Structure Formation with Warm White Noise: Effects of Finite Number Density and Velocity Dispersion in Particle and Wave Dark Matter

Abstract: We investigate the evolution of density perturbations in dark matter with finite number density and velocity dispersion. Using a truncated BBGKY hierarchy, we derive analytical expressions for the dark matter power spectrum during radiation and matter domination. A component of {\it warm white noise} emerges in our analysis, which arises due to the finite number density and undergoes scale-dependent evolution because of the velocity dispersion. Although free streaming erases adiabatic initial perturbations on small scales, warm white noise persists below the free-streaming length and grows during matter domination, with growth suppressed below the dark matter Jeans length. Our calculated power spectra agree with $N$-body simulations in the linear regime and accurately predict halo mass functions in the nonlinear regime. Effects of warm white noise can emerge on observable quasi-linear scales for ultralight dark matter produced after inflation with a subhorizon correlation length. Our formalism is applicable to these scenarios (with de Broglie-scale quasi-particles), to cases in which dark matter includes macroscopic structures (such as primordial black holes), and to traditional warm and cold dark matter scenarios.