Virtual absorption modes of Schwarzschild-de Sitter spacetimes in semi-open systems

Abstract

We present a study of virtual absorption modes (VAMs) in Schwarzschild-de Sitter (SdS) spacetime under semi-open boundary conditions, where the VAMs correspond to total transmission modes (TTMs) with the reflection amplitude being vanished. Our numerical analysis reveals that as the reflectivity $|\mathcal{K}|$ decreases, the VAM spectra migrate systematically toward regions of less negative imaginary parts, with each overtone exhibiting a critical reflectivity at which $\text{Im}(ω_{\text{VAM}})=0$. Using simulations based on spectral collocation methods, it is demonstrated that excitation precisely at a VAM spectrum leads to coherent perfect absorption (CPA). These results establish VAMs as the spectrum signatures of CPA for exotic compact objects (ECOs).

Figures (5)

• Figure 1: The migrations of the VAM spectra with $\mathcal{K}$ varying with $r_\text{c}=1.1$. For four panels, the positions of the reflective walls are $x_{\text{w}}=-100$, $x_{\text{w}}=-200$, $x_{\text{w}}=-300$, and $x_{\text{w}}=-800$, respectively. Different color lines represents different modes. Real lines are for $\mathcal{K}>0$ while dashed lines are for $\mathcal{K}<0$.
• Figure 2: The migrations of the VAM spectra with $\mathcal{K}$ varying with $r_\text{c}=2$. For four panels, the positions of the reflective walls are $x_{\text{w}}=-20$, $x_{\text{w}}=-50$, $x_{\text{w}}=-80$, and $x_{\text{w}}=-100$, respectively. Different color lines represents different modes. Real lines are for $\mathcal{K}>0$ while dashed lines are for $\mathcal{K}<0$.
• Figure 3: The VAM eigenfunctions are shown. For the three top panels, we choose the modes $n=6$ and parameters being $r_\text{c}=1.1$, $x_{\text{w}}=-100$ with $\mathcal{K}>0$. For the three bottom panels, we choose the modes $n=4$ and parameters being $r_\text{c}=2$, $x_{\text{w}}=-20$ with $\mathcal{K}<0$. The solid line represents the real part of the eigenfunction, while the dashed line represents the imaginary part of the eigenfunction. The black thick dotted line is used to depict the position of the reflective wall.
• Figure 4: The time evolution of $\Psi(t,x)$ and energy are shown, where the parameters are $r_\text{c}=1.1$, $x_{\text{w}}=-100$ and $\mathcal{K}=0.03162$. The parameters of the initial conditions are $\mathcal{A}=1$, $x_{\text{t}}=800$ and $\sigma=30$. In the top panels, the overtone $n=6$ mode $\omega_{\text{VAM}}=0.1921+0.0181\mathrm{i}$ is excited (purple solid line in Fig. \ref{['TTM_Spectra_rc_1p1_a']}), while in the middle and bottom panels, non-VAM spectrum with $\Omega_0=0.9\omega_{\text{VAM}}=0.1729+0.0163\mathrm{i}$ and non-VAM spectrum with $\Omega_0=0.6\omega_{\text{VAM}}=0.11525+0.01084\mathrm{i}$ are excited for comparison. In addition, numerical parameters are $\mathrm{d}t=0.1$ and $N=350$, and $x_{\text{cut}}=1.76$ is used to distinguish between the inside and outside aspects of effective potential. The black thick dotted line is used to depict the position of the reflective wall.
• Figure 5: The time evolution of $\Psi(t,x)$ and energy are shown, where the parameters are $r_\text{c}=2$, $x_{\text{w}}=-20$ and $\mathcal{K}=-0.09261$. The parameters of the initial conditions are $\mathcal{A}=1$, $x_{\text{t}}=600$ and $\sigma=18$. In the top panels, the overtone $n=4$ mode $\omega_{\text{VAM}}=0.7520+0.0304\mathrm{i}$ is excited (cyan dashed line in Fig. \ref{['TTM_Spectra_rc_2_a']}), while in the middle and bottom panels, non-VAM spectrum with $\Omega_0=0.9\omega_{\text{VAM}}=0.6768+0.0273\mathrm{i}$ and non-VAM spectrum with $\Omega_0=0.6\omega_{\text{VAM}}=0.4512+0.0182\mathrm{i}$ are excited for comparison. In addition, numerical parameters are $\mathrm{d}t=0.1$ and $N=390$, and $x_{\text{cut}}=100$ is used to distinguish between the inside and outside aspects of effective potential. It is worth mentioning that in this simulation, the real part of mode is larger, so a higher resolution is required. The black thick dotted line is used to depict the position of the reflective wall.