Testing gravitational wave polarizations with LISA

In this paper we quantify the ability of the Laser Interferometer Space Antenna (LISA) to test the presence of non-tensorial polarizations as well as modifications to the tensor ones in gravitational waves emitted from massive black hole binaries. We employ the Parametrized Post-Einsteinian (PPE) formalism to model deviations from General Relativity (GR) for tensor, vector, and scalar polarizations. Our PPE parametrization is inspired by post-Newtonian waveforms from four modified gravity theories: Horndeski, Einstein-æther, Rosen's bimetric, and Lightman-Lee. We consistently implement these modifications during the inspiral, ensuring proper waveform alignment and tapering at merger. Subsequently, we perform Fisher forecasts to derive expected constraints on deviations from General Relativity and map these constraints to the parameter spaces of the four gravity theories. For tensor polarizations, LISA achieves constraints on amplitude modifications ranging between ∼ 10 -4 - 10 -2 precision level, depending on the frequency evolution of the modifications, for systems with 10 5 - 10 7 M ⊙ at z = 1. We find that LISA can distinguish breathing and longitudinal scalar polarizations, a capability that current ground-based detectors do not possess, only for relatively light binaries with M ≲ 10 4 M ⊙ , beyond which these modes become degenerate in the detector response. Importantly, constraints on vector polarizations are approximately 2–3 times more precise than for scalar polarizations. For both vector and scalar modes, amplitude measurements reach precisions ranging between ∼ 10 -8 - 10 -2 , depending on the frequency evolution of the modifications, for systems with 10 5 - 10 7 M ⊙ at z = 1. These results demonstrate LISA's potential to probe gravity in the emission process of gravitational waves.