eLISA eccentricity measurements as tracers of binary black hole formation

Up to hundreds of black hole binaries individually resolvable by eLISA will coalesce in the Advanced LIGO/Virgo band within ten years, allowing for multi-band gravitational wave observations. Binaries formed via dynamical interactions in dense star clusters are expected to have eccentricities $e_0\sim {10}^{-3}$-${10}^{-1}$ at the frequencies $f_0={10}^{-2}$ Hz where eLISA is most sensitive, while binaries formed in the field should have negligible eccentricity in both frequency bands. We estimate that eLISA should always be able to detect a nonzero $e_0$ whenever $e_0\gtrsim {10}^{-2}$; if $e_0\sim 10^{-3}$,eLISAshoulddetectnonzeroeccentricityforafraction$\sim 90%$($\sim 25%$)ofbinarieswhentheobservationtimeis$T_{\rm obs}=5$($2$) years, respectively. Therefore eLISA observations of BH binaries have the potential to distinguish between field and cluster formation scenarios.