Matter power spectrum

Development according to gravitational expansion

If the autocorrelation function describes the probability of a galaxy at a distance ${\displaystyle r}$ from another galaxy, the matter power spectrum decomposes this probability into characteristic lengths, ${\displaystyle k\approx 2\pi /L}$, and its amplitude describes the degree to which each characteristic length contributes to the total over-probability.

The overall shape of the matter power spectrum is best understood in terms of the linear perturbation theory analysis of the growth of structure, which predicts to first order that the power spectrum grows according to:

$${\displaystyle P(\mathbf{k},t)=D_{+}^2(t)P(\mathbf{k},t_0)=D_{+}^2(t)P_0(\mathbf{k})}$$

Where ${\displaystyle D_{+}(t)}$ is the linear growth factor in the density, that is to first order ${\displaystyle \delta (r,t)=D_{+}(t){\delta }_0(r)}$, and ${\displaystyle P_0(\mathbf{k})}$ is commonly referred to as the primordial matter power spectrum. Determining the primordial ${\displaystyle P_0(\mathbf{k})}$ is a question that relates to the physics of inflation.

The simplest ${\displaystyle P_0(\mathbf{k})}$ is the Harrison–Zeldovich spectrum (named after Edward R. Harrison and Yakov Zeldovich), which characterizes ${\displaystyle P_0(\mathbf{k})}$ according to a power law, ${\displaystyle P_0(\mathbf{k})=Ak}$. More advanced primordial spectra include the use of a transfer function which mediates the transition from the universe being radiation dominated to being matter dominated.

The broad shape of the matter power spectrum is determined by the growth of large-scale structure, with the turnover (the point where the spectrum goes from increasing with k to decreasing with k) at ${\displaystyle k_{eq}\approx 2\times {10}^{-2}h{\text{Mpc}}^{-1}}$, corresponding to ${\displaystyle {\lambda }_{eq}=350h^{-1}\text{Mpc}}$ (where h is the dimensionless Hubble constant). The co-moving wavenumber corresponding to the maximum power in the mass power spectrum is determined by the size of the cosmic particle horizon at the time of matter-radiation equality, and therefore depends on the mean density of matter and to a lesser extent on the number of neutrino families (${\displaystyle N\ge 3}$), ${\displaystyle k_{\text{eq}}=(2{\mathrm{\Omega }}_M{H}_0^2{z}_{\text{eq}}/c^2)=7.3\times {10}^{-2}{\mathrm{\Omega }}_M{h}^2{\text{Mpc}}^{-1}}$, for ${\displaystyle T_{\text{cmb}}=2.728\mathrm{K}}$. The ${\displaystyle P_0(k)}$ at smaller k (equivalently, larger scales) corresponds to scales which were larger than the particle horizon at the time of the transition from the regime of radiation dominance to that of matter dominance. At linear order in perturbations ${\displaystyle \delta }$, the power spectrum's broad shape follows

$${\displaystyle P(\mathbf{k}\mathbf{,}\mathbf{t}\mathbf{)}\propto \{\begin{array}{ll}{k}^{n_s}& k\ll k_{eq}\\ k^{n_s-1}& k\gg k_{eq}\end{array}\mathbf{,}}$$

where ${\displaystyle n_s\simeq 1}$ is the scalar spectral index.