Bispectrum measurements & Precise modelling

Transcription

1 Discussion on Bispectrum measurements & precise modelling Héctor Gil-Marín Laboratoire de Physique Nucleaire et de Hautes Energies Institut Lagrange de Paris CosKASI-ICG-NAOC-YITP Workshop 8th September 206

2 Outline Bispectrum estimator 2 Modelling the dark matter bispectrum in real space 3 Modelling the dark matter bispectrum in redshift space 4 Other issues on modelling

3 Estimation of the Bispectrum Efficient algorithm for computing the Bispectrum Sefusatti E., 2005, PhD thesis, New York University, New York, USA (no public access) Baldauf et al. 205 (arxiv: ) Based on Fourier Transforms and not in previous brute force approaches.

4 Definition of the Bispectrum Probability of finding 3 galaxies separated by r, s and t: P 3 (r, s, t) = [ + ξ 2 (r) + ξ 2 (s) + ξ 2 (t) + ζ(r, s, t)]dv dv 2 dv 3 The bispectrum is defined as the FT of ζ, B(k, k 2 ) dr ds ζ(r, s)e ir k e is k 2 Since, ζ(r, s, t) δ(x + r)δ(x + t)δ(x) x B(k, k 2, k 3 ) = δ(k )δ(k 2 )δ(k 3 ) δ D (k + k 2 + k 3 )

5 Algorithm Proposed estimator, ˆB 23 = k3 f dq δ(q ) dq 2 δ(q 2 ) V 23 S S 2 S 3 dq 3 δ(q 3 )δ D (q + q 2 + q 3 ), CosKASI-ICG-NAOC-YITP Workshop, 8th aaaa September 206 S i stands for S(k i k), which is the k-region contained by k i k/2 k k i + k/2, given a k-bin, k

6 Algorithm Proposed estimator, ˆB 23 = k3 f dq δ(q ) dq 2 δ(q 2 ) V 23 S S 2 S 3 dq 3 δ(q 3 )δ D (q + q 2 + q 3 ), CosKASI-ICG-NAOC-YITP Workshop, 8th aaaa September 206 V 23 is the associated volume ( total number of triangles ), V 23 = k3 f dq dq 2 V 23 S S 2 S 3 dq 3 δ D (q + q 2 + q 3 ),

7 Algorithm We can write the Dirac delta in its exponential form, δ D (q, q 2, q 3 ) dr e ir q e ir q 2 e ir q 3 and re-write the estimator as, ˆB 23 dq δ(q ) dq 2 δ(q 2 ) S S 2 dq 3 δ(q 3 ) dr e ir q e ir q 2 e ir q 3 S 3 }{{} δ D (q +q 2 +q 3 ) Then, the estimator is partly separable, ˆB 23 dr dq δ(q )e ir q... {q 2 }... {q 3 } S } {{} FT-like of δ(q )

8 Algorithm In the end the algorithm looks like, ˆB 23 dr D S (r)d S2 (r)d S3 (r) where, D Sj (r) dq j δ(q j )e iq j r. S j D Sj can be though as the Fourier Transform of F(q) where { δ(q) if q S j F(q) = 0 otherwise The main complication when coding is to find an efficient way to select the elements of δ(q i ) S i.

9 Algorithm Features, Fast. 850 triangles in 2 hours using 20 processors. Speed can be tuned by increasing the size of k-bins (i.e. reducing the number of total triangles). Easy-to-code. It relies on FT libraries such as fftw, which are optimised and widely used. General. No limitation on scales or survey geometry other than the power spectrum estimator

10 Algorithm Recyclable. No need to recompute the 3 D new terms for each new triangle, as many triangles share one or two sides. Higher orders. The algorithm can be trivially generalised to higher order functions, such as the Trispectrum. Publicity. Might be a release of a public code in the near(?) future.

11 Modelling the Bispectrum in real space Different models (full shape), Perturbation Theory: SPT, RPT, EFT,... Phenomenological: Halo model PT, Benchmark model, 2-halo boost, Fitting formulae Having a working model for the bispectrum is much more complicated than the power spectrum..., but for BAO-feature only may not be that difficult.

12 Modelling the Bispectrum in real space Tree level approach based on Perturbation Theory, B(k, k 2 ) = 2P lin (k )P lin (k 2 )F 2 (k, k 2 ) + cyc., where F 2 is the two-point kernel of the density of the form, F 2 (k i, k j ) = ( 2 cos(α ki ij) + k ) j + 2 k j k i 7 cos2 (α ij ) This kernel has a weak dependence with the cosmological parameters but may be sensible to modifications of GR. Here we follow the empirical approach of taken the tree level formula and modify the kernels (Scoccimarro & Couchman 200, HG-M et al. 202): F 2 F eff 2 P lin P nl

13 Modelling the Bispectrum in real space F eff 2 (k i, k j ) = 5 7 a(n i, k i, a F )a(n j, k j, a F ) + ( 2 cos(α ki ij) + k ) j b(n i, k i, a F )b(n j, k j, a F ) k j k i cos2 (α ij )c(n i, k i, a F )c(n j, k j, a F ) where a, b, c are empirical functions that depend on k i, on the local slope of the power spectrum, n, and on a set of 9 free parameters, a F {a F,..., af 9 } to be fitted from N-body simulations.

14 Modelling the Bispectrum in real space Comparing different models with Nbody... Lanzau et al. 206; arxiv:

15 Modelling the Bispectrum in real space: PT Tree Btree SPT = 2P lin P lin F 2 + cyc. SPT B loop SPT = BSPT tree + B B (I ) 32 + B(II ) 32 + B 4 EFT B EFT = B SPT + B cs [no extra free parameters wrt P EFT ] MPTBreeze B = [B SPT tree +B 222 +B (I ) RLPT [ B = k 2 exp +k2 2 +k2 3 2π dp 2 Plin (p) ] [ 32 ] exp[f (k )+f (k 2 )+f (k 3 )] Btree SPT dp Plin (p)] B SPT loop tree + k2 +k2 2 +k3 3 2π 2

16 Modelling the Bispectrum in real space: PT Tree Btree SPT = 2P lin P lin F 2 + cyc. SPT B loop SPT = BSPT tree + B B (I ) 32 + B(II ) 32 + B 4 EFT B EFT = B SPT + B cs [no extra free parameters wrt P EFT ] MPTBreeze B = [B SPT tree +B 222 +B (I ) RLPT [ B = k 2 exp +k2 2 +k2 3 2π dp 2 Plin (p) ] [ 32 ] exp[f (k )+f (k 2 )+f (k 3 )] Btree SPT dp Plin (p)] B SPT loop tree + k2 +k2 2 +k3 3 2π 2

17 Modelling the Bispectrum in real space: PT Tree Btree SPT = 2P lin P lin F 2 + cyc. SPT B loop SPT = BSPT tree + B B (I ) 32 + B(II ) 32 + B 4 EFT B EFT = B SPT + B cs [no extra free parameters wrt P EFT ] MPTBreeze B = [B SPT tree +B 222 +B (I ) RLPT [ B = k 2 exp +k2 2 +k2 3 2π dp 2 Plin (p) ] [ 32 ] exp[f (k )+f (k 2 )+f (k 3 )] Btree SPT dp Plin (p)] B SPT loop tree + k2 +k2 2 +k3 3 2π 2

18 Modelling the Bispectrum in real space: Halo Model Halo Model B HM = B h + B 2h + B 3h Halo Model PT Valageas and Nishimichi 20a, 20b Benchmark model B = f (K)Stree SPT ; S is shape function 2-halo boost.

19 Modelling the Bispectrum in real space: Halo Model Halo Model B HM = B h + B 2h + B 3h Halo Model PT Valageas and Nishimichi 20a, 20b Benchmark model B = f (K)Stree SPT ; S is shape function 2-halo boost.

20 Modelling the Bispectrum in real space: Halo Model Halo Model B HM = B h + B 2h + B 3h Halo Model PT Valageas and Nishimichi 20a, 20b Benchmark model B = f (K)Stree SPT ; S is shape function 2-halo boost.

21 Modelling the Bispectrum in real space: Summary

22 Modelling the Bispectrum in redshift space The tree level prediction for the redshift-space bispectrum, B s (k, k 2 ) = D B fog [2P lin(k ) Z (k ) P lin (k 2 ) Z (k 2 ) Z 2 (k, k 2 ) + cyc.] Z (k i ) ( + f µ 2 i ) [ Z 2 (k, k 2 ) F 2 (k, k 2 ) + f µk ( µ + µ )] 2 + f µ 2 G 2 (k, k 2 ) + 2 k k 2 + f 3 ( µk 2 µ µ2 µ 2 + µ ) k k 2 f d ln δ/d ln a, µ (µ k + µ 2 k 2 )/k, k 2 = (k + k 2 ) 2. G 2 is the second-order real-space kernels of the velocities. Non-linear damping due to intra-halo velocity dispersion (FoG) D B fog = DB fog (k, k 2, k 3, σ B fog [z])

23 Modelling the Bispectrum in redshift space The tree level prediction for the redshift-space bispectrum, B s (k, k 2 ) = D B fog [2P lin(k ) Z (k ) P lin (k 2 ) Z (k 2 ) Z 2 (k, k 2 ) + cyc.] Z (k i ) ( + f µ 2 i ) [ Z 2 (k, k 2 ) F 2 (k, k 2 ) + f µk ( µ + µ )] 2 + f µ 2 G 2 (k, k 2 ) + 2 k k 2 + f 3 ( µk 2 µ µ2 µ 2 + µ ) k k 2 f d ln δ/d ln a, µ (µ k + µ 2 k 2 )/k, k 2 = (k + k 2 ) 2. G 2 is the second-order real-space kernels of the velocities. Non-linear damping due to intra-halo velocity dispersion (FoG) D B fog = DB fog (k, k 2, k 3, σ B fog [z])

24 Modelling the Bispectrum in redshift space Inspired by the real space modification, we perform a similar transformation to G 2 (HG-M et al. 204), G eff 2 (k i, k j ) = 3 7 a(n i, k i, a G )a(n j, k j, a G ) + ( 2 cos(α ki ij) + k ) j b(n i, k i, a G )b(n j, k j, a G ) k j k i cos2 (α ij )c(n i, k i, a G )c(n j, k j, a G ) where a G = {a G,..., ag 9 } to be fitted from N-body simulations. Z 2 Z eff 2 We will work with the bispectrum monopole (angle averaged), B (0) (k, k 2 ) = dµ dµ 2 B s (k, k 2 )

25 Modelling the Bispectrum in redshift space 2 k 2 /k = k 2 /k =.5 k 2 /k =2 z = 0 We fit a G to B (0) for k /k 2 =.0,.5, 2.0, 2.5, for z = 0, 0.5,.0,.5. We allow the σfog B [z] to be free as well. F2 eff & G2 eff in Z2 eff F2 eff & G 2 in Z2 eff F 2 & G 2 in Z 2 B 0 sim / B0 model B 0 / B B 0 sim / B0 model B 0 / B B 0 sim / B0 model B 0 / B B 0 sim / B0 model B 0 real space real space real space / B real space k 3 [h/mpc] k 3 [h/mpc] k 3 [h/mpc] k 2 =0.05 h/mpc k 2 =0.0 h/mpc k 2 =0.5 h/mpc k 2 =0.20 h/mpc

26 Modelling the Bispectrum in redshift space k 2 /k = k 2 /k =.5 k 2 /k =2 z = 0.5 We fit a G to B (0) for k /k 2 =.0,.5, 2.0, 2.5, for z = 0, 0.5,.0,.5. We allow the σfog B [z] to be free as well. F2 eff & G2 eff in Z2 eff F2 eff & G 2 in Z2 eff F 2 & G 2 in Z 2 B 0 / B real space B 0 sim / B0 model B 0 / B real space B 0 sim / B0 model B 0 / B real space B 0 sim / B0 model B 0 / B real space B 0 sim / B0 model k 3 [h/mpc] k 3 [h/mpc] k 3 [h/mpc] k 2 =0.05 h/mpc k 2 =0.0 h/mpc k 2 =0.5 h/mpc k 2 =0.20 h/mpc

27 Modelling the Bispectrum in redshift space 3.2 k 2 /k = k 2 /k =.5 k 2 /k =2 B 0 / B real space k 2 =0.05 h/mpc z =.0 We fit a G to B (0) for k /k 2 =.0,.5, 2.0, 2.5, for z = 0, 0.5,.0,.5. We allow the σfog B [z] to be free as well. F2 eff & G2 eff in Z2 eff F2 eff & G 2 in Z2 eff F 2 & G 2 in Z 2 B 0 sim / B0 model B 0 / B real space B 0 sim / B0 model B 0 / B real space B 0 sim / B0 model B 0 / B real space B 0 sim / B0 model k 3 [h/mpc] k 3 [h/mpc] k 3 [h/mpc] k 2 =0.0 h/mpc k 2 =0.5 h/mpc k 2 =0.20 h/mpc

28 Modelling the Bispectrum in redshift space 3.4 k 2 /k = k 2 /k =.5 k 2 /k =2 z =.5 We fit a G to B (0) for k /k 2 =.0,.5, 2.0, 2.5, for z = 0, 0.5,.0,.5. We allow the σfog B [z] to be free as well. F2 eff & G2 eff in Z2 eff F2 eff & G 2 in Z2 eff F 2 & G 2 in Z 2 B 0 sim / B0 model B 0 / B B 0 sim / B0 model B 0 / B B 0 sim / B0 model B 0 / B B 0 sim / B0 model B 0 real space real space real space / B real space k 3 [h/mpc] k 3 [h/mpc] k 3 [h/mpc] k 2 =0.05 h/mpc k 2 =0.0 h/mpc k 2 =0.5 h/mpc k 2 =0.20 h/mpc

29 Dependence with Cosmology? The best-fitting a F & a G parameters present a weak dependence with σ 8 and Ω m. B 0 i / B0 B 0 model i / B0 fiducial k 2 /k = k 2 /k = k 2 /k =2 Ω m =0.2 Ω m =0.4 k 2 =0.035 /Mpc 5 5 Ω m = 0.2, 0.27(fid), 0.4. Ω m h 2 = constant (SV cancellation). σ 8 adjusted to have same amplitude in PS B 0 B 0 i / B0 B 0 model i / i / B0 B 0 model i / B0 fiducial B0 fiducial k 2 =0.070 /Mpc k 2 =0.05 /Mpc B 0 i / B0 fiducial k 2 =0.4 /Mpc B 0 i / B0 model k 3 [/Mpc] k 3 [/Mpc] k 3 [/Mpc]

30 Modelling the Bispectrum in redshift space Other phenomenological approaches for redshift space bispectrum based on halo model (Smith, Sheth & Scoccimarro 2008 )

31 Other issues There are other issues concerning the bispectrum modelling, Galaxy Bias. Non-linear and non-local bias model seems to be needed (McDonald & Roy 09). How to model the survey selection function? Wilson et al. 206 seems the best approach but needs to be extended to the bispectrum. Is Poisson shot noise prediction enough? How the bispectrum shot noise is modified by halo exclusion? Bispectrum BAO. How much information is it there? How much does the reconstruction solve this problem? After reconstruction is it worth to look and use bispectrum BAO?...