Multi-species Dark Matter with Warmth and Randomness

TL;DR

This work introduces a general analytic framework for the growth of cosmic structure in a multi-species dark matter sector that simultaneously accounts for finite velocity dispersion (warmth) and Poisson (shot-noise) fluctuations. Building on a truncated BBGKY hierarchy, the authors derive transfer-function formalism governed by Volterra integral equations to compute the total matter power spectrum and the full set of inter- and intra-species spectra for arbitrary numbers of components with distinct properties, under both adiabatic and isocurvature initial conditions. They present a fast numerical algorithm and public code, validate results against analytic estimates and N-body simulations, and illustrate the physics with two-component examples highlighting how warmth and discreteness imprint scale-dependent features through Jeans and free-streaming effects. The framework unifies descriptions of cold and warm, as well as discrete-population DM like primordial black holes or solitons, and provides a pathway to confront mixed-dark-matter scenarios with small-scale structure observations while clarifying the limits of the approach (subhorizon, non-relativistic, gravitational dynamics).

Abstract

We present a general analytic framework for the evolution of cosmic structure in multi-species dark matter models that simultaneously incorporates finite velocity dispersion and Poisson fluctuations. Our approach accommodates arbitrary numbers of dark matter components with distinct mass fractions, velocity distributions, and number densities -- ranging from cold particles to warm species and sparse populations such as primordial black holes or solitons. The framework is based on solving a truncated BBGKY hierarchy, whose solution is obtained by solving Volterra integral equations. We provide an efficient algorithm to solve for the total, as well as inter- and intra-species power spectra. Worked examples with two-component mixtures illustrate how isocurvature (initially Poisson) and adiabatic spectra evolve differently depending on the properties of the warm or sparse fraction. This evolution is controlled by the free-streaming and Jeans scales, and the results match analytic estimates and $N$-body simulations.

Figures (5)

• Figure 1: The isocurvature (left) and adiabatic (right) growth for 2-component dark matter compared to single-component CDM. The 2-component DM consists of a dominant CDM component without significant Poisson fluctuations (component 1) and a 1% component that is warm and has significant Poisson fluctuations (component 2). For the isocurvature part of the power spectrum, the suppression due to the warm component begins at the Jeans scale at equality, corresponding to $\alpha_{k\, 2}\simeq \sqrt{3/2}$ (see equation \ref{['eq:alpha_scales']}). For $y\gg 1$, suppression for larger $\alpha_{k\,2}$ scales as $(4/9)\alpha_{k\,2}^{-1}$. For the adiabatic part, the suppression begins at the free-streaming scale $\alpha_{\mathrm{fs}\,2}(y)$. For $y\gg1$ the suppression begins around $\alpha_{k\, 2}\simeq 0.15$ (see equation \ref{['eq:alpha_scales']}) and plateaus at the current Jeans scale $\alpha_{k\, 2}\simeq \sqrt{3y/2}$ with a plateau depth of $\approx (2/5)\mathfrak{f}_2(8+ 3\ln y)$. To convert the horizontal axis to wave number, use $k\approx 10^2\,{\rm Mpc}^{-1}\left({22\,{\rm km}\,{\rm s}^{-1}}/{\sigma_{{\rm eq}\,S}}\right)\alpha_{k\,S}.$
• Figure 2: The Isocurvature (left) and Adiabatic (right) growth of PS compared to CDM for 2-component dark matter, with 1% cold with significant Poisson fluctuations (rest warm dark matter). For the isocurvature part, the suppression from unity due to the warm component begins at the Jeans scale for the warm component at equality $\alpha_{\mathrm{J}\,1}(y=1)=\sqrt{3/2}$ and plateaus $\alpha_{\mathrm{J}\,1}(y)=\sqrt{3y/2}\gg 1$. The depth of the suppression is $\approx (4y^{-2}/9)[1+6 \mathfrak{f}_2\ln(y/4)]$ at large $y$. For the adiabatic part, the suppression begins at the free-streaming scale $\alpha_{\mathrm{fs}\,1}=\mathcal{F}^{-1}(y,y_0)$. For $y\gg1$ the suppression plateaus at the current Jeans scale $\alpha_{\mathrm{J}\,1}(y)$. The height of this suppressed part $\sim \mathfrak{f}_2^2 4y^{-2}/9$. To convert the horizontal axis to wave number, use $k\approx 10^2\,{\rm Mpc}^{-1}\left({22\,{\rm km} s^{-1}}/{\sigma_{{\rm eq}\,1}}\right)\alpha_{k\,1}.$
• Figure 3: Examples of dark matter power spectra at $y\sim 10^3$ ($z\sim 2$) in dark matter models with two components. The mass fractions, velocity dispersion, and Poisson noise levels are varied. We assume that the second component is always subdominant and has the Poisson noise, but either component may be warm. We use the (approximately) parameter independent transfer functions in Fig. \ref{['fig:Case1']} and \ref{['fig:Case2']} to construct the above examples by appropriate scalings. The wavenumber of departure from CDM power spectra, and the amplitude and shape of the departure, can be controlled by choosing the mass fractions $\mathfrak{f}_S$, the velocity dispersions at equality $\sigma_{\rm eq,S}$ and the number densities $\bar{n}_{S}$.
• Figure 4: Growth of structure in a two-component dark matter model. The subdominant component $\mathfrak{f}_2=0.03$ is warm ($\sigma_{{\rm eq}\,2}\approx 65\,{\rm km}\, {\rm s}^{-1}$) and has massive particles ($m_2\approx 2\times 10^4M_{\odot}$), with correspondingly significant Poisson fluctuations. The dominant component is usual CDM. The subdominant component seeds structure (above its Jean's length) in the dominant cold component. The growth of structure in each species and total is captured well by our analytic framework. The above simulation considers Poisson initial conditions deep in the radiation era for the subdominant component, and assumes no initial density perturbations in the dominant component.
• Figure 5: The scale-dependent isocurvature growth in a two species dark matter model where the first component is cold and without significant Poisson fluctuations, whereas the subdominant second component ($\mathfrak{f}_2=0.03$) is warm ($\sigma_{{\rm eq}\,2}\approx 65\,{\rm km}\, {\rm s}^{-1}$) and has large Poisson fluctuations ($\bar{n}_2\approx 4.5\times 10^4\,{\rm Mpc}^{-3}$). The solid curves are from our analytic calculations, while the dots are based on an $N$-body simulation. The gray shaded region delineates nonlinear evolution. Along with the total power spectrum, we also show the intra- and inter-species power spectra. The analytics and $N$-body simulation results agree in the linear regime (and the total matter and species 1 power spectra surprisingly even agree in the nonlinear regime).