Gravitational dark matter: free streaming and phase space distribution

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 decaying inflaton 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. Ignoring any other internal parameters except the DM mass and spin, a particular inflation model such as $\alpha$-attractor, with a specific scalar spectral index $(n_s)$ has been shown to uniquely fix the dark matter mass. For fermion type dark matter we found the mass $m_f$ should be within $(10^4 - 10^{13})$ GeV, and for boson type DM, the mass $m_{s/X}$ turned out to be within $(10^{-8}-10^{13})$ GeV. Interestingly, if the inflaton equation of state $\omega_{\phi}\rightarrow 1/3$, the DM mass also approaches towards unique value, $m_f \sim 10^{10}$ GeV and $m_{s/X} \sim 10^3\,(\,8\times 10^3\,)$ GeV irrespective of the value of $\omega_\phi$. We further analyzed the phase space distribution $(f_Y)$, and free streaming length $(\lambda_{fs})$ of these gravitationally produced DM. $f_Y$, which is believed to encode important information about DM, is shown to contain a characteristic primary peak at the initial time where the gravitational production is maximum for both fermion/boson. Apart from this fermionic phase-space distribution function contains an additional peak near the inflaton and fermion mass equality ($m_Y=m_\phi$) arising for $\omega_\phi>5/9$. Since dark matter is produced during the reheating phase, gravitational instability forming small-scale DM structures during this period will encode those phase space information and be observed at present. Crucial condition $\lambda_{fs} <\lambda_{re}$ of forming such a small scale DM structure has been analyzed in detail.