Multiparameter tests of general relativity using a principle component analysis with next-generation gravitational-wave detectors

Principal component analysis (PCA) is an efficient tool to optimize multiparameter tests of general relativity (GR), wherein one looks for simultaneous deviations in multiple post-Newtonian phasing coefficients. This is accomplished by introducing non-GR deformation parameters in the phase evolution of the gravitational-wave templates used in the analysis. A PCA is performed to construct the ``best-measured'' linear combinations of the deformation parameters. This helps to set stringent limits on deviations from GR and to more readily detect possible beyond-GR physics. In this paper, we study the effectiveness of this method with the proposed next-generation gravitational-wave detectors, Cosmic Explorer (CE) and Einstein Telescope (ET). For compact binaries at a luminosity distance of 500 Mpc and the detector-frame total mass in the range $20--200{M}_{\ensuremath{\bigodot}}$, CE can measure the most dominant linear combination with a $1\text{\ensuremath{-}}\ensuremath{\sigma}$ uncertainty $\ensuremath{\sim}0.1%$ and the next two subdominant linear combinations with a $1\text{\ensuremath{-}}\ensuremath{\sigma}$ uncertainty of $\ensuremath{\le}10%$. For a specific range of masses, constraints from ET are better by a factor of a few than CE. This improvement is because of the improved low frequency sensitivity of ET compared to CE (between 1--7 Hz). In addition, we explain the sensitivity of the PCA parameters to the different post-Newtonian deformation parameters and discuss their variation with total mass. We also discuss a criterion for quantifying the number of most dominant linear combinations that capture the information in the signal up to a threshold.