NumCosmoGrowthFunc

Growth function of linear perturbations.

This object implements the integration of second order differential equation (ODE) for the matter (baryons + cold dark matter: $\Omega_m$, see nc_hicosmo_E2Omega_m() and nc_hicosmo_E2()) density contrast, $\delta$, in the linear regime of perturbations, see for instance [Matínez and Saar (2002)][XMartinez2002]. The equation is given by \begin{equation} \ddot{ \delta } + 2 \frac{\dot{a}}{a} \dot{ \delta } - 4\pi G \bar{\rho}(a)\delta = 0, \end{equation} where, $a$ is the scale factor of the universe, $G$ is the universal gravitational constant, $\bar{\rho}$ is the mean matter density at $a$, and the derivatives are taken with respect to the cosmic time, $t$. By changing the variable from cosmic time to $x=(1+z)$, the ODE becomes, \begin{equation}\label{eq:mov} \delta” + \left( \frac{E’(x)}{E(x)} - \frac{1}{x} \right) \delta’ - \frac{3}{2} \frac{\Omega_{m}(x)}{x^2}\delta = 0. \end{equation} Where $’$ denote derivatives with respect to $x$ and $\Omega_{m}(x)$ is the matter density as a function of the redshift $z$, \begin{equation} \Omega_{m}(z) = \frac{(1+z)^3}{E(z)^{2}} \Omega_{m,0} \,\, , \end{equation} and $E$ is the normalized Hubble function [nc_hicosmo_E()].

The ODE initial conditions are defined at $a_i / a_0 = 10^{-12}$ (NcGrowthFunc:x-i), where the universe is well approximated by a radiation and matter model. Therefore, within this assumption the growing mode is simply \begin{equation} \delta(a) \propto 1 + \frac{3\Omega_{m,0}}{2\Omega_{r,0}}\frac{a}{a_0}, \end{equation} Note that it was chosen a large enough redshift ($z \approx 10^{12}$) such that it is safe to assume that the dark energy component and curvature are negligible.

As usual, the growth function is set to unit at the present time, $D(a_0) = 1$.