I Primordial Power Spectrum

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1. Nov 22, 2016

shinobi20

Why is the power spectrum defined as
$P(k) = \frac{k^3}{2π^2} |w_k|^2$
where $w_k$ is the mode function?

Cosmology books and papers just states that it is defined that way but there are no details on why.

2. Nov 23, 2016

bapowell

The power spectrum is defined as the Fourier transform of the correlation function of density perturbations. Does this help? Can you write down the expression of the correlation function of density (of curvature) perturbations?

3. Nov 23, 2016

shinobi20

Let $ξ(r)$ be the two point correlation function, then as you said, we define (why?) the power spectrum as

$P(k) = C∫ ξ(r) e^{-ikx} d^3r$

where C is a constant

4. Nov 24, 2016

bapowell

Because the power spectrum tells you how the various frequencies contribute to the spatial density perturbation. That's generally what the Fourier transform tells you.