Time-domain analysis of scalar perturbations of Kerr--Newman black holes with Chern--Simons and Maxwell couplings

To study the onset of scalarization, we investigate linear scalar perturbations around Kerr–Newman black holes in the Einstein-Chern-Simons-Maxwell-scalar theory with quadratic Chern–Simons (CS) coupling $\gamma$ ($\widehat{\gamma }=\gamma \mathrm{/}{M}^2$) and Maxwell coupling $\eta$. For this purpose, we employ hyperboloidal Rácz–Tóth evolution and a stationary axisymmetric zero-mode computation. The resulting thresholds are compared with the time-domain response at representative points. For $\widehat{\gamma }=0$, we find two asymmetric threshold branches in $\eta$, while for $\widehat{\gamma }=\eta$, the two branches have opposite signs and unequal magnitudes. For unequal CS and Maxwell couplings, the two real threshold branches merge as the spin or CS coupling increases.