Enhancing the performance of multiparameter tests of general relativity with LISA using Principal Component Analysis

The Laser Interferometer Space Antenna (LISA) will provide us with a unique opportunity to observe the early inspiral phase of supermassive binary black holes (SMBBHs) in the mass range of $10^5-10^6\,M_{\odot}$, that lasts for several years. It will also detect the merger and ringdown phases of these sources. Therefore, such sources are extremely useful for multiparameter tests of general relativity (GR), where parametrized deviations from GR at multiple post-Newtonian orders are simultaneously measured, thus allowing for a rigorous test of GR. However, the correlations of the deviation parameters with the intrinsic parameters of the system make multiparameter tests extremely challenging to perform. We demonstrate the use of principal component analysis (PCA) to obtain a new set of deviation parameters, which are best-measured orthogonal linear combinations of the original deviation parameters. With the observation of an SMBBH of total redshifted mass, $\sim\mathrm{7\times10^5\,M_{\odot}}$ at a luminosity distance of 3 Gpc, we can estimate the five most dominant PCA parameters, with 1-$σ$ statistical uncertainty of $\lesssim 0.2$. The two most dominant PCA parameters can be bounded to $\sim \mathcal{O}(10^{-4})$, while the third and fourth-dominant ones to $\sim \mathcal{O}(10^{-3})$. Measurement of the PCA parameters with such unprecedented precision with LISA makes them an excellent probe to test the overall PN structure of the GW phase evolution.