Measuring dark matter spikes around primordial black holes with Einstein Telescope and Cosmic Explorer

Future ground-based gravitational wave observatories will be ideal probes of the environments surrounding black holes with masses 1 - 10M_⊙. Binary black hole mergers with mass ratios of order q = m₂ / m₁ ≲ 10^-3 can remain in the frequency band of such detectors for months or years, enabling precision searches for modifications of their gravitational waveforms with respect to vacuum inspirals. As a concrete example of an environmental effect, we consider here a population of binary primordial black holes which are expected to be embedded in dense cold dark matter spikes. We provide a viable formation scenario for these systems compatible with all observational constraints and predict upper and lower limits on the merger rates of small-mass-ratio pairs. Given a detected signal of one such system by either Einstein Telescope or Cosmic Explorer, we show that the properties of the binary and of the dark matter spike can be measured to excellent precision with one week's worth of data, if the effect of the dark matter spike on the waveform is taken into account. However, we show that there is a risk of biased parameter inference or missing the events entirely if the effect of the predicted dark matter overdensity around these objects is not properly accounted for.