Spectral instability of horizonless compact objects within astrophysical environments

TL;DR

The paper investigates how environmental effects, modeled as a localized Gaussian bump outside the light ring, interact with spectral instabilities in the quasinormal-mode (QNM) spectrum of horizonless exotic compact objects (ECOs) that have a purely reflecting surface. Using both a hyperboloidal approach and a Leaver continued-fraction method, the authors show that environmental bumps can destabilize the fundamental and overtone QNMs depending on the ECO compactness, with loosely-compact objects displaying significant fundamental-mode instability and ultra-compact objects exhibiting remarkable robustness of the fundamental mode but instability in overtones via an overtaking mechanism. Although environmental perturbations can amplify spectral instabilities, they do not trigger genuine modal instabilities within the parameter range explored; the overtaking dynamics reveal a rich restructuring of the mode hierarchy, particularly for less compact ECOs. The findings have implications for gravitational-wave spectroscopy, suggesting that ringdown signals could bear environmental imprints without forcing ECOs into unstable modal regimes, and point toward future work incorporating rotation and more realistic environmental models to sharpen observational tests.

Abstract

Recent non-modal analyses have uncovered spectral instabilities in the quasinormal-mode spectrum of black holes; a phenomenon that intriguingly extends to spherically-symmetric exotic compact objects. These results point to a sensitivity of the spectrum with potentially far-reaching implications for black-hole spectroscopy. At the same time, growing attention has turned to astrophysical environments around compact objects and their role in shaping gravitational-wave astrophysics. In this work, we establish a direct link between spectral instabilities and environmental effects by modeling matter as a localized bump outside the light ring of a spectrally-unstable exotic compact object with a purely reflective surface. We find that while such environments can destabilize the fundamental quasinormal modes of loosely-compact exotic objects, the fundamental modes of ultra-compact horizonless objects remain remarkably robust. In contrast, overtones are shown to develop spectral instabilities in the presence of the bump. By tracking both interior modes, trapped between the light ring and the surface of the exotic compact object, and exterior modes, confined between the bump and the light ring, we uncover an overtaking instability in which ``unperturbed'' exterior overtones metamorphose into ``perturbed'' fundamental modes as the bump moves outward. Finally, we demonstrate that environmental effects, while capable of further amplifying spectral instabilities, cannot induce next-to-leading-order perturbations strong enough to trigger a modal instability.

Figures (8)

• Figure 1: Illustrative picture of the integration procedure. We start at the surface of the compact object, represented by the vertical solid red line, using the imposed boundary conditions, directly integrating to the region $I$ (DI path). At the region $I$, the solution is matched with two continued fraction expansions that carry the information of ingoing and outgoing wave behavior. Finally, we integrate again through the bump, matching the solution in the region $II$ with a purely outgoing one.
• Figure 2: Top row: Convergence tests of the fundamental $\ell=2$ mode ($n=0$, top left) and first overtone ($n=1$, top right) for an ECO with $\mathcal{E}/r_h=10^{-3}$ and a bump at varying positions $a_0$ and amplitude $\epsilon=10^{-6}$. Due to the absence of of an exact spectrum, we have used the $N=300$-grid-points value of QNMs as our baseline and compared its convergence by varying $N$. We find that the absolute difference of the imaginary parts between the $N=300$ and, say, $N=250$ is of order $\mathcal{O}(10^{-3}-10^{-4})$. The same order of magnitudes are found for the real part of the QNMs compared. Bottom row: Same as above but for an ECO with $\mathcal{E}/r_h=10^{-2}$. We find that the absolute difference of the imaginary parts between the $N=300$ and, say, $N=250$ is of order $\mathcal{O}(10^{-8}-10^{-10})$ for the fundamental mode (bottom left) and $\mathcal{O}(10^{-4}-10^{-8})$. The same order of magnitudes are found for the real part of the QNMs compared. The exponential convergence in all plots confirms the expected behavior from the employed spectral scheme.
• Figure 3: Scalar massless $\ell=2$ QNMs of an ECO, with a perfectly reflective surface at $r=r_s$. By varying the surface position from $\mathcal{E}/r_h=10^{-3}$ to $0.5$ we plot the evolution of the first two modes, i.e. the fundamental $n=0$ and first overtone $n=1$, in the complex plane. The change of the color from blue to red designates the increment of $\mathcal{E}/r_h$.
• Figure 4: Scalar massless fundamental ($n=0$) $\ell=2$ QNMs of an ECO, with a perfectly reflective surface at $r=r_s$, and a Gaussian bump with $\epsilon=10^{-6}$ and $\varrho=1$ center at $a_0$. By varying the position of the bump from $a_0/r_h=2$ to $60$ we plot the real (left panel) and imaginary part (right panel) of the fundamental mode, for three different compactness, i.e. $\mathcal{E}/r_h=10^{-1}$ (red), $\mathcal{E}/r_h=10^{-2}$ (green) and $\mathcal{E}/r_h=10^{-3}$ (blue).
• Figure 5: Scalar massless first overtone ($n=1$) $\ell=2$ QNMs of an ECO, with a perfectly reflective surface at $r=r_s$, and a Gaussian bump with $\epsilon=10^{-6}$ and $\varrho=1$ center at $a_0$. By varying the position of the bump from $a_0/r_h=2$ to $60$ we plot the real (left panel) and imaginary part (right panel) of the fundamental mode, for three different compactness, i.e. $\mathcal{E}/r_h=10^{-1}$ (red), $\mathcal{E}/r_h=10^{-2}$ (green) and $\mathcal{E}/r_h=10^{-3}$ (blue).