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

35min

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)

(SDSS survey)• DE is constrained by 1D scale: 3/1 2

A2

V )()()1()( zHczzDzzD

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

Power spectrum,N-body & 1-loop PTMeiksin et al. (1999)

Power spectrum,large N-body simulationSeo et al. (2008)

Correlation function,large N-body simulationEisenstein et al. (2007)

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, Sanchez et al. (2008)N-body, Angulo et al. (2005)

Theories for nonlinear dynamics• Recent developments: nonlinearity in dynamics

– Renormalized perturbation theory and its variants– Infinitely higher-order perturbations are reorganized and partially

resummed

“Renormalized perturbation theory” Crocce & Scoccimarro (2008)

“Renormalizationgroup method”Matarrese & Pietroni(2008)

“Closure theory”Taruya & Hiramatsu(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

Smith et al. (2007) Jeong & Komatsu (2008)

} both in real space

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 !!

q

),( tqΨ : displacement vector

： initial position

),( tqx : 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:

x

z : line of sight

vz/(aH)

s

xs

det

11 s

The halo bias in the Lagrangian picture• Halo bias

– (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

)( )( 3L

Lagrangianhalo

3Eulerhalo qΨqxqx nqd

1~~

4L2L1

2halo1halo2halo

2213

halo0qk0qqkk iii enn

nddqedP

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

(Points from N-body simulation of Eisenstein & Seo 2005)

Linear theory

1-loop SPT

N-bodyThis work

This work

N-body

Linear theory

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

– 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)