Thermodynamics of self-gravitating fermions as a robust theory for dark matter halos: Stability analysis applied to the Milky Way

TL;DR

This work develops a fully general-relativistic, thermodynamically grounded framework for self-gravitating fermionic dark matter halos by adopting a truncated Fermi-Dirac (fermionic King) distribution with particle evaporation. Using a maximum entropy principle and the Poincaré-Katz turning-point criterion, it derives stable core–halo configurations and maps their existence onto a calori curve in the Milky Way context, constraining the DM particle mass to $mc^2 oughly 194$–$387$ keV and linking a $4.2 imes 10^6 M_\odot$ quantum core to a $oxed{M_{ m DM} o O(10^{11}) M_\odot}$ halo. The outer halo adopts a polytropic tail with index $5/2$, and the model reproduces Gaia DR3 rotation curves, a local DM density near $0.4 ext{ GeV}/ ext{cm}^3$, and a total MW mass around $2 imes 10^{11} M_\odot$, while predicting a maximal halo mass beyond which a fermionic core cannot be stable. The results imply that MW-like galaxies can host a stable fermion ball rather than a central black hole, and they provide explicit mass-relations and bounds that generalize to other galaxies, suggesting a transition to black holes in sufficiently massive halos. Overall, the paper demonstrates that thermodynamic stability analyses of relativistic, degenerate fermionic halos yield robust constraints on DM properties with direct observational relevance.

Abstract

We present a framework for dark matter (DM) halo formation based on a kinetic theory of self-gravitating fermions together with a solid connection to thermodynamics. Based on maximum entropy arguments, this approach predicts a most likely phase-space distribution which takes into account the Pauli exclusion principle, relativistic effects, and particle evaporation. The most general equilibrium configurations depend on the particle mass and develop a degenerate compact core embedded in a diluted halo, both linked by their fermionic nature. By applying such a theory to the Milky Way we analyze the stability of different families of equilibrium solutions with implications on the DM distribution and the mass of the DM candidate. We find that stable core-halo profiles, which explain the DM distribution in the Galaxy, exist only in the range $mc^2 \approx 194 - 387\,\rm{keV}$. The lower bound is a consequence of imposing thermodynamical stability on the core-halo solutions having a $4.2\times 10^6 M_\odot$ quantum core mass alternative to the black hole hypothesis at the Galaxy center. The upper bound is solely an outcome of general relativity when the quantum core reaches the Oppenheimer-Volkoff limit and undergoes gravitational collapse towards a black hole. We demonstrate that there exists a set of stable core-halo profiles which are astrophysically relevant in the sense that their total mass is finite, do not suffer from the gravothermal catastrophe, and agree with observations. The morphology of the outer halo tail is described by a polytrope of index $5/2$, developing a sharp decline of the density beyond $25\,\rm{kpc}$ in excellent agreement with the latest Gaia DR3 rotation curve data. Moreover, we obtain a total mass of about $2\times 10^{11} M_\odot$ including baryons and a local DM density of about $0.4\,\rm{GeV}\,c^{-2}\,\rm{cm}^{-3}$ in line with recent independent estimates.

Figures (12)

• Figure 1: Stability analysis for a particle mass of 56k, 250k and 387k, each with specific constraints $N_{\rm DM}$ and $\mu=\mathrm{e}^{W_0 - \theta_0}$. See \ref{['sec:method']} how the constraints are obtained from MW observations and how caloric curves are computed. From the caloric curves we extract the Keplerian (core) mass $M_K$ and the DM halo mass $M_{\rm DM}$. Stable core-halo solutions are represented by the green area while unstable core-halo solutions are represented by the gray area. Dot-dashed lines show the computed values. All other values are interpolated. The vertical dashed line represents a MW-like galaxy with a core mass ${M_K}^\text{\tiny (MW)} = 4.2E6\Msun$. All plots show the same ($M_K$, $M_{\rm DM}$)-window for better comparison how the areas change for different particle masses. The green triangle (stable region) shifts to the left as $m$ increases until it saturates to ${M_K}^\text{\tiny (max)}={M_K}^\text{\tiny (MW)}$ at the maximum particle mass $mc^2=387k\eV$. The core-halo relation found by 2002ApJ...578...90F is shown as purple dashed line together with the bounds as purple solid lines.
• Figure 2: Caloric curves corresponding to the three examples (a, b, c) with fixed DM halo masses in \ref{['fig:mkms:rhop-10']} for the case of 250k. Thin lines represent unstable solutions. Thick lines represents stable solutions. Points A, B and C indicate a change of stability. The additional window in the bottom panel clarifies that solution (c) belongs to the unstable branch. The extension of the stable core-halo branch (between B and C) becomes smaller and smaller from top to bottom panel.
• Figure 3: Illustration of three density profiles for the case of 250k, see colored squares in \ref{['fig:mkms:rhop-10']}. (a) represents a stable solution (thick solid line), (b) is in the unstable region (thin dot-dashed line), (c) is in the unstable region with $M_{\rm DM} \lesssim {M_{\rm DM}}^\text{\tiny (max)}$ (thin dashed line).
• Figure 4: $W_p$-$M_K$ curve corresponding to selected horizontal lines of the second panel in \ref{['fig:mkms:rhop-10']}. Shown are the stable branches bounded by the points B (green circle) and C (black dot) and the unstable branches before point B. The value of $W_p$ allows us to describe the characteristics of the outer halo and it is sufficient to show only up to $W_p = 10$. For clarity, we do not show the unstable branch after point C.
• Figure 5: Caloric curve for a DM distribution described by the parameters $mc^2=250k\eV$, $M_K = 4.2E6\Msun$, $M_{\rm DM} = 1.4E11\Msun$ and $\rho_p = 2E-2\Msun\per\parsec\cubed$. These parameters represent a stable DM configuration for the MW.