Eccentricity of Long Inspiraling Compact Binaries Sheds Light on Dark Sirens

The localization and distance inference of gravitational waves are two crucial factors for dark sirens as precise probes of cosmology, astrophysics, and fundamental physics. In this Letter, for the first time we investigate the parameter estimation of gravitational waves emitted by the eccentric compact binaries in the mid-frequency (0.1--10 Hz) band. Based on the configuration of one cluster of DECIGO (B-DECIGO), we simulate five types of typical compact binaries in GWTC-3 with component mass ranging from $\mathcal{O}(1\sim100)~M_{\odot}$. For each type of binaries, we assign discrete eccentricities from 0 to 0.4 at 0.1 Hz in $10^3$ random orientations. The multiple harmonics induced by eccentricity can break the degeneracy between parameters. We find that with eccentricity $e_0=0.4$, these typical binaries can achieve $\mathcal{O}(10^2-10^4)$ improvement for the distance inference in the near face-on orientations, compared to the circular case. More importantly, a nonvanishing eccentricity ($0.01\sim0.4$) can significantly improve the source localization of the typical binary black holes, most by $1.5\sim{3.5}$ orders of magnitude. Our result shows the remarkable significance of eccentricity for dark sirens in the mid-band as precise probes of the Universe.