I Fourier transform of the density fluctuation

[happyparticle]

TL;DR

Computing the Fourier transform of the density fluctuation.

There is a Fourier transform that I don't really understand in my textbook.

I have the following equation:
$\ddot{\delta} + 2H\dot{\delta} -\frac{3}{2} \Omega_m H^2 \delta = 0$

Then using the Fourier transform:
$\delta_{\vec{k}} = \frac{1}{V} \int \delta(\vec{r}) e^{i \vec{k} \cdot \vec{r}} d^3 r$

Where $\delta(\vec{r})$ is the density fluctuation.

We get
$\ddot{\delta_{\vec{k}}} + 2H\dot{\delta_{\vec{k}}} -\frac{3}{2} \Omega_m H^2 \delta_{\vec{k}} = 0$


The only function that does that is a gaussian function, I guess. I don't understand the process here.

Thank you

 

[PeterDonis]

my textbook

Which textbook?

 

[happyparticle]

Which textbook?

Introduction to Cosmology second edition by Barbara Ryden (p.215-219)