Why is the Primordial Power Spectrum Defined as P(k) = (k^3)/(2π^2)|w_k|^2?

[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.

 

[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?

 

[shinobi20]

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?

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

 

[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.