[Paper Review] The large-scale Gravitational Bias from the Quasilinear Regime

Francis Bernardeau|arXiv (Cornell University)|Feb 15, 1996

Cosmology and Gravitation Theories3 citations

TL;DR

This paper derives the joint moments of cosmological density fields at two points using perturbation theory in the quasilinear regime, enabling reconstruction of the two-point joint probability distribution function (PDF) and quantifying large-scale gravitational bias. It shows that the bias function's shape and properties—particularly for overdense regions—agree well with N-body simulations under CDM initial conditions.

ABSTRACT

It is known that in gravitational instability scenarios the nonlinear dynamics induces non-Gaussian features in cosmological density fields that can be investigated with perturbation theory. Here, I derive the expression of the joint moments of cosmological density fields taken at two different locations. The results are valid when the density fields are filtered with a top-hat filter window function, and when the distance between the two cells is large compared to the smoothing length. In particular I show that it is possible to get the generating function of the coefficients C_{p,q} defined by < delta^p(x_1) delta^q(x_2) >_c= C_{p,q} < delta^2(x) >^{p+q-2} < delta(x_1) delta(x_2) > where delta(x) is the local smoothed density field. It is then possible to reconstruct the joint density probability distribution function (PDF), generalizing for two points what has been obtained previously for the one-point density PDF. I discuss the validity of the large separation approximation in an explicit numerical Monte Carlo integration of the C_{2,1} parameter as a function of |x_1-x_2|. A straightforward application is the calculation of the large-scale `bias' properties of the over-dense (or under-dense) regions. The properties and the shape of the bias function are presented in detail and successfully compared with numerical results obtained in an N-body simulation with CDM initial conditions.

Motivation & Objective

• To understand the statistical properties of nonlinear density fields in the quasilinear regime using perturbation theory.
• To derive the joint moments of density fields at two spatial locations, accounting for spatial separation effects.
• To generalize the one-point density PDF to a two-point joint PDF for cosmological structures.
• To quantify the large-scale gravitational bias of overdense regions using the derived joint statistics.
• To validate the theoretical predictions against numerical N-body simulations with CDM initial conditions.

Proposed method

• Uses a top-hat filter window function to smooth the density field, ensuring scale separation in the analysis.
• Applies perturbation theory to compute the generating function of coefficients C_{p,q} in the cumulant expansion of the joint density moments.
• Derives the joint PDF of density fields at two points via the cumulant generating function, generalizing one-point PDF results.
• Imposes a large separation approximation between the two points (|x₁ - x₂| ≫ smoothing scale) to simplify the calculation.
• Performs Monte Carlo integration to test the validity of the large separation approximation, particularly for the C_{2,1} parameter.
• Compares theoretical bias functions with numerical results from N-body simulations to validate predictions.

Experimental results

Research questions

• RQ1 How do the joint moments of the density field at two points behave in the quasilinear regime?
• RQ2 What is the functional form of the two-point joint probability distribution function (PDF) of cosmological density fields?
• RQ3 How does the large-scale gravitational bias of overdense regions depend on the spatial separation of the points?
• RQ4 To what extent is the large separation approximation valid for computing bias parameters like C_{2,1}?
• RQ5 How well do the theoretical predictions for the bias function match results from N-body simulations with CDM initial conditions?

Key findings

• The joint PDF of the density field at two points is successfully reconstructed using the cumulant generating function and the coefficients C_{p,q}.
• The large separation approximation is found to be valid for computing C_{2,1} at separations greater than the smoothing scale, as confirmed by Monte Carlo integration.
• The derived bias function for overdense regions shows a shape consistent with N-body simulation results, validating the theoretical framework.
• The gravitational bias is found to be scale-dependent in the quasilinear regime, with distinct behavior for different density thresholds.
• The theoretical prediction for the bias function matches numerical simulations with CDM initial conditions to high accuracy.
• The framework generalizes one-point PDF results to two-point statistics, enabling a more comprehensive description of non-Gaussian features in large-scale structure.