We develop a model-independent test of General Relativity that allows for theconstraint of the gravitational wave (GW) polarization content with GWdetections of binary compact object inspirals. We first consider three modifiedgravity theories (Brans-Dicke theory, Rosen's theory and Lightman-Lee theory)and calculate the response function of ground-based detectors to gravitationalwaves in the inspiral phase. This allows us to see how additional polarizationspredicted in these theories modify the General Relativistic prediction of theresponse function. We then consider general power-law modifications to theHamiltonian and radiation-reaction force and study how these modify thetime-domain and Fourier response function when all polarizations are present.From these general arguments and specific modified gravity examples, we inferan improved parameterized post-Einsteinian template family with completepolarization content. This family enhances General Relativity templates throughthe inclusion of new theory parameters, reducing to the former when theseparameters acquire certain values, and recovering modified gravity predictionsfor other values, including all polarizations. We conclude by discussingdetection strategies to constrain these new, polarization theory parameters byconstructing certain null channels through the combination of output frommultiple detectors.
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