Long-lived modes and grey-body factors of massive fields in quantum-corrected (Hayward) black holes

TL;DR

We investigate the dynamics of a massive scalar field in the Hayward black hole geometry, using Padé–WKB and time-domain Prony analysis to compute quasinormal frequencies $\omega$ and a WKB framework to estimate grey-body factors $\Gamma_\ell(\omega)$. We find that increasing the field mass $\mu$ suppresses damping and produces long-lived quasi-resonances, while time-domain signals develop oscillatory tails with a power-law envelope; grey-body factors shift their peak to higher frequencies and exhibit low-frequency suppression. The QNM–GBF correspondence remains highly accurate for large multipole number $\ell$ and moderate masses, but its precision diminishes as $\mu$ grows or $\ell$ decreases. These results illuminate how quantum corrections encoded in the Hayward/AS geometry influence black-hole perturbations and could guide observational probes of regular and quantum-corrected black holes.

Abstract

We study the dynamics of a massive scalar field in the background of the Hayward black hole, which can be interpreted both as a regular spacetime and as an effective geometry arising from Asymptotically Safe gravity. The quasinormal spectrum and grey-body factors are computed using the WKB method with Padé improvements and confirmed through time-domain integration followed by Prony analysis. We find that the mass of the field significantly suppresses the damping rate of quasinormal oscillations, giving rise to long-lived modes that continuously approach arbitrarily long-lived states (quasi-resonances) at certain critical field masses. In the time domain, the standard exponentially decaying ringdown is replaced by oscillatory tails with a power-law envelope. The corresponding grey-body factors reveal a pronounced shift of the transmission peak toward higher frequencies and a suppression of the low-frequency part of the spectrum. Finally, we show that the correspondence between quasinormal modes and grey-body factors remains valid for massive fields, being highly accurate for large multipole numbers and gradually losing precision as either the field mass increases or the multipole number decreases.

Figures (7)

• Figure 1: Potential as a function of the tortoise coordinate of the $\ell=0$, $\mu=0.1$ scalar field for the Hayward black hole ($M=1$): $\gamma=0$ (black) (green) $\gamma=0.9$ (orange), and $\gamma=1.1$ (blue).
• Figure 2: Potential as a function of the tortoise coordinate of the $\ell=0$, $\mu=0.4$ scalar field for the Hayward black hole ($M=1$): $\gamma=0$ (black) $\gamma=0.9$ (orange), and $\gamma=1.1$ (blue). One can see that at sufficiently large mass of the field, the effective potential does not have a peak.
• Figure 3: Semi-lograrithmic time-domain profile for $\gamma=1.18$, $\ell=0$, $\mu=0$, $M=1$. The Prony method allows one to extract the fundamental mode $\omega = 0.110201 - 0.0881366 i$, and the 6th order WKB method gives $\omega = 0.110839 - 0.088116 i$. The difference between the quasinormal modes obtained by the two methods is $0.000638 + 0.000021 i$, which is considerably smaller than one tenth of a percent.
• Figure 4: Semi-lograrithmic time-domain profile for $\gamma=1.18$, $\ell=1$, $\mu=0.1$, $M=1$. The Prony method allows one to extract the fundamental mode $\omega = 0.311752 - 0.079697 i$, and the 6th order WKB method gives $\omega = 0.311568 - 0.080557 i$. The difference between the quasinormal modes obtained by the two methods is $-0.000184 - 0.0008598 i$, which is considerably smaller than one tenth of a percent. After the ringdown phase one can see the beginning of the intermediate oscillatory tail with the power-law envelope.
• Figure 5: Semi-lograrithmic time-domain profile for $\gamma=1.18$, $\ell=2$, $\mu=0.1$, $M=1$. The Prony method allows one to extract the fundamental mode $\omega = 0.512372 - 0.0811234 i$, and the 6th order WKB method gives $\omega = 0.512393 - 0.081119 i$. The difference between the quasinormal modes obtained by the two methods $0.0000212532 + 4.43302*10^-6 I$ is negligible. After the ringdown phase one can see the beginning of the intermediate oscillatory tail with the power-law envelope.