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Existing constraints on the graviton mass from gravitational-wave detectionsrely on the phase difference developed between different frequencies during thepropagation. Effects on the quasinormal-mode frequencies of the black-holeringdown due to the graviton mass are often ignored. While perturbationtheories of black holes have been well developed in the context of generalrelativity, this is not the case for modified gravity theories. We propose aphenomenological modification to the Teukolsky equation of perturbed blackholes to include the dispersion relation due to a gravitational field ofnonzero mass. Solving this modified Teukolsky equation by logarithmicperturbation theory, we compute the shift of the quasinormal-mode frequenciesdue to the presence of a graviton mass. This hypothetical shift can be used toconstrain the graviton mass with ringdown signals, either standalone or inconjunction with the phase difference accumulated due to the wave propagation.We estimate that constraints on the graviton mass of $m_g \lesssim10^{-15}\,\textrm{eV}$ can be put with a detection of the ringdown signal aloneby second generation gravitational-wave detectors.

physical review. d/physical review. d.

Phenomenological Inclusion of Alternative Dispersion Relations to the  Teukolsky Equation and its Application to Bounding the Graviton Mass with  Gravitational-wave Measurements