Massive fields affected by echoes: New physics vs. astrophysical environment

TL;DR

This work demonstrates that perturbations of massive fields around compact objects produce oscillatory, slowly decaying tails that are highly sensitive to environmental or near-horizon deformations of the effective potential. Using time-domain simulations across Schwarzschild, squashed Kaluza-Klein, and Schwarzschild-like wormhole geometries, the authors show that distant bumps induce late-time echoes that modify the asymptotic tail, while near-horizon bumps primarily affect the ringing phase; in wormholes, large masses can substantially amplify tail signals. These findings highlight a qualitative difference from massless-field echoes and suggest that environmental effects could render massive-tail features detectable by future observations, such as Pulsar Timing Arrays. The work provides a concrete framework for predicting how effective-potential deformations influence massive-field dynamics in diverse geometries.

Abstract

Unlike the perturbations of massless fields, the asymptotic tails of massive fields exhibit oscillations and decay slowly, following a power-law envelope. In this work, considering various scenarios admitting (either fundamental or effective) massive scalar and gravitational fields, we demonstrate that bump deformations in the effective potential, either in the near-horizon or far-field regions, modify these asymptotic oscillatory tails. Specifically, the power-law envelope transitions to a more complex oscillatory pattern, which cannot be easily fitted to a simple formula. This behavior is qualitatively different from the echoes of massless fields, which appear mainly during the quasinormal ringing stage and are considerably suppressed at the asymptotic tails. We show that in some models echoes may considerably amplify the signal at the stage of asymptotic tails.

Figures (7)

• Figure 1: Effective potential and time-domain profile for a massive scalar field perturbations around the Schwarzschild black hole with a bump: $r_0=1$, $\ell=1$, $r_{m}=100$, $\kappa =1$, $\mu=1$, $A=1/100$ (blue) and $A=1/12$ (red). The observer is situated at $r_p=3$, and the center of the Gaussian wave-package is between the observer and the event horizon, $p=16$. The bump changes the asymptotic late-time behavior only.
• Figure 2: Effective potential and time-domain profile for a massive scalar field perturbations around the Schwarzschild black hole with a bump: $r_0=1$, $\ell=1$, $r_{m}=8$, $\kappa =1$, $\mu=1$, $A=1/12$ (blue) and $A=1/4$ (red). The observer is situated at $r_p=3$, and the center of the Gaussian wave-package is between the observer and the event horizon, $p=16$. The bump changes intermediate tails, but not the asymptotic late-time behavior.
• Figure 3: Semi-logarithmic plot of the time-domain profiles for a massive scalar field perturbations around the Schwarzschild black hole with a bump: $r_0=1$, $\ell=1$, $r_{m}=8$, $\kappa =1$, $\mu=1$ (blue) and $\mu=0$ (red), $A=1/100$ (top panel) and $A=1/12$ (bottom panel). The observer is situated at $r_p=3$, and the center of the Gaussian wave-package is between the observer and the event horizon, $p=16$. The black slopes show exponential decay of the echoes' amplitudes: $\propto\exp(-0.02t)$ for $A=1/100$ (top) and $\propto\exp(-0.01t)$ for $A=1/12$ (bottom). The orange slope in the top panel shows the final signal after relaxation, governed by the quasinormal modes of the composite potential. The higher the bump, the later the time at which full relaxation occurs.
• Figure 4: Logarithmic plot of the time-domain profiles for a massive scalar field perturbations around the Schwarzschild black hole with a bump: $r_0=1$, $\ell=1$, $r_{m}=8$, $\kappa =1$, $\mu=2$, $A=1/100$ (left panel) and $A=1/12$ (right panel). The black slopes show power-law decay of the intermediate tails ($\propto t^{-2}$) and the orange slopes show power-law decay of the amplitudes of the echoes at late times ($\propto t^{-5/6}$).
• Figure 5: Effective potential and time-domain profile for $K=1$ (upper panels) $K=2$ (lower panels) perturbations around the squashed Kaluza-Klein black hole with a bump: $\rho_0=\rho_+=1$, $r_m=100$, $\kappa=1$, $A=1/12$. The central panels are for the semi-logarithmic plots, while the right panels are for the logarithmic plots. he observer is situated at $r_p=3$, and the center of the Gaussian wave-package is between the observer and the event horizon, $p=16$.