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

TL;DR

This work develops a BBGKY-based analytic framework to study dark-matter structure formation when both finite number density and non-zero velocity dispersion are important, revealing a warm white-noise component in the density power spectrum. The authors derive growth functions that couple adiabatic evolution, free streaming, and gravity, yielding a compact expression for P_δ(y,k) that combines an adiabatic piece with a scale-dependent white-noise term that grows below the Jeans scale during matter domination. The methodology produces power spectra that agree with N-body simulations in the linear regime and provide accurate halo-mass functions in the nonlinear regime, enabling applications to ultralight wave dark matter and macroscopic dark-matter constituents. The paper also offers numerical tools and discusses generalizations to wave dark matter, non-gravitational interactions, fractional dark matter components, and relativistic effects, highlighting the broad relevance to small-scale structure and observational probes.

Abstract

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.

Figures (15)

• Figure 1: During radiation domination, free streaming erases existing correlations below the free-streaming length, revealing more of the underlying Poisson distribution of particles: $(a)\rightarrow (b)$. However, free streaming cannot change the underlying Poisson distribution: $(b)\rightarrow (c)$. During matter domination, gravitational clustering builds correlations above the Jeans length: $(c)\rightarrow (d)$. The points could represent de Broglie-scale quasi-particles in wave dark matter, composite/macroscopic dark matter (including primordial black holes or substructures) or "usual" particle dark matter.
• Figure 2: Small length-scale features in the dimensionless density power spectrum depend on the co-moving number $\bar{n}$ (setting the amplitude of the white noise contribution) and 1D velocity dispersion $\sigma_{\rm eq}$ (at matter-radiation equality, which determines the free-streaming and Jeans scales). For the left panel, we keep $\bar{n}$ fixed, whereas on the right $\sigma_{\rm eq}$ is held fixed. We show linear-theory power spectra at $y=a/a_{\rm eq}=100$ evaluated using the results of this work. The filled dots indicate the Jeans scales $k_{\rm J}$ at this time, whereas the stars are the Jeans scales $k_{\rm J,eq}$ at $y=1$. The open circles indicate the free-streaming scale $k_{\rm fs}$. The dotted curve is the "usual" cold dark matter (CDM) without any significant velocity dispersion or white noise $(\bar{n},\sigma_{\rm eq})\rightarrow(\infty,0)$. Existing observations in the quasi-linear regime only permit major deviations from the dotted curve for $k\gtrsim 10\,\,{\rm Mpc}^{-1}$.
• Figure 3: Evolution of the white-noise power spectrum as a function of wavenumber horizontal axis) and scale factor (colors). The colored lines are based on evaluation of our expression \ref{['eq:PwEvolution']} for the evolution of the power spectrum. The dots indicate the co-moving Jeans scale $k_{\rm J}(a)$. The left edge of the plot evolves upwards as $(1+\frac{3}{2}y)^2$ in accordance with standard expectations, with growth at higher $k$ being Jeans-suppressed. The above calculations are done for a Maxwellian initial momentum distribution $f_0(q)=A e^{-q^2/2q_*^2}$, and the 1D velocity dispersion at equality is $\sigma_{\rm eq}=q_*/a_{\rm eq}m\approx 22\,{\rm km}\, {\rm s}^{-1}$. The dimensionless scale ($\alpha_k$) on the bottom axis is converted to $k$ in $\,{\rm Mpc}^{-1}$ (top axis) using this $\sigma_{\rm eq}$. Note that in terms of $\alpha_k$, the shape of $\bar{n}P_{\delta_{\rm wn}}(y,k)$ does not depend on $\sigma_{\rm eq}$.
• Figure 4: Evolution of the total dimensionless power spectrum (solid lines). The lighter curves are the separate adiabatic (dashed) and white-noise (solid) contributions. The open circles on the adiabatic part indicate the free-streaming wavenumber $k_{\rm fs}(y)$, while the closed circles on the white-noise part are the Jeans wavenumber $k_{\rm J}(y)$. We adopt a Maxwellian initial velocity distribution, with $\sigma_{\rm eq}\approx 22\,\textrm{km} \,\textrm{s}^{-1}$, and $\bar{n}\approx 5\times 10^7/\,{\rm Mpc}^3$. The adiabatic part has an amplitude consistent with Planck (2018) observations. The chosen parameters are motivated by ref. Amin:2022nlh, with $k_{\rm fs}\sim 10/\,{\rm Mpc}$ and $k_{\rm wn}=(2\pi^2\bar{n})^{1/3}\sim 10^3/\,{\rm Mpc}$ -- at the boundary of being consistent with observations in the quasi-linear regime.
• Figure 5: Gravitational clustering of a warm, Poisson distributed collection of particles (without initial adiabatic perturbations). We show, with a logarithmic color scale, the projected density across the $1.38\,\,{\rm Mpc}$ (co-moving) simulation volume. Note the lack of clustering during radiation domination ($a<a_{\rm eq}$) and significant clustering during matter domination. For the scenario shown, the co-moving Jeans length during matter domination is $2\pi/k_{\rm J}\approx 0.05\,\,{\rm Mpc} \times \sqrt{a_{\rm eq}/a}$. Clustering is suppressed below this length scale.