Primary hairs may create echoes

TL;DR

This work demonstrates that black holes endowed with primary Proca-Gauss-Bonnet hair can exhibit late-time echoes in the gravitational-wave ringdown without external environmental effects, owing to a nonmonotonic metric that creates a secondary peak in the perturbation potential. The authors analyze scalar and Dirac perturbations on these backgrounds using time-domain integration and a high-order Padé-resummed WKB method to obtain the quasinormal spectrum, showing that the echoes arise from intrinsic geometry changes rather than horizon-scale or environmental physics. The study maps the parameter space where a second potential peak exists and characterizes the corresponding quasinormal modes, highlighting a novel mechanism by which primary hair imprints on observable ringdown signals. These results have implications for testing modified gravity and near-horizon physics with future detectors such as LISA and third-generation ground-based observatories, which may constrain or detect such echoes with improved templates and stacking techniques.

Abstract

In most scenarios studied so far, the appearance of echoes in the ringdown signal requires modifications external to the black hole itself, such as the presence of matter in the near-horizon region, quantum field clouds, or exotic compact objects like wormholes that effectively introduce additional peaks in the effective potential. In this work we show that echoes can naturally arise in a different setting: black holes endowed with primary Proca-Gauss-Bonnet hair. We demonstrate that the primary hair modifies the effective potential in such a way that a second peak is formed, giving rise to late-time echoes without invoking any external environment or exotic horizon-scale physics. Using both the higher-order WKB method with Padé resummation and time-domain integration, we compute the quasinormal spectrum for scalar and Dirac test fields and show the appearance of these echoes. Our results highlight a novel mechanism by which primary hairs alone can leave observable imprints on the ringdown signal of black holes in modified gravity.

Figures (6)

• Figure 1: Distance between the main and secondary peak of the effective potential $s=\ell=0$ as a function of $Q$. Left panel for fixed $\beta=1.05M^2$ (from left to right): $\alpha=-M^2$ (magenta), $\alpha=-0.2M^2$ (red), $\alpha=-0.1M^2$ (blue). Right panel (from left to right): $\alpha=-1.1M^2$ and $\beta=1.05M^2$ (orange), $\alpha=-M^2$ and $\beta=1.05M^2$ (magenta), $\alpha=-1.1M^2$ and $\beta=1.1M^2$ (pink), $\alpha=-M^2$ and $\beta=1.1M^2$ (purple).
• Figure 2: Dominant quasinormal frequency of the scalar field ($\ell=1$) for $\alpha=-1.1M^2$, $\beta=1.03M^2$, $Q=0.56M$ calculated with the help of the WKB formula with the Padé approximants vs the Prony fit of the time-domain profile at late times of the ringdown (solid line), resulting to an accurate value of the fundamental frequency, $\omega M\approx0.3104732-0.0791347i$. The right panel shows absolute deviation from the time-domain result on semilogarithmic scale.
• Figure 3: Metric and time-domain profiles of the test scalar field for the black holes with $\alpha=-1.1M^2$ and $\beta=1.03M^2$: $Q=0.54M$ (red), $Q=0.55M$ (blue), and $Q=0.56M$ (black). The upper-left panel shows the metric function starting from the horizon. For the first two black-hole configurations, the nonmonotonic behavior of the metric function produces an additional (smaller) peak in the effective potential. The upper-right panel illustrates this feature for $s=\ell=0$; for other fields and multipole numbers, the picture is qualitatively similar. The middle panels display $\ell=0$ perturbation profiles: at early times, the perturbations exhibit a ringdown governed by the main peak, which is nearly identical for the black holes considered. For the second configuration, echoes appear only at very late times, consistent with the large separation between the two peaks. In the first configuration, the shorter separation leads to more frequent echoes, followed by the final relaxation governed by a dominant quasinormal frequency different from that of the ringdown phase. The bottom panels present $\ell=1$ perturbation profiles: the echoes have nearly the same time interval, since the peak separation is similar; however, because the main peak is higher than in the $\ell=0$ case, the ringdown stage lasts longer and the final relaxation occurs much later. For the third configuration, there is no additional peak, and no echoes are observed: after the ringdown, the signal transitions into the asymptotic power-law tail.
• Figure 4: Real and imaginary parts for the fundamental frequency of the test scalar field $\ell=1$ as a function of the hair parameter $Q$ of the black hole in the vector-tensor theory ($\beta=0$). From bottom to top: $\alpha=-M^2$ (cyan), $\alpha=-0.5M^2$ (blue), $\alpha=-0.1M^2$ (purple), $\alpha=0.1M^2$ (magenta), $\alpha=0.5M^2$ (red), $\alpha=M^2$ (orange). $Q=0$ corresponds to the Schwarzschild black hole.
• Figure 5: Real and imaginary parts for the fundamental frequency of the Dirac field $\kappa=1$ as a function of the hair parameter $Q$ of the black hole in the vector-tensor theory ($\beta=0$). From bottom to top: $\alpha=-M^2$ (cyan), $\alpha=-0.5M^2$ (blue), $\alpha=-0.1M^2$ (purple), $\alpha=0.1M^2$ (magenta), $\alpha=0.5M^2$ (red), $\alpha=M^2$ (orange). $Q=0$ corresponds to the Schwarzschild black hole.