Nonradial oscillation modes of axion condensed neutron stars in the Cowling approximation

In light QCD axion models, finite-density corrections to the axion potential can source a nonvanishing axion field inside neutron stars. Extending our previous analysis of radial oscillations, we study the nonradial ($l=2$) oscillation spectrum of such axion condensed neutron stars in the relativistic Cowling approximation, keeping the fluid and axion perturbations fully coupled and dynamical. The condensate modifies the fluid spectrum in two ways: the altered equilibrium structure raises the $f$- and $p$-mode frequencies by up to $\sim20\%$, while the dynamical coupling to the axion field partially compensates this shift, almost entirely for the $f$-mode. Fluid modes with frequencies above the axion mass acquire a new damping channel through axion emission. The $f$-mode remains below this threshold for all configurations supporting a condensate, whereas the $p$-modes are damped on timescales that can be as short as $\sim10$ ms, shorter than their gravitational-wave damping times, except for certain axion masses where the outgoing axion amplitude nearly vanishes. The spectrum also contains a family of strongly damped axion-dominated quasinormal modes, with imaginary parts of order kHz, and a sequence of gravitationally bound axion states analogous to gravitational atom levels. Below the propagation threshold, the bound $p$-modes undergo avoided crossings with these levels, exchanging their fluid- and axion-dominated character.