Tests of General Relativity with Einstein Telescope

TL;DR

This paper addresses how the Einstein Telescope can perform precision tests of General Relativity using gravitational-wave signals from BBH mergers. It develops a Fisher-matrix–based forecast with a Bayesian hierarchical model for inspiral PN deviations, parameterized by $\delta\varphi_p$, and analyzes ET layouts $\Delta$, $2\mathrm{L}_0$, and $2\mathrm{L}_\!45$ with higher-mode waveform models IMRPhenomHM/D. The authors validate their method against LVK O3b results and forecast ET’s 90% upper bounds on PN deviations, finding improvements of 2–4 orders of magnitude over current bounds, especially at the lowest PN orders, with a catalog of ~$10^4$ BBH events collected in a few months. They also study ET’s ability to falsify GR by injecting Gaussian population deviations and estimating the number of detections needed to recover the hyperparameters, showing that hundreds of events could tightly constrain or reveal beyond-GR effects within days to weeks. Overall, the work demonstrates ET’s potential to dramatically tighten GR tests and informs detector design and waveform modeling, while noting limitations of the Fisher approach and the need for fuller Bayesian analyses and broader source classes.

Abstract

Gravitational wave signals from compact binary coalescences offer a powerful and reliable probe of General Relativity. To date, the LIGO-Virgo-KAGRA collaboration has provided stringent consistency tests of General Relativity predictions. In this work, we present forecasts for the accuracy with which General Relativity can be tested using third-generation ground-based interferometers, focusing on Einstein Telescope (ET) and binary black hole mergers. Given the expected high detection rate, performing full Bayesian analyses for each event becomes computationally challenging. To overcome this, we adopt a Fisher matrix approach, simulating parameter estimation in an idealized observation scenario, which allows us to study large populations of compact binary coalescences with feasible computational efforts. Within this framework, we investigate the constraints that ET, in its different configurations, can impose on inspiral post-Newtonian coefficients, by jointly analyzing events using a Bayesian hierarchical methodology. Our results indicate that ET could in principle achieve an accuracy of $\mathcal{O}(10^{-7})$ on the dipole radiation term and $\mathcal{O}(10^{-3})$ on higher-order post-Newtonian coefficients, for both the triangular and the two L-shaped designs, with $10^4$ catalog events. We also assess the number of detections required to confidently identify deviations from General Relativity at various post-Newtonian orders and for different detector configurations.

Figures (9)

• Figure 1: Forecast of the 90% upper bounds on the magnitude of the restricted PN deformation coefficients $|\delta\varphi_p|$ for a LVK O3b-like observing run, for the different PN orders considered. The bounds are obtained from the distribution $\mathring{P}\left(\delta\varphi_p \,\middle| \, D^{N_\mathrm{obs}}, I\right)$ in \ref{['eq:conditioned_deviation_posterior']}, assuming that deviations take on the same value for all observed events. The label $\varphi_p$ refers to the $\frac{p}{2}$-PN order coefficient, $\varphi_{p\ell}$ refers to the $\frac{n}{2}$-PN order coefficient with the log term. Each marker and its corresponding error bar represent the mean and 90% confidence interval, for $|\delta\varphi_p|$ calculated for $\mathcal{O}(50)$ independent realizations of the observing run and different noise realizations. The color-coded violin plots show the distribution of the single event $90\%$ upper bounds on $|\delta\varphi_i|$. We show the results of the forecast employing two different waveform models: IMRPhenomHM and IMRPhenomD. For comparison, we report the results obtained by the LVK collaboration, performing a full Bayesian analysis with the SEOBNRv4_ROM waveform template on the data of 9 observed real events, see Figure 6 in LIGOScientific:2021test. The present FIM analysis provides forecast in good agreements with the GWTC-3 results.
• Figure 2: Forecast for the 90% upper bounds on the magnitude of the post-Newtonian inspiral deviation coefficients $\delta\varphi_i$ for the three different ET configurations considered: triangular $\Delta$, two L-shaped interferometers with 45$^\circ$ misalignment 2L_45, two aligned L-shaped interferometers 2L_0. The bounds are obtained from the distribution $\mathring{P}\left(\delta\varphi_p \,\middle| \, D^{N_\mathrm{obs}}, I\right)$ in \ref{['eq:conditioned_deviation_posterior']}, assuming that deviations take on the same value for all observed events, and are represented by three different markers, respectively. The estimate used $\mathcal{O}(8000)$ events observed by each detector configuration in a fixed period of approximately 4 months. Each marker represents the average upper bound, obtained by averaging over 20 independent realizations of the simulated observations, while the error bars represent the 90% confidence intervals of the mean. The color-coded violin plots show the distribution of the single event $90\%$ upper bounds on $|\delta\varphi_i|$. For comparison, as in the previous Figure, the blue diamonds represent the current GWTC-3 constraints LIGOScientific:2021test.
• Figure 3: Plot showing the 90% credible upper bounds on the magnitude of the post-Newtonian inspiral deformation coefficients $N_\mathrm{obs}$. The solid lines show the average combined upper bounds on $|\delta\varphi_p|$ obtained from the distribution $\mathring{P}\left(\delta\varphi_p \,\middle| \, D^{N_\mathrm{obs}}, I\right)$ in \ref{['eq:conditioned_deviation_posterior']}. The average is taken over several catalogs and noise realizations, with the faint color band representing the resulting 90% confidence intervals. The dash-dotted lines show the average value of the tightest single event upper bounds on $|\delta\varphi_p|$, among $N_\mathrm{obs}$ events drawn from the catalog. The fainter colored bands quantify the corresponding 90% confidence intervals of the best single event constraints, obtained by drawing multiple catalog subsets of size $N_\mathrm{obs}$ and identifying the best bound. The detector was fixed to ET in the 2L_45 configuration.
• Figure 4: Number of detections needed to claim a deviation from GR with 90% confidence level, given fiducial deviation parameter $\delta\varphi^0_p$, drawn from a Gaussian with mean $\mu_{\delta \varphi_p}^{0}$ and standard deviation $\sigma_{\delta \varphi_p}^{0}$. We plot the results for $\delta\varphi_{-2}$ on the left and for $\delta\varphi_2$ on the right, for the 2L_45 configuration. The numbers plotted are the median of 150 repetitions of the forecast, with different catalogs and noise realizations.
• Figure 5: Plot of the detector amplitude spectral densities (ASD) for detectors used in this work: $\mathtt{LHO}$, $\mathtt{LLO}$, and $\mathtt{V}$ during the O3b observation run and ET with $10$km and $15$km arm length. The faint parts of the ASDs mark the regions below $f_\mathrm{min}$ and above $f_\mathrm{max}$, omitted in the Fisher analysis.