Methods for detecting gravitational waves from mini-extreme-mass-ratio inspirals. I. Statistics based on time-frequency signal tracks

Mini-extreme-mass-ratio inspirals (mini-EMRIs), composed of a stellar-mass compact object and a much lighter companion, are promising sources of continuous gravitational waves in the frequency band of ground-based interferometers such as LIGO-Virgo-KAGRA. Such systems, consisting of sub-solar-mass compact objects, provide a unique probe of exotic compact objects, including primordial black holes. Detecting such long-lived signals, however, remains challenging. Here, we adapt standard methods used in searches for quasi-monochromatic signals to search for mini-EMRIs, and derive a statistical framework that explicitly handles spectral leakage. In particular, we introduce a new method that sums along the tracks in the time-frequency plane carved out by possible mini-EMRI signals, which we call $Σ$Track. This refinement establishes a general basis for analyzing long-duration transient signals with rapid frequency evolutions, regardless of the underlying mechanism for gravitational-wave emission. We also compute a new semi-analytic sensitivity estimate within our new statistical framework, which is valid under the assumption that the signal is weak with respect to the noise level. We then establish a statistic that quantifies how to discretize the search parameter space for our method, which works for mini-EMRIs, as well as arbitrary signal types. Our results provide a foundation for mini-EMRI searches and demonstrate the potential of current ground-based detectors to probe the existence of sub-solar-mass compact objects.