Constraining ultra light fermionic dark matter with Milky-Way observations

TL;DR

The paper investigates whether ultra-light fermionic dark matter, treated as a semi-degenerate, non-interacting gas, can describe Milky Way halo properties. It solves the Tolman-Oppenheimer-Volkoff equations with a non-relativistic polytropic equation of state $p \sim \rho^{5/3}/m_F^{8/3}$ to connect microphysical parameters, notably the fermion mass $m_F$, to galactic-scale structure, predicting core-like halos rather than cuspy profiles. Milky-Way rotation-curve data constrain $m_F$ to about $32$–$34$ eV and central densities $\rho_0$ to roughly $1.3$–$1.6$ GeV cm$^{-3}$ (68–90% CL), yielding a local DM density consistent with observations and a core-shaped DM distribution distinct from NFW, while also enabling an estimate of the central Fermi energy. Complementarily, the Sgr A* spectral energy distribution provides stringent bounds on the annihilation cross section $\langle \sigma v\rangle$, implying that such ultra-light fermionic DM would have suppressed or negligible annihilation signals. Together, these results indicate that while ultra-light fermion DM can produce core halos and address some small-scale problems, its viability is tightly constrained and likely requires a joint treatment with other astrophysical and cosmological data.

Abstract

The equation of state for a degenerate gas of fermions at zero temperature in the non-relativistic case is a polytrope, i.e. $p \simρ^{5/3}/m_F^{8/3}$. If dark matter is modeled by such a non-interacting fermion, this dependence in the mass of the fermion $m_F$ explains why if dark matter is very heavy the effective pressure of dark matter is negligible. Nevertheless, if the mass of the dark matter is very small, the effective pressure can be very large, and thus a system of self-gravitating fermions can be formed. In this work we model the dark matter halo of the Milky-Way by solving the Tolman-Oppenheimer-Volkoff equations, with the equation of state for a partially degenerate ultralight non-interacting fermion. We found that to fit the rotational velocity curve of the Milky-Way, the mass of the fermion should be in the range $31.5 ~\mbox{eV} < m_F < 35~$eV at $90\%$ C.L. Moreover, the central density is restricted to be in the range of $1.2 < ρ_0<1.7$ GeV/cm$^3$ at $90\%$ C.L. The fermionic dark matter halo has a very different profile as compared with the standard Navarro-Frenk-White profile, thus, the possible indirect signals for annihilating dark matter may change by orders of magnitude. We found bounds for the annihilation cross section in this case by using the Saggitarius A* spectral energy distribution.

Figures (7)

• Figure 1: Self-gravitating structures made of non-interacting, non-relativistic semi-degenerate fermions at zero temperature. In the upper panel it is shown typical density and mass profiles for a central density of $\rho_0=47 ~\hbox{GeV/cm}^3$ for two different masses of the fermion, namely $m_F=30$ eV (solid) and $m_F=100$ eV (dashed). The flat central region of the density profile correspond to the degenerate gas. Lower panel shows the mass function.
• Figure 2: Iso-curves of the total mass $M(\rho_0,m_F)$ (left) and the radius of the core $R_{Core}(\rho_0,m_F)$ (right) for the full set of self-gravitating configurations obtained by varying the free parameters $m_F$ and $\rho_0$. The gray band corresponds to the allowed region for $\rho_0$ and $m_F$ needed to have configurations that fullfill the condition of constant dark matter surface density eq. \ref{['salucci']}.
• Figure 3: Milky-Way rotational velocities and theoretical rotational curve obtained for the best fit $\rho_0=1.45~\hbox{GeV cm}^{-3},m_F=33.23~\hbox{eV}$ for a dark matter halo model made of semi-degenerate non-interacting fermions )
• Figure 4: Iso-curves at $68\%$ C.L (cyan) and $95\%$ C.L. (brown) for the mass of the fermion and central densities that fit the Milky-Way rotational curve data. The allowed region in the parameter space $(m_F,\rho_0)$ that adjust the rotational curve of the Milky-Way. In gray it is shown the region in $(m_F,\rho_0)$ that fulfills eq. \ref{['salucci']}.
• Figure 5: Dark matter profile of the Milky Way halo made by fermionic dark matter with $31.5 ~\hbox{eV} < m_F < 35~$eV and $1.2 < \rho_0<1.7$ GeV/cm$^3$ in the neighborhood of the Solar System.