Hierarchical Test of General Relativity with Gravitational Waves

We propose a hierarchical approach to testing general relativity with multiple gravitational wave detections. Unlike existing strategies, our method does not assume that parameters quantifying deviations from general relativity are either common or completely unrelated across all sources. We instead assume that these parameters follow some underlying distribution, which we parametrize and constrain. This can be then compared to the distribution expected from general relativity, i.e., no deviation in any of the events. We demonstrate that our method is robust to measurement uncertainties and can be applied to theories of gravity where the parameters beyond general relativity are related to each other, as generally expected. Our method contains the two extremes of common and unrelated parameters as limiting cases. We apply the hierarchical model to the population of 10 binary black hole systems so far detected by LIGO and Virgo. We do this for a parametrized test of gravitational wave generation, by modeling the population distribution of beyond-general-relativity parameters with a Gaussian distribution. We compute the mean and the variance of the population and show that both are consistent with general relativity for all parameters we consider. In the best case, we find that the population properties of the existing binary signals are consistent with general relativity at the $\sim 1\%$ level. This hierarchical approach subsumes and extends existing methodologies and is more robust at revealing potential subtle deviations from general relativity with increasing number of detections.

Figure 1

Expected behavior of the population hyperparameters vs number of detections. We show the width of the 90% credible interval for ${\mu }_i$ (top) and the 90% upper limit on ${\sigma }_i$ (bottom). In both panels, we average over 200 population realizations and shaded regions correspond to $1\sigma$ uncertainty. The dotted line show the mean over populations. The dashed line is proportional to $1/\sqrt{N}$; the bounds follow the expected scaling with the number of detections.

Figure 2

Example hyperparameter posteriors when GR is not the correct theory of gravity. The deviation is only present at $\delta {\widehat{p}}_2^{(j)}=0.1$, but it is recovered both in ${\mu }_2$ and ${\sigma }_0$. All other hyperparameters are consistent with GR.

Figure 3

The inferred population distribution $dN/d(\delta {\widehat{p}}_i)$ for the beyond-GR parameters $\delta {\widehat{p}}_i$’s given the 10 confirmed binary black hole signals observed to date. (The scale of $\delta {\widehat{\phi }}_{-2}$ has been expanded by a factor of 20.) All population distributions are consistent with $\delta {\widehat{p}}_i=0$, the GR prediction. With more observed signals, and under the assumption that they will obey GR, the population distributions are expected to become more narrowly centered around the origin, approximating a $\delta$ function. These distributions are defined in Eq. (5) of the Supplemental Material [21].