Power Spectrum

This post goes through calculating the matter power spectrum. I found the notes by Cui et al. 2008 useful.

Theoretical Power Spectrum

The power spectrum can be written as:

\[P(k,a) = A k^n T^2(k) D^2(z)\]

where \(k\) is the wavenumber, \(T\) is the transfer function, and \(D\) is the growth factor and \(A\) is the amplitude.

I calculated \(A\) from calculating \(\sigma(8 h^{-1}\, {\rm Mpc})\) as in this post (note that there is a factor of \(1/(2 \pi^2)\) that I absorbed into the constant \(A\) in the previous post),

\(T\) was calculated from Equation 29 of Eisenstein & Hu 1998.

\(D(z)\) is equal to:

\[D(z)= A_0 E(z) \int_{z}^{\infty} \dfrac{(1+z')}{E^{3}(z')} d z'\]

Assuming a flat universe, \(E(z)=\sqrt{\Omega_{\Lambda}+\Omega_{M}(1+z)^{3}}\), and setting \(D(0)=1\), \(A_0=\left(E(z) \int_{0}^{\infty} \frac{(1+z')}{E^{3}(z')} d z' \right)^{-1}\)

Power Spectrum From \(N\)-Body Simulation

The power spectrum can be calculated from the overdensities:

\[\delta(x) = \dfrac{\rho(x)-\bar{\rho}}{\bar{\rho}}\]

I binned the particles in 3D to calculate the density—this corresponds to nearest grid point (NGP) method.

The power spectrum is \(P(k) = \langle \mid \delta_k \mid^2 \rangle\). However, this is for continuous functions. You need to correct for (1) the shot noise and (2) the window function.

The steps I followed are:

(1) find the density on a grid, \(\mathbf{r}_g\)

(2) calculate the density contrast at every position $\mathbf{r}_g$

(3) take the FFT, and calculate \(\mid\delta_k\mid^2\) in each cell

(4) bin \(\mid\delta_k\mid^2\) in \(k\)

(5) find the average in each bin

(6) subtract \(1/N\): this should account for the shot noise (this is effectively zero though, so doesn’t matter)

(7) multiply by the volume, and divide by \(N^2\) (I think this is the right normalization?)

Results

Here are my results:

The simulation does look like it is above the theoretical power spectrum, but it is very possible I messed up the normalization on the theoretical and/or the measured power spectrum.

I want to generate a couple more realizations of the ICs and see if they look closer to the theoretical curve.

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