Accurate Modeling of the Redshift-Space Distortions of Biased Tracers +α Takahiro Nishimichi (IPMU, the Univ. of Tokyo) PRD, 84(2011), 043526 with Atsushi Taruya (RESCEU, the Univ. of tokyo) PTchat @ IPhT CEA Saclay, Sep. 21 2011 2011年9月21日水曜日 Dark Energy? Modified Gravity? observations → acceleration of cosmic expansion ✓type-1a supernova ✓BAOs ✓CMB ✓... 8πG Gµν = f(R), DGP, ... modified gravity? c4 Tµν SNLS3, Conley+’11 dark energy? expansion history in a DE model may be mimicked by a MG model geometrical + growth tests are essential! 2011年9月21日水曜日 Anisotropies in galaxy clustering distance measurements in z-space BAOs + Alcock & Paczynski test power spectrum (normalized) → H(z) & DA(z) © A. Taruya f σ8 z-space distiontions → d ln D(z) γ f (z) ≡ " [Ωm (z)] d ln a Blake+‘11a Blake+‘11b redshift z space WiggleZ Dark Energy Survey wavenumber k [h/Mpc] 2011年9月21日水曜日 © A. Taruya real space peculiar velocity Redshift-space distortions z-space Kaiser Effect r-space r-space peculiar velocity large-scale coherent motion → enhancement of clustering k|| μ=1 Finger-of-God Effect small-scale random motion → suppression of clustering μ=0 e.g., Scoccimarro’04 streaming model vel. divergence: 2011年9月21日水曜日 k⊥ vel. dispersion: σv Redshift-space distortions z-space Kaiser Effect r-space r-space peculiar velocity large-scale coherent motion → enhancement of clustering k|| μ=1 Finger-of-God Effect small-scale random motion → suppression of clustering μ=0 e.g., Scoccimarro’04 streaming model vel. divergence: 2011年9月21日水曜日 k⊥ vel. dispersion: σv Redshift-space distortions z-space Kaiser Effect r-space z-space r-space peculiar velocity large-scale coherent motion → enhancement of clustering k|| μ=1 Finger-of-God Effect small-scale random motion → suppression of clustering μ=0 e.g., Scoccimarro’04 streaming model vel. divergence: 2011年9月21日水曜日 k⊥ vel. dispersion: σv Redshift-space distortions z-space Kaiser Effect r-space r-space peculiar velocity large-scale coherent motion → enhancement of clustering k|| μ=1 Finger-of-God Effect small-scale random motion → suppression of clustering μ=0 e.g., Scoccimarro’04 streaming model vel. divergence: 2011年9月21日水曜日 k⊥ vel. dispersion: σv Redshift-space distortions z-space Kaiser Effect r-space r-space peculiar velocity large-scale coherent motion → enhancement of clustering k|| μ=1 Finger-of-God Effect small-scale random motion → suppression of clustering μ=0 e.g., Scoccimarro’04 streaming model vel. divergence: 2011年9月21日水曜日 k⊥ vel. dispersion: σv Redshift-space distortions z-space Kaiser Effect r-space z-space r-space peculiar velocity large-scale coherent motion → enhancement of clustering k|| μ=1 Finger-of-God Effect small-scale random motion → suppression of clustering μ=0 e.g., Scoccimarro’04 streaming model vel. divergence: 2011年9月21日水曜日 k⊥ vel. dispersion: σv Redshift-space distortions (contd.): TNS model Exact formula for the z-space P(k) with a help of cumulant expansion theorem ∝ cross-bispectrum of δ & θ new terms! B term ∝ sum of convolutions of Pδθ & Pθθ A term 2011年9月21日水曜日 quadrupole monopole A notice !eA BC" = # !e "!BC" Taruya, Nishimichi, Saito (‘10) w/o correction Redshift-space distortions (contd.): TNS model Exact formula for the z-space P(k) with a help of cumulant expansion theorem ∝ cross-bispectrum of δ & θ new terms! B term ∝ sum of convolutions of Pδθ & Pθθ A term 2011年9月21日水曜日 quadrupole monopole A notice !eA BC" = # !e "!BC" Taruya, Nishimichi, Saito (‘10) w/o w/ correction RSDs for biased tracers? Okumura & Jing’11 Many people are working hard on this! e.g., Okumura & Jing’11 Tang, Kayo & Takada’11 Reid & White’11 Sato & Matsubara ’11 biased tracer? k [h/Mpc] k [h/Mpc] assume δg = bδ β=f/b ∝ cross-bispectrum of δ & θ B term ∝ sum of convolutions of Pδθ & Pθθ A term 2011年9月21日水曜日 Are correction terms enhanced by bias? Analysis T.N. & Taruya ’11 Large N-body simulations (L=1.14Gpc/h, N=1,2803) starting with 2LPT initial conditions x 15 realizations ✓ 9 halo catalogs over a wide mass range @ z=0.35 ✓ volume & number density ≒ SDSS DR7 LRGs ✓ b(k) is directly measured from r-space clustering ✓ σv is treated as a free fitting parameter mass: h-1Msun, density: h3Mpc-3 2011年9月21日水曜日 k [h/Mpc] Result T.N. & Taruya ’11 N-body simulations dark matter 2011年9月21日水曜日 light halos heavy halos dark matter light halos Result TNS model dark matter 2011年9月21日水曜日 heavy halos T.N. & Taruya ’11 light halos heavy halos dark matter light halos Result streaming TNS modelmodel dark matter 2011年9月21日水曜日 heavy halos T.N. & Taruya ’11 light halos heavy halos dark matter light halos Result streaming TNS modelmodel dark matter 2011年9月21日水曜日 heavy halos T.N. & Taruya ’11 light halos heavy halos Goodness of fits best-fit values of σv: • smaller for streaming model • consistent with 0 for massive halos • consistent with the linear theory for TNS • does not depend the halo mass goodness of fit: • worse for streaming model • especially for massive halos • reduced χ2 are close to 1 for TNS • independent of the halo mass 2011年9月21日水曜日 T.N. & Taruya ’11 FoG function: Lorentzian (L), Gaussian (G) Recovery of f(z) 2 parameter fit to N-body data: f and σv ✓ streaming model • seams OK only at kmax<0.1h/Mpc • typically ~5% underestimate of f ✓ TNS model • gives unbiased estimate of f • up to kmax ~ 0.2 h/Mpc 2011年9月21日水曜日 T.N. & Taruya ’11 Recovery of f(z) 2 parameter fit to N-body data: f and σv ✓ streaming model • seams OK only at kmax<0.1h/Mpc • typically ~5% underestimate of f ✓ TNS model • gives unbiased estimate of f • up to kmax ~ 0.2 h/Mpc 2011年9月21日水曜日 T.N. & Taruya ’11 Recovery of f(z) 2 parameter fit to N-body data: f and σv ✓ streaming model • seams OK only at kmax<0.1h/Mpc • typically ~5% underestimate of f ✓ TNS model • gives unbiased estimate of f • up to kmax ~ 0.2 h/Mpc 2011年9月21日水曜日 T.N. & Taruya ’11 Recovery of f(z) 2 parameter fit to N-body data: f and σv ✓ streaming model • seams OK only at kmax<0.1h/Mpc • typically ~5% underestimate of f ✓ TNS model • gives unbiased estimate of f • up to kmax ~ 0.2 h/Mpc 2011年9月21日水曜日 T.N. & Taruya ’11 Recovery of f(z) 2 parameter fit to N-body data: f and σv ✓ streaming model • seams OK only at kmax<0.1h/Mpc • typically ~5% underestimate of f ✓ TNS model • gives unbiased estimate of f • up to kmax ~ 0.2 h/Mpc 2011年9月21日水曜日 T.N. & Taruya ’11 Recovery of f(z) 2 parameter fit to N-body data: f and σv ✓ streaming model • seams OK only at kmax<0.1h/Mpc • typically ~5% underestimate of f ✓ TNS model • gives unbiased estimate of f • up to kmax ~ 0.2 h/Mpc 2011年9月21日水曜日 T.N. & Taruya ’11 Future/on-going surveys T.N. & Taruya ’11 Fisher matrix analysis with 5 parameters (b, σv, H, DA, f) Assumption TNS model is true, but adopt streaming model h-3Gpc3 h3Mpc-3 h Mpc-1 γ f (z) = [Ωm (z)] γ = 0.77 ± 0.04 2011年9月21日水曜日 (short) summary ・tested the clustering of halos in z-space by N-body simulations ... ・frequently used phenomenological model is not sufficient ~5% systematic bias in f(z) ・correction terms in TNS model more prominent for more massive halos or more biased objects ・codes for our model are publicly available!! visit CPT library: http://www-utap.phys.s.u-tokyo.ac.jp/~ataruya/cpt_pack.html 2011年9月21日水曜日 Beyond TNS model... Lesson from the success of RPT ... Crocce & Scoccimarro 06, 08 ✓decay of BAO bump by propagator ✓shift of peak position by MC term propagator tree level 2011年9月21日水曜日 mode coupling Beyond TNS model... T.N. in prep. A simple extension of the idea of the propagator to halos in z-space: !δh (k; z)δ0 (k! )" Gh (k, µ; z) = !δ0 (k)δ0 (k! )" Ph (k, µ; z) = 2011年9月21日水曜日 2 Gh (k, µ; z)P0 (k) + PMC (k; z) z-space propagator of halos T.N. in prep. preliminary Mh > 1.8x1012 Msun/h 2011年9月21日水曜日 z-space propagator of halos T.N. in prep. ξh (s⊥ , s|| ) tree-level in RPT preliminary 2011年9月21日水曜日 z-space propagator of halos T.N. in prep. ξh (s⊥ , s|| ) mode-coupling term preliminary 2011年9月21日水曜日 Appendix 2011年9月21日水曜日 signal-noise decomposition 2011年9月21日水曜日
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