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

Abstract: 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 modelling for data analysis.