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
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, 2015 1/18
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, 2015 2/18
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, 2015 3/18
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, 2015 3/18
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, 2015 4/18
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 (%) 50 40 30 20 10 0 l=2 l=4 z=0.32 β=0.35 [Samushia et al. 15] 0.05 0.10 0.15 wavenumber (h/mpc) Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, 2015 4/18
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, 2015 5/18
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, 2015 6/18
FFT-based Clustering Wedges Estimation k P 3w,i (k) [(Mpc/h) 2 ] 2500 2000 1500 1000 QPM CMASS NGC - no FC cubes FFT Yamamoto sum Yamamoto FFT 0.05 0.10 0.15 0.20 0.25 0.30 k [h/mpc] P 3w,i /P 3w,i,sum 1.01 1.00 0.99 0.98 1.01 1.00 0.99 0.98 1.01 1.00 0.99 0.98 Yamamoto direct sum Yamamoto FFT (Bianchi et al. '15) Yamamoto FFT (Scoccimarro '15) 0.05 0.10 0.15 0.20 0.25 0.30 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, 2015 7/18
A First Look at the Data: BOSS DR12 sample k P(k) [(Mpc/h) 2 ] 2500 2000 1500 1000 0.2 z < 0.5 P 3w,1 (k) P 3w,2 (k) P 3w,3 (k) k P(k) [(Mpc/h) 2 ] 2500 2000 1500 1000 0.5 z < 0.75 P 3w,1 (k) P 3w,2 (k) P 3w,3 (k) [JG et al. 15b (in prep.)] 0.05 0.10 0.15 0.20 0.25 0.30 k [h/mpc] 0.05 0.10 0.15 0.20 0.25 0.30 k [h/mpc] Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, 2015 8/18
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) ]. 2600 2400 2200 QPM CMASS NGC - no FC boxes P(k) conv. boxes cutsky P(k) k P(k) 2000 1800 1600 1400 1200 1000 0.05 0.1 0.15 0.2 0.25 k [h/mpc] Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, 2015 9/18
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) 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10 1 10 0 10-1 10-2 SubFind haloes Tinker et al. 10 13 10 14 10 15 M [M sun /h] N cen (M) N sat (M) 10 13 10 14 10 15 M [M sun /h] N(M) Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, 2015 10/18
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] 4 3 2 1 6 5 4 3 2 1 σ P σ P 0.05 0.10 0.15 0.20 0.25 σ P3w,1 σ P3w,2 σ P3w,3 0.05 0.10 0.15 0.20 0.25 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, 2015 11/18
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 2500 2000 DR12 catalog Patchy V6C mocks k P(k) [(Mpc/h) 2 ] 1500 1000 0.05 0.10 0.15 0.20 0.25 k [h/mpc] Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, 2015 12/18
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 ) 60000 50000 40000 30000 20000 10000 0 60000 50000 40000 30000 20000 10000 0 0.2 z < 0.5 3 wedges 2 wedges P 0 0.05 0.10 0.15 0.20 0.5 z < 0.75 3 wedges 2 wedges P 0 0.05 0.10 0.15 0.20 k max [h/mpc] 0.2 z < 0.5 k j [h/mpc] 0.2 0.1 0.2 0.1 0.2 0.1 P 3w,1 P 3w,2 P 3w,3 0.0 0.0 0.1 0.2 0.1 0.2 0.1 0.2 k i [h/mpc] 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.2 3w,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] 0.2 0.1 0.2 0.1 0.2 0.1 P 3w,1 P 3w,2 P 3w,3 0.0 0.0 0.1 0.2 0.1 0.2 0.1 0.2 k i [h/mpc] 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.2 3w,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, 2015 13/18
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 1.10 1.05 α α fσ 8 1.00 0.95 0.90 1.04 1.02 1.00 0.98 0.96 0.54 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.16 0.18 0.20 0.22 0.16 0.18 0.20 0.22 0.16 0.18 0.20 0.22 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, 2015 14/18
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 8 0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04 2 wedges SN fixed, k max =0.25 SN varied, k max =0.2 N <i> fσ 8 0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04 3 wedges SN fixed, k max =0.25 SN varied, k max =0.2 N <i> Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, 2015 15/18
Ready to fit the DR12 galaxy catalog k P(k) [(Mpc/h) 2 ] 2500 2000 1500 0.2 z < 0.5 ΛCDM fit P 3w,1 (k) P 3w,2 (k) P 3w,3 (k) k P(k) [(Mpc/h) 2 ] 2500 2000 1500 0.5 z < 0.75 ΛCDM fit P 3w,1 (k) P 3w,2 (k) P 3w,3 (k) 1000 1000 0.05 0.10 0.15 0.20 0.25 0.30 k [h/mpc] 0.05 0.10 0.15 0.20 0.25 0.30 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, 2015 16/18
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, 2015 17/18
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, 2015 18/18
References Backup slides References NOT UP TO DATE! http://lambda.gsfc.nasa.gov/, http://wmap.gsfc.nasa.gov/ L. Anderson et al. (BOSS Collaboration), MNRAS 441 (1) 24-62 (2013), arxiv:1312.4877 R. Angulo, C. Baugh, C. Frenk, and C. Lacey, Mon.Not.Roy.Astron.Soc., 383, 755 (2008), arxiv:astro-ph/0702543 F. Beutler et al. (BOSS Collaboration) (2013), arxiv:1312.4611 J. Hartlap, P. Simon, and P. Schneider Astron.Astrophys. (2006), arxiv:astro-ph/0608064 Komatsu, E. et al. ApJS, 192, 18 (2011), arxiv:1001.4538 [astro-ph.co] L. Samushia, E. Branchini, and W. Percival, (2015), arxiv:1504.02135 A. G. Sánchez, E. A. Kazin, F. Beutler, et al. (BOSS Collaboration) MNRAS 433 (2) 1202-1222 (2013), arxiv:1303.4396 [astro-ph.co] A. G. Sánchez, F. Montesano, E. A. Kazin, et al. (BOSS Collaboration) MNRAS 440 (3) 2692-2713 (2013), arxiv:1312.4854 [astro-ph.co] Jan Grieb (MPE, Garching) Fourier Space Wedges Jul 20th, 2015 I/IV
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
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
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