# 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

\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

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

\delta_m=C_1 H+C H\int^a_0 \frac{1}{a’^3H(a’)^3}\mathrm da’

Since $$H$$ is decaying, drop $$C_1 H$$, then

\delta_m=C H\int^a_0 \frac{1}{a’^3H(a’)^3}\mathrm da’

Apply initial condition $$\delta_m(a_{i})=\delta_{i}$$

C=\frac{\delta_i}{H(a_i)}\frac{1}{\int^{a_i}_0\frac{1}{a’^3H(a’)^3}}\mathrm d a’

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

C=\delta_i \frac {5\Omega_{m0}H_0^2}{2} \frac1{D_+(a_i)}

With this, we have

\delta_m(a)=\delta_i \frac{D_+(a)}{D_+(a_i)}

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