A Novel Density Profile for Isothermal Cores of Dark Matter Halos

TL;DR

The paper addresses the need for a closed-form analytic density profile that faithfully captures the isothermal-core region of self-interacting dark matter halos while preserving a simple tail to NFW-like outskirts. It derives ρ_T25, an analytic profile with a core where the velocity dispersion is nearly constant, by enforcing an isothermal condition within a Jeans-equation framework and fixing γ=2 to stabilize the core, yielding ρ_T25(r) = ρ_c ( tanh(r/r_c)/(r/r_c) )^n / (1+(r/r_s)^2)^{(3-n)/2}. Validation against high-resolution SIDM N-body simulations shows that ρ_T25 (and the Yang profile) reproduce the density structure well across evolutionary stages, with the reconstructed velocity-dispersion profiles closely matching the isothermal-core configuration, especially for late-core-collapse phases. The study demonstrates that ρ_T25 provides a robust, analytic, and computationally efficient tool for analyzing SIDM halo evolution, generating initial conditions for deep-core-collapse runs, and facilitating comparisons across SIDM scenarios through stable, self-similar parameter evolution (notably n ≈ 2.5). Overall, the work offers a practical framework for probing SIDM core physics and reduces reliance on costly simulations for certain core-regime analyses.

Abstract

We present a novel analytic density profile for halos in self-interacting dark matter (SIDM) models, which accurately captures the isothermal-core configuration, i.e. where both the density and velocity dispersion profiles exhibit central plateaus in the halo innermost region. Importantly, the profile retains a simple and tractable functional form. We demonstrate analytically how our density profile satisfies the aforementioned conditions, with comparisons to other contemporary functional choices. We further validate the profile using idealized N-body simulations, showing that it provides excellent representations of both the density and velocity dispersion profiles across a broad range of evolutionary stages, from the early thermalization phase to the late core-collapse regime. As a result of its accuracy and simplicity, the proposed profile offers a robust framework for analyzing halo evolution in a variety of SIDM scenarios. It also holds practical utility in reducing simulation needs and in generating initial conditions for simulations targeting the deep core-collapse regime.

Figures (7)

• Figure 1: The density profiles of the Read (top left), Robertson-Fischer (top right), Yang (bottom left), and our (bottom right) profiles. Densities and radii are scaled with the characteristics core density $\rho_{\rm{c}}$ and core radius $r_{\rm{c}}$. Each profile family is shown with three choices of $r_{\rm{s}} / r_{\rm{c}} = 3, 10, 30$ (in pink, green, and blue) and three choices of $n = 1, 1.75, 2.5$ (in dotted, dashed, and solid lines). The choice of $n$ is not relevant in the context of the Read profile.
• Figure 2: The velocity dispersion profiles reconstructed from the density profiles presented in Figure \ref{['fig:density_profiles']} using the spherically symmetric Jeans equation (Equation \ref{['eqn:Jeans_equation']}). Velocity dispersions and radii are scaled with $\sqrt{G \, \rho_{\rm{c}} \, r_{\rm{c}}^2}$ and $r_{\rm{c}}$, respectively. Again, the Read (top left), Robertson-Fischer (top right), Yang (bottom left), and our (bottom right) profiles are shown with three choices of $r_{\rm{s}} / r_{\rm{c}} = 3, 10, 30$ (in pink, green, and blue), as well as three choices of $n = 1, 1.75, 2.5$ (in dotted, dashed, and solid lines) for the relevant profiles. The sub-panels present a zoomed-in view of the velocity dispersion profiles within the expected isothermal region of the core $r \lesssim 3 \, r_{\rm{c}}$.
• Figure 3: The density profile fits (left) and reconstructed one-dimensional velocity dispersion profiles (right) at three different snapshots for the halo of mass $10^8{\,\rm M_\odot}$ evolved under the cross section of $21.94\,{\rm cm}^2\,{\rm g}^{-1}$. The first snapshot, $T \simeq 0.02\,\tau$ (top), corresponds to the earliest phase of core formation. The second, $T \simeq 0.45\,\tau$ (middle), represents a typical snapshot during core collapse, while the final snapshot, $T \simeq 0.94\,\tau$ (bottom), shows the halo deep into core collapse, during the gravothermal catastrophe. The scatter points and associated error bars represent measurements obtained from N-body simulations. The uncertainties for the density profile are derived from Poisson statistics, while the uncertainties in the velocity dispersion are quantified as the standard error of the standard deviation of the one-dimensional particle velocities. The fitted and reconstructed profiles for the Read (orange), Robertson-Fischer (purple), Yang (cyan), and our (pink) profiles are shown in solid lines. The dashed lines represent the initial NFW configuration for the halo of interest. The horizontal red and blue lines (and corresponding shaded areas) display the directly measured core density $\rho_{\rm{core}}$ and velocity dispersion $\sigma_{\rm{core}}$, respectively.
• Figure 4: Comparisons between direct measurements (shaded areas) and fitted parameters using our density profile $\rho_{\rm{T25}}$ (black lines) for the core density $\rho_{\rm{c}}$ (top, left y-axis), core half-density radius $r_{\rho/2}$ (top, right y-axis), and core velocity dispersion $\sigma_{\rm{c}}$ (bottom) in the halo of mass $10^8{\,\rm M_\odot}$ evolved under the cross section of $21.94\,{\rm cm}^2\,{\rm g}^{-1}$. The direct measurements are displayed with uncertainties, while the fitting errors of the parameters are negligible and therefore not shown.
• Figure 5: The evolutions of the fitted transition index $n$ (top) and scale radius $r_{\rm{s}}^{\prime}$ (bottom) for the Robertson-Fischer (purple), Yang (cyan) and our (pink) density profiles. Each data point corresponds to a single snapshot from an individual halo. All simulated halos are included in the figures. The dashed horizontal line represents a constant value of $n=2.5$, observed throughout most evolutionary stages in both the Yang and our profiles.