  Recent gravitational wave observations have allowed stringent new constraints
on modifications to General Relativity (GR) in the extreme gravity regime.
Although these observations were consistent with compact binaries with no
orbital eccentricity, gravitational waves emitted in mildly eccentric binaries
may be observed once detectors reach their design sensitivity. In this paper,
we study the effect of eccentricity in gravitational wave constraints of
modified gravity, focusing on Jordan-Brans- Dicke-Fierz theory as an example.
Using the stationary phase approximation and the post-circular approximation
(an expansion in small eccentricity), we first construct an analytical
expression for frequency-domain gravitational waveforms produced by inspiraling
compact binaries with small eccentricity in this theory. We then calculate the
overlap between our approximate analytical waveforms and an eccentric numerical
model (TaylorT4) to determine the regime of validity (in eccentricity) of the
former. With this at hand, we carry out a Fisher analysis to determine the
accuracy to which Jordan-Brans-Dicke-Fierz theory could be constrained given
future eccentric detections consistent with General Relativity. We find that
the constraint on the theory initially deteriorates (due to covariances between
the eccentricity and the Brans-Dicke coupling parameter), but then it begins to
recover, once the eccentricity is larger than approximately 0.03. We also find
that third-generation ground-based detectors and space-based detectors could
allow for constraints that are up to an order of magnitude more stringent than
current Solar System bounds. Our results suggest that waveforms in modified
gravity for systems with moderate eccentricity should be developed to maximize
the theoretical physics that can be extracted in the future.
