Popcorn EMRIs: Transient Gravitational Wave Signals and Their Analysis in Schwartz Space

This study investigates extreme-mass ratio inspirals (EMRIs) with orbital periods exceeding the observation windows of mHz gravitational wave detectors. During their early, highly eccentric stages, these systems produce periodic gravitational wave eruptions at pericentre, separated by long quiescent periods. We define these systems as “popcorn EMRIs”–bound, coherent binaries in a deterministic evolutionary phase. This distinguishes them from stochastic, non-coherent bursts associated with parabolic fly-bys. Using a steady-state model based on the phase-space continuity equation, we estimate the population within a Milky Way-like galaxy. Because these estimates depend on sensitive astrophysical variables like stellar relaxation and density profiles, our figures are illustrative rather than absolute. For a one-year observation baseline, we estimate a steady-state population of 200 to 2,000 popcorn EMRIs, significantly outnumbering the 10 to 88 continuously observable EMRIs. This yields an expected rate of 5 to 44 detectable eruptions per year. Despite high signal-to-noise ratios (exceeding 700 for Galactic Centre events), the low duty cycle confirms their transient nature. Standard data processing techniques, designed for continuous signals, often employ windowing that distorts eruption morphology and artificially reduces the signal-to-noise ratio. To address this, we establish a mathematical apparatus using based on standard smoothing techniques, in the theory of tempered distributions. We demonstrate that standard windowing techniques irreversibly degrade the signal-to-noise ratio of transient sources when the taper overlaps the eruption, and we utilize Schwartz space regularization to justify the direct Fourier transform as the energy-preserving estimator.