Muhammad Junaid, Fazeel Mahmood Khan

In this study, we investigate the dynamical evolution of a massive binary black hole (MBHB) in the Leo I dwarf spheroidal galaxy model and examine how dark matter along with stellar matter's gravitational interactions influence its long-term behavior. Using high-resolution direct N-body simulations, we follow the orbital evolution of the binary within a realistic model of the Leo I stellar and dark matter distribution. We found that the binary separation decreases from an initial 300-parsec orbit to roughly 1 parsec over a period of about 2 Gyr, primarily driven by dynamical friction and stellar hardening. The orbital evolution then stalls at this scale, illustrating the well-known final parsec problem. During this phase, the binary also develops increasing orbital eccentricity and produces a modest redistribution of the inner mass profiles in some cases. We then further estimate the final stage of the system's evolution using gravitational-wave emission models and find that the binary is unlikely to merge within a Hubble time. The prolonged dynamical friction phase appears to be related to the low stellar and dark matter densities in Leo I. These results suggest that massive binary black holes in dwarf spheroidal galaxies such as Leo I will not contribute to the gravitational-waves detectable from LISA even if dark matter is considered.