Long-lived quasinormal modes and echoes in the Einstein-Gauss-Bonnet-Proca theory

TL;DR

This paper investigates the quasinormal modes and echo phenomena of black holes in Einstein-Gauss-Bonnet-Proca gravity, revealing long-lived states, potential echoes, and universal late-time tails through advanced numerical methods.

Contribution

It provides the first detailed analysis of quasinormal modes and echoes in Einstein-Gauss-Bonnet-Proca black holes, highlighting the effects of primary Proca hair and massive scalar fields.

Findings

Long-lived quasi-resonances for increasing scalar mass.

Presence of echoes when the potential develops an additional peak.

Universal late-time tails with specific power-law decay.

Abstract

We study quasinormal modes and time-domain profiles of a massive scalar field in the background of black holes arising in Einstein--Gauss--Bonnet--Proca gravity. The black holes in this theory possess \emph{primary Proca hair}, which modifies the effective potential and gives rise to distinctive dynamical phenomena. Using three complementary numerical techniques -- the WKB method with Pad\'e resummation, time-domain integration, and the Frobenius (Leaver) method -- we obtain accurate spectra of quasinormal frequencies. Our analysis shows that increasing the scalar-field mass leads to arbitrarily long-lived states, known as quasi-resonances, a behavior shared by both the fundamental mode and the first overtone. The real part of the quasinormal mode decreases for the first and higher overtones and starting from the second overtone it reaches zero at some critical value of mass of the field. When the effective potential develops an additional peak, the late-time signal exhibits a sequence of echoes. Furthermore, the time-domain evolution reveals distinct regimes of intermediate power-law tails $\sim t^{-4/3-\ell}\sin(A(\mu)t)$ and a universal asymptotic tails $\sim t^{-5/6}\sin(\mu t)$.