Superradiant anomaly magnification in evolution of vector bosonic condensates bounded by a Kerr black hole with near-horizon reflection

Ultralight vector particles can form evolving condensates around a Kerr black hole (BH) due to superradiant instability. We study the effect of near-horizon reflection on the evolution of this system; by matching three pieces of asymptotic expansions of the Proca equation in Kerr metric and considering the leading order in the electric mode, we present explicit analytical expressions for the corrected energy level shifts and the superradiant instability rates. Particularly, in high-spin BH cases, we identify an anomalous situation where the superadiance rate is temporarily increased by the reflection parameter $\mathcal{R}$, which also occurs in the scalar scenario, but is largely magnified in vector condensates due to a faster growth rate in dominant mode; we constructed several featured quantities to illustrate this anomaly, and formalized the magnification with relevant correction factors, which may be of significance in future studies of gravitational waveforms of this monochromatic type. In addition, the duration of superradiance for the whole evolution is prolonged with a delay factor, which is calculated to be $(1+\mathcal{R})/({1-\mathcal{R}})$ approximately.