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The effect of primordial non-Gaussianity on halo bias

Sabino Matarrese, Licia Verde

TL;DR

This work derives an analytic, peak-theory based halo bias formula for primordial non-Gaussianity, using high-peak and large-separation approximations to relate halo clustering to the underlying density field. Focusing on a local f_NL model and its generalizations through the transfer function M_R(k) and a form factor F_R(k), it yields a scale-, mass-, and redshift-dependent correction to the halo power spectrum: Delta P_h(k) = b_{0,L}^2 4 f_NL Delta_c P_{phi delta}(k) F_R(k), leading to a scale-dependent bias b_h^{f_NL} = 1 + [Delta_c(z) / (sigma_R^2 D^2(z))] [1 + 2 f_NL (Delta_c(z)/D(z)) (F_R(k)/M_R(k))]. The results highlight that primordial non-Gaussianity imprints a pronounced large-scale bias, especially for massive halos, and the framework can be generalized to non-local NG models, providing a practical path to constrain f_NL with future large-volume surveys.

Abstract

It has long been known how to analytically relate the clustering properties of the collapsed structures (halos) to those of the underlying dark matter distribution for Gaussian initial conditions. Here we apply the same approach to physically motivated non-Gaussian models. The techniques we use were developed in the 1980s to deal with the clustering of peaks of non-Gaussian density fields. The description of the clustering of halos for non-Gaussian initial conditions has recently received renewed interest, motivated by the forthcoming large galaxy and cluster surveys. For inflationary-motivated non-Gaussianites, we find an analytic expression for the halo bias as a function of scale, mass and redshift, employing only the approximations of high-peaks and large separations.

The effect of primordial non-Gaussianity on halo bias

TL;DR

This work derives an analytic, peak-theory based halo bias formula for primordial non-Gaussianity, using high-peak and large-separation approximations to relate halo clustering to the underlying density field. Focusing on a local f_NL model and its generalizations through the transfer function M_R(k) and a form factor F_R(k), it yields a scale-, mass-, and redshift-dependent correction to the halo power spectrum: Delta P_h(k) = b_{0,L}^2 4 f_NL Delta_c P_{phi delta}(k) F_R(k), leading to a scale-dependent bias b_h^{f_NL} = 1 + [Delta_c(z) / (sigma_R^2 D^2(z))] [1 + 2 f_NL (Delta_c(z)/D(z)) (F_R(k)/M_R(k))]. The results highlight that primordial non-Gaussianity imprints a pronounced large-scale bias, especially for massive halos, and the framework can be generalized to non-local NG models, providing a practical path to constrain f_NL with future large-volume surveys.

Abstract

It has long been known how to analytically relate the clustering properties of the collapsed structures (halos) to those of the underlying dark matter distribution for Gaussian initial conditions. Here we apply the same approach to physically motivated non-Gaussian models. The techniques we use were developed in the 1980s to deal with the clustering of peaks of non-Gaussian density fields. The description of the clustering of halos for non-Gaussian initial conditions has recently received renewed interest, motivated by the forthcoming large galaxy and cluster surveys. For inflationary-motivated non-Gaussianites, we find an analytic expression for the halo bias as a function of scale, mass and redshift, employing only the approximations of high-peaks and large separations.

Paper Structure

This paper contains 5 sections, 16 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Halos as peaks of the density field
  3. Application to a local non-Gaussian model
  4. Results
  5. Discussion and conclusions

Figures (3)

  • Figure 1: The function ${\cal F}_R(k)$ for three different masses: $1\times 10^{14}$ M$_{\odot}$ (solid), $2\times 10^{14}$ M$_{\odot}$ (dotted), $1\times 10^{15}$ M$_{\odot}$ (dashed).
  • Figure 2: The redshift dependence of $\Delta b_h/b_h$.
  • Figure 3: The scale dependence of $\Delta b_h/b_h$ for three different masses: $1\times 10^{14}$ M$_{\odot}$ (solid), $2\times 10^{14}$ M$_{\odot}$ (dotted), $1\times 10^{15}$ M$_{\odot}$ (dashed).