1. 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

2. 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)

3. 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

4. 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)

5. 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)

6. 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)

7. 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)

8. 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)

9. 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

10. 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

11. 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

12. 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

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

14. 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)

15. 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)

16. 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

17. 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)