How is the horizon length related to the power law spectrum

[thinkLamp]

Homework Statement

For a power spectrum density fluctuations $P(k) \propto k^n$, I need to find the scaling (with respect to $a$) of the horizon wavenumber $\frac{2\pi}{\chi_H}$ in a matter dominated universe in terms of $n$. $\chi_H(a)$ is the evolving particle horizon, in a flat universe.

Homework Equations

$$
\chi_H = \int_0^t \frac{c \; \mathrm{d}t'}{a(t')} \;,
$$

The Attempt at a Solution

I know that the horizon distance is the comoving radius of the particle horizon

$$
\chi_H = \int_0^t \frac{c \; \mathrm{d}t'}{a(t')} \;,
$$

which I can also write as

$$
\chi_H = \int_0^a \frac{\mathrm{d}a'}{a'} \left( \frac{8 \pi G \rho(a') a'^2}{3 c^2} - K \right)^{-1/2}\;,
$$
and that the matter-dominated universe part is expressed in terms of $\rho(a')$'s scaling with $a$. But I'm not sure how this relates to the power law spectrum of density fluctuations.

How is the power spectrum related to $\chi_H$?

 

[member 553083]

Is there a specific way to express the scaling of $\chi_H$ with respect to $a$ and $n$? Any help is appreciated!