Rogue echoes from exotic compact objects

Binary systems containing exotic compact objects may emit repeated bursts of gravitational waves (GWs) following coalescence. Such GW echoes would provide a clear signature of new physics, but searches for them have not yielded a convincing detection. Here we argue that the typical time delay between a GW event and its echoes is much greater than generally expected, due to long propagation times through objects that mimic black holes. We provide a simple recipe for computing the time delay and several examples. These time delays can be billions of years, resulting in rogue echoes that are not correlated with GW events and evade all current constraints. They would be detectable only by searches for individual echoes or GW bursts.