Testing general relativity with gravitational waves -- improving and extending Modified Dispersion Relation tests

Searching for a modified dispersion relation is one of the general relativity tests performed by the LIGO-Virgo-KAGRA collaboration with each new cumulative Gravitational Wave Transient Catalog (GWTC). It considers classes of theories that modify the dispersion of gravitational waves by introducing a massive graviton or breaking Lorentz invariance. The symmetry breaking is parameterized phenomenologically by a momentum power law term $p^\alpha$ added to the dispersion relation, with the test placing constraints on the amplitude $A_\alpha$ of the introduced deviation. In this work, we implement improvements to the test, chief among them group velocity parametrization, a better sampling procedure, and extension to negative exponents $\alpha$ of $p$. We then reanalyze the events from the third catalog, GWTC-3, with our improved method. Compared with GWTC-3 results, we find significant improvement, mostly from the improved sampling method, in the posteriors obtained by analyzing individual event and more modest improvements in the combined bounds on amplitude parameters $A_\alpha$ -- on average, we observe 19% shrinking of posterior width. The 90% upper bound on the graviton mass changes from $2.42 \times 10^{-11}$ peV to $2.21 \times 10^{-11}$ peV. For the extension of our test to $\alpha \in \{-1, -2, -3\}$, we find no evidence in favor of general relativity violation.