  Dynamically formed compact object binaries may still be eccentric while in
the LIGO/Virgo band. For a neutron star (NS) in an eccentric binary, the
fundamental modes (f-modes) are excited at pericenter, transferring energy from
the orbit to oscillations in the NS. We model this system by coupling the
evolution of the NS f-modes to the orbital evolution of the binary as it
circularizes and moves toward coalescence. NS f-mode excitation generally
speeds up the orbital decay and advances the phase of the gravitational wave
signal from the system. We calculate how this effect changes the timing of
pericenter passages and examine how the cumulative phase shift before merger
depends on the initial eccentricity of the system. This phase shift can be much
larger for highly eccentric mergers than for circular mergers, and can be used
to probe the NS equation of state.
