Cross-Correlation of CFHTLenS Galaxy Catalogue and Planck CMB Lensing

Cross-Correlation of CFHTLenS Galaxy Catalogue and Planck CMB Lensing A 5-month internship under the direction of Simon Prunet Adrien Kuntz Ecole Normale Supérieure, Paris July 08, 2015

Outline 1 Introduction 2 Theoretical Cosmology Cosmological Perturbation Theory Gravitational Lensing and Galaxy Density Cross-Correlation The Halo Model 3 Data Maps Galaxies Convergence 4 Results Bayesian Analysis Consistency Checks 5 Conclusions A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 2 / 57

The Initial Motivation Introduction A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 3 / 57

Aims of my Internship Introduction Achieve a good comprehension of the cosmological perturbation theory and of the halo model (1 month1/2) Implement a python program to compute covariances from the halo model (1 month 1/2) Retrieve data from both the CFHTLenS and the Planck collaboration, format them in a sky map, and perform simulations (1 month) Use a bayesian analysis to fit galaxy bias from the correlation of the two maps (1 week) Write an insternship report and a presentation(2 weeks) A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 4 / 57

Outline 1 Introduction Theoretical Cosmology Cosmological Perturbation Theory 2 Theoretical Cosmology Cosmological Perturbation Theory Gravitational Lensing and Galaxy Density Cross-Correlation The Halo Model 3 Data Maps Galaxies Convergence 4 Results Bayesian Analysis Consistency Checks 5 Conclusions A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 5 / 57

Theoretical Cosmology Friedmann-Lemaître Model Cosmological Perturbation Theory ds 2 = c 2 dt 2 a 2 (t) dx 2 Physical coordinates : r = a (t) x Friedmann equations : ȧ 2 +kc 2 a 2 ä a = 4πG 3 = 8πGρ+Λc2 3 ( ) ρ + 3p + Λc2 c 2 3 Solution : p = wρc 2 ρ a 3(1+w) A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 6 / 57

Perturbation Theory Theoretical Cosmology Cosmological Perturbation Theory ρ (x, t) ρ (t) (1 + δ (x, t)) Linear solution : δ (x, t) = D + 1 (t) A (x) Non-linear solution : δ (k, t) = + n=1 an (t) δ n (k) A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 7 / 57

Theoretical Cosmology Linear and Non-Linear Scales Cosmological Perturbation Theory Credit : Baghram et al (2011) A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 8 / 57

Theoretical Cosmology Cosmological Perturbation Theory Correlation Function and Power Spectrum Primordial quantum fluctuations and inflation Gaussian initial conditions Gravitational instability ξ (r) = δ (x) δ (x + r) δ (k) δ (k ) = (2π) 3 δ D (k + k ) P (k) and higher order : B, T... A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 9 / 57

Outline 1 Introduction Theoretical Cosmology Gravitational Lensing and Galaxy Density 2 Theoretical Cosmology Cosmological Perturbation Theory Gravitational Lensing and Galaxy Density Cross-Correlation The Halo Model 3 Data Maps Galaxies Convergence 4 Results Bayesian Analysis Consistency Checks 5 Conclusions A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 10 / 57

Lens Equation Theoretical Cosmology Gravitational Lensing and Galaxy Density S α α given in General Relativity D LS L D OS θ D OL θ s O Lens equation : θ = f (θ s ) A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 11 / 57

Illustration Theoretical Cosmology Gravitational Lensing and Galaxy Density Figure: Distortion of an extended source by a point-like lens A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 12 / 57

Illustration Theoretical Cosmology Gravitational Lensing and Galaxy Density A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 13 / 57

Illustration Theoretical Cosmology Gravitational Lensing and Galaxy Density A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 14 / 57

Theoretical Cosmology Gravitational Lensing and Galaxy Density Effect on the CMB Deformation of the photon paths κ (θ) = A. Kuntz (ENS Ulm) χcmb 0 Correlation of CFHTLenS and Planck W κ (χ) δ (χθ, χ) dχ 07/08/2015 15 / 57

Galaxy Overdensity Theoretical Cosmology Gravitational Lensing and Galaxy Density δ gal (χθ, χ) = b (χ) δ (χθ, χ) g (θ) = χ CMB 0 W g (χ) δ (χθ, χ) dχ W g (χ) b (χ) A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 16 / 57

Lensing kernels Theoretical Cosmology Gravitational Lensing and Galaxy Density A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 17 / 57

Outline 1 Introduction Theoretical Cosmology Cross-Correlation 2 Theoretical Cosmology Cosmological Perturbation Theory Gravitational Lensing and Galaxy Density Cross-Correlation The Halo Model 3 Data Maps Galaxies Convergence 4 Results Bayesian Analysis Consistency Checks 5 Conclusions A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 18 / 57

Theoretical Cosmology Cross-Correlation Spherical Harmonics and Correlation κ (θ) lm κ lmyl m (θ) and g (θ) lm g lmyl m (θ) κlm g l m = δll δ mm C κg l C κg l = ) dχ W κ (χ)w g (χ) P (k = l χ 2 χ, z (χ) A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 19 / 57

Covariance Theoretical Cosmology Cross-Correlation Covariance : Σ κg ij = C κg κg l i C l j C κg l i C κg l j Σ κg ij = Gaussian Term + Non-Gaussian Term Non-Gaussian term involves T = δ (k 1 ) δ (k 2 ) δ (k 3 ) δ (k 4 ) A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 20 / 57

Correlation Matrix Theoretical Cosmology Cross-Correlation A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 21 / 57

Outline 1 Introduction Theoretical Cosmology The Halo Model 2 Theoretical Cosmology Cosmological Perturbation Theory Gravitational Lensing and Galaxy Density Cross-Correlation The Halo Model 3 Data Maps Galaxies Convergence 4 Results Bayesian Analysis Consistency Checks 5 Conclusions A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 22 / 57

Spherical Collapse Theoretical Cosmology The Halo Model Expansion δ sc = 1.69 Turnaround ρ halo = sc ρ background, sc 340 Gravitational collapse Virialization A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 23 / 57

Ingredients Theoretical Cosmology The Halo Model n (m, z) Press & Schechter (1974) ρ (r; m) NFW profile (1997) δ h (x,z; m) = k>0 b k (m, z) δ (x) k /k! Mo & White (1995) A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 24 / 57

Correlation Function Theoretical Cosmology The Halo Model 1-halo ρ tot (x) = i ρ (x x i, m i ) ξ (r) = ρ(x)ρ(x+r) ρ 2 1 2-halo ρ (x) ρ (x + r) = 1-halo term + 2-halo term A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 25 / 57

Power Spectrum Theoretical Cosmology The Halo Model Credit : Cooray & Sheth (2002) A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 26 / 57

Trispectrum Theoretical Cosmology The Halo Model T hhhh (k 1, k 1, k 2, k 2 ) = T 1h + T 2h + T 3h + T 4h A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 27 / 57

Outline 1 Introduction 2 Theoretical Cosmology Data Maps Galaxies Cosmological Perturbation Theory Gravitational Lensing and Galaxy Density Cross-Correlation The Halo Model 3 Data Maps Galaxies Convergence 4 Results Bayesian Analysis Consistency Checks 5 Conclusions A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 28 / 57

Data Maps The CFHTLenS Catalogue Galaxies Analysis of the CFHT Legacy Survey specialized in the production of a galaxy catalogue for lensing measurement CFHTLS : 171 MegaCam pointings 2003-2008, more than 2300 hours 4 patches, 154 deg 2 Galaxy catalogue and accurate photometric redshifts : σ 0.04 (1 + z) 0.2 < z < 1.3 and 18 < i AB < 24 A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 29 / 57

Redshift Distribution Data Maps Galaxies dn dz = α z 0 ( z z 0 ) λ exp ( ( z z 0 ) β ) A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 30 / 57

Healpix Data Maps Galaxies JPL / NASA Pixels of equal area Iso-latitude Interface in Python 1 pixel = 1.7 arcmin A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 31 / 57

Data Maps Galaxies Mask Healpix pixel Mask Weight : w i = S unmasked S tot Galaxy overdensity : δ i = N i /w i N N n = 15.1 gal / arcmin 2 A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 32 / 57

Mask Data Maps Galaxies A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 33 / 57

Galaxy Map Data Maps Galaxies A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 34 / 57

Galaxy Map Data Maps Galaxies A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 35 / 57

Noise Data Maps Galaxies Poisson process : shot noise C gg l C gg l + N gg l, N gg l = 1/ n ( n in galaxies / steradian!) Small effect A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 36 / 57

Outline 1 Introduction 2 Theoretical Cosmology Data Maps Convergence Cosmological Perturbation Theory Gravitational Lensing and Galaxy Density Cross-Correlation The Halo Model 3 Data Maps Galaxies Convergence 4 Results Bayesian Analysis Consistency Checks 5 Conclusions A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 37 / 57

Convergence Map Data Maps Convergence A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 38 / 57

Noise Data Maps Convergence Obtained from the Planck simulations Dominant over C κκ l! A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 39 / 57

Data Maps Convergence A Consequence of the High Level of Noise Σ κg ij = 1 [ (2π) 2 ( (C δ K κκ ij l 4πf sky Ω i i + Nl κκ ) ( ) ( ) ) 2 i C gg l j + N gg l i + C κg l i + ] κg T ij A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 40 / 57

Outline 1 Introduction 2 Theoretical Cosmology Results Bayesian Analysis Cosmological Perturbation Theory Gravitational Lensing and Galaxy Density Cross-Correlation The Halo Model 3 Data Maps Galaxies Convergence 4 Results Bayesian Analysis Consistency Checks 5 Conclusions A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 41 / 57

Results Bayesian Analysis Bayes Theorem C κg l C gg l = A b C κg l = b 2 C gg l P (A, b C) = P(A,b)P(C A,b) P(C) { P (C A, b) = N exp 1 ( ) ( ) } C C (A, b) Σ 1 C C (A, b) 2 A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 42 / 57

MCMC 2013 Results Bayesian Analysis A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 43 / 57

MCMC 2015 Results Bayesian Analysis A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 44 / 57

Best-Fit Values Results Bayesian Analysis 2013 2015 Patch A b χ 2 /ν A b χ 2 /ν All 1.05 +0.15 0.15 0.93 +0.02 0.02 47.2/36 0.86 +0.15 0.16 0.92 +0.02 0.02 37.4/36 A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 45 / 57

C κg l Results Bayesian Analysis A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 46 / 57

C gg l Results Bayesian Analysis A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 47 / 57

Outline 1 Introduction 2 Theoretical Cosmology Results Consistency Checks Cosmological Perturbation Theory Gravitational Lensing and Galaxy Density Cross-Correlation The Halo Model 3 Data Maps Galaxies Convergence 4 Results Bayesian Analysis Consistency Checks 5 Conclusions A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 48 / 57

κg, simu Cl Results Consistency Checks A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 49 / 57

gg, simu Cl Results Consistency Checks A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 50 / 57

Recovered A and b Results Consistency Checks A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 51 / 57

Outline Conclusions 1 Introduction 2 Theoretical Cosmology Cosmological Perturbation Theory Gravitational Lensing and Galaxy Density Cross-Correlation The Halo Model 3 Data Maps Galaxies Convergence 4 Results Bayesian Analysis Consistency Checks 5 Conclusions A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 52 / 57

Summary Conclusions Learned Cosmological Perturbation Theory and Halo Model Implemented a Python program to compute the trispectrum in the halo model Created a galaxy and convergence map Used a Bayesian analysis to fit a galaxy bias Performed consistency checks A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 53 / 57

Conclusions Conclusions b = 0.92 +0.02 0.02 : good tracers of matter A 2013 = 1.05 +0.15 0.15 and A2015 = 0.86 +0.15 0.15 : compatible but different Partly confirm Omori & Holder (2015) A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 54 / 57

Forthcoming Work Conclusions SDSS data : much better coverage of the sky If still a difference : investigate! A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 55 / 57

Questions? Comments? Conclusions A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 56 / 57

Conclusions Thank You CFHT! A. Kuntz (ENS Ulm) Correlation of CFHTLenS and Planck 07/08/2015 57 / 57