  Future ground-based gravitational wave observatories will be ideal probes of
the environments surrounding black holes with masses $1 -
10\,\mathrm{M_\odot}$. Binary black hole mergers with mass ratios of order
$q=m_2/m_1\lesssim10^{-3}$ 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.
