Characterization of echoes: A Dyson-series representation of individual pulses

The ability to detect and scrutinize gravitational waves from the merger and coalescence of compact binaries opens up the possibility to perform tests of fundamental physics. One such test concerns the dark nature of compact objects: are they really black holes? It was recently pointed out that the absence of horizons—while keeping the external geometry very close to that of General Relativity—would manifest itself in a series of echoes in gravitational wave signals. The observation of echoes by LIGO/Virgo or upcoming facilities would likely inform us on quantum gravity effects or unseen types of matter. Detection of such signals is in principle feasible with relatively simple tools but would benefit enormously from accurate templates. Here we analytically individualize each echo waveform and show that it can be written as a Dyson series, for arbitrary effective potential and boundary conditions. We further apply the formalism to explicitly determine the echoes of a simple toy model: the Dirac delta potential. Our results allow to read off a few known features of echoes and may find application in the modeling for data analysis.

Figure 1

QNM frequencies of the membrane-mirror system for different values of $V_{0}L=1$, 2, 4, 8 from bottom (blue) to top (red), respectively.

Figure 2

Time evolution of the waveform using initial conditions (57) and parameters (58). The plot shows the decay with the fundamental mode of the system (with mirror on the left), $\Im {\omega }_{\pm 1}\approx -0.00205$ (in red), for large $t$. The early echoes decay in a way that is governed by the QNMs of the pure delta. The high-frequency component is filtered out and progressively the signal is described by the modes of the composite system at late times, as it should.

Figure 3

Snapshots of the scalar profile at $t=0,6,10.5,16,20,28,36,58,74$ (top to bottom, left to right) in the presence of a deltalike potential at $x=0$ and a mirror at $x=-10$. The initial profile (57) quickly gives way to two pulses traveling in opposite directions at $t=6$, as described by Eq. (27); the left propagating pulse interacts with the (delta) potential at $t=10.5$ and gives rise to a transmitted pulse and a reflected one ($t=16$). The reflected pulse eventually reaches the boundary, at $t=20$, and will cross the potential at around $t=36$ giving rise to the first echo. The wave confined to the cavity ($\mathrm{mirror}+\mathrm{potential}$) will produce all subsequent echoes. At $t=58$, after $2L=20$ time units the second echo emerges out of the cavity and at $t=74$ a third echo is about to be produced. These snapshots were obtained by adding three “echoes,” and coincides up to numerical error, with the waveform obtained via numerical evolution of the initial data. In the central panel, the red line shows $-e^{V_0(x-10)}$, confirming that the initial decay is described by the QNMs (44) of the pure delta function (no mirror).