  A space-based interferometer such as eLISA could observe few to few thousands
progenitors of black hole binaries (BHBs) similar to those recently detected by
Advanced LIGO. Gravitational radiation circularizes the orbit during inspiral,
but some BHBs retain a measurable eccentricity at the low frequencies where
eLISA is most sensitive. The eccentricity of a BHB carries precious information
about its formation channel: BHBs formed in the field, in globular clusters, or
close to a massive black hole (MBH) have distinct eccentricity distributions in
the eLISA band. We generate mock eLISA observations, folding in measurement
errors, and using Bayesian model selection we study whether eLISA measurements
can identify the BHB formation channel. We find that a handful of observations
would suffice to tell whether BHBs were formed in the gravitational field of a
MBH. Conversely, several tens of observations are needed to tell apart field
formation from globular cluster formation. A five-year eLISA mission with the
longest possible armlength is desirable to shed light on BHB formation
scenarios.
