Eccentricity-induced systematic error on parametrized tests of general relativity: Hierarchical Bayesian inference applied to a binary black hole population

One approach to testing general relativity (GR) introduces free parameters in the post-Newtonian (PN) expansion of the gravitational-wave (GW) phase. If systematic errors on these testing GR (TGR) parameters exceed the statistical errors, this may signal a false violation of GR. Here, we consider systematic errors produced by unmodeled binary eccentricity. Since the eccentricity of GW events in ground-based detectors is expected to be small or negligible, the use of quasicircular waveform models for testing GR may be safe when analyzing a small number of events. However, as the catalog size of GW detections increases, more stringent bounds on GR deviations can be placed by combining information from multiple events. In that case, even small systematic biases may become significant. We apply the approach of hierarchical Bayesian inference to model the posterior probability distributions of the TGR parameters inferred from a population of eccentric binary black holes (BBHs). We assume each TGR parameter value varies across the BBH population according to a Gaussian distribution. The means and standard deviations that parametrize these Gaussians are related to the statistical and systematic errors measured for each BBH event. We compute the posterior distributions for these Gaussian hyperparameters, characterizing the BBH population via three possible eccentricity distributions. This is done for LIGO and Cosmic Explorer (CE, a proposed third-generation detector). We find that systematic biases from unmodeled eccentricity can signal false GR violations for both detectors when considering constraints set by a catalog of events. We also compute the projected bounds on the 10 TGR parameters when eccentricity is included as a parameter in the waveform model. This is done via multiplying the individual likelihoods for each event in the catalog, and also by combining them via hierarchical inference. We find that the first four dimensionless TGR deformation parameters can be bounded at 90% confidence to $\delta {\hat{\phi }}_i\lesssim {10}^{-2}$ for LIGO and $\lesssim {10}^{-3}$ for CE [where $i=(0,1,2,3)$; GR predicts zero for all values of the $\delta {\hat{\phi }}_i$]. The most stringent bound applies to the $-1\mathrm{PN}$ (dipole) parameter: it is constrained to $\left|\delta {\hat{\phi }}_{-2}\right|\lesssim {10}^{-5}$ ($\lesssim 4\times {10}^{-7}$) by LIGO (CE). In comparison to the circular orbit case, the combined bounds on the TGR parameters worsen by a modest factor of $\lesssim 2$ when eccentricity is included in the waveform. (The dipole parameter bound degrades by a factor of $\sim 3–4$ in this case).