The matter power spectrum is a statistical description of the spatial distribution of matter density fluctuations in the Universe. It quantifies how the variance of the matter overdensity field, $\delta(\mathbf{x}) = \frac{\rho(\mathbf{x}) - \bar{\rho}}{\bar{\rho}}$, is distributed as a function of spatial scale, or equivalently as a function of wavenumber $k$ in Fourier space. Formally, the power spectrum $P(k)$ is defined by
$$ \langle \tilde{\delta}(\mathbf{k}) \tilde{\delta}^{*}(\mathbf{k}') \rangle = (2\pi)^3 , \delta_{\mathrm{D}}(\mathbf{k}-\mathbf{k}') , P(k), $$
where $\tilde{\delta}(\mathbf{k})$ is the Fourier transform of $\delta(\mathbf{x})$, $\langle \cdot \rangle$ denotes an ensemble average, and $\delta_{\mathrm{D}}$ is the Dirac delta function. The power spectrum is thus the Fourier counterpart of the two‑point correlation function $\xi(r)$, linked via a Fourier transform.
Physical significance
- Structure formation: In the framework of linear perturbation theory, the shape and amplitude of $P(k)$ determine the growth of cosmic structures from primordial fluctuations to the present‑day distribution of galaxies, clusters, and the intergalactic medium.
- Cosmological parameters: The matter power spectrum depends on fundamental parameters such as the total matter density $\Omega_m$, the baryon density $\Omega_b$, the spectral index $n_s$ of the primordial perturbations, the Hubble constant $H_0$, and the nature of dark energy. Measurements of $P(k)$ therefore provide constraints on these quantities.
- Features: Characteristic features include a turnover at $k \sim 0.02,h,\text{Mpc}^{-1}$ corresponding to the horizon size at matter–radiation equality, baryon acoustic oscillations (BAO) that appear as a series of small amplitude wiggles, and a small‑scale suppression due to processes such as free‑streaming of massive neutrinos or the effects of dark matter particle physics.
Theoretical modeling
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Linear regime: For scales larger than a few tens of megaparsecs ($k \lesssim 0.1,h,\text{Mpc}^{-1}$), density fluctuations remain small ($|\delta| \ll 1$) and evolve linearly. The linear power spectrum can be written as
$$ P_{\mathrm{lin}}(k, z) = P_{\mathrm{prim}}(k) , T^2(k) , D^2(z), $$
where $P_{\mathrm{prim}}(k) \propto k^{n_s}$ is the primordial spectrum, $T(k)$ is the transfer function that encodes the physical processes affecting perturbations before recombination, and $D(z)$ is the linear growth factor.
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Non‑linear regime: On smaller scales ($k \gtrsim 0.1,h,\text{Mpc}^{-1}$), gravitational collapse leads to non‑linear evolution. Semi‑analytic approaches (e.g., halo model, fitting formulas such as HALOFIT) and numerical N‑body simulations are employed to predict the non‑linear power spectrum.
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Baryonic effects: Hydrodynamical processes (e.g., gas cooling, star formation, feedback from supernovae and active galactic nuclei) modify the matter distribution on scales of a few megaparsecs, requiring either calibrated corrections or dedicated simulations to incorporate these effects into $P(k)$.
Observational determination
- Galaxy redshift surveys: Large‑scale surveys (e.g., SDSS, BOSS, DESI) map the three‑dimensional positions of galaxies, allowing estimation of the galaxy power spectrum $P_g(k)$. Bias models relate $P_g(k)$ to the underlying matter power spectrum.
- Weak gravitational lensing: Measurements of coherent distortions in the shapes of background galaxies trace the projected matter distribution, providing constraints on the matter power spectrum integrated along the line of sight.
- Lyman‑$\alpha$ forest: Absorption features in quasar spectra probe the intervening intergalactic medium, offering sensitivity to the matter power spectrum at high redshift and small scales.
- Cosmic microwave background (CMB) lensing: The CMB temperature and polarization fields are distorted by intervening mass, and reconstruction of the lensing potential yields a measurement of the matter power spectrum at redshifts $z \sim 2$.
Units and conventions
The power spectrum is commonly expressed in units of $(\text{Mpc}/h)^3$. In practice, the dimensionless quantity
$$ \Delta^2(k) = \frac{k^3}{2\pi^2} P(k) $$
is used to represent the contribution to the variance per logarithmic interval in $k$.
Current status
High‑precision observations have measured the matter power spectrum over a wide range of scales and redshifts, confirming the predictions of the spatially flat $\Lambda$CDM cosmological model to within a few percent. Ongoing and upcoming surveys (e.g., Euclid, the Rubin Observatory LSST, the Nancy Grace Roman Space Telescope) aim to improve measurements of $P(k)$ and thereby tighten constraints on dark energy, neutrino masses, and possible deviations from General Relativity.