Co-evolution of Nuclear Star Clusters and Massive Black Holes: Extreme Mass-Ratio Inspirals

We explore extreme mass-ratio inspirals (EMRIs) in the co-evolution of massive black holes (MBHs) and nuclear star clusters (NSCs), which host diverse stellar populations across a wide range of masses. The dynamics are simulated self-consistently with GNC, which we have updated to incorporate gravitational wave orbital decay, the loss cone of a spinning MBH, and stellar evolution. Over $12$ Gyr, we investigate the evolution of the NSC with a mass-growing MBH, as well as the EMRIs of stellar black holes, neutron stars, white dwarfs, brown dwarfs (BDs), and low-mass main-sequence stars (MSs), along with tidal disruption events (TDEs) involving MSs, BDs, and post-MSs. The mass growth of the MBH contributed by TDEs is typically $\sim 10^7\,M_{\odot}$, $\sim 10^6\,M_{\odot}$, and $\sim 5\times10^4\,M_{\odot}$ for massive, Milky-Way-like, and smaller NSCs, respectively. Between $40\%$ and $70\%$ of the stellar mass is lost during stellar evolution, which dominates the mass growth of the MBH if a significant fraction of the lost mass is accreted. The evolution of EMRI rates is generally affected by the cluster's size expansion or contraction, stellar population evolution, MBH mass growth, and the stellar initial mass function. The EMRI rates for compact objects peak at early epochs ($\lesssim 1$ Gyr) and then gradually decline over cosmic time. LISA-band ($0.1$ mHz) EMRIs involving compact objects around Milky-Way-like MBHs tend to have high eccentricities, while those around spinning MBHs preferentially occupy low-inclination (prograde) orbits. In contrast, MS- and BD-EMRIs usually have eccentricity and inclination distributions that are distinct from those of compact objects.