Dark matter mounds from the collapse of supermassive stars: a general-relativistic analysis

TL;DR

This work develops a fully general-relativistic framework to track the DM distribution function during the non-adiabatic collapse of a supermassive star into a black hole, producing a relativistic DM mound rather than a steep adiabatic spike. By combining relativistic adiabatic invariants for the SMS phase, the OS collapse model, and Liouville evolution of DM geodesics, the authors compute the post-collapse phase-space distribution and density profile with self-consistent mapping from pre- to post-collapse states. They find that rapid collapse depletes low-binding-energy DM orbits, yielding a central density enhancement that is milder than in adiabatic scenarios, and they quantify how subsequent adiabatic regrowth can erase these features depending on the growth factor. These results have direct implications for EMRI gravitational-wave dephasing and offer a pathway to use future GW observations to constrain DM properties and SMBH formation histories.

Abstract

Recent work has highlighted the importance of a fully relativistic treatment of the dephasing of gravitational waves induced by dark-matter overdensities in extreme mass-ratio inspirals (EMRIs). However, a general-relativistic description of the dark matter phase-space distribution is currently available only for the case of a dark matter "spike" arising from adiabatic black hole growth. Here we develop a fully general-relativistic formalism for the more realistic scenario in which a supermassive stellar progenitor collapses to a black hole and produces a shallower dark matter overdensity, or "mound". We follow self-consistently the evolution of the supermassive star, its collapse, and the subsequent growth of the resulting black hole, together with the collisionless dark matter orbits. We find that in the regime where the collapse becomes non-adiabatic, the dark matter distribution function is significantly reshaped, with a clear depletion in the low-binding-energy region of phase space. Our results provide a more realistic prediction for the dark matter phase-space distribution around supermassive black holes, which is an essential step in our programme to use future EMRI observations to extract information about both the nature of dark matter and the formation history of the black hole.

Figures (7)

• Figure 1: Top panel: DM rest mass density profile pre- and post- SMS collapse; these are labeled, respectively, by SMS and Relativistic Mound. The case of a SMBH growing adiabatically from a light seed to a mass of $10^5\,M_\odot$ is also shown for comparison and it is labeled Relativistic Spike. Bottom panel: Profile slope, $\gamma\equiv-\dd \log \rho/\dd r$, for each case.
• Figure 2: Left: Coarse-grained distribution function of DM after OS collapse projected in the phase space $(\mathcal{L},1-\mathcal{E})$. Right: Same quantity but for a spike grown adiabatically. In both panels, the top blue curve corresponds to circular orbits and the bottom orange curve corresponds to the minimum angular momentum required to avoid capture by the BH for the given energy.
• Figure 3: Left: Distribution function of DM particles in the projected phase space of velocities (radial and tangential) as measured in the LIF at $3\,R_\star$ for a spike following OS collapse. Right: Same quantity but for a spike grown adiabatically.
• Figure 4: DM rest mass density profiles after the SMS collapse (Mound) and various factors of regrowth $n=M_f/M$ (Regrowth $\times$$n$). The adiabatic equivalents for the initial BH mass (Spike) and each regrowth factor (Spike $\times$$n$) are also shown as dotted lines.
• Figure 5: Distribution function of DM particles for their radial and tangential velocity, as perceived by an observer in the LIF at $3\, R_\star$, for a spike post OS collapse and a subsequent regrowth by a factor 2.