New Parametrization for the Scale Dependent Growth Function in General Relativity

TL;DR

This paper introduces a new scale-dependent parametrization for the growth function of cosmological perturbations in General Relativity, improving accuracy on large scales compared to traditional models.

Contribution

It derives a generalized growth function parametrization that accounts for scale dependence, fitting full relativistic results up to horizon scales.

Findings

The new parametrization accurately models growth on horizon scales.

Traditional scale-independent models are insufficient beyond 5% of the horizon scale.

The approach applies to both DM and dynamical dark energy models.

Abstract

We demonstrate the scale dependence of the growth function of cosmological perturbations in dark energy models based on General Relativity. This scale dependence is more prominent on cosmological scales of $100h^{-1}Mpc$ or larger. We derive a new scale dependent parametrization which generalizes the well known Newtonian approximation result $f_0(a)\equiv \frac{d\ln \delta_0}{d\ln a}=\om(a)^\gamma$ ($\gamma ={6/11}$ for \lcdm) which is a good approximation on scales less than $50h^{-1}Mpc$. Our generalized parametrization is of the form $f(a)=\frac{f_0(a)}{1+\xi(a,k)}$ where $\xi(a,k)=\frac{3 H_{0}^{2} \omm}{a k^2}$. We demonstrate that this parametrization fits the exact result of a full general relativistic evaluation of the growth function up to horizon scales for both \lcdm and dynamical dark energy. In contrast, the scale independent parametrization does not provide a good fit on scales beyond 5% of the horizon scale ($k\simeq 0.01 h^{-1}Mpc$).