Non-Equilibrium Relativistic Core Collapse of Self-Interacting Dark Matter Halos -- Limits On Seed Black Hole Mass

TL;DR

This work develops a fully general-relativistic, non-equilibrium framework based on the Misner-Sharp formalism to model the gravothermal collapse of self-interacting dark matter halos, including heat conduction. It follows the evolution from an initial NFW-like halo through a long-lived LMFP phase and a rapid SMFP-driven non-equilibrium phase, to the formation of an apparent horizon and a seed black hole. The key finding is that pure SIDM with constant cross-section produces only a light seed, $M_{ m BH}\napprox 200\,M_igodot$, at horizon formation, with $M_{ m BH}/M_{ m halo} \sim 3\times10^{-8}$, because intense outward heat flux from the core drives mass loss and stalls collapse. This implies that additional physics—such as baryonic cooling and accretion, a velocity-dependent cross-section, or pre-existing central BHs—are likely required to generate the heavy seeds needed to explain high-redshift SMBHs. The results place quantitative constraints on SIDM-only seeding and motivate future extensions incorporating baryons and variable cross-sections.

Abstract

Recent observations of supermassive black holes (SMBHs) at high redshifts pose challenges to standard seeding mechanisms. Among competing models, the collapse of self-interacting dark matter (SIDM) halos provide a plausible explanation for early SMBH formation. While previous studies on modeling the gravothermal collapse of SIDM halos have primarily focused on non-relativistic evolution under the assumption of hydrostatic equilibrium, We advance this framework by relaxing the equilibrium assumption and additionally incorporating general-relativistic effects. To this end, we introduce the Misner-Sharp formalism to the SIDM context for the first time. Our model reproduces the standard hydrostatic models in the early long-mean-free-path (LMFP) regime, but displays interesting distinct behavior in the late short-mean-free-path (SMFP) regime, where intense outward heat flux drives a rapid expansion of the outer envelope, removing mass from the core and significantly decelerating the collapse. Our general relativistic treatment enables us to follow halo evolution to the final stage when the apparent horizon forms. Our simulation yields a seed black hole mass of approximately $3\times10^{-8}$ of the halo mass at horizon formation, suggesting that additional mechanisms such as baryonic effects are critical for seeding black holes that are sufficiently massive to account for SMBHs in the early Universe.

Figures (5)

• Figure 1: The evolution of the central density with time, comparing the traditional method (assuming hydrostatic equilibrium, red line) with our proposed method (non-equilibrium, black line). Our non-equilibrium treatment agrees well with the equilibrium fluid model during core formation and the early stages of core collapse, but predicts faster collapse in the late stages.
• Figure 2: Evolution of the specific internal energy (left column) and density profiles (right column) for the fiducial halo. The upper panels illustrate the core expansion phase, where the halo evolves from an initial NFW profile ($t=0$) towards an isothermal core. The lower panels depict the subsequent gravothermal collapse phase. The red dashed line at $t=63.2t_0$ marks the formation of the isothermal core, serving as the transition boundary between the expansion and collapse regimes. The color gradient of the solid lines represents the progression of time. Note that $\epsilon$ denotes the specific internal energy normalized by $c^2$
• Figure 3: The evolution of the bulk velocity profile (upper panel) and the density profile (lower panel) during the SMFP core collapse stage. The notations follow those in Figure \ref{['fig:fig2']}. In the upper panel, the y-axis represents the ratio of the bulk velocity $U$ to the velocity dispersion $v$, which serves as a diagnostic for deviations from hydrostatic equilibrium. The black-edged dots mark the boundary of the SMFP regime at each evolutionary snapshot. The density depletion in the outer region arises from mass outflow, which is induced by the intense outward heat flux characteristic of the SMFP regime.
• Figure 4: Evolution of the SMFP mass fraction with central density, comparing the traditional method (assuming hydrostatic equilibrium, red line) with our proposed method (non-equilibrium, black line). In the non-equilibrium case, the SMFP regime emerges at $t=445.0t_0$, reaches its maximum enclosed mass at $t=445.6t_0$, and gradually stabilizes after $t=445.8t_0$. In contrast, the equilibrium calculation yields a substantially larger SMFP mass.
• Figure 5: The evolution of the mass profile at BH formation stage. The upper panel traces the $m-R$ evolution in the non-relativistic regime ($m_\mathrm{core}/R_\mathrm{core} \lesssim 10^{-2}$), while the lower panel displays the strong-gravity regime approaching the horizon ($m_\mathrm{core}/R_\mathrm{core} \gtrsim 10^{-2}$). The violet dashed lines represent the apparent horizon condition $R=2m$. At $t=580t_0$, the mass profile (red dashed line) intersects $R=2m$, marking the formation of the central BH. Other notations follow those in Figure \ref{['fig:fig2']}. The seed BH mass at horizon formation is about $3\times10^{-8}$ of the halo mass ($\approx 200\ \mathrm{M}_{\odot}$).