  The origin and formation of stellar-mass binary black holes remains an open
question that can be addressed by precise measurements of the binary and
orbital parameters from their gravitational-wave signal. Such binaries are
expected to circularize due to the emission of gravitational waves as they
approach merger. However, depending on their formation channel, some binaries
could have a non-negligible eccentricity when entering the frequency band of
current gravitational-wave detectors. In order to measure eccentricity in an
observed gravitational-wave signal, accurate waveform models that describe
binaries in eccentric orbits are necessary. In this work we demonstrate the
efficacy of the improved TEOBResumS waveform model for eccentric coalescing
binaries with aligned spins. We first validate the model against mock signals
of aligned-spin binary black hole mergers and quantify the impact of
eccentricity on the estimation of other intrinsic binary parameters. We then
perform a fully Bayesian reanalysis of GW150914 with the eccentric waveform
model. We find (i) that the model is reliable for aligned-spin binary black
holes and (ii) that GW150914 is consistent with a non-eccentric merger although
we cannot rule out small values of initial eccentricity at a reference
frequency of $20$ Hz. Finally, we present a systematic method to measure the
eccentricity and its evolution directly from the gravitational-wave posterior
samples. Such an estimator is useful when comparing results from different
analyses as the definition of eccentricity may differ between models. Our
scheme can be applied even in the case of small eccentricities and can be
adopted straightforwardly in post-processing to allow for direct comparison
between models.
