Cosmologically degenerate fermions

Abstract

Dark matter (DM) with a mass below a few keV must have a phase space distribution that differs substantially from the Standard Model particle thermal phase space: otherwise, it will free stream out of cosmic structures as they form. We observe that fermionic DM $\ensuremath{\psi}$ in this mass range will have a non-negligible momentum in the early Universe, even in the total absence of thermal kinetic energy. This is because the fermions were inevitably more dense at higher redshifts, and thus experienced Pauli degeneracy pressure. They fill up the lowest-momentum states, such that a typical fermion gains a momentum $\ensuremath{\sim}\mathcal{O}({p}_{F})$ that can exceed its mass ${m}_{\ensuremath{\psi}}$. We find a simple relation between ${m}_{\ensuremath{\psi}}$, the current fraction ${f}_{\ensuremath{\psi}}$ of the cold DM energy density in light fermions, and the redshift at which they were relativistic. Considering the impacts of the transition between nonrelativistic and relativistic behavior as revealed by constraints on $\mathrm{\ensuremath{\Delta}}{N}_{\mathrm{eff}}$ and the matter power spectrum, we derive qualitatively new bounds in the ${f}_{\ensuremath{\psi}}\ensuremath{-}{m}_{\ensuremath{\psi}}$ plane. We also improve existing bounds for ${f}_{\ensuremath{\psi}}=1$ to be ${m}_{\ensuremath{\psi}}\ensuremath{\ge}2\text{ }\text{ }\mathrm{keV}$. We remark on implications for direct detection and suggest models of dark sectors that may give rise to cosmologically degenerate fermions.