  We study generic tests of strong-field General Relativity using gravitational
waves emitted during the inspiral of compact binaries. Previous studies have
considered simple extensions to the standard post-Newtonian waveforms that
differ by a single term in the phase. Here we improve on these studies by (i)
increasing the realism of injections and (ii) determining the optimal waveform
families for detecting and characterizing such signals. We construct waveforms
that deviate from those in General Relativity through a series of
post-Newtonian terms, and find that these higher-order terms can affect our
ability to test General Relativity, in some cases by making it easier to detect
a deviation, and in some cases by making it more difficult. We find that simple
single-phase post-Einsteinian waveforms are sufficient for detecting deviations
from General Relativity, and there is little to be gained from using more
complicated models with multiple phase terms. The results found here will help
guide future attempts to test General Relativity with advanced ground-based
detectors.
