- #1

thinkLamp

- 16

- 0

## 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##?