Gravitational dark matter: Free streaming and phase space distribution

Abstract

Gravitational dark matter (DM) is the simplest possible scenario that has recently gained interest in the early Universe cosmology. In this scenario, DM is assumed to be produced from the scattering of inflatons through the gravitational interaction during reheating. Gravitational production from the radiation bath will be ignored as our analysis shows it to be suppressed for a wide range of reheating temperatures $({T}_{\mathrm{re}})$. Ignoring any other internal parameters except the DM mass $({m}_{Y})$ and spin, a particular inflation model such as $\ensuremath{\alpha}$ attractor, with a specific scalar spectral index $({n}_{s})$ has been shown to uniquely fix the DM mass of the present Universe. For fermion type DM, we found the mass ${m}_{f}$ should be within $({10}^{4}--{10}^{13})\text{ }\text{ }\mathrm{GeV}$, and for boson/vector type DM, the mass ${m}_{s}/{m}_{X}$ turned out to be within $(1.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}/9.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}--{10}^{13})\text{ }\text{ }\mathrm{GeV}$. Interestingly, if the inflaton equation of state ${\ensuremath{\omega}}_{\ensuremath{\phi}}\ensuremath{\rightarrow}1/3$, the DM mass also approaches towards unique value, ${m}_{f}\ensuremath{\sim}{10}^{10}\text{ }\text{ }\mathrm{GeV}$ and ${m}_{s}({m}_{X})\ensuremath{\sim}{10}^{3}(8\ifmmode\times\else\texttimes\fi{}{10}^{3})\text{ }\text{ }\mathrm{GeV}$. We further analyzed the phase space distribution ${f}_{Y}$ and free streaming length ${\ensuremath{\lambda}}_{\mathrm{fs}}$ of these gravitationally produced DM. ${f}_{Y}$, which is believed to encode important information about DM, contains a characteristic primary peak at the initial time where the gravitational production is maximum for both fermionic and bosonic DM. Apart from this fermionic phase space, the distribution function contains an additional peak near the inflaton and fermion mass equality (${m}_{Y}\ensuremath{\sim}{m}_{\ensuremath{\phi}}$) arising for ${\ensuremath{\omega}}_{\ensuremath{\phi}}>5/9$. Furthermore, the height of this additional peak turned out to be increasing with decreasing ${T}_{\mathrm{re}}$ and at some point, dominates over the primary one. Since reheating is a causal process and DM is produced during this phase, gravitational instability forming small-scale DM structures during this period will encode those phase space information and be observed at present. The crucial condition of forming such a small-scale DM structure during reheating suggests that the ${\ensuremath{\lambda}}_{\mathrm{fs}}$ must be less than the length scale associated with the mode reentering the Hubble radius at the end of reheating ${\ensuremath{\lambda}}_{\mathrm{re}}$, which has been analyzed in detail. We further estimate in detail the range of scales within which the above condition will be satisfied for different masses of scalar, fermionic, and vector DM. Finally, we end by stating the fact that all our results are observed to be insensitive on the parameter $\ensuremath{\alpha}$ of the inflaton potential within the allowed range set by the latest Planck and BICEP/Keck results.