Reflectionless and echo modes in asymmetric Damour-Solodukhin wormholes

TL;DR

This work investigates the relationship between reflectionless scattering modes (RSMs) and echo modes in asymmetric Damour-Solodukhin wormholes. By extending the RSM concept to fully reflectionless, complex-frequency modes and applying both transfer-matrix and Green's-function methods, the authors reveal a strong spectral resemblance between RSMs and echo modes in the high-overtone regime, with the real parts aligning and a uniform spacing set by ${\Delta\omega}={\pi}/{(2x_c)}$. Symmetric DS wormholes yield purely real RSMs, while asymmetry introduces imaginary parts that measure the degree of asymmetry and affect the time-domain echoes. The study includes detailed analytical derivations and numerical demonstrations with double-$\delta$ and double-square barrier models, illustrating how echo periods and spectral features emerge from the geometry and barrier parameters. Overall, RSMs and echo modes offer complementary, robust tools for understanding gravitational-wave echoes and their observational significance, with greybody factors providing stable observables in the presence of spectral instability.

Abstract

It is understood that the echo waveforms in ultracompact objects can be regarded as composed mainly of the asymptotic high-overtone quasinormal modes, dubbed echo modes, which predominantly lie parallel to the real frequency axis. Alternatively, Rosato {\it et al.} recently suggested that high-frequency quasi-reflectionless scattering modes are primarily responsible for the echo phenomenon. This identification relies on greybody factors as stable observables, despite the apparent spectral instability of quasinormal modes. In this work, by extending the definition of quasi-reflectionless modes to reflectionless ones and generalizing symmetric Damour-Solodukhin wormholes to asymmetric cases, we examine the underlying similarity between the reflectionless and echo mode spectra in the complex frequency plane. Through a primarily analytical treatment, we demonstrate that the asymptotic properties of these two spectra exhibit a strong resemblance, featuring an approximately uniform distribution parallel to the real frequency axis with the same spacing between successive modes. Specifically, the real parts of echo modes coincide with those of reflectionless modes at the limit $|\mathrm{Re}ω| \gg |\mathrm{Im}ω|$. While echo modes typically possess non-vanishing imaginary parts, the reflectionless modes of symmetric Damour-Solodukhin wormholes lie precisely on the real frequency axis, with any deviation serving as a measure of the degree of asymmetry of the wormhole. We support our derivations by employing two complementary approaches, based on the scattering matrix and the Green's function, and argue that both perspectives provide effective tools for describing the echo phenomenon.

Figures (6)

• Figure 1: An illustration of the effective potential in an asymmetric Damour-Solodukhin wormhole in the tortoise coordinate $x$. The effective potential consists of two distinct black hole effective potentials separated by a distance $2x_c$, where the black hole effective potential on the l.h.s. is spatially reflected.
• Figure 2: The RSM and quasi-RSM modes and their asymptotic values for symmetric Damour-Solodukhin wormholes consisting of two identical delta effective potential barriers. The calculations are carried out using the parameters $V_0^\mathrm{L}=V_0^\mathrm{R}=1$ and $x_c=\frac{1}{2}$. Upper row: The RSMs and quasi-RSMs coincide and are shown in empty red squares. They are compared with the asymptotic values given by Eq. \ref{['rootRSMread']}, represented by empty blue diamonds. Lower row: The reflection amplitude evaluated as a function of the frequency. It vanishes identically at their local minima, which coincide with the RSMs. The left column show the low-lying modes ($0\le \mathrm{Re}\omega\le 20$) and the right column illustrates the asymptotic modes ($180\le \mathrm{Re}\omega\le 200$).
• Figure 3: The RSM and quasi-RSM modes and their asymptotic values for asymmetric Damour-Solodukhin wormholes consisting of two delta effective potential barriers with different magnitudes. The calculations are carried out using the parameters $V_0^\mathrm{L}=2$, $V_0^\mathrm{R}=1$, and $x_c=\frac{1}{2}$. Upper row: The RSMs and quasi-RSMs are shown in empty red squares and empty purple triangles. They are compared with the asymptotic values given by Eq. \ref{['rootRSM']}, represented by empty green diamonds, and their real parts, indicated by empty blue diamonds. Lower row: The reflection amplitude evaluated as a function of the frequency. It does not vanish at its local minima, which reflects that the RSMs are complex. The left column show the low-lying modes ($0\le \mathrm{Re}\omega\le 20$) and the right column illustrates the asymptotic modes ($180\le \mathrm{Re}\omega\le 200$).
• Figure 4: The echo modes and their asymptotic values for symmetric and asymmetric Damour-Solodukhin wormholes consisting of two square barriers. The calculations are carried out by using $x_c=\frac{1}{2}$ and the parameters $V_0^\mathrm{L}=V_0^\mathrm{R}=1$ for symmetric case and $V_0^\mathrm{L}=2$, $V_0^\mathrm{R}=1$ for asymmetric case. The numerical results for QNMs are shown in empty red squares (for symmetric wormhole) and empty green triangles (for asymmetric wormhole). Those obtained by employing the estimated expression Eq. \ref{['exEstiQNM']} are indicated by empty blue diamonds (for symmetric wormhole) and empty purple inverted-triangles (for asymmetric wormhole). The left column show the low-lying modes ($0\le \mathrm{Re}\omega\le 20$) and the right column illustrates the asymptotic modes ($180\le \mathrm{Re}\omega\le 200$).
• Figure 5: The RSM and quasi-RSM modes and their asymptotic values for asymmetric Damour-Solodukhin wormholes consisting of two square barriers with different magnitudes. The calculations are carried out using the parameters $V^\mathrm{L}=1$, $V^\mathrm{R}=\frac{2}{3}$, $W^\mathrm{L}=W^\mathrm{R}=\frac{1}{10}$, and $x_c=1$. Top row: The RSMs are shown in empty red squares, and they are compared with the asymptotic values given by Eq. \ref{['rootRSM']}, represented by empty blue diamonds. Middle row: The reflection amplitude evaluated as a function of the frequency. It does not vanish at their local minima, which reflects that the RSMs are complex. The quasi-RSMs and the real parts of the RSMs are represented by empty blue diamonds and empty purple triangles. The left column show the low-lying modes ($0\le \mathrm{Re}\omega\le 20$) and the right column illustrates the asymptotic modes ($180\le \mathrm{Re}\omega\le 200$). Bottom row: An overhead shot of RSMs for the asymmetric wormhole. The deviations from the estimation Eq. \ref{['rootRSM']} decrease with increasing frequency, and are observed to have a period of $10\pi$, which reflects the finite size of the square barriers.