Quantum-Degenerate Fermion Dark Matter Cores

• Quantum-degenerate fermion dark matter cores are compact, high-density regions upheld by the quantum (Pauli) exclusion principle, preventing gravitational collapse.
• They manifest unique core–halo scaling relations, such as M_c ∝ M_v^(1/2) and R_c ∝ M_c^(-1/3), emerging from polytropic equilibrium models.
• These cores offer an alternative to supermassive black holes, addressing the core–cusp dilemma and influencing galaxy evolution through dynamic mass distributions.

Quantum-degenerate fermion dark matter cores are compact, high-density central regions within dark matter halos where Pauli exclusion pressure (degeneracy pressure) from massive fermionic dark matter particles supports the structure against gravitational collapse. These cores exist in equilibrium with an extended, thermally-dominated halo and are characterized by a polytropic equation of state and distinctive mass–radius–density scaling relations. The study of these cores is central to quantum-inspired models of galactic structure, alternatives to central supermassive black holes, and understanding the solutions to small-scale structure problems in cosmology such as the core–cusp dilemma.

1. Core–Halo Structure and Fundamental Equations

The core–halo decomposition models a dark matter (DM) halo as a quantum-degenerate core of mass $M_c$ and radius $R_c$, embedded in a much larger isothermal halo of mass $M_v$ and radius $r_v$ ($R_c \ll r_v$). The core is described as a $T=0$ fully degenerate Fermi gas in hydrostatic equilibrium, with a polytropic equation of state,

$$P_c = K \rho_c^{5/3}, \quad K = \frac{1}{20} \left( \frac{3}{\pi} \right)^{2/3} \frac{h^2}{m^{8/3}}$$

where $m$ is the fermion mass. The isothermal halo follows $P_v = \rho_v k_B T / m$ and exhibits approximately flat rotation curves at large radii.

The velocity dispersions in each region are

$$v_c^2 \simeq \frac{G M_c}{R_c}, \qquad v_v^2 \simeq \frac{G M_v}{r_v}$$

where $G$ is Newton's constant. The equilibrium (core–halo morphology) emerges as the maximum-entropy solution of the Fermi–Dirac–Poisson or general-relativistic Tolman–Oppenheimer–Volkoff (TOV) system, enforced by the Pauli exclusion principle and the requirement of an extremum of free energy at fixed mass and temperature (Chavanis, 2019, Argüelles et al., 2020, Ruffini et al., 2014, Chavanis, 2021, Argüelles et al., 2019).

2. Core Mass–Radius and Core–Halo Scaling Relations

The mass–radius relation for a zero-temperature degenerate Fermi core is

$$M_c R_c^3 = \frac{9 \omega_{3/2}}{8192\pi^4} \frac{h^6}{G^3 m^8} \Rightarrow R_c(M_c) = \left[ \frac{C_F}{M_c} \right]^{1/3}$$

where $C_F$ is a constant determined by $m$. For the composite system, the steady-state (maximum-entropy) condition arising from extremizing the total free energy $F(M_c)$ leads to "velocity-dispersion tracing": $GM_c/R_c \simeq GM_v/r_v$ This relation yields the core–halo scaling law

$M_c \propto M_v^{1/2}$

for quantum-degenerate fermion dark matter halos, in agreement with core–halo solutions found numerically in general-relativistic and kinetic equilibrium treatments (Chavanis, 2019, Ruffini et al., 2014, Argüelles et al., 2020).

The implications are:

• Core masses scale sublinearly with halo mass, $M_c \propto M_v^{1/2}$.
• Core radii shrink with increasing $M_c$: $R_c \propto M_c^{-1/3}$.
• Minimum halo mass for nonrelativistic, degenerate cores is $M_v \sim 10^8\,M_\odot$.
• Degenerate core sizes for keV–100 keV $m$ range from $10^{-4}$ pc (Milky Way) to $\sim$kpc (dwarf spheroidals).

3. Stability, Critical Mass, and Relativistic Collapse

In the full general-relativistic (TOV) analysis, each value of central density corresponds to a unique equilibrium configuration. As central density increases, one reaches a maximum core mass,

$$M_\mathrm{crit} \simeq 0.38\,\frac{m_\mathrm{Pl}^3}{m^2}$$

($m_\mathrm{Pl}$ is the Planck mass). For $m \sim 100\,\mathrm{keV}$, $M_\mathrm{crit} \sim 6.3 \times 10^7\,M_\odot$. Cores above this mass are gravitationally unstable and collapse to black holes (Arguelles et al., 2023). In typical galactic environments, meta-stable core–halo solutions are long-lived, with entropy barriers giving lifetimes far exceeding the Hubble time.

External baryonic accretion can drive sub-critical cores to collapse. The critical baryon-to-dark-matter mass ratio $\chi_\mathrm{BH}$ modifies the threshold mass for collapse,

$$M_\mathrm{crit}(\chi) \approx M_\mathrm{crit}^{(0)} [1+1.47\chi+0.46\chi^2]$$

Environments with high enough $\rho_b$ and modest $v_b$ can trigger collapse on Gyr timescales, seeding supermassive black holes as observed in high-$z$ quasars (Arguelles et al., 2023, Wang et al., 4 Feb 2026).

4. Physical Implications and Astrophysical Significance

Quantum-degenerate cores have profound astrophysical signatures:

• Central core–halo morphology: Galaxies exhibit a nearly constant-density quantum core, an intermediate semi-degenerate "plateau," and an asymptotic $\rho \sim r^{-2}$ isothermal halo.
• Resolution of the core–cusp problem: The quantum pressure from the Pauli exclusion principle strictly prevents the central density from diverging, ensuring a cored density profile even at small radii (core radii $\sim 0.1$–$1$ kpc for $m\sim$ few keV, $\sim 10^{-4}$ pc for $m\sim 100$ keV).
• Core–halo scaling and dynamical fits: The predicted $M_c \propto M_v^{1/2}$ and $R_c \propto m^{-8/3} M_c^{-1/3}$ scaling explain the sublinear growth of central masses with host halo and the strong dependence of core properties on $m$ (Chavanis, 2019, Argüelles et al., 2020, Ruffini et al., 2014, Destri et al., 2012).
• Mimicking supermassive black holes: For $m\sim 50$–$100$ keV, degenerate cores with $M_c \sim 10^6$–$10^8\,M_\odot$ and $R_c$ only a few times $R_S$ can dynamically match the central mass profiles in the Milky Way and other galaxies without an event horizon (Argüelles et al., 2016, Argüelles et al., 2019, Mestre et al., 2024).
• Early black hole seeds: Collapse of degenerate cores at high redshift ($z\sim 20-30$) produces massive seeds ($M_0\sim 10^6\,M_\odot$) that explain $z>6$ quasars, consistent with the timescales imposed by cosmic accretion history (Wang et al., 4 Feb 2026).

5. Particle Mass Constraints and Viable Parameter Space

The size and mass of quantum-degenerate cores depend sensitively on the fermion mass $m$:

• Lower limits arise from the requirement that core radii do not exceed observed cored structures in dwarf galaxies and from the Tremaine–Gunn and Lyman-$\alpha$ bounds. For typical central densities, $m \gtrsim 0.4$–$1$ keV ensures core radii $\lesssim$kpc and cold-enough velocity dispersions (Destri et al., 2012, Randall et al., 2016, Argüelles et al., 2014).
• Upper limits stem from the necessity to avoid over-compact, cusp-like cores and from core masses not exceeding the observed central masses or the Oppenheimer–Volkoff instability threshold; typically, $m \lesssim 10^3$ keV (Chung et al., 2018, Argüelles et al., 2016).
• Milky Way fits: modeling of the Galactic center and halo yields $mc^2=48$–$345$ keV as the viable window for S2-star constraints (Argüelles et al., 2016, Argüelles et al., 2019), with similar ranges favored by fits to Gaia rotation curves and streams (Mestre et al., 2024).

Distributed masses outside this window cannot replicate both observed core sizes in low-mass galaxies and the compact dynamical mass at the Galactic center. For sub-keV models in dwarfs, core radii are $\sim$100–300 pc, matching data for classical dwarf spheroidals (Choi et al., 2020, Randall et al., 2016).

6. Observational Probes and Theoretical Generalizations

Quantum-degenerate cores present testable predictions:

• Rotation curve "kinks" at $r\simeq R_c$ and transitions to flat isothermal behavior.
• Distinguishing true black holes: Core radii for $m\sim 100$ keV are only a small factor above the Schwarzschild radius, implying possible observable differences from event horizons at sub-milliarcsecond or pulsar-timed orbital scales (Argüelles et al., 2020, Mestre et al., 2024).
• Scaling laws and core–halo relations derived from first principles—especially the $M_c \propto m^{-2}$, $R_c \propto m^{-2}$ at fixed central degeneracy—provide a direct connection from microphysics to macroscopic galactic structure (Ruffini et al., 2014, Chavanis, 2021, Hernandez-Gutierrez et al., 2 Dec 2025).
• Extensions: The formalism is generalizable to include self-interacting fermions, multi-component halos, and scenarios with coupled ordinary/dark matter fluids, relevant for dark-matter admixed stellar remnants and potential ultra-low-energy supernovae (Parmar et al., 25 Jul 2025, Leung et al., 2013).

Ongoing and future high-resolution studies of galactic centers, stellar kinematics, lensing, and early-universe quasars will continue to test and constrain the parameter space and physical viability of quantum-degenerate fermion dark matter cores.

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The theoretical framework for quantum-degenerate fermion dark matter cores thus rests on polytropic hydrostatic equilibrium, entropy maximization under Fermi–Dirac statistics, and detailed matching to astrophysical observables across a broad mass and scale range (Chavanis, 2019, Argüelles et al., 2020, Arguelles et al., 2023, Destri et al., 2012, Argüelles et al., 2016, Argüelles et al., 2020, Ruffini et al., 2014, Chavanis, 2021, Argüelles et al., 2019, Hernandez-Gutierrez et al., 2 Dec 2025, Lin, 2024, Mestre et al., 2024, Wang et al., 4 Feb 2026).