Matter Perturbations and Growth Function

This post gives a simple derivation of the matter perturbation evolution at late times and the growth function.

In this article, prime (‘) stands for the derivative with time \(t\), over dot stands for the derivative with scale factor \(a\).

Use Navier Stokes equations for matter (with pressure \(p=0\)) as the basic equation set.

Using \(\delta_m=\frac{\rho_m-\bar\rho_m}{\bar\rho_m}\), the equations become
\begin{eqnarray}
\frac{\partial {\delta_m}}{\partial t}+a^{-1}\vec{v}_m\cdot \delta_m &=& -a^{-1}(1+\delta_m)\nabla\cdot\vec{v}_m \\
\frac{\partial \vec{v}_m}{\partial t}+ a^{-1}(\vec{v}_m\cdot \nabla)\vec v_m &=& -\frac{\nabla\Phi}{a}-H\vec v_m \\
a^{-2}\nabla^2\Phi &=& 4\pi G\bar\rho_m\delta_m
\end{eqnarray}

In these equations, \(\Phi\) is the gravitational potential.

This is too complex to solve. So we only use the linear approximation,that is, no second or higher order of \(\delta_m\), \(\vec v_m\), \(\Phi\) appear, because we already use those as perturbations.
Finally,
\begin{eqnarray}
\frac{\partial {\delta_m}}{\partial t}&=&-a^{-1}\nabla \cdot\vec{v}_m \\
\frac{\partial \vec{v}_m}{\partial t}&=& -\frac{\nabla\Phi}{a}-H\vec v_m \\
a^{-2}\nabla^2\Phi &=& 4\pi G\bar\rho_m\delta_m
\end{eqnarray}

Easily, one can use the well known trick for this kind of equations to find the second derivative equation for \(\delta_m\), or the basic function for this problem,
\begin{eqnarray}
\delta_{m}”+2H\delta_m’-4\pi G \bar\rho_m \delta_m=0
\end{eqnarray}

However, we usually use scale factor \(a\) as a measurement of time. Thus we would transform it the following form
\begin{equation}
\ddot \delta_m+(\frac{\mathrm d\ln{H}}{\mathrm da}+\frac{3}{a})-\frac{3\Omega_{m0}H_0^2}{2a^5H^2}\delta_m =0
\end{equation}

Almost each book of ODE will tell us how to solve such an equation.

After a little work, we will get two special solutions:
\begin{eqnarray}
\delta_{m,1}&=&H \\
\delta_{m,2}&=&H\int^a_0 \frac1{a’^3H(a’)^3}\mathrm d a’
\end{eqnarray}

General solution is
\begin{equation}
\delta_m=C_1 H+C H\int^a_0 \frac{1}{a’^3H(a’)^3}\mathrm da’
\end{equation}

Since \(H\) is decaying, drop \(C_1 H\), then
\begin{equation}
\delta_m=C H\int^a_0 \frac{1}{a’^3H(a’)^3}\mathrm da’
\end{equation}

Apply initial condition \(\delta_m(a_{i})=\delta_{i}\)
\begin{equation}
C=\frac{\delta_i}{H(a_i)}\frac{1}{\int^{a_i}_0\frac{1}{a’^3H(a’)^3}}\mathrm d a’
\end{equation}

To simplify these expressions, we now define \(D_+=\frac 5 2 \Omega_{m0} H(a) \int^a_0\frac{1}{a’^3H(a’)^3}\mathrm d a’\). (One might be confused with this defination at first. The constant multiplied the the original part is to ensure one condition: at matter domination,\(D_+\) is \(a\). This condition endow \(D_+\) with more physics.) Then
\begin{equation}
C=\delta_i \frac {5\Omega_{m0}H_0^2}{2} \frac1{D_+(a_i)}
\end{equation}

With this, we have
\begin{equation}
\delta_m(a)=\delta_i \frac{D_+(a)}{D_+(a_i)}
\end{equation}

Actually, this \(D_+(a)\) is a growing mode of the equation, and we all call it growth function.