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

TL;DR

The paper investigates the quasinormal modes (QNMs) and time-domain behavior of a massive scalar field on black holes in Einstein-Gauss-Bonnet-Proca gravity with primary Proca hair. It combines Leaver/Frobenius, WKB-Padé, and time-domain integration to compute QNM spectra and waveforms, revealing quasi-resonances where the fundamental and first overtone dampings vanish as the field mass $\mu$ increases. When the effective potential acquires a second peak, the late-time signal features a sequence of echoes, a distinctive signature of the Proca hair. The study also identifies intermediate and universal late-time tails, $|\Psi| \sim t^{-\frac{4}{3}-\ell} \sin(A(\mu)t)$ and $|\Psi| \sim t^{-\frac{5}{6}} \sin(\mu t)$ respectively, highlighting new dynamical features in this modified-gravity setup with potential observational relevance for gravitational-wave phenomenology.

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 Padé 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 $\sim t^{-4/3-\ell}\sin(A(μ)t)$ and a universal asymptotic tails $\sim t^{-5/6}\sin(μt)$.

Figures (6)

• Figure 1: Effective potentials for $\ell=0$, $\alpha=-\beta=0.1$, $Q=0.1$, $r_{h}=1$, $\mu=0$ (blue, bottom), $\mu=0.3$ (red, middle) and $\mu=0.6$ (green, top).
• Figure 2: Quasinormal modes for the fundamental mode and first three overtones at $r_{h}=1$, $Q=0.1$, $\alpha=-\beta=0.1$ found by the precise Leaver method.
• Figure 3: Time-domain profile for $\ell=0$, $r_{h}=1$, $Q=0.1$, $\alpha=-\beta =0.1$, $\mu=0.1$. The intermediate envelope for the tails $\approx t^{-4/3}$. The first two oscillations represent period of dominance of quasinormal modes. The Prony method gives $\omega = 0.23 - 0.24 i$, while the precise Frobenius method gives $\omega =0.227367 - 0.2412166 i$.
• Figure 4: Effective potential (left) and time-domain profile (right) for $\ell=0$, $r_{h}=1$, $Q=1.16366$, $\alpha=-\beta =-4.13757$, $\mu=0.1$.
• Figure 5: Effective potential (left) and time-domain profile (right) for $\ell=0$, $r_{h}=1$, $Q=1.7$, $\alpha=-\beta =0.1$, $\mu=10$.