Black Hole Cold Brew: Fermi Degeneracy Pressure

TL;DR

This work extends previous gravothermal analyses by incorporating quantum degeneracy through a truncated Fermi–Dirac description and solving the Tolman–Oppenheimer–Volkoff equations to study dynamical instability in self-gravitating fermionic systems. It demonstrates that Fermi pressure can, in general relativity, promote collapse at lower temperatures, yielding a quantum-limit critical mass set by the particle mass rather than the thermal state. The authors map stability across classical, mixed, and quantum regimes, deriving a two-regime behavior for the black-hole-mass threshold and highlighting implications for seed black holes formed from degenerate dark matter cores in the early Universe. These results provide a particle-mass–driven pathway for black hole formation and offer connections to observational constraints from JWST and dark matter bounds, with future work planned to include interactions and self-forces in the EOS.

Abstract

We investigate the dynamical instability of a self-gravitating thermal system in the quantum regime, where Fermi degeneracy pressure becomes significant. Using a truncated Fermi-Dirac distribution and solving the Tolman-Oppenheimer-Volkoff equation, we identify marginally stable configurations following Chandrasekhar's criterion. While Fermi pressure stabilizes a system against gravitational collapse in Newtonian gravity, in general relativity it can instead drive the instability, enabling collapse even at low temperatures. We discuss implications for the formation of massive black holes in the early Universe through the gravothermal collapse of dark matter.

Figures (4)

• Figure 1: The normalized classical Maxwell–Boltzmann distribution (top) and quantum Fermi--Dirac distribution with $\alpha(R)=0$ (bottom). The cutoff energy is fixed at $bw= 0.3$, and the boundary temperature varies as $b = k_B T(R) / mc^2 = 10^{-3}$ (blue), $10^{-2}$ (orange), and $10^{-1}$ (red). In the classical limit, the energy peak shifts toward higher energies as the temperature increases, whereas in the quantum limit, the trend reverses as the fermions become fully degenerate.
• Figure 2: Three-dimensional velocity dispersion $v/c$ (top) and adiabatic index $\gamma$ (bottom) as functions of the cutoff energy $bw$ for the Maxwell–Boltzmann distribution (left) and Fermi--Dirac distribution with $\alpha(R)=-10$ (middle) and $0$ (right), spanning the classical to quantum regimes. Results are shown for boundary temperatures $b = k_B T(R)/mc^2 = 10^{-3}$ (blue), $10^{-2}$ (orange), and $10^{-1}$ (red). The horizontal dashed lines in the top and bottom panels indicate $v/c = 0.57$ and $\gamma = 1.59$, respectively. Dynamical instability occurs when $v/c \gtrsim 0.57$ and $\gamma \lesssim 1.59$Feng:2021rst.
• Figure 3: Parameter space for stable ($\langle \gamma \rangle > \gamma_{\rm cr}$, blue shaded) and unstable ($\langle \gamma \rangle < \gamma_{\rm cr}$, red shaded) configurations in the plane of central cutoff energy $bw(0)$ versus boundary temperature $b$ for the Maxwell–Boltzmann distribution and the Fermi--Dirac distribution with $\alpha(R)=-10$, $-5$ and $0$, spanning the transition from classical to quantum regimes. In the classical regime, instability occurs only when $b \gtrsim 0.1$. When Fermi pressure becomes important, instability can also arise in the low-temperature regime $b \lesssim 10^{-2}$. A gap remains between the low- and high-temperature instability regions until $\alpha(R) \gtrsim -5$. The gray dashed line marks the boundary between classical ($n\lambda_{\rm dB}^3<g$) and quantum ($n\lambda_{\rm dB}^3>g$) regimes based on wavefunction overlap.
• Figure 4: Critical mass $M$ as a function of particle mass $m$ for the degeneracy $\alpha(R)=-50$ (left), $-10$ (middle), and $0$ (right), obtained by varying the boundary temperature $b=0.5$ (red), $10^{-1}$ (orange), and $10^{-2}$ (blue). At the low temperature $b=10^{-2}$, marginally stable configurations are not found for $\alpha(R)=-50$, where the classical Maxwell–Boltzmann distribution is recovered, while the critical mass converges for $\alpha(R)=-10$ and $0$ as the core becomes completely degenerate.