Mimicker Quasi-Normal Modes

• Mimicker quasi-normal modes are spectral features produced by partial reflection at near-horizon surfaces, distinguishing black hole mimickers from classical black holes.
• The resonance comb observed at low frequencies, with spacing δω = π/(2M|logε|), provides a measurable probe into the reflective properties of these compact objects.
• The high-frequency spectral break, resulting from interference effects, offers a pathway to enhance signal detection through stacking multiple EMRI plunge observations.

Mimicker quasi-normal modes (MNMs) arise in gravitational wave astronomy and black hole spectroscopy as distinct spectral features associated with “black hole mimickers”—compact objects that lack an event horizon or have modified near-horizon reflectivity. Unlike standard black hole quasi-normal modes (QNMs), which are determined by the interior's absorbing boundary, MNMs result from the partial reflection of perturbations at a surface arbitrarily close to the would-be horizon. These modes imprint a characteristic structure on the gravitational wave energy spectrum, especially in the context of plunge events, and provide powerful discriminants for testing the nature of observed compact objects.

1. Spectral Structure of the Plunge: Resonance Comb and Spectral Break

The gravitational wave (GW) energy spectrum generated during the plunge of a compact object into a Schwarzschild-like black hole mimicker exhibits two robust features (Nair, 12 Sep 2025):

• Low-Frequency Resonance Comb: The spectrum displays a sequence of sharp, narrowly spaced resonances at frequencies equal to the real parts of the mimicker’s QNMs. Mathematically, these peaks are produced when the complex frequency $\omega$ aligns with one of the MNM frequencies; the small imaginary part (indicating a long-lived mode) ensures the Wronskian in the spectral representation vanishes, maximizing the energy output. The spacing between resonances is inversely related to the logarithm of the proximity $\epsilon$ of the reflective surface to the Schwarzschild radius:

$\delta\omega = \frac{\pi}{2M|\log\epsilon|}$

with the surface at $r_s = 2M(1+\epsilon),\ \epsilon\ll1$.

• High-Frequency Spectral Break: Above a threshold frequency, approximately $M\omega_{\mathrm{th}}\simeq 0.39$ for the dominant mode, the spectrum undergoes a qualitative change: it no longer follows the black hole’s exponential suppression and instead develops a substantial high-frequency tail:

$$\left(\frac{dE}{d\omega}\right)_{\text{mimicker}} \simeq \left(\frac{dE}{d\omega}\right)_{\text{BH}} + E_r(\omega), \qquad E_r(\omega) = A_m e^{-k_m\omega}$$

The amplitude $A_m$ is set by the amplitude-squared of the reflectivity $|R|^2$, and $k_m>0$. This exponential tail arises from high-frequency waves probing the near-horizon region and reflecting off the mimicker’s surface.

2. Theoretical Origin and Mode Structure

MNMs originate from the modified boundary conditions at the would-be event horizon. For a standard black hole, the condition is purely ingoing (perfect absorption). For a mimicker, partial or full reflection is permitted. When solving the Regge–Wheeler or Zerilli equations for gravitational perturbations, these reflective conditions lead to a spectrum of sharp, long-lived quasi-normal mode resonances—a “comb” in frequency space. Each resonance corresponds to a pole in the frequency domain Green function at a complex frequency set by the MNM spectrum, with real parts associated to oscillation and imaginary parts (typically small) governing the decay rate.

The energy per mode is given (for the dominant mode) by

$$\frac{dE}{d\omega} = \frac{\omega^2}{64\pi^2}\frac{(\ell+2)!}{(\ell-2)!}\left( |C^{(+),\text{out}}_{\ell m \omega}|^2 + \frac{4}{\omega^2} |C^{(-),\text{out}}_{\ell m \omega}|^2 \right)$$

where $C^{(\pm),\text{out}}_{\ell m \omega}$ are the mode amplitudes for each parity. The peak structure in $dE/d\omega$ aligns with the real parts of the MNMs.

3. Threshold Frequency and High-Frequency Behavior

A critical observation is the existence of a frequency threshold $M\omega_{\mathrm{th}} \simeq 0.39$. This value demarcates two regimes in the spectrum (Nair, 12 Sep 2025):

• Below $\omega_{\mathrm{th}}$: The effective potential $V_\ell^{(\pm)}(r)$ for linear perturbations is greater than $\omega^2$, so outgoing waves are exponentially suppressed. The resulting emission shows the QNM resonance comb.
• Above $\omega_{\mathrm{th}}$: $V_\ell^{(\pm)}(r) < \omega^2$ for all $r$; thus, waves can freely penetrate the potential, reach the near-horizon region, and be (partially) reflected. This process leads to increased energy output and the non-BH exponential spectral tail, a direct result of mode superposition and interference originating from the mimicker surface.

The existence of both spectral features arises generically for any compact object with a near-horizon reflecting surface and low absorption in the intermediate region.

4. Observational Discriminants and Stacking Strategies

Although the plunge waveform’s signal-to-noise ratio (SNR) is low for individual extreme mass ratio events (EMRIs), the coherent structure of the resonance comb and high-frequency tail provides a unique observational discriminant. The positions and regular spacing of the low-frequency MNM resonances, as well as the amplitude and decay rate of the high-frequency tail, are set solely by the mimicker’s interior and the location/reflection property of the surface ($\epsilon$ and $|R|^2$) (Nair, 12 Sep 2025).

Stacking multiple EMRI plunge observations—a process made feasible by the coherent and repeatable nature of these features and by the frequency invariance across similar events—can significantly enhance the cumulative SNR. Methods of coherent mode stacking previously proposed for other GW searches are especially suited to this spectral regime.

5. Comparative Table: Black Hole vs. Mimicker Spectral Features

Feature	Classical Black Hole	Black Hole Mimicker (MNMs)
Low-frequency spectrum	Smooth, no sharp resonances	“Comb” of sharp MNM resonances
High-frequency behavior	Strong exponential suppression	Elevated tail: $E_r(\omega) \propto e^{-k_m \omega}$
Threshold behavior	No qualitative break	Break at $M\omega_{\mathrm{th}}\simeq0.39$
QNM lifetimes	Short (rapid decay)	Long-lived (small $\operatorname{Im}\omega$)
Origin	Purely absorbing horizon	Partially/fully reflecting surface

6. Significance in Gravitational-Wave Spectroscopy

Mimicker quasi-normal modes enable GW spectroscopy to perform qualitative tests of the presence of event horizons. The resonance comb is a generic consequence of reflection at a near-horizon surface, not tied to a particular mimicker model. The sharpness of the comb and the spectral break provide direct probes of boundary conditions at the would-be horizon and can, in principle, distinguish between true black holes and horizonless compact objects or exotic states with modified near-horizon physics. Detection or stringent constraints on these features would have important implications for quantum gravity, compact object models, and the endpoint of gravitational collapse.

By accurately modeling plunge spectra and stacking multiple observations, it may be possible to rule out or provide evidence for black hole mimickers in EMRI mergers, advancing the empirical reach of gravitational-wave astronomy into regimes sensitive to fundamental questions about strong gravity and the nature of compact objects.