Gravitational waves from extreme-mass-ratio inspirals in the semiclassical gravity spacetime

More recently, Fernandes \cite{Fernandes:2023vux} discovered analytic stationary and axially-symmetric black hole solutions within semiclassical gravity, driven by the trace anomaly. The study unveils some distinctive features of these solutions. In this paper, we compute the gravitational waves emitted from the \ac{EMRI} around these quantum-corrected rotating black holes using the kludge approximate method. Firstly, we derive the orbital energy, angular momentum and fundamental frequencies for orbits on the equatorial plane. We find that, for the gravitational radiation described by quadrupole formulas, the contribution from the trace anomaly only appears at higher-order terms in the energy flux when compared with the standard Kerr case. Therefore, we can compute the EMRI waveforms from the quantum-corrected rotating black hole using the Kerr fluxes. We assess the differences between the EMRI waveforms from rotating black holes with and without the trace anomaly by calculating the dephasing and mismatch. Our results demonstrate that space-borne gravitational wave detectors can distinguish the EMRI waveform from the quantum-corrected black holes with a fractional coupling constant of $\sim 10^{-3}$ within one year observation. Finally, we compute the constraint on the coupling constant using the Fisher information matrix method and find that the potential constraint on the coupling constant by LISA can be within the error $\sim 10^{-4}$ in suitable scenarios.