General-relativistic instability in rapidly accreting supermassive stars in the presence of dark matter

Abstract: The collapse of supermassive stars (SMSs) via the general-relativistic (GR) instability would provide a natural explanation to the existence of the most extreme quasars. The presence of dark matter in SMSs is thought to potentially impact their properties, in particular their mass at collapse. Dark matter might be made of weakly interacting massive particles (WIMPs) that can be captured by the gravitational potential well of SMSs due to interaction with the baryonic gas. The annihilation of WIMPs can provide fuel to support the star before H-burning ignition, favouring low densities of baryonic gas, long stellar lifetimes and high final masses. Here, we estimate the impact of dark matter on the GR dynamical stability of rapidly accreting SMSs. We add a dark matter term to the relativistic equation of adiabatic pulsations and apply it to hylotropic structures in order to determine the onset point of the GR instability. We find that, in principle, the dark matter gravitational field can remove completely the GR instability. However, for SMSs fuelled by H-burning, the dark matter densities required to stabilise the star against GR are orders of magnitude above the values that are expected for the dark matter background. On the other hand, for SMSs fuelled by WIMP annihilation, we find that the low densities of baryonic gas inhibit the destabilising GR corrections, which shifts the stability limit by typically an order of magnitude towards higher masses. As long as central temperatures $\lesssim10^7$ K are maintained by WIMP annihilation, the GR instability is reached only for stellar masses $>10^6$ M$\odot$. Dark matter can impact the GR dynamical stability of SMSs only in the case of energetically significant WIMP annihilation. The detection of a SMS with mass $>10^6$ M$\odot$ in an atomically cooled halo can be interpreted as an evidence of WIMP annihilation in the star's core.