Exotic compact object behavior in black hole analogues

Classical phenomenological aspects of acoustic perturbations on a draining bathtub geometry where a surface with a reflectivity $\mathcal{R}$ is set at a small distance from the would-be acoustic horizon, which is excised, are addressed. Like most exotic compact objects featuring an ergoregion but not a horizon, this model is prone to instabilities when $|\mathcal{R}{|}^2\approx 1$. However, stability can be attained for sufficiently slow drains when $|\mathcal{R}{|}^2\lesssim 70\%$. It is shown that the superradiant scattering of acoustic waves is more effective when their frequency approaches one of the system’s quasinormal mode frequencies.

Figure 1

Real (top) and imaginary (bottom) parts of the fundamental $m=1$ QNM frequencies of ECO-like vortices with a perfectly reflecting surface at $r_0\equiv r_H(1+\delta )$, $0<\delta \ll 1$, where $r_H$ is the would-be acoustic horizon of the corresponding draining bathtub, as a function of the rotation parameter $B$, for DBCs (left) and NBCs (right). The left (right) arms of the interpolating functions refer to negative (positive) frequencies. The colored dots are zero-frequency (marginally stable) QNMs.

Figure 2

Ergoregion instability rotation parameter threshold of the fundamental $m=1$ QNMs of perfectly reflecting ECO-like vortices for both DBCs ($\mathcal{R}=-1$) and NBCs ($\mathcal{R}=1$). The shaded regions refer to the domain of the instability.

Figure 3

Plot of $(1+Z_0{)}^{-1}$ as a function of the rotation parameter $B$ and of the frequency $\omega$ of acoustic perturbations. When $|\mathcal{R}{|}^2<(1+Z_0{)}^{-1}$, ECO-like vortices are dynamically stable. Note that the upper limit on $|\mathcal{R}{|}^2$ decreases as $B$ increases.

Figure 4

Numerical and analytical values of the amplification factors for superradiant ($0<\omega <mB$) acoustic perturbations with $m=1$ scattered off a ECO-like vortex with $B=0.6$ and featuring a surface with reflectivity $\mathcal{R}$ at $r_0=r_H(1+\delta )$, where $r_H$ is the would-be acoustic horizon of the corresponding draining bathtub and $\delta ={10}^{-6}$. The resonance matches the real part of the fundamental QNM frequency of the vortex-like ECO (dotted vertical lines).

Figure 5

Imaginary part of the QNM frequency which sets the peaks of the amplification factors plotted in Fig. 4, as a function of the reflectivity $\mathcal{R}$. ${\omega }_I$ vanishes at $\mathcal{R}\approx 0.964$ (dotted vertical line).

Figure 6

Analytical approximation for the amplification factors plotted in Fig. 4 in the neighborhood of the fundamental $m=1$ QNM frequency for reflectivities near $\mathcal{R}\approx 0.964$.