Unveiling Primordial Black Hole Relics Through Induced Gravitational Waves

TL;DR

This work investigates Planck-mass primordial black hole (PBH) relics as dark matter candidates and shows that, if PBHs evaporate incompletely leaving relics, the current relic abundance and relic number density imprint a distinctive peak in the induced gravitational wave (IGW) spectrum generated by the initial PBH isocurvature perturbations during a PBH-dominated epoch. The authors derive analytic relations linking the initial PBH mass $M_{\mathrm{PBH,f}}$ and fraction $\beta$ to the relic mass $m_{\text{relic}}=rM_{\mathrm{Pl}}$ and to the present relic abundance $f_{\text{relic}}$, establishing a one-to-one mapping between early-Universe PBH parameters and today’s relic DM. The peak frequency today scales as $f_{\mathrm{uv}} \approx 61.9\,\mathrm{Hz}\,(r/f_{\text{relic}})^{-1/3}$ and the peak amplitude scales steeply with $\beta$, offering a testable GW signature in the tens-to-hundreds-of-Hz band for future detectors. If observed, this IGW peak would provide a powerful probe of PBH relic scenarios, the early thermal history, and potential quantum-gravity signatures, while also enabling constraints on non-monochromatic mass spectra and PBH clustering. The work highlights a concrete, physics-rich pathway to use gravitational waves as a diagnostic of PBH relic dark matter.

Abstract

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.

Figures (4)

• Figure 1: A schematic diagram of PBH-dominated era, or early matter dominated era (EMD), when $H_{\mathrm{eva}}\to H_{\mathrm{eeq}}$, the PBH-dominated era duration would become shorter and shorter. Inf: inflation, ERD: early radiation dominated era, RD: radiation dominated era, MD: matter dominated era, DED: dark energy dominated era, eeq: early rad-matter equilibrium, eq: rad-matter equilibrium.
• Figure 2: Energy fraction in two examples described by \ref{['eq:drhoPBH']} and \ref{['eq:drhor']}. In all lines, $M_{\mathrm{PBH,f}}=1\mathrm{~g}$, starting from $t_{\mathrm{f}}$, ending with $t_{\mathrm{eva}}$ (vertical line). The blue lines are the time evolution of energy fraction $\Omega_{\mathrm{r}}$, the orange ones are for $\Omega_{\mathrm{PBH}}$. The solid lines is an example for $\beta=10^{-2}$, it appears a PBH-dominated era. The dashed lines is an example for $\beta=10^{-4}$\ref{['def:betamin']}, PBH-dominated era is shorter.
• Figure 3: Contour plot of $\Omega_{\mathrm{GW}, 0}^{\text{peak }} h^2$ in the $(\beta,n_{\mathrm{PBH}, 0})$ space. We focused on the region $10^{-16} < \Omega_{\mathrm{GW}, 0}^{\text{peak }} h^2 < 10^{-5}$.
• Figure 4: Contour plot of $\beta$ in the $(f_{\text{uv}},\Omega_{\mathrm{GW}, 0}^{\text{peak }} h^2)$ plane.