  We present $\texttt{ENIGMA}$, a time domain, inspiral-merger-ringdown
waveform model that describes non-spinning binary black holes systems that
evolve on moderately eccentric orbits. The inspiral evolution is described
using a consistent combination of post-Newtonian theory, self-force and black
hole perturbation theory. Assuming eccentric binaries that circularize prior to
coalescence, we smoothly match the eccentric inspiral with a stand-alone,
quasi-circular merger, which is constructed using machine learning algorithms
that are trained with quasi-circular numerical relativity waveforms. We show
that $\texttt{ENIGMA}$ reproduces with excellent accuracy the dynamics of
quasi-circular compact binaries. We validate $\texttt{ENIGMA}$ using a set of
$\texttt{Einstein Toolkit}$ eccentric numerical relativity waveforms, which
describe eccentric binary black hole mergers with mass-ratios between $1 \leq q
\leq 5.5$, and eccentricities $e_0 \lesssim 0.2$ ten orbits before merger. We
use this model to explore in detail the physics that can be extracted with
moderately eccentric, non-spinning binary black hole mergers. We use
$\texttt{ENIGMA}$ to show that GW150914, GW151226, GW170104, GW170814 and
GW170608 can be effectively recovered with spinning, quasi-circular templates
if the eccentricity of these events at a gravitational wave frequency of 10Hz
satisfies $e_0\leq \{0.175,\, 0.125,\,0.175,\,0.175,\, 0.125\}$, respectively.
We show that if these systems have eccentricities $e_0\sim 0.1$ at a
gravitational wave frequency of 10Hz, they can be misclassified as
quasi-circular binaries due to parameter space degeneracies between
eccentricity and spin corrections. Using our catalog of eccentric numerical
relativity simulations, we discuss the importance of including higher-order
waveform multipoles in gravitational wave searches of eccentric binary black
hole mergers.
