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 $e_0leq {0.175,, 0.125,,0.175,,0.175,, 0.125}$, respectively. We show that if these systems have eccentricities $e_0sim 0.1$ 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.
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