Testing Gravity with Gravitational Wave Source Counts

We show that the gravitational wave source counts distribution can test how gravitational radiation propagates on cosmological scales. This test does not require obtaining redshifts for the sources. If the signal-to-noise ratio (SNR, $\rho$) from a gravitational wave source is proportional to the strain then it falls as $R^{-1}$, thus we expect the source counts to follow $dN/d\rho \propto \rho^{-4}$. However, if gravitational waves decay as they propagate or propagate into other dimensions, then there can be deviations from this generic prediction. We consider the possibility that the strain falls as $R^{-\gamma }$, where $\gamma =1$ recovers the expected predictions in a Euclidean uniformly-filled universe, and forecast the sensitivity of future observations to deviations from standard General Relativity. We first consider the case of few objects, 7 sources, with a signal-to-noise from 8 to 24, and impose a lower limit on $\gamma$, finding $\gamma >0.33$ at $95\mathrm{\%}$ confidence level. The distribution of our simulated sample is very consistent with the distribution of the trigger events reported by Advanced LIGO. Future measurements will improve these constraints: with 100 events, we estimate that $\gamma$ can be measured with an uncertainty of $15\mathrm{\%}$. We generalize the formalism to account for a range of chirp masses and the possibility that the signal falls as $\exp (-R\mathrm{/}{R}_0)\mathrm{/}{R}^{\gamma }$.