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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.

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Nonlinear Perturbation Theory with Halo Bias and Redshift-space Distortions via the Lagrangian Picture

Taka Matsubara (Nagoya Univ.)


Oct. 27 – 28, 2008, KIAS, Seoul


precision cosmology with galaxy clustering
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
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
Nonlinearity in dynamics
  • Nonlinear dynamics distorts the BAO signature
    • N-bodyexperiments
  • 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
Nonlinearity in redshift-space distortions
  • Redshift-space distortions change the nonlinear effects on BAO
    • P(k): Small-scale enhancement relative to the large-scalepower 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
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
Theories for nonlinear dynamics
  • Recent developments: nonlinearity in dynamics
    • Renormalized perturbation theory and its variants
    • Infinitely higher-order perturbations are reorganized and partially resummed


group method”

Matarrese & Pietroni


“Closure theory”

Taruya & Hiramatsu


“Renormalized perturbation theory”

Crocce & Scoccimarro (2008)

theory for nonlinear halo bias
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
Nonlinear redshift distortions and bias
  • Redshift distortions & bias
    • Standard Eulerianperturbation 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 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:




z : line of sight

the halo bias in the lagrangian picture

1-halo term

2-halo term

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
perturbation theory via the lagrangian picture
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 Lagrangianperturbation theory

diagrammatic representations are useful
Diagrammatic representations are useful
  • Feynman rules
  • Relevant diagrams up to one-loop PT
result nonlinear redshift space distortions
Result: nonlinear redshift-space distortions
  • Comparison of the one-loop PT to a N-body simulation

Linear theory


1-loop SPT

This work

This work


Linear theory

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

result halo bias in redshift space
Result: halo bias in redshift space
  • The one-loop perturbation theory via the Lagrangian picture
    • Nonlinear dynamics + nonlinear halo bias + nonlinear redshift-space distortions



  • 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 BAOscale
    • Scale dependence of the nonlinear halo bias
      • Smooth function, no characteristic scale
      • Shift of the BAO scale is correctable
  • P(k) vsx(r)
    • Not equivalent in data analysis with finite procedures
  • 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)