Small scale induced gravitational waves from primordial black holes, a stringent lower mass bound, and the imprints of an early matter to radiation transition

In all inflationary scenarios of primordial black holes (PBH) formation, amplified scalar perturbations inevitably accompany an induced stochastic gravitational waves background (ISGWB) at smaller scales. In this paper, we study the ISGWB originating from the inflationary model, introduced in our previous paper [1] wherein PBHs can be produced with a nearly monochromatic mass fraction in the asteroid mass window accounting for the total dark matter in the universe. We numerically calculate the ISGWB in our scenario for frequencies ranging from nanoHz to KHz that covers the observational scales corresponding to future space based GW observatories such as IPTA, LISA, DECIGO and ET. Interestingly, we find that ultralight PBHs ($M_{\rm PBH} \sim 10^{-20} M_\odot$) which shall completely evaporate by today with exceedingly small contribution to dark matter, would still generate an ISGWB that may be detected by a future design of the ground based Advanced LIGO detector. Using a model independent approach, we obtain a stringent lower mass limit for ultralight PBHs which would be valid for a large class of ultra slow roll inflationary models. Further, we extend our formalism to study the imprints of a reheating epoch on both the ISGWB and the derived lower mass bound. We find that any non-instantaneous reheating leads to an even stronger lower bound on PBHs mass and an epoch of a prolonged matter dominated reheating shifts the ISGWB spectrum to smaller frequencies. In particular, we show that an epoch of an early matter dominated phase leads to a secondary amplification of ISGWB at much smaller scales corresponding to the smallest comoving scale leaving the horizon during inflation or the end of inflation scale. Finally, we discuss the prospects of the ISGWB detection by the proposed and upcoming GW observatories.