  Using the effective field theory framework for extended objects we describe
the evolution of spinning compact objects in the late inspiral of the
coalescence of a binary, before the plunge and merger, by including leading
order corrections due to spin, tides, dissipation and gravitational wave
radiation. Our implementation is of particular relevance for probing the
stellar structure of compact objects with gravitational wave observations. A
spinning compact object in the effective field theory framework is described as
a spinning point particle, with its finite size effects encoded in higher order
operators in the effective action, operators which have coefficients that
encapsulates the internal structure of the star. For the inspiral regime
described by non-relativistic general relativity, post-Newtonian corrections to
each term of the action can be obtained in a diagrammatic approach, including
gravitational radiation effects. Taking into account the aforementioned
ingredients of the effective theory, we solve for the dynamics of the inspiral
of binary systems using an algorithm for point particle simulations. We extract
the gravitational wave as a function of the orbital frequency, input that is
generated numerically and then evaluated in the analytic function of the
waveform. By performing illustrative numerical experiments of systems that the
LIGO-Virgo observatories have already detected, we show the role of the stellar
structure and its coefficients in the phase evolution of the waveform, as well
as the order in which they arise and the sensitivity required for the
gravitational wave observatories to measure them. If these coefficients are to
be measured, tight constraints on fundamental physics can be placed.
