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

We investigate the evolution of density perturbations in dark matter, including the new combined effects of finite number density and non-zero 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 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.