Gravitational Atom Spectroscopy

TL;DR

The paper investigates gravitational atoms, i.e., self-gravitating clouds of ultralight scalar fields around black holes, and their imprint on the gravitational-wave ringdown. It derives and solves the axial perturbation equations in a fully relativistic, spherically symmetric background to obtain the fundamental quasi-normal mode in both frequency and time domains, revealing shifts from the vacuum spectrum that scale with the cloud compactness $\mathcal{C}_{\rm sc}$ and depend on the dimensionless mass parameter $\alpha=\mu_{\rm s} M_{\rm BH}$. The results show that, in the small-$\mathcal{C}_{\rm sc}$ regime, both the real and imaginary parts of the QNM frequency shift nearly linearly with $\mathcal{C}_{\rm sc}$, with slopes that differ with $\alpha$, and that LVK-like sensitivities could detect such shifts for plausible compactness values. The analysis also demonstrates a breakdown of the eikonal photon-ring–QNM correspondence in the presence of the cloud, while near-horizon geometry, as encoded in the surface gravity, remains a robust indicator of the environment. These findings motivate further study of polar perturbations and inspiral/postmerger effects to assess the observability of gravitational atoms with current and future gravitational-wave detectors.

Abstract

Black holes in our Universe are rarely truly isolated, being instead embedded in astrophysical environments such as plasma or dark matter. A particularly intriguing possibility is that light scalar fields form bound states around black holes, producing extended ''clouds'' known as gravitational atoms. When these clouds become sufficiently compact, the spacetime can no longer be described by a vacuum solution of General Relativity. In this regime, one can construct quasi-stationary, spherically symmetric, self-gravitating scalar gravitational-atom configurations. Here, we explore an observationally relevant aspect of these systems by computing their fundamental quasi-normal mode. We present a fully relativistic calculation of the axial modes in both the time and frequency domains, finding frequency shifts relative to the vacuum case that depends mostly on the compactness of the gravitational atom. For sufficiently compact configurations, these shifts may be detectable by current or future gravitational wave detectors.

Figures (6)

• Figure 1: Background solution for $\alpha = 0.249$ and $\mathcal{A} = 0.401$. Shown are the mass function $m/M_{\rm BH}$ (solid red), which asymptotes to the constant value $M_{\mathrm{ga}} = 1.913 M_{\mathrm{BH}}$; the metric function $a$ (dashed blue), which approaches unity at large radii; and the real (solid dark green) and imaginary (dashed light green) parts of the normalised scalar field $\psi/\mathcal{A}$, both decaying to zero asymptotically. The vertical line marks the characteristic radius of the gravitational atom, $R_{\mathrm{ga}} = 22.92 M_{\mathrm{BH}}$. The corresponding eigenfrequency is $M_{\mathrm{BH}}\Omega = 0.2161 + 0.0188i$, and the compactness is $\mathcal{C}_{\rm sc}=0.03486$.
• Figure 2: Comparison between time-domain (TD) and frequency-domain (FD) results. The top panels show the real and imaginary parts of the fundamental mode as functions of the compactness $\mathcal{C}_{\rm sc}$ for two values of $\alpha$. The Schwarzschild (vacuum) limit is marked by a black dot. The bottom panels display the relative differences between TD and FD computations (half-filled markers), which typically remain below $10^{-3}$ for the real part and $10^{-3}$ for the imaginary part.
• Figure 3: Percentage relative shift of the real ( top panel) and imaginary ( bottom panel) parts of the fundamental QNM as functions of the compactness for two values of $\alpha$: $\alpha=0.249$ (red circles) and $\alpha=0.110$ (dark blue squares). For $\mathcal{C}_{\rm sc}\lesssim 3\times 10^{-2}$, the shift varies linearly with compactness; the solid lines show the results of a linear fit. The insets show the same data on a linear scale, highlighting the deviation from the linear trend for $\mathcal{C}_{\rm sc}>10^{-2}$. The red dashed horizontal line marks the 90% credible LVK constraint from the GW250114 event KAGRA:2025oiz. Configurations above this line could, in principle, produce QNM shifts observable with the LVK detectors.
• Figure 4: Imaginary versus real part of the fundamental mode for different scalar-cloud configurations. Results are shown for $\alpha=0.249$ (red circles) and $\alpha=0.110$ (dark blue squares). The black dot marks the vacuum limit, $M_{\rm BH}\Upsilon_{\rm vac}$. The inset shows that for $\mathcal{C}_{\rm sc}<0.02$, the ratio $\mathscr{L}$\ref{['eq:L_def']} remains approximately constant.
• Figure 5: Relative percentage shifts of the light-ring frequency $\Omega_{\rm LR}$, Lyapunov exponent $\lambda_{\rm LR}$ and surface gravity $\kappa$ induced by the presence of a scalar cloud, shown relative to the vacuum case for $\alpha=0.110$. Solid lines denote linear fits valid for $\mathcal{C}_{\rm sc}\lesssim 10^{-4}$.