The Growth of the Central Black Holes in Quasi-stars

TL;DR

This work investigates whether quasi-stars can funnel rapid growth of central black holes in the early universe, using the MESA stellar evolution code to simulate time-dependent BH growth under two complementary inner-boundary prescriptions (Ball and Coughlin) and with/without winds. The Ball-based approach yields an analytic BH-mass cap $M_{ m crit}(N)$ that constrains $M_{ m BH}$ as the envelope evolves, while the Coughlin-based framework allows the interior to be convective with an outer adiabatic envelope, yielding a quasi-self-similar limit of $M_{ m BH}/M_{ m star}\approx 0.33$, largely independent of $M_{ m star}$. Introducing dynamical opacity and mass loss via eruptive winds shows winds can dramatically erode the envelope and shorten the quasi-star lifetime, yet the final BH-to-envelope ratios remain close to self-similar values, suggesting robust pathways for growing substantial BH seeds even in the presence of outflows. Overall, the results illuminate how inner Boundary conditions and winds shape BH growth in quasi-stars, offering insights into the origins of high-redshift SMBHs and potential connections to observed faint red sources in JWST data.

Abstract

Observations by JWST have confirmed the presence of supermassive black holes (BHs) at redshifts $z\gtrsim10$, lending support to scenarios in which BHs experience rapid growth through intense gas accretion. Here we investigate the growth of a BH embedded at the center of a quasi-star, a theoretically predicted object formed via direct collapse. In a quasi-star, the central BH accretes at a highly super-Eddington rate, while the excess energy is transported outward by convection and radiated at approximately the Eddington luminosity of the entire star. We employ the open-source stellar evolution code \texttt{MESA} to construct quasi-star models and follow the time-dependent growth of the central BH under different prescriptions for the accretion rate at the inner boundary $R_i$, and further considering the effect of winds. For the case $R_i=NR_{\rm B}$, where $N$ is a constant and $R_{\rm B}$ is the Bondi radius corresponding to the mass of the BH and the gas infalling onto it, our models terminate when the BH mass reaches a critical value $M_{\mathrm{crit}}(N)=c_{s,i}^3/(12\sqrt{N^3G^3πρ_i})$ (where $c_{s,i}$ and $ρ_i$ are the sound speed and density at $R_i$, respectively), a limit we also derive analytically. Models that feature an inner convective region matched to an outer adiabatic envelope exhibit BH growth up to approximately $M_{\mathrm{BH}}/M_\star\simeq 0.33$, largely independent of the stellar mass $M_\star$ itself. This ratio is approximately preserved even in the presence of mass loss, as several properties of the model are independent of the quasi-star's total mass.

Figures (12)

• Figure 1: The radii of the photosphere (solid) and interior region $R_i$ (dashed) for a $M_\star=10^5 M_\odot$ quasi-star as a function of the BH mass normalized by the instantaneous stellar mass, with our MESA implementation of the Ball2011 model for different values of the parameter $N$ (cf. Eq. \ref{['eq:bondi']}). All models are initialized with the same properties and an initial BH mass of $M_{\rm BH} = 100 M_\odot$.
• Figure 2: Solid lines: Ratio between BH and star mass as a function of the dimensionless entropy, and for different values of the parameter $N$ in our MESA implementation of the Ball model. The numerical curves are plotted until the end of the MESA simulations. Dotted horizontal lines indicate our analytically derived maximum mass in the Ball model, $M_{\rm crit}(N)$ (Eq. \ref{['eq:Mcrit']}). Also shown with dashed curves are the analytical solutions of Coughlin2024, with the maximum BH masses predicted by their solutions marked with filled circles.
• Figure 3: Comparison between various quasi-star properties (from top right, clockwise: density $\rho$, pressure ratio $p_{\rm gas}/p_{\rm rad}$ and adiabatic index $\gamma_{\rm ad}$, relevant radii $R$, temperature $T$) obtained with our MESA implementation of the baseline model of Coughlin2024 (solid lines) and the corresponding analytical predictions from the same work (dashed lines), shown over the course of the quasi-star’s evolution. The total quasi-star mass is $M_\star=10^5M_\odot$. For a consistent comparison, the mass $M_c$ in the analytical model (i.e. the mass at which $L = L_{\rm conv,max}$; see Eq. \ref{['eq:stopping']}) is gradually adjusted to track the time-dependent value from the simulation.
• Figure 4: Comparison of $R_c$ with $R_i$ (top) and $M_c$ with $M_i$ (bottom), both when using $\kappa=\kappa_{\rm es}=0.34 \; \rm cm^2 \; g^{-1}$ (left) and using $\kappa=\kappa_{\rm IC}$ (right) for our calculation of $L_{\rm Edd}$, where $\kappa_{\rm IC}$ is the opacity taken at the base of the IC layer. All values shown here are normalized by the total radius or mass of the quasi-star. $R_c$ and $M_c$ are defined as the radius and mass at the point where $L_{\rm conv,max} = L_{\rm Edd}$.
• Figure 5: Top panel: Ratio of $R_c$ and $R_{\rm IC}$ to the photospheric radius, as a function of the BH mass normalized to the quasi-star total mass. We show this radius using the definitions for the IC layer from both Cheng2024 (indicated with $M_{\rm IC,C24}$) and Begelman2025 (indicated with $M_{\rm IC,B25}$). The model here is a $M_\star = 10^5 M_\odot$ quasi-star with opacity taken from the base of the IC layer in the Eddington luminosity calculation. Bottom panel: Same as above, but showing the ratio of the masses below these points to the quasi-star's total mass.