Unveiling Primordial Black Hole Relics Through Induced Gravitational Waves

Black hole relics are of significant interest in cosmology and theoretical physics. In this work, we consider tiny primordial black holes (PBHs) ( $M_{\text {PBH }} \lesssim 10^7 \mathrm{~g}$ ) which are generated soon after the end of inflation and evaporate and reheat the Universe before big bang nucleosynthesis (BBN), but leave their remnants due to incomplete evaporation. These PBHs remnants may contribute as part or all of the dark matter (DM) today. Assuming that there exist PBH relics, we point out that the number density of PBH today can be directly read from the peak positions of the induced gravitational waves due to the inhomogeneous PBH distribution. If PBH relics are of Planck mass and they forms all the DM today, the PBH number density would be of $10^{-25} \mathrm{~cm}^{-3}$ with the peak frequency 60 Hz . The peak frequency scales as $f_{\text {relic }}^{1 / 3}$ where $f_{\text {relic }}$ is the fraction of the PBH relics in the total DM density. The peak amplitude carries the information of initial PBH abundance. For monochromatic-mass PBH with the current number density $10^{-41} \sim 10^{-25} \mathrm{~cm}^{-3}$ and initial abundance $10^{-13} \sim 10^{-7}$, the amplitude may be large enough to be detected by planned gravitational wave experiments in the near future.