Observations of gravitational radiation from compact binary systems provide an unprecedented opportunity to test General Relativity in the strong field dynamical regime. In this paper, we investigate how future observations of gravitational radiation from binary neutron star mergers might provide constraints on finite-range forces from a universally coupled massive scalar field. Such scalar degrees of freedom are a characteristic feature of many extensions of General Relativity. For concreteness, we work in the context of metric $f(R)$ gravity, which is equivalent to General Relativity and a universally coupled scalar field with a non-linear potential whose form is fixed by the choice of $f(R)$. In theories where neutron stars (or other compact objects) obtain a significant scalar charge, the resulting attractive finite-range scalar force has implications for both the inspiral and merger phases of binary systems. We first present an analysis of the inspiral dynamics in Newtonian limit, and forecast the constraints on the mass of the scalar and charge of the compact objects for the Advanced LIGO gravitational wave observatory. We then perform a comparative study of binary neutron star mergers in General Relativity with those of a one-parameter model of $f(R)$ gravity using fully relativistic hydrodynamical simulations. These simulations elucidate the effects of the scalar on the merger and post-merger dynamics. We comment on the utility of the full waveform (inspiral, merger, post-merger) to probe different regions of parameter space for both the particular model of $f(R)$ gravity studied here and for finite-range scalar forces more generally.
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