Detecting the tidal heating with the generic extreme-mass-ratio inspirals

The horizon of a classical black hole, functioning as a one-way membrane, plays a vital role in the dynamic evolution of binary objects, capable of absorbing fluxes entirely. Tidal heating, stemming from this phenomenon, exerts a notable influence on the production of gravitational waves (GWs).This impact can be utilized for model-independent investigations into the nature of massive objects. In this paper, assuming that the extreme-mass-ratio inspiral (EMRI) contains a stellar-mass compact object orbiting around a massive exotic compact object (ECO) with a reflective surface, we compute the GWs from the generic EMRI orbits. Using the accurate and analytic flux formulas in the black hole spacetime, we adapted these formulas in the vicinity of the ECO surface by incorporating a reflectivity parameter. Under the adiabatic approximation, we can evolve the orbital parameters and compute the EMRI waveforms. The effect of tidal heating for the spinning and non-spinning objects can be used to constrain the reflectivity of the surface at the level of O(10^-4) by computing the mismatch and Fisher information matrix.