Taka Matsubara (Nagoya Univ.) 35min Nonlinear Perturbation Theory with Halo Bias and Redshift-space Distortions via the Lagrangian Picture Taka Matsubara (Nagoya Univ.) “The Third KIAS Workshop on COSMOLOGY AND STRUCTURE FORMATION” Oct. 27 – 28, 2008, KIAS, Seoul 10/28/2008

Precision cosmology with galaxy clustering BAO as a probe of dark energy In correlation function In power spectrum Eisenstein et al. (SDSS, 2005) Percival et al. (SDSS, 2007) DE is constrained by 1D scale: (SDSS survey)

Theoretical modeling The BAO dynamics is qualitatively captured by linear theory, but... Nonlinearity in various aspects should be theoretically elucidated, otherwise the estimation of dark energy would be biased. Nonlinearity in dynamics Nonlinearity in redshift-space distortions Nonlinearity in halo/galaxy bias

Nonlinearity in dynamics Nonlinear dynamics distorts the BAO signature N-body experiments Simple nonlinear perturbation theory does not work well at relevant redshift z < 3 Correlation function, large N-body simulation Eisenstein et al. (2007) Power spectrum, large N-body simulation Seo et al. (2008) Power spectrum, N-body & 1-loop PT Meiksin et al. (1999)

Nonlinearity in redshift-space distortions Redshift-space distortions change the nonlinear effects on BAO P(k): Small-scale enhancement relative to the large-scale power is much less (but overall Kaiser enhancement) x(r): Nonlinear degradation is larger N-body, Seo et al. (2005) N-body, Eisenstein et al. (2007)

Nonlinearity in bias Effects of nonlinear (halo) bias P(k): Scale-dependent bias is induced by nonlinearity x(r): Linear bias seems good for r > 60 h-1Mpc N-body, Angulo et al. (2005) N-body, Sanchez et al. (2008)

Theories for nonlinear dynamics Recent developments: nonlinearity in dynamics Renormalized perturbation theory and its variants Infinitely higher-order perturbations are reorganized and partially resummed “Renormalization group method” Matarrese & Pietroni (2008) “Closure theory” Taruya & Hiramatsu (2008) “Renormalized perturbation theory” Crocce & Scoccimarro (2008)

Theory for nonlinear halo bias Nonlinear perturbation theory with simple local bias is not straightforward Smith et al. (2007): 1-loop PT + halo-like bias McDonald (2006): bias renormalization } both in real space Smith et al. (2007) Jeong & Komatsu (2008)

Nonlinear redshift distortions and bias Redshift distortions & bias Standard Eulerian perturbation theory + local bias model do not give satisfactory results… Lagrangian picture is useful for these issues !! ： initial position : displacement vector : final position

Redshift distortions in the Lagrangian picture Redshift-space mapping is exactly “linear” even in the nonlinear regime c.f.) In the Eulerian picture, the mapping is fully nonlinear: vz/(aH) x s z : line of sight

The halo bias in the Lagrangian picture (extended) Press-Schechter theory Halo number density is biased in Lagrangian space Lagrangian picture is natural for the halo bias No need for assuming the spherical collapse model as in the usual halo approach 1-halo term 2-halo term

Perturbation theory via the Lagrangian picture Nonlinear dynamics + nonlinear halo bias + nonlinear redshift-space distortions (T.M. 2008) Relation between the power spectrum and the displacement field Fourier transf. & Ensemble average Evaluation by adopting Lagrangian perturbation theory

Diagrammatic representations are useful Feynman rules Relevant diagrams up to one-loop PT

Result: nonlinear redshift-space distortions Comparison of the one-loop PT to a N-body simulation Linear theory N-body 1-loop SPT This work This work N-body Linear theory (Points from N-body simulation of Eisenstein & Seo 2005)

Result: halo bias in redshift space The one-loop perturbation theory via the Lagrangian picture Nonlinear dynamics + nonlinear halo bias + nonlinear redshift-space distortions P(k) x(r)

Discussion Galaxy bias Determination of the BAO scale P(k) vs x(r) On large scales, halo bias ~ galaxy bias (2-halo term) On small scales, 1-halo term should be included 1-halo term in redshift space (White 2001; Seljak 2001;…) Determination of the BAO scale Scale dependence of the nonlinear halo bias Smooth function, no characteristic scale Shift of the BAO scale is correctable P(k) vs x(r) Not equivalent in data analysis with finite procedures

Conclusions Nonlinear modeling of the galaxy clustering is crucial for precision cosmology Three main sources of nonlinear effects on LSS Nonlinearity in dynamics Nonlinearity in redshift-space distortions Nonlinearity in halo/galaxy bias Lagrangian picture is useful to elucidate above nonlinear effects (with perturbation theory)