  Mergers of Primordial Black Holes (PBHs) may contribute to the gravitational
wave mergers detected by the LIGO-Virgo-KAGRA (LVK) Collaboration. We study the
dynamics of PBH binaries dressed with dark matter (DM) spikes, for PBHs with
extended mass functions. We analyze the impact of DM spikes on the orbital
parameters of the PBH binaries formed in the early Universe and calculate their
merger rates at the age of the Universe today. We consider two possible
scenarios for the dynamics of the dressed binaries: assuming that either the DM
spikes are completely evaporated from the binaries before merger or they remain
static until the merger. Contrary to previous studies, we find that the
presence of spikes may increase or decrease the present-day PBH merger rates,
in some cases dramatically. Comparing with merger rates reported by the LVK
Collaboration in the third Gravitational Wave Transient Catalog (GWTC-3), we
derive approximate constraints on the fraction of Solar-mass PBHs in cold dark
matter as $f_\mathrm{pbh}\leq \mathcal{O}(10^{-5} - 10^{-3})$, depending on the
mass function. Our calculations are valid only for the idealized scenarios in
which the DM spikes are either evaporated or static. However, they suggest that
the impact of DM spikes on PBH merger rates may be more complicated than
previously thought and motivate the development of a more general description
of the merger dynamics, including feedback of the DM spikes in highly eccentric
PBH binaries.
