Gravitational Waves from Primordial Black Hole Dark Matter Spikes

The origin of the binary black hole mergers observed by LIGO--Virgo--KAGRA remains an open question. We calculate the merger rate from primordial black holes (PBHs) within the density spike around supermassive black holes (SMBHs) at the centers of galaxies. We show that the merger rate within the spike is comparable to that within the wider dark matter halo. We also calculate the extreme mass ratio inspiral (EMRI) signal from PBHs hosted within the density spike spiralling into their host SMBHs due to gravitational wave emission. We predict that LISA may detect $\sim10^4$ of these EMRIs with a signal-to-noise ratio threshold of 20 within a 4 yr observation run, if all dark matter is made up of $\sim30{\rm\,M}_\odot$ PBHs. Uncertainties in our rates come from the uncertain mass fraction of PBHs within the dark matter spike, relative to the host central SMBHs, which defines the parameter space LISA can constrain.