Extreme mass-ratio inspirals around rotating accelerating black holes
Extreme mass-ratio inspirals (EMRIs) can magnify small departures from Kerr dynamics into appreciable gravitational-wave phase shifts accumulated over many orbital cycles. We exploit this sensitivity to investigate the imprint of a rotating black hole's acceleration on an EMRI waveform. The spinning C metric poses two obstacles to the standard Kerr flux framework: the spacetime is not asymptotically flat, and the acceleration breaks the reflection symmetry that supports exactly equatorial circular timelike orbits. For sufficiently small acceleration $AM$, we therefore formulate the calculation in an intermediate Kerr-like wave zone satisfying $M/r\ll1$ and $Ar\ll1$, and construct a near-equatorial circular orbit by examining its coupled radial--polar stability. We derive the separated point-particle source for the spin$-2$ radial Teukolsky equation, construct a regular normalized angular solution, solve the radial equation using the Sasaki--Nakamura transformation and the Green function method, and couple the resulting horizon and far-zone fluxes to the adiabatic evolution of stable near-equatorial circular orbits. The framework recovers the Kerr limit and reproduces the dominant $l=2$ Kerr fluxes with relative errors of order $10^{-7}$. Acceleration modifies both radiation reaction and the orbital frequency, producing a characteristic nonmonotonic accumulated dephasing. For $M=10^6M_\odot$, $m_s/M=10^{-5}$, $a/M=0.7$, and $AM=3\times10^{-7}$, the dominant-mode dephasing slightly exceeds $1$ rad over one year. Thus even weak acceleration can generate an order-radian secular phase imprint on long-duration EMRIs within the controlled regime of the present approximation.