Eccentricity of the long inspiraling compact binaries sheds light on
  dark sirens
  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 binaries with component mass greater than
$30~M_{\odot}$, most by a factor of $1.5\sim{3.5}$ orders of magnitude. Our
result shows the remarkable significance of eccentricity for dark sirens in the
mid-band as precise probes of the Universe.
