Anisotropic Clustering Measurements using Fourier Space Wedges

Transcription

1 Anisotropic Clustering Measurements using Fourier Space Wedges and the status of the BOSS DR12 analysis Jan Niklas Grieb Max-Planck-Institut für extraterrestrische Physik, Garching bei München Universitäts-Sternwarte München, Ludwig-Maximilians-Universität München July 20th, 2015 collaborators: A. Sánchez, F. Montesano, S. Salazar, R. Scoccimarro, M. Crocce, C. Dalla Vechia, and the Galaxy Clustering working group

2 Outline 1 Introduction and Motivation 2 Anisotropic Clustering in Fourier Space 3 Covariance Matrices for Cubes and Cut-Sky Catalogs 4 Verification of the new RSD Model 5 BOSS DR12 status 6 Conclusions Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

3 Introduction and Motivation Anisotropic Clustering Covariance Estimation Model Verification BOSS DR12 status Conclusions Motivation: Anisotropic Analysis of Galaxy Clustering Aim for the BOSS Analysis Line-of-Sight Decomposition Excellent large spectroscopic galaxy sample z -space matter clustering is inherently anisotropic s Baryonic Acoustic Oscillations imprint in galaxy clustering signal s Δz Δα constrain separately DA ( z ) = source: [F. Montesano] BAO serves as standard ruler and H (z ) = s α(1 + z ) c z s probe of expansion history Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

4 Extend Clustering Wedges to Fourier Space The LOS parameter µ r B B r A B r A LOS θ A µ = cos(θ) P(k, µ) = δ(k, µ)δ (k, µ) bad S N for fine µ-bins! Power Spectrum Wedges P(µ, k) averaged over wide bins in µ harmonized S/N 1 µ2 Pµ 1,µ 2 (k) P(µ, k) dµ µ 2 µ 1 µ 1 simple window function description transverse projection P (k) P 0, 1 (k) 2 line-of-sight projection P (k) P 1 2,1(k) µ = cos(θ) µ = 0.5 µ = 0 θ µ = 1 µ = 0 Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

5 Extend Clustering Wedges to Fourier Space The LOS parameter µ r B B r A B r A LOS θ A µ = cos(θ) P(k, µ) = δ(k, µ)δ (k, µ) bad S N for fine µ-bins! Power Spectrum Wedges P(µ, k) averaged over wide bins in µ harmonized S/N 1 µ2 Pµ 1,µ 2 (k) P(µ, k) dµ µ 2 µ 1 µ 1 simple window function description S/N even high enough for three wedges P 3w,i(k) P i 1 3, (k) i 3 µ = cos(θ) µ = 1/3 µ = 2/3 µ = 0 µ = 1 θ µ = 0 Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

6 Measurements of Anisotropic Clustering Yamamoto estimator pairwise LOS depends on observer and galaxy pair double sum over objects [Yamamoto et al. 05] impossible scaling N k (N 2 gal + N2 rnd )! [Samushia et al. 15] Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

7 Measurements of Anisotropic Clustering Yamamoto-Blake estimator per-object-los approximation instead of pairwise LOS single direct sum [Blake et al. 11] wide-angle bias for low-z and l 4 [Samushia et al. 15] Fractional Bias in APS (%) l=2 l=4 z=0.32 β=0.35 [Samushia et al. 15] wavenumber (h/mpc) Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

8 Yamamoto estimator for Fourier space wedges I Yamamoto Estimator for Clustering Wedges extend Yamamoto estimator to any number of wedges replace Legendre polynomials by µ-top-hat functions wedge (or multipole) overdensity field F a = 1 [D a (k) αr a (k)] A weighted sum over galaxies and randoms (1/α more numerous): D a (k) = i w i e ik x i Θ a (µ ki ), R a (k) = j w j e ik x j Θa (µ kj ) θ a (µ): top-hat for this wedge, with argument µ ki := k x i k x i. spoils use of FFTs!? Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

9 Yamamoto estimator for Fourier space wedges II wedge power spectrum computed as: P a (k) = F a (k)f 0 (k) S a normalization A := α j n j w 2 (just as for FKP), j n j : the estimated number density of galaxies. shot noise S a (k) = α(α + 1) j w 2 j Θ a (µ kj ) for polynomial µ dependence: fast FFT-scheme for Pl(µ) developed [Bianchi et al. 15, Scoccimarro 15] µ 2 = x i x j k i k j 6 combinations ij x 2 k 2 unbeatable scaling 6 Nfft log Nfft instead of N k (Ngal + Nrnd) A Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

10 FFT-based Clustering Wedges Estimation k P 3w,i (k) [(Mpc/h) 2 ] QPM CMASS NGC - no FC cubes FFT Yamamoto sum Yamamoto FFT k [h/mpc] P 3w,i /P 3w,i,sum Yamamoto direct sum Yamamoto FFT (Bianchi et al. '15) Yamamoto FFT (Scoccimarro '15) k [h/mpc] Pl(k) by Yamamoto-FFT estimator (EUCLID comparison project) transform to wedges by P µ 2 µ 1 (k) = 1 µ 2 µ 1 l {0,2,4} P l(k) µ 2 µ 1 L l (µ) dµ Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

11 A First Look at the Data: BOSS DR12 sample k P(k) [(Mpc/h) 2 ] z < 0.5 P 3w,1 (k) P 3w,2 (k) P 3w,3 (k) k P(k) [(Mpc/h) 2 ] z < 0.75 P 3w,1 (k) P 3w,2 (k) P 3w,3 (k) [JG et al. 15b (in prep.)] k [h/mpc] k [h/mpc] Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

12 The Effect of the Window Function Convolution with wedge window function (assuming isotropy) in analogy to monopole: P conv a (k) = [ d 3 k Pa model (k ) Wa 2 ( kê z k ) W a 2 (k) W0 2 (0) Pmodel 0 (k ) W0 2(k ) (second term: integral constraint) ] QPM CMASS NGC - no FC boxes P(k) conv. boxes cutsky P(k) k P(k) k [h/mpc] Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

13 Covariance estimation for Clustering Wedges Estimate P a (k i )-covariance C ab (k i, k j ) either 1 theoretically derived (smooth, model required) or 2 measured from a large set of synthetic catalogues (noisy) Full N-body Minerva simulations Verification of covariance estimate (and RSD modelling) 100 realizations, V = 3.37 (Gpc/h) 3 HOD galaxies at z = 0.57 mimicking CMASS sample (similar n and clustering) dn/dlnm h N(M) SubFind haloes Tinker et al M [M sun /h] N cen (M) N sat (M) M [M sun /h] N(M) Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

14 The Covariance Matrix for Fourier-Space Wedges k 2 i σ P, (k i ) [Mpc/h] k 2 i σ P3w,n (k i ) [Mpc/h] σ P σ P σ P3w,1 σ P3w,2 σ P3w, k i [h/mpc] For a cubic box, Fourier modes P(k, µ) are uncorrelated on large scales Variance can be constructed by a Gaussian model using an RSD power spectrum [JG et al. 15a (in prep.)] volume-average for each power spectrum bin k2 k1 d3 k... Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

15 Synthetic Catalogues as Covariance Estimate noise in covariance propagates to the final constraints [Percival et al. 14] accurate constraints require O ( 10 3) of synthetic catalogs (mocks) quick generation: non-linear evolution w/ fast approximative schemes mimicking full survey including veto regions and fibre collisions DR12 catalog Patchy V6C mocks k P(k) [(Mpc/h) 2 ] k [h/mpc] Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

16 The Covariance Matrix for Fourier-Space Wedges the survey geometry introduces correlations on the off-diagonals fibre collisions also correlate distant bins (S/N) 2 (0.02 h/mpc; k max ) z < wedges 2 wedges P z < wedges 2 wedges P k max [h/mpc] 0.2 z < 0.5 k j [h/mpc] P 3w,1 P 3w,2 P 3w, k i [h/mpc] w,n (k i) σ P nl 3w,m (k j)) C P,3w n,m (k i,k j )/(σ P nl 0.5 z < 0.75 k j [h/mpc] P 3w,1 P 3w,2 P 3w, k i [h/mpc] w,n (k i) σ P nl 3w,m (k j)) C P,3w n,m (k i,k j )/(σ P nl Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

17 Verification of the modelling Validation of the new RSD model (to Ariel s talk) Verify the modelling of PS wedges with Minerva simulations Smallest possible modes kmax to get unbiased parameters? mean Minerva HOD galaxies α α fσ k max unbiased f σ 8 sets limit kmax = 0.2 h/mpc varying the shot noise (prepare for catalogues fits) introduces small α, bias tighter constraints for 3 wedges Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

18 BOSS Mock Challenge Model performance compared in a blind challenge Blind results handed in and analyzed New Results for Cutsky Mocks Too optimistic choice of kmax Need to vary the shot noise fσ wedges SN fixed, k max =0.25 SN varied, k max =0.2 N <i> fσ wedges SN fixed, k max =0.25 SN varied, k max =0.2 N <i> Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

19 Ready to fit the DR12 galaxy catalog k P(k) [(Mpc/h) 2 ] z < 0.5 ΛCDM fit P 3w,1 (k) P 3w,2 (k) P 3w,3 (k) k P(k) [(Mpc/h) 2 ] z < 0.75 ΛCDM fit P 3w,1 (k) P 3w,2 (k) P 3w,3 (k) k [h/mpc] k [h/mpc] PS fits not ready for the public yet, but... model predictions using Ariel s preliminary 2PCF fits good agreement between Fourier and configuration space be patient until the release! Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

20 Conclusions i) new RSD model for galaxy clustering Major improvement, state-of-the art modelling for analysis both in configuration and Fourier space Tested and validated with large-scale simulations ii) BOSS Power Spectrum Wedges largest volume probed so far for galaxy clustering analysis, optimized data processing and fitting intensive work on final analysis highest demands: complementary analysis for multipoles and wedges in conf. and Fourier space µ = cos(θ) µ = 0.5 µ = 0 θ µ = cos(θ) θ µ = 1 µ = 0 µ = 1/3 µ = 2/3 µ = 0 µ = 1 µ = 0 Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

21 Outlook! Questions? Outlook 1 Analysis is tremendous team effort 2 Consistency check: configuration and Fourier space 3 Unprecedented accuracy can be expected Thank you for your attention! Time for all your questions! Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, /18

22 References Backup slides References NOT UP TO DATE! L. Anderson et al. (BOSS Collaboration), MNRAS 441 (1) (2013), arxiv: R. Angulo, C. Baugh, C. Frenk, and C. Lacey, Mon.Not.Roy.Astron.Soc., 383, 755 (2008), arxiv:astro-ph/ F. Beutler et al. (BOSS Collaboration) (2013), arxiv: J. Hartlap, P. Simon, and P. Schneider Astron.Astrophys. (2006), arxiv:astro-ph/ Komatsu, E. et al. ApJS, 192, 18 (2011), arxiv: [astro-ph.co] L. Samushia, E. Branchini, and W. Percival, (2015), arxiv: A. G. Sánchez, E. A. Kazin, F. Beutler, et al. (BOSS Collaboration) MNRAS 433 (2) (2013), arxiv: [astro-ph.co] A. G. Sánchez, F. Montesano, E. A. Kazin, et al. (BOSS Collaboration) MNRAS 440 (3) (2013), arxiv: [astro-ph.co] Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, 2015 I/IV

23 References Backup slides Angular Diameter Distance and the BAO Angular Diameter Distance, D A (z) = c Sound Horizon, z 0 dz H(z ) t dec c s (t ) dt r s = 0 a(t ) known from CMB measurements (r s = 147 Mpc [Komatsu et al. 11]) From the BAO position, we can get (rab = r s ) θ BAO = 1 r s 1 + z D A (z) zbao = r s H(z) c, θ z 1 z r 2 AB θ=r AB /r =L AB /D A (z) go back Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, 2015 II/IV

24 References Backup slides Dependence of Geometry on Cosmology Fiducial cosmology of simulations: w = wtrue = 1 Assumed cosmology from measurement: wassumed = wtrue + w Mismatch causes geometry of the late universe to be misinterpreted Relates to change α = kapp/ktrue [Angulo et al. 08] α = D A(z, wassumed), α D A (z, wtrue) = H(z, w true) H(z, wassumed) α α 2/3 α 1/3 D A angular diameter distance, H Hubble parameter D A and the BAO Goals: α = 1 (no bias), α 1 (high precision) α and w of same magnitude go back Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, 2015 III/IV

25 References Backup slides Estimation of Model Parameters using MCMC Likelihood function for mean power spectrum wedges P, (k), measured at wavenumber bins k i : P( P θ) exp[ χ 2 ( P θ)/2], where χ 2 ( P θ) = [ [ Px (k i ) Px,rpt(k i )] C 1 Py (k xyij j ) Py,rpt(k j )] x,y,i,j covariance matrix estimated [ from set ] of[ realizations ] C xyij = P x (k i ) Px (k i ) P y (k j ) Py (k j ) inverse corrected for noise [Hartlap et al. 06] step through parameter space using Markov chain Monte Carlo go back Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, 2015 IV/IV