n e (r)dV = N e , in which N e is the total number of electrons within the cluster given by: N e = 1+f H 2m p f gas M cl (B.5) where f H is the hydrogen fraction, f gas = Ω b /Ω m is the baryonic gas mass fraction of the total cluster mass. Here we take f gas = 0.168 from WMAP 5 year results and f H = 0.76. We find, n e0 = N e 4πr 3 c [ 1 (0.14(1+z) 1/5 ) +tan −1 (0.14(1+z) 1/5 − π 2 )] (B.6) The Comptonization parameter is proportional to the product of the electron temper- ature and density, and is given by y = R dl(k B T e )/(m e c 2 )n e σ T . Performing this integral along the line-of-sight of the cluster, we obtain: y(θ)=2 k B T e m e c 2 σ T r c n e0 q 1+ θ 2 θ 2 c tan −1 v u u t 1 (0.14(1+z) 2 − θ 2 θ 2 c 1+ θ 2 θ 2 c (B.7) 123 where θ c = r c /D A . The SZ signal is then calculated using this compton y parameter for each cluster within the virial radius. It is easy to show from Eq. (B.7) that y drops to zero whenθ≥θ vir . The analytic total Comptonization parameter Y is obtained simply by integration Eq. (B.7) over the cluster size: Y = Z dΩy(θ)=D −2 A (z) k B σ T m e c 2 Z dVn e T e = k B T e σ T N e m e c 2 D 2 A (B.8) where D A is the angular diameter distance to the cluster. We then scale the pixels within the disc of angular size θ vir by a factor N = Y P i y i dΩ pix , wherey i is the y value in the center of pixel i and dΩ pix is the pixel solid angle. There is an expected cluster mass–optical depth scatter in nature due to the fact that unknownclusterphysicscanaffecttheexactformandnormalizationofthescalingrelations, e.g. Y−M relation. [167]have estimated thetypical intrinsicscatter intheY−M relation with simulations of the ACT clusters and found that this is about 15% in Y. While ACT clusters represented in those simulations have masses that are lower than those expected in our samples (in particularly Planck), we do not expect bigger objects which are more virializedtohavealargerscatter. Wethereforeintroducea15%scatterintheY parameter inoursimulationsforallclustersamples. Inprinciple,thereisalsoameasurementerrorfor optical depth due to inaccuracy of SZ photometry. However, we ignore this effect because such uncertainty is not yet well characterized for Planck. 124 Appendix C Cosmological Constraints from SZ cluster survey using Fisher matrix (FM) Analysis The Fisher matrix analysis is widely used in cosmology as a forecasting tools to estimate cosmological parametersfromobservationsandastronomicalsimulations. Herewedescribe theformalismanditsapplicationtoestimatecosmological parametersfromclusternumber counts and power spectrum. C.1 Formalism The Fisher information Matrix (FM hereafter) is defined as F αβ ≡− ∂ 2 lnL ∂p α p β (C.1) where L is the likelihood of a data set, e.g. a cluster sample, written as a function of the parameters p α describing the model. The parameters p α comprise the cosmological parametersaswellas“nuisance”parametersrelatedtothedataset(e.g., masscalibration). The inverse of the Fisher matrix is the covariance matrix of the parameters. The Cramer-Rao theorem states that any unbiased estimator for the parameters would have a 125 covariancematrixontheparametersnobetterthanF −1 . TheFishermatrixthereforeoffers the best-case scenario to constrain cosmological parameters given a set of observations. C.2 Cosmological parameter estimation In Chapter. 3, Chapter. 4, and Chapter. 5, we consider two observables that can be extracted from cluster surveys: cluster number counts and power spectrum. Here we will describe their analytical forms which were employed in the analysis described in those chapters. We also specify the fiducial model that is common in the analysis. C.2.1 FM of Cluster Number counts The FM for the cluster count N l,m within the l-th redshift bin and m-th mass bin is F αβ = X l,m ∂N l,m p α ∂N l,m p β 1 N l,m (C.2) where the sum over l and m runs over intervals in the redshift and mass respectively. Throughout,wedividetheredshiftrangez =0−1intobinsl ofwidthΔz =0.02. Further, we bin clusters in logarithmic mass binsm of width ΔlnM =0.3 from the minimum mass M lim (z) for each survey (Sec. 1.3) up to a large cut-off mass of M max =10 16 M ⊙ . Since the mass limit varies with redshift, the number of mass bins thus also varies somewhat across the redshift range. Instead of using Eq. (1.10) which calculates the total number count expected in asurvey, we writethe numbercount withina given redshiftand mass intervals due to the binning as: N l,m =ΔΩΔz d 2 V dzdΩ Z M l,m+1 M l,m dM ob Z ∞ 0 dlnM dn dM (M,z)p(M ob |M) (C.3) 126 whereΔΩisthesolidanglecoveredbytheclustersurvey, lnM l,m =lnM lim (z l )+mΔlnM, and dn/dM(M,z) is the mass function given in Eq. (1.7). Following [104], we take into account the intrinsic scatter in the relation between true and observed mass, as inferred fromagivenmassproxy,bythefactorp(M ob |M)whichistheprobabilityforagivencluster mass with M of having an observed mass M ob . Under the assumption of a log–normal distribution for the intrinsic scatter, with variance σ 2 lnM , the probability is p(M ob |M)= exp[−x 2 (M ob )] q 2πσ 2 lnM (C.4) where x(M ob )= lnM ob −B M −lnM q 2σ 2 lnM . (C.5) With these notations, we parameterize the M ob −M relation, in addition to the intrin- sic scatter, by a systematic fractional mass bias B M . With this prescription, the final expression for the number count FM is: N l,m = ΔΩΔz 2 d 2 V dzdΩ × Z ∞ 0 dlnM n(M,z)(erfc[x m ]−erfc[x m+1 ]) (C.6) where erfc(x) is the complementary error function. C.2.2 FM of Cluster Power spectrum We derive the Fisher matrix element for the cross- and auto-power spectra of clusters binned in mass. Let P mn (k) denote the cross-power spectrum between mass bins m and n, calculated for the given redshift and wavenumber through P mn h (k i ,z l )=b m eff (z l )b n eff (z l )P L (k i ,z l ). (C.7) 127 where, b m eff is the mass function weighted effective bias, b m eff (z)= R ∞ 0 dM n(M,z)b L (M,z)(erfc[x m ]−erfc[x m+1 ]) R ∞ 0 dM n(M,z)(erfc[x m ]−erfc[x m+1 ]) (C.8) The variance of the cross-power spectrum measured in a narrow k range is given by (the explicit redshift-dependence is suppressed for clarity): σ 2 (P mn (k)) = 1 N mod " P mm (k)+ 1 n m P nn (k)+ 1 n j + P mn (k)+ δ mn n m 2 # (C.9) Here, n m denotes the comoving number density of clusters in mass bin i, and the number of modes is given by N mod = 1 2 V k 2 Δk 2π 2 , where the factor of 1/2 in front accounts for the fact that the density field is real, reducing the number of independent modes by one half. This factor is sometimes neglected in the literature. The volume is given by V(z)=Ω s χ 2 (z) c H(z) Δz. (C.10) Using this, we can derive the general power spectrum Fisher matrix as F αβ = 1 (2π) 2 X m,n X l,i ∂lnP mn h (k i ,z l ) ∂p α ∂lnP mn h (k i ,z l ) ∂p β ×V mn,eff l,i k 2 i Δk wherethesumoverm,nrunsover massbins,whilethesuminl andirunsover intervalsin thewholeredshiftrangeandwavenumber0.01hMpc −1 ≤k≤0.1hMpc −1 withΔlog 10 k = 0.017 respectively. The effective volume for mass bins m,n, wave number k i , and redshift z l is given by 128 Figure C.1: The dependence on redshift (left) and wavenumber (right) of the effective vol- ume(Eq.(C.11))forasinglemassbinandeachsurvey: Planck (black), SPT(green), SPT- Pol(magenta), andACTPol(blue). Theeffectivevolumeisaweakfunctionofwavenumber k but strongly depends on the redshift. V mn,eff (k i ,z l ) V 0 (z l ) =P mn (k i ,z l )] 2 n m (z l )n n (z l ) × h (n m P mm +1)(n n P nn +1)+n m n n (P nm +δ nm n −1 m ) 2 i −1 (C.11) where V 0 (z) is the comoving volume of the redshift slice [z l −0.01,z l +0.01] covered by the given survey. The effective volume gives the weight carried by each bin in the (z,k) spacetothepowerspectrumFishermatrix,andhencequantifiestheamountofinformation contained in a given redshift- and k-bin. Fig. C.1 shows the redshift and scale dependence of the effective volume for the four cluster surveys. We find that V eff . 0.3V 0 for all redshifts and surveys considered, even when not binning in mass, hence the cluster power spectrum is shot-noise dominated for all surveys. As the right panel of Fig. C.1 illustrates, Planck is most limited by shot noise, while SPTPol is least limited, as expected from their respective mass limits and coverage. 129 C.2.3 Fiducial Model We assume a spatially flat (Ω k = 0) ΛCDM cosmology that comprises of a total of seven cosmological parameters. The seven parameters and their fiducial values (in parenthesis, taken from the best-fit flat ΛCDM model from WMAP 7yr data, BAO and H 0 mea- surements [92]) are: baryon density parameter Ω b h 2 (0.0245); matter density parameter ω m ≡Ω m h 2 (0.143); dark energy density Ω Λ =1−Ω m (0.73); power spectrum normaliza- tion σ 8 (0.809); indexof power spectrumn s (0.963); effective dark energy equation of state through w(z) = w 0 +(1−a)w a , with fiducial values w 0 = −1 and w a = 0. The Hubble parameter is then a derived parameter given by h= p ω m /(1−Ω Λ )=0.73 in the fiducial case. C.3 CMB Prior This Fisher matrix approach can best approximate the likelihood when the fiducial model is close to the true, underlying model and the likelihood is close to gaussian. Typically, the gaussian approximation is more accurate, and the use of the Fisher matrix is better justified, when the likelihood is peaked and the parameter in hand has little degeneracies with other parameters. In order to achieve this goal, the use of external priors can be beneficial. In addition to observables measured from cluster surveys, we consider external infor- mation from Cosmic Microwave Background. We present results with the Fisher matrix for the Planck CMB temperaturepower spectrumC l addedto the constraints from cluster counts and power spectrum. We calculate the full CMB Fisher matrix with CAMB [102] and method described in [148]. For the Planck experiment, we use the three frequency bands 100, 143 and 217 GHz, and the C l are calculated up to l max = 2500. Our fidu- cial parameter set for the CMB experiment is, as described in the DETF report [2], 130 θ = (n s ,Ω b h 2 ,Ω Λ ,Ω m h 2 ,w 0 ,A s ,τ), where A s is the primordial amplitude of scalar per- turbations and τ is the optical depth due to reionization. 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Abstract (if available)

Abstract In this work, we present various studies to forecast the power of the galaxy clusters detected by the Sunyaev-Zel'dovich (SZ) effect in constraining cosmological models. The SZ effect is regarded as one of the new and promising technique to identify and study cluster physics. With the latest data being released in recent years from the SZ telescopes, it is essential to explore their potentials in providing cosmological information and investigate their relative strengths with respect to galaxy cluster data from X-ray and optical, as well as other cosmological probes such as Cosmic Microwave Background (CMB). ❧ One of the topics regard resolving the debate on the existence of an anomalous large scale bulk flow as measured from the kinetic SZ signal of galaxy clusters in the WMAP CMB data. We predict that if such measurement is done with the latest CMB data from the Planck satellite, the sensitivity will be improved by a factor of > 5 and thus be able to provide an independent view of its existence. As it turns out, the Planck data, when using the technique developed in this work, find that the observed bulk flow amplitude is consistent with those expected from the ΛCDM, which is in clear contradiction to the previous claim of a significant bulk flow detection in the WMAP data. ❧ We also forecast on the capability of the ongoing and future cluster surveys identified through thermal SZ (tSZ) in constraining three extended models to the ΛCDM model: modified gravity f(R) model, primordial non-Gaussianity of density perturbation, and the presence of massive neutrinos. We do so by employing their effects on the cluster number count and power spectrum and using Fisher Matrix analysis to estimate the errors on the model parameters. We find that SZ cluster surveys can provide vital complementary information to those expected from non-cluster probes. Our results therefore give the confidence for pursuing these extended cosmological models with SZ clusters.

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Creator Mak, Suet-Ying (author)

Core Title Cosmological study with galaxy clusters detected by the Sunyaev-Zel'dovich effect

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 08/02/2013

Defense Date 05/10/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag cosmology,galaxy clusters,OAI-PMH Harvest,Sunyaev-Zel'dovich effect

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Pierpaoli, Elena (committee chair), Bars, Itzhak (committee member), Bonahon, Francis (committee member), Columbo, Loris (committee member), Daeppen, Werner (committee member), Däppen, Werner (committee member), Johnson, Clifford V. (committee member)

Creator Email daisymak@yahoo.com,suetyinm@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-312976

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Legacy Identifier etd-MakSuetYin-1937.pdf

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Document Type Dissertation

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