Quasinormal-mode redshifts of black holes in galactic halos with radial pressures

Abstract

We investigate the black-hole quasinormal mode redshifts induced by a galactic dark-matter halo with nonvanishing radial pressure. The halo is described by an anisotropic fluid with a generalized density profile and a constant radial equation-of-state parameter $w$. We construct the corresponding static, spherically symmetric spacetime and study the resulting corrections to the unstable light ring and quasinormal-mode spectrum. For dilute halos and $w\neq -1$, the light-ring angular frequency and Lyapunov exponent acquire the same leading-order gravitational redshift, implying a universal shift of the real and imaginary parts of the quasinormal frequencies. The magnitude of this effect depends on both the halo structure and the radial pressure. This universal redshift relation holds not only in the case of $w=0$ that has been discovered in the literature but also in a much wider scenario, supporting its importance in black hole spectroscopy. On the other hand, the case $w=-1$ is qualitatively different: the factor that corresponds to the leading common redshift vanishes, the corrections are further suppressed by the hierarchy between the black-hole and halo scales, and the universal relation is broken, with the dominant contribution controlled by the inner halo profile. Our numerical computations of scalar-field quasinormal mode spectra confirm the analytic eikonal results.