biasing: what is the consistent formulation in quasi- nonlinear...

Biasing:What is the consistent formulation in quasi-

nonlinear regime

Taka Matsubara (Nagoya U.)

@Cahill Center, Caltech2010/10/8

TM 2010 in prep.

2010年10月19日火曜日

Biasing and Galaxy Survey

• For the precision cosmology with large-scale galaxy surveys, we should identify the effects of biasing

• Linear regime: linear bias is probably OK

• But, nonlinear effects are not negligible in precision cosmology even on large scales

• How can we characterize the biasing effects in nonlinear regime?


Biasing and Galaxy Surveys

• In quasi-nonlinear regime probed by large-scale galaxy surveys, the nonlinear perturbation theory should be quite useful.

• Let’s consider how we should formulate the galaxy biasing in quasi-nonlinear regime, using the nonlinear perturbation theory


Bias between mass and objects

• Mass density 　number density in general

• Densities of both mass and astronomical objects are determined by initial density field

• There should be a relation between them


Eulerian Local Bias model

• Local bias: A simple model usually considered with the perturbation theory

• The number density is assumed to be locally determined by (smoothed) mass density

• Apply the Taylor expansion

• Phenomenological model, just for simplicity


Eulerian Local bias is not physical

initial density field

linear evolution: local

mass density fieldnumber density field

linear density field

nonlinear evol.: nonlocal

galaxy formation: local or nonlocal nonlinear evol.: nonlocal

Nonlocal relation in generalLocal only in linear regime & local galaxy formation


Nonlocal biasing

• The local biasing with (Eulerian) nonlinear perturbation theory involves an inconsistent picture

• Local approximation is good in limited cases

• tree-level PT, high-peak limit,...

• For a consistent picture, nonlocal formulations of bias model is necessary


Nonlocal bias• The number density is a nonlocal

functional of the mass density field, the generic expansion is:

Fry & Gaztanaga (1993)


Nonlocal bias in Fourier space

• In Fourier space, the expansion reduces to:

• “Nonlocal bias => Scale-dependent bias” in Fourier space

• c.f.) local bias => scale-independent bias

• Scale-dependent bias naturally arise in generally nonlocal biasing

Scherrer & Weinberg (1998)


Perturbation theory with nonlocal bias

• Perturbative expansions in Fourier space:

• nonlocal bias:

• nonlinear dynamics

• It is straightforward to calculate observables, such as power spectrum, bispectrum, etc.


The problem in Eulerian bias

• The problem is: We don’t have a physical model of Eulerian nonlocal bias !!

• Physical models of bias known so far is provided in Lagrangian space

• e.g., Halo bias model, Peak bias model,...

• In those models, conditions of galaxy formation are imposed on initial (Lagrangian) density field

• What is the relation between Eulerian bias and Lagrangian bias?


Eulerian bias and Lagrangian bias

mass density fieldnumber density field

linear density field

nonlinear evol.

Lagrangian bias

nonlinear evol.

Eulerianbias

In nonlocal bias scheme, Eulerian and Lagrangian biases are equivalent, only representations are different


Eulerian and Lagrangian bias• Equivalence of Eulerian and Lagrangian

nonlocal bias

• Nonlocal Eulerian and Lagrangian biases are equivalent. Only representations are different

• The relations can be explicitly derived in perturbation theory:

local biases are incompatible! At least one must be nonlocal


Redshift-space distortions

• Perturbation theory and redshift-space distortions

• In the framework of standard perturbation theory, redshift-space distortions are straightforwardly included

• Including the nonlocal bias in the above method is also possible both in Eulerian and Lagrangian PT

e.g., Scoccimarro, Couchman & Frieman (1999) for the Eularian local biasing


Application:Effects of halo bias on BAO

• Apply halo bias (local Lagrangian bias)

• redshift-space distortions also included

TM (2008)


Application:Scale-dependent bias and prim.nG

real space:comparison with simple formula

redshift space

• Previous methods are not accurate enough

!! PRELIMINARY

!!

is accurate only in a high-peak limit

Figs removed from

the

Presentation


Summary• Consistent inclusion of bias in Eulerian PT is

possible only in the framework of nonlocal biasing scheme

• Eulerian bias is scale-dependent in general

• The scale-dependence comes from the nonlocality of biasing

• Perturbative relations between Eulerian and Lagrangian biases are derived

• both biasing schemes are compatible only in nonlocal framework


biasing: what is the consistent formulation in quasi- nonlinear...

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