  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.
