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The determination of the polarization modes of gravitational waves (GWs), andof their dispersion relations is decisive to scrutinize the viability ofextended theories of gravity. A tool to investigate the polarization states ofGWs is the Newman-Penrose (NP) formalism. However, if the speed of GWs issmaller than the speed of light, the number of NP variables is greater than thenumber of polarizations. To overpass this inconvenience we use the Bardeenformalism to describe the six possible polarization modes of GWs consideringdifferent general dispersion relations for the modes. The definition of a newgauge-invariant quantity enables an unambiguous description of the scalarlongitudinal polarization mode. We apply the formalism to General Relativity,scalar-tensor theories, and $f(R)$-gravity. To obtain a bridge between theoryand experiment, we derive an explicit relation between a physical observable(the derivative of the frequency shift of an electromagnetic signal) with thegauge-invariant variables. From this relation, we find an analytical formulafor the Pulsar Timing rms response to each polarization mode. To estimate thesensitivity of a single Pulsar Timing we focus on the case of a dispersionrelation of a massive particle. The sensitivity curves of the scalarlongitudinal and vector polarization modes change significantly depending onthe value of the effective mass. The detection (or absence of detection) of thepolarization modes using the Pulsar Timing technique has decisive implicationsfor alternative theories of gravity. Finally, the investigation of a cutofffrequency in the Pulsar Timing band can lead to a more stringent bound on thegraviton mass than that presented by ground-based interferometers.

Testing gravity with gauge-invariant polarization states of gravitational waves