An alternative approach to the Finger of God in large scale structures

TL;DR

This paper proposes a method to accurately map the two-point correlation function from real to redshift space, revealing that the Finger of God effect can be explained by mapping artifacts rather than solely non-linear dynamics.

Contribution

It introduces a way to compute the real-to-redshift space mapping function, showing that the Finger of God can arise from approximation errors rather than purely non-linear effects.

Findings

The observed structure depends on parameters $eta$, $\gamma$, and $AP$.

Proper mapping can distinguish between cosmological and gravitational redshift effects.

FoG-like features can appear in linear theory due to mapping artifacts.

Abstract

It is generally accepted that linear theory of growth of structure under gravity produces a squashed structure in the two-point correlation function (2PCF) along the line of sight (LoS). On the other hand, the observed radial spread out structure known as Finger of God (FoG) is attributed to non-linear effects, like virial relaxation or random motions in the inner regions of galaxy clusters, or non-linear terms in collapse theory. In this paper we argue that the squashed structure associated with the redshift-space ($s-$) linear theory 2PCF is obtained only when this function is displayed in real-space ($r-$), or when the mapping from $r-$ to $s-$space is trivialized by one of a series of approximations: the "wide separation" sometimes mistaken by the "distant observer", the $\mu(r) \sim \mu(s)$ in the cosine of the position vector with the LoS, or plainly by approximating $s \sim r$. To display the $s-$space 2PCF properly in a grid in $s-$space requires one to solve for the mapping function $\bm s(\bm r)$ and its inverse, which exists in the small gravitational disturbance case. We find this mapping by using plane of the sky projections of the $r-$ and $s-$ 2PCFs in the simplest case of the linear Kaiser spectrum with a conservative power-law $r-$space 2PCF. We show that even in this simple case, a structure quite similar to the FoG is observed in the small scale region, while in the large scale the expected squashed structure is obtained. We show that the observed structure is a function of three parameters: the dimensionless growth rate for visible matter ($\beta$), the power-law exponent ($\gamma$) and the cosmological distortion given by the Alcock-Paczy{\'n}ski parameter ($AP$). We conclude that there appears to be the way of discerning between cosmological and gravitational redshift distortions avoiding known degeneracies.