Constraining initial orbital eccentricity of inspiral-dominated gravitational-wave events with an analytic approximant

The LIGO-Virgo-KAGRA consortium has sporadically detected inspiral-dominated gravitational-wave events such as GW170817 and GW190425. These events offer an opportunity to constrain possible initial (residual) orbital eccentricities using purely inspiral template families. We detail the implementation of an LALSuite approximant, TaylorF2Ecck, which analytically models inspiral gravitational waves from non-spinning compact binaries in Post-Newtonian-accurate eccentric orbits and restricts the initial-eccentricity contributions to leading order. Specifically, our frequency-domain approximant consistently incorporates orbital, advance of periastron, and gravitational-wave emission effects fully up to 3PN order. We conduct detailed parameter-estimation studies of GW170817 and GW190425 using TaylorF2Ecck, following comprehensive sanity checks to validate model performance and investigate the influence of eccentricity and periastron advance in the relevant parameter space. The results indicate that the initial eccentricity at 20 Hz is negligible within the 90 percent credible intervals, and Bayes factors show no strong evidence favoring the eccentric waveform over the quasi-circular waveform. At such negligible initial eccentricities, comparisons between eccentric models with and without periastron advance show no clear signature of this effect, with no significant model-dependent shifts in the posterior distributions and no strong Bayes-factor evidence favoring one model over the other. Additionally, these detailed studies reveal the importance of incorporating initial-eccentricity contributions at least up to 3.5PN order and discuss its implications. We substantiate this inference by employing versions of the quasi-circular TaylorF2 approximant that incorporate Fourier phase contributions beyond the conventional 3.5PN order.