Testing general relativity with gravitational waves: A reality check

The observations of gravitational-wave signals from astrophysical sources such as binary inspirals will be used to test general relativity for self-consistency and against alternative theories of gravity. I describe a simple formula that can be used to characterize the prospects of such tests, by estimating the matched-filtering signal-to-noise ratio required to detect non-general-relativistic corrections of a given magnitude. The formula is valid for sufficiently strong signals; it requires the computation of a single number, the fitting factor between the general-relativistic and corrected waveform families; and it can be applied to all tests that embed general relativity in a larger theory, including tests of individual theories such as Brans-Dicke gravity, as well as the phenomenological schemes that introduce corrections and extra terms in the post-Newtonian phasing expressions of inspiral waveforms. The formula suggests that the volume-limited gravitational-wave searches performed with second-generation ground-based detectors would detect alternative-gravity corrections to general-relativistic waveforms no smaller than 1%–10% (corresponding to fitting factors of 0.9 to 0.99).

Figure 1

SNR required for AG detection with efficiency $E=1/2$, with false-alarm probability $F={10}^{-4}$ and ${10}^{-8}$, as a function of FF. The right-side vertical axis shows the number of events required in a volume-limited search with detection threshold of 8 to yield a loudest event with the (median) SNR on the left-side vertical axis.