Probing parity-breaking black-hole geometry with extreme-mass-ratio inspirals

We investigate the gravitational-wave signatures of equatorial reflection symmetry breaking in extreme-mass-ratio inspirals (EMRIs) around a parity-violating conformal Kerr black hole. For the circular-orbit branch connected to equatorial Kerr orbits, we derive the dynamics through second order in the dimensionless deformation parameter $\epsilon$ and construct adiabatic inspirals and gravitational-wave signals using a mass-quadrupole prescription. Equatorial reflection maps $\epsilon$ to $-\epsilon$, making the polar displacement odd and the orbital frequency and energy even in the deformation. The induced first harmonic has a small direct contribution for the configurations studied, while second-order dynamical corrections accumulate over many cycles to produce appreciable dephasing. We find that rapid black-hole rotation enhances these phase effects and the associated noise-weighted waveform mismatch. For a one-year inspiral of a $10M_{\odot }$ object into a ${10}^6{M}_{\odot }$ black hole with $a\mathrm{/}M=0.9$ and viewing inclination $\iota =2\pi \mathrm{/}3$, the fixed-parameter mismatch reaches the comparison benchmark $\mathcal{M}=0.01$ at approximately $\epsilon ={10}^{-3}$, without optimization over time, phase, or Kerr parameters. A single-parameter Fisher analysis also indicates improved local sensitivity to $\epsilon$ at high spin. These results connect the reflection properties of the circular branch to accumulated source-phase effects and complement tests of propagation birefringence. The sensitivity estimates remain conditional on the adiabatic quadrupole model and the adopted comparison procedure; robust observational constraints require more accurate radiation modeling and a full parameter-estimation analysis.