Statistics of peaks of weakly non-Gaussian random fields: Effects of bispectrum in two- and three-dimensions

Abstract

Analytic expressions for the statistics of peaks of random fields with weak non-Gaussianity are provided. Specifically, the abundance and spatial correlation of peaks are represented by formulas which can be evaluated only by virtually one-dimensional integrals. We assume the non-Gaussianity is weak enough such that it is represented by linear terms of the bispectrum. The formulas are formally given in N-dimensional space, and explicitly given in the case of N = 1, 2, 3. Some examples of peak statistics in cosmological fields are calculated for the cosmic density field and weak lensing field, assuming the weak non-Gaussianity is induced by gravity. The formulas of this paper would find a fit in many applications to statistical analyses of cosmological fields.

Application of peaks theory to the abundance of primordial black holes

“…Whilst we have here assumed Gaussian statistics for the density contrast δ, it has recently been discussed in a number of papers [8,9,16] that δ will not have a Gaussian distribution even if the curvature perturbation ζ does. However, the easiest way to account for this is to note that high peaks in the smoothed Gaussian field correspond to peaks in the smoothed non-Gaussian field [9,16], although the recent paper [24] may be used to describe the statistics of peak of non-Gaussian fields. In order to calculate the abundance and mass function of PBHs, the expression derived here for number density of peaks can be substituted into the existing calculation (for example, that given in [9]) in place of the previous expression for npk (more details for the calculation in the Gaussian case are given in appedix E).…”

“…Let us now consider the integral over the PDF which appears in equation (24). Using equation (C5), then substituting in ν, J1 = (4σ 0 /R 2 σ 2 )ν and ζ 00 into the expressions for ν, x and z gives, after some simple algebra, This is a relatively complicated formula, but in the next section we will discuss the high-peak limit relevant for PBH formation, where the benefit of expressing the PDF in this form will become clear.…”

“…00 | δ D (ζ 00 − αν) = αν. (D2)Substituting this into equation(24), we arrive at the following expression for the number density of high peaks, n hi−pk (ν)…”

Probing the Anisotropy and Non-Gaussianity in the Redshift Space through the Conditional Moments of the First Derivative

“…In the high-precision cosmology era, drastic attention should be paid to the various robust measures constructed for extracting information from random cosmological fields as accurately as possible, particularly from large-scale structures of the matter distribution in the universe (Bernardeau et al 2002;Peebles 2020). On the other hand, discrepancies between what we observe through various surveys and theoretical counterparts essentially persuade researchers to include the stochastic notion (Kaiser 1984;Bardeen et al 1986;Bernardeau et al 2002;Matsubara 2003;Codis et al 2013;Matsubara 2020). It is supposed that on sufficiently large scales, the distribution of galaxies in the real space is homogeneous and isotropic, while such an assumption is no longer satisfied in the redshift space when the position of structures is plotted as a function of the redshift rather than their distances.…”

The integrated perturbation theory for cosmological tensor fields I: Basic formulation

“…The procedure of deriving the non-Gaussian corrections are found in Ref. [23,70], which is straightforward to apply in this case. However, for the illustrative examples below of this paper, we do not need to include the non-Gaussian corrections in the evaluations of renormalized bias functions.…”