An Efficient Signal-to-noise Approximation for Eccentric Inspiraling Binaries

Eccentricity has emerged as a potentially useful tool for helping to identify the origin of black hole mergers. However, eccentric templates can be computationally very expensive owing to the large number of harmonics, making statistical analyses to distinguish formation channels very challenging. We outline a method for estimating the signal-to-noise ratio (S/N) for inspiraling binaries at lower frequencies such as those proposed for LISA and DECIGO. Our approximation can be useful more generally for any quasi-periodic sources. We argue that surprisingly, the S/N evaluated at or near the peak frequency (of the power) is well approximated by using a constant-noise curve, even if in reality the noise strain has power-law dependence. We furthermore improve this initial estimate over our previous calculation to allow for frequency dependence in the noise to expand the range of eccentricity and frequency over which our approximation applies. We show how to apply this method to get an answer accurate to within a factor of 2 over almost the entire projected observable frequency range. We emphasize this method is not a replacement for detailed signal processing. The utility lies chiefly in identifying theoretically useful discriminators among different populations and providing fairly accurate estimates for how well they should work. This approximation can furthermore be useful for narrowing down parameter ranges in a computationally economical way when events are observed. We furthermore show a distinctive way to identify events with extremely high eccentricity where the signal is enhanced relative to naive expectations on the high-frequency end.