Testing Super-Heavy Dark Matter from Primordial Black Holes with Gravitational Waves

Ultra-light primordial black holes with masses $M_{BH}<10^9$~g evaporate before big-bang nucleosynthesis producing all matter fields, including dark matter, in particular super-heavy dark matter: $M_{DM}\gtrsim 10^{10}$ GeV. If the dark matter gets its mass via $U(1)$ symmetry-breaking, the phase transition that gives a mass to the dark matter also produces cosmic strings which radiate gravitational waves. Because the symmetry-breaking scale $\Lambda_{CS}$ is of the same order as $M_{DM}$, the gravitational waves radiated by the cosmic strings have a large enough amplitude to be detectable across all frequencies accessible with current and planned experimental facilities. Moreover, an epoch of early primordial black hole domination introduces a unique spectral break in the gravitational wave spectrum whose frequency is related to the super-heavy dark matter mass. Hence, the features of a stochastic background of primordial gravitational waves could indicate that super-heavy dark matter originated from primordial black holes. In this perspective, the recent finding of a stochastic common-spectrum process across many pulsars by two nano-frequency pulsar timing arrays would fix the dark matter mass to be $3\times 10^{13}~\text{GeV} \lesssim M_{DM} \lesssim 10^{14}~\text{GeV}$. The (non-)detection of a spectral break at $0.2~\text{Hz} \lesssim f_* \lesssim 0.4~\text{Hz}$ would (exclude) substantiate this interpretation of the signal.