Distinguishing scale-dependent Planck stars from renormalization group improved Schwarzschild black holes by Gravitational waves

Abstract: Extreme mass-ratio inspirals (EMRIs), consisting of a stellar-mass black hole orbiting a supermassive black hole, are among the primary targets for future space-based gravitational wave detectors. By analyzing the emitted gravitational wave signals, we can probe the nature of compact objects in the strong-field region. To achieve this, we examine the effects of gravitational radiation. In this work, we base our calculations on the general relativistic Schwarzschild background and calculate the energy and angular momentum fluxes of gravitational waves. We perform a theoretical analysis of the equations of motion and the orbital evolution equations for EMRIs. The gravitational waveforms generated by the different periodic orbits of timelike test particles around scale-dependent Planck stars or renormalization group improved Schwarzschild black holes are investigated using both time-domain and frequency-domain methods. The time-domain method employs the ``analytic kludge" (AK) approach, while the frequency-domain method utilizes the discrete Fourier transform. We calculate the characteristic strain of the corresponding gravitational waves and compare them with the sensitivity curves of both ground-based and space-based detectors. These gravitational wave sensitivity curves can be experimentally tested for both spacetimes considered. Additionally, we use two approximate methods--the large eccentricity (EL) method and the small eccentricity (ES) method--to study the orbital evolution of EMRIs and compare the results with equatorial orbits derived from geodesic equations. Our findings will contribute to a deeper understanding of the nature of spacetime.