Testing gravity with gauge-invariant polarization states of gravitational waves

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