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

TL;DR

This work identifies popcorn EMRIs as a transient gravitational-wave population arising when the orbital period exceeds the observation time, estimating their Galactic Centre numbers via a steady-state phase-space continuity model tied to Fokker-Planck relaxation. It forecasts observable burst rates of a few to several dozen per year with very low duty cycles, and demonstrates that individual bursts near the Galactic Centre can be detected with high SNR by LISA using a Peters quadrupole framework. Crucially, it develops a Schwartz-space (tempered-distributions) formalism to define the Fourier transform of intrinsically transient bursts, avoiding spectral leakage and energy loss introduced by conventional windowing. Numerical illustrations with rectangular pulses and truncated sinusoids validate the convergence of smoothed approximations to the true spectra, and Peters-based harmonic analysis confirms strong detectability for representative sources, establishing popcorn EMRIs as a potentially significant, rigorously analyzable GW transient population.

Abstract

We investigate extreme-mass ratio inspirals (EMRIs) with orbital periods exceeding the observational timescale of mHz gravitational wave observatories. In their early, highly eccentric phases, these systems generate transient gravitational wave bursts during pericentre passages, separated by long quiescent intervals; we designate these signals ``popcorn EMRIs.'' We utilize a steady-state analytical model based on the continuity equation in phase space to estimate the population in a Milky Way-like galaxy. The normalization of this model is linked to the solution of the Fokker-Planck equation describing stellar relaxation. Adopting a conservative one-year observation baseline ($P>1$ year), we estimate the steady-state population of popcorn EMRIs. We forecast an observable burst rate of 5 to 44 events per year. The low duty cycle ($\sim 10^{-4}$) confirms their manifestation as isolated transients. Individual bursts from the Galactic Centre exhibit high detectability. Analyzing these intrinsically transient signals demands a rigorous mathematical framework, as standard windowing techniques distort burst morphology. We establish an analytical foundation using standard smoothing techniques commonly used in real analysis. This yields the mathematically correct definition for the Fourier transform of transient signals, justifying the use of the direct Fourier transform without ad hoc windowing and ensuring the integrity of spectral analysis.

Figures (3)

• Figure 1: Time domain smoothing of the transient gravitational wave signal. The original discontinuous signal $h(t)$ (black dashed line) is approximated by smooth functions $h_n(t)$ for $n=5\times 10^4$ (red solid line) and $n=5\times 10^5$ (blue solid line). Convergence to $h(t)$ improves as $n$ increases.
• Figure 2: Frequency domain convergence of the Fourier transform magnitude. The analytical spectrum (black dashed line) is compared with the spectra of the smoothed signals for $n=5\times 10^4$ (red solid line) and $n=5\times 10^5$ (blue solid line). Attenuation caused by the smoothing decreases as $n$ increases.
• Figure 3: Cumulative SNR accumulation as a function of frequency for the illustrative popcorn EMRI system ($10 M_\odot$, $P=1$ yr, $e\approx 0.9974$) calculated using the Peters formalism. The characteristic frequency $f_{\rm burst}$ (red dashed line) is located where the noise curve is steep. 90% of the total SNR is accumulated by $f\approx 0.87$ mHz (green dotted line), where the noise level is significantly lower.