Angular power spectra of large scale structures in the context of wide photometric surveys

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1 Angular power spectra of large scale structures in the context of wide photometric surveys Jérémy Neveu IN2P3/LAL in collaboration with Jean-Eric Campagne, Stéphane Plaszczynski March 22nd, rd Rencontres de Moriond

2 Cosmological context From the statistical distribution of galaxies [in positions, shapes, magnitudes], extract information about the cosmological parameters More precision => wider and deeper photometric and spectroscopic galaxy surveys BOSS / eboss DES Y1 / KIDs-45 DESI LSST / Euclid Type of survey Spectroscopic Photometric Spectroscopic Photometric Area [1 3 deg 2 ] 1 1 / Depth [z] 3.5 / / / 2 & 6 # Galaxies 1 6 / 1 6 3x1 8 / 15x1 6 3x / 1 9 Colors 36-1 nm 5 [grizy] / 4 [ugri] nm 6 [ugrizy] / 4 [{riz}yjh] + 1-2um

3 Photometric surveys Photometric surveys: lots of galaxies but with photo-z Repartition of the galaxies in redshift bins Cosmological parameter estimation with angular matter power spectrum C l (z) or angular correlation functions ξ(θ,z) Observational space (θ,z) or spherical harmonics space Cross-correlations between probes l BAO C(θ; z) ßà C l (z) θ BAO Galaxy z α δ

4 The power of cross-correlations Within a single probe: Recover some information from the structure evolution along the line of sight (despite photo-z) Control of survey and catalogue building systematics [McLeod et al., 216, Rhodes et al., 213] With multiple probes: Cosmological constraints more powerful than the product of the likelihoods [Krause & Eifler., 216] [Nicola et al., 216] Fit nuisance parameters / systematics (galaxy bias, shear bias, intrinsic alignment ) n Multi-probe fits n 3x2pt analysis from DES [DES Y1, ; Baxter et al., 217] Refs and plots. Depth variations controlled by crosscorrelations? [Doux et al., 217]

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b(z)j`(kr(z)) + f(z)j ` (kr(z)) 3(2 5s) m H 2 + `(` + 1) 2 (ck) 2 + subdominant terms Matter density with bias Bessel functions order l! Z r(z) RSD term with growth rate f(z) s r dr r(z) r (1 + z(r )) r(z) [adapted from Bonvin, 214] P (k, z(r )) P (k, z) Magnification lensing term with luminosity function slope s j`(kr ) #

6 ! <latexit sha1_base64="a33pdmadmkoxvkptinorwipey=">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</latexit> <latexit sha1_base64="a33pdmadmkoxvkptinorwipey=">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</latexit> <latexit sha1_base64="a33pdmadmkoxvkptinorwipey=">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</latexit> <latexit sha1_base64="a33pdmadmkoxvkptinorwipey=">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</latexit> Angpow C l (z 1,z 2 ) equations implemented in CLASSgal but slow Angpow: code co-developed with J.E. Campagne and S. Plaszczynski [Dio et al., 213] [Campagne et al., 217] Optimised integral of Bessel functions : Quadrature in redshift space and Fourier space: C`(z 1,z 2 ) Split the k integral in subspaces of N k ~1 Bessel roots k p l and truncate in k max! Computation of I l using Chebyshev/Clenshaw-Curtis quadrature (3C-algorithm) n N cc sampling points defined as x k = cos(k /N cc ) n T n (cos ) = cos n n N X z1 N X z2 i= j= w i w j W 1 (z i,z 1 )W 2 (z j,z 2 ) I`(k`p,k`p+1; r i,r j )= Z k` p+1 NX k 1 p= Use properties of Chebyshev polynomials Discrete Cosine Transform of type I (DCT-I) => FFT and matrix multiplications k`p dk f`(k, z i )f`(k, z j ) I`(k`p,k`p+1; z i,z j )

7 Testing Angpow x 1-6 C (z 1,z 2 ) (Angpow/CLASSgal): Gaussian selection x 1-7 σ z =.1 Orange points for negative correlations Z 1 =1. Z 2 =1.1 Z 1 =1. Z 2 =1.3 x 1-7 x 1-7 Z 1 =1. Z 2 =1.6 Z 1 =1. Z 2 =1.1 Angpow CLASSgal

8 Testing Angpow Computation times in second Several tests with OpenMP : Test 1 : auto-correlation with Dirac windows Test 2 : cross-correlation with two Dirac windows Test 3 : auto-correlation with Gaussian window Test 4 : cross-correlation with two Gaussian windows Without RSD or magnification effects With CLASSgal, same computations can last several minutes Computation times <~1s : compatible with MCMC exploration of cosmological parameters

9 Testing Limber approximation Limber approximation : j`(kr(z)) r 2` +1 r(z) ` +1/2 k Angpow : no Limber approximation j l (x) 1..5 ` = x Commonly used approximation Using this approximation makes the computation easier and faster But breaks beats between j`(kr(z 1 )) j`(kr(z 2 )) Negative C l (z 1,z 2 ) values are impossible with this approximation

10 Testing Limber approximation Two Gaussian windows z 1 =1. and z 2 =1., 1.1, 1.6, 1.1 (σ z =.1) ) W(z) 3 2 W(z) z z Limber approximation not good at large scales for thin windows (σ z <.1) W(z) Limber approximation breaks cross-correlations W(z) z z

11 RSD and magnification effects on C l 1 8 W(z) z auto cross auto [plots by Nick Samaras] Two Gaussian shells overlapping: RSD effect: enhance auto-correlations, reduce cross-correlation Magnification effect: negligible

12 RSD and magnification effects on C l 1 8 W(z) z auto cross auto [plots by Nick Samaras] Two Gaussian shells well separated: RSD effect: enhance auto-correlations, negligible in cross-correlation Magnification effect: weak anti-correlation on large scales

13 From P(k,z) to Cl (z1,z2) Zero-th order galaxy counting RSD effect (main correction) Redshift bin 1 Redshift bin 2 Magnification effect (small correction) Falling structures distort the shells: auto-correlations increase at every scales Effect reduces with shell thickness Low-z structures imprint their statistical distribution on the high-z density field Effect increases with distances

14 C l (z 1,z 2 ) for DES Y1 galaxy clustering W(z) z 5 Gaussian bins: 5 autocorrelations 1 cross-correlations n almost null but can help control survey systematics and photo-z [Krause & Eifler., 216]

15 Summary About the code: Angpow is a fast and accurate code to compute C l (z 1,z 2 ) Angular power spectrum are hard to compute because of Bessel function integrals: the 3C-algorithm is adapted to this problem Angpow now includes RSD and magnification lensing effects Interfaces with other codes possible (CCL for instance) arxiv: , arxiv: About cosmology: Limber approximation can fail for thin redshift windows and cross-correlations RSD effect enhance auto-correlations, magnification lensing introduces power at large scales in cross-correlation between independent shells Null cross-correlations are information on cosmology and systematics Muti-probe angular power spectrum analysis are more than the product of the probes to constrain cosmology and systematics