An Enhanced Isothermal Jeans Approach to Constraining Dark Matter Self-Interactions from Galactic Kinematics

TL;DR

This work advances the semi-analytic isothermal Jeans framework for self-interacting dark matter (SIDM) halos by incorporating velocity-dependent cross sections and an empirical gravothermal-core-collapse treatment, enabling robust fits to a large SPARC rotation-curve sample. The analysis reveals an L-shaped degeneracy in the cross-section parameter space, with viable constant and velocity-dependent regimes, and finds that about one-sixth of galaxies show bimodal posteriors consistent with core-collapse scenarios. Best-fit parameters cluster near σ0 ≈ 5–6 cm^2/g with ω ≈ 220–250 km/s, broadly aligning with prior independent constraints and tolerating both core-growth and core-collapse evolutions. The results suggest SIDM can reproduce galactic kinematic diversity without strong ties to baryonic feedback, while also impacting halo concentrations and SHMRs, and they underscore the need for low-mass kinematic data to break degeneracies in velocity-dependent models.

Abstract

We present an improved semi-analytical model to predict density profiles of self-interacting dark matter (SIDM) halos and apply it to constrain the self-scattering cross section using SPARC galaxy rotation curves. Building on the isothermal Jeans approach, our model incorporates (i) velocity-dependent cross sections, (ii) an empirical treatment of core collapse, and (iii) enhanced robustness for identifying solutions. These advances allow us to fit a large sample of galaxies, including systems with baryon-dominated centers often excluded in earlier studies. We find that roughly 1/6 of galaxies admit both a core-growth and a core-collapse solution, while the rest favor a unique evolutionary state. Joint constraints across the sample reveal clear velocity dependence: the allowed parameter space forms an L-shaped degeneracy, where both nearly constant, low cross sections ($σ_0\sim2\,{\rm cm}^2$/g, $ω\gtrsim500\,$km/s) and strongly velocity-dependent models ($σ_0\sim100\,{\rm cm}^2$/g, $ω\sim60\,$km/s) are viable. Adopting the core-growth interpretation yields best-fit values $σ_0\simeq5\,{\rm cm}^2$/g and $ω\simeq250\,$km/s. Our constraints are remarkably consistent with previous results derived from a variety of independent probes. Compared to cold dark matter (CDM) models, SIDM outperforms simple adiabatic-contraction profiles and rivals empirical feedback-based CDM profiles, yet shows no correlation with stellar-to-halo mass ratio, a proxy for feedback strength, offering a distinct explanation for dwarf galaxy diversity. Moreover, SIDM does not affect galaxy-halo scaling relations significantly and makes concentration systematically lower. Our results highlight SIDM as a compelling framework for small-scale structure, while future low-mass kinematic data will be crucial for breaking degeneracies in velocity-dependent cross-section models.

Figures (13)

• Figure 1: Illustration of effective cross section $\sigma_{\rm eff}$ as a function of the maximum circular velocity $V_{\rm max}$ of a halo, for velocity-dependent cross section models with different $\sigma_0$ and $\omega$, as defined in eq. (\ref{['eq:diff-cross-section']}).
• Figure 2: Gravothermal evolution of an SIDM halo modeled with the improved isothermal Jeans method, including an empirical treatment of core-collapse. The example halo has a virial mass of $M_{\rm vir}=10^{12}M_\odot$, concentration of $c=10$, and a baryonic component described by a Hernquist profile with a total mass of $M_{\rm b}=10^{10}M_\odot$ and a scale radius of $a=5\,{\rm kpc}$. Left: Evolution of the central density from the low-density (solid black) and high-density (dashed black) solutions. Density and time are shown in dimensionless form $\hat{\rho}_{\rm c}=\rho_{\rm c}/\rho_{\rm s}$, and $\hat{t}=8\sqrt{G\rho_{\rm s}}\rho_{\rm s} r_{\rm s} \sigma_{\rm eff} t_{\rm age}$, where $\rho_{\rm s}$ and $r_{\rm s}$ are the scale density and radius of the target NFW halo. The black triangle marks the merger of the two solutions, indicating the onset of core-collapse. The mirrored high-density branch (grey curve) provides an empirical approximation to the post-collapse evolution, as shown by YangJiang24. Together, the solid curves trace the full gravothermal history. Middle: Density profiles along the evolutionary track, marked by crosses in the left panel. As $\hat{t}$ increases, the central density initially decreases as the core expands, reaching its maximum size at $\hat{t}\sim100$, before shrinking as the central density rises. Right: Comparison of core-expanding and core-collapsing halos with identical central density. For example, at $p\equiv\sigma_{\rm eff}t_{\rm age}=1 \,\text{cm}^2/\text{g}\,\text{Gyr}$ and $308.5 \,\text{cm}^2/\text{g}\,\text{Gyr}$, the central densities conincide and the profiles are similar. This degeneracy underlies the bimodal posteriors found in MCMC fits of SIDM mass models to galaxy rotation curves (see Section \ref{['sec:MassModel']}).
• Figure 3: Representative case studies of SIDM rotation-curve fitting. Left: Observed circular-velocity profiles (black points with 1$\sigma$ error bars) from the SPARC survey Lelli16. The baryonic contribution (green curve) is derived from $\mathrm{H\textsc{i}}$ intensity and stellar light profiles, with mass-to-light ratios determined as described in Section \ref{['sec:BaryonModel']}. The offset between the data points and the baryonic curve corresponds to the DM contribution. The total mass model, combining the baryonic distribution with the best-fit SIDM halo, is shown by the blue (and red) curves. Middle: Evolution of the central density of the best-fit SIDM halos. The filled circles indicate the evolutionary stage that provides the best fit. Right: Posterior distributions of the model parameters: halo virial mass $M_{\rm vir}$, concentration $c$, and the parameter $p=t_{\rm age}\sigma_{\rm eff}$, i.e. the product of halo age and effective cross section. The upper row illustrates a case with bimodal posterior: both a core-growth solution ($M_{\rm vir}=10^{11.4}M_\odot$, $c=13.5$, and $p=1.31\,{\rm cm}^2\,{\rm g}^{-1}\,{\rm Gyr}$) and a core-collapse solution ($M_{\rm vir}=10^{11.2}M_\odot$, $c=14.6$, and $p=202\,{\rm cm}^2\,{\rm g}^{-1}\,{\rm Gyr}$) provide comparably good fits, differing by more than an order of magnitude in $p$. This bimodality reflects degeneracy in the cross section rather than halo age (see Section \ref{['sec:bimodality']}). The lower row shows an example with a unimodal posterior: a halo with $M_{\rm vir}=10^{12.0}M_\odot$, $c=8.41$, and $p=11.4\,{\rm cm}^2\,{\rm g}^{-1}\,{\rm Gyr}$. This halo lies near the maximal-core stage of gravothermal evolution, as indicated by the circle in the central-density track.
• Figure 4: Maximum circular velocities ($V_{\rm max}$) and effective cross sections ($\sigma_{\rm eff}$) inferred from SIDM mass-model fits to galaxy rotation curves ( left), and the resulting joint constraints on the cross-section parameters $\sigma_0$ and $\omega$ ( right). Blue symbols denote galaxies with unimodal posterior. Orange and green symbols represent the core-growth and core-collapse solutions, respectively, for galaxies with bimodal posteriors. Error bars indicate 68% confidence intervals. The joint constraints on $\sigma_0$ and $\omega$ are obtained by fitting the effective cross section defined in eq. (\ref{['eq:EffectiveCrossSection']}) to the ($V_{\rm max}$, $\sigma_{\rm eff}$) data. In the right panels, we adopt the core-growth and core-collapse solutions, respectively, for bimodal galaxies, while including unimodal galaxies in both analyses. The color scale indicates the $\chi^2$ values relative to the minimum in the $\sigma_0$-$\omega$ space (eq. (\ref{['eq:chisquare']})). Constraints on the parameters do not significantly depend on the choice of solutions. With either core-growth or core-collapse solutions, the allowed region forms a clear L-shaped band, where both high-$\sigma_0$ low-$\omega$ and low-$\sigma_0$ high-$\omega$ models are comparably viable. Three representative examples (A, B, and C) illustrate this degeneracy. When adopting core-growth solutions, the best-fit MCMC model lies between (B) and (C), with $\sigma_0 = 4.9\,{\rm cm}^2/\,{\rm g}$ and $\omega=250\,{\rm km/s}$, marked by an orange cross and a surrounding contour approximating the 3$\sigma$ posterior interval; the corresponding $\sigma_{\rm eff}(V_{\rm max})$ relation is shown by the orange curve. When adopting core-collapse solutions instead, the parameter constraints tighten and the L-shaped allowed region narrows slightly. The best-fit model is similar, given by $\sigma_0 = 6.0\,{\rm cm}^2/\,{\rm g}$ and $\omega=220\,{\rm km/s}$, marked by a green cross with a $3\sigma$ contour, and its $\sigma_{\rm eff}(V_{\rm max})$ relation is shown by the green curve. Without kinematics data for ultra-faint dwarfs ($V_{\rm max}\lesssim 30\,{\rm km/s}$), these degeneracies remain difficult to break.
• Figure 5: Comparison of constraints on the velocity-dependent SIDM cross section from this work with other studies across various scales. Left: The inferred effective cross section versus maximum circular velocity $V_{\rm max}$ for the SPARC sample, taking the core-growth solution for bimodal cases. Blue (orange) points represent unimodal galaxies (the core-growth solution of bimodal galaxies) used in this study. Three of our galaxies are highlighted with green outlines, representing 1/3 of the sample analyzed by YangJiang24. Points in other colors show constraints from previous studies using other probes: the stellar cores in ultra faint dwarfs SA25, Milky Way satellites Correa21, strong lensing and stellar kinematics of groups and clusters Sagunski21O'Donnell25. Right: Joint constraints in the $\sigma_{\rm eff}-V_{\rm max}$ plane. The blue L-shaped region is from this work. Overlaid for comparison are the constraints from YangJiang24 and Slone23. The orange cross marks the best-fit parameters adopting unimodal galaxies and the core-growth solutions of bimodal galaxies, with the surrounding ellipse approximating the $3\sigma$ posterior interval, the same as in Fig. \ref{['fig:constraints']}. The Slone23 result (regions surronded by purple dotted lines) has a significant overlap with our preferred region. Offset exists between our work and YangJiang24, shown in green solid line with dashed lines indicating 1$\sigma$ scatter. We refrain from over-interpreting this difference, as the YangJiang24 sample is small and biased. The cyan point represents the cross sections required to reproduce the SMBH mass functions inferred for the Little Red Dots in the early Universe Jiang25, which lies closely to our allowed region.