Probing Gravitational-Wave Extra Polarizations with Ground-Based Interferometers

Gravitational waves (GWs) were first predicted by Albert Einstein in his theory of General Relativity (GR) back in 1916 and were directly observed in 2016 by the Ligo-Virgo collaboration for the first time, opening up new horizons to make us better understand our Universe. Moreover, interplays among upcoming detectors are expected to resolve GW sources with incredible precision and the Stochastic Gravitational Wave Background (of astrophysical and/or of cosmological origin) is expected to be detected in the future, making the following decades an exciting period for the study of GWs. When general metric theories of gravity are considered, at most 6 gravitational wave polarization modes are allowed: 2 tensor modes, already predicted by General Relativity, 2 scalar modes and 2 vector modes. Therefore the importance of testing for the presence of such extra-polarization modes is clear: if they are detected new physics is discovered and GR needs to be extended. In this thesis we consider a cosmological stochastic background of gravitational waves (SGWB) involving a mixture of all possible modes and we discuss their detectability and separation by cross-correlating $2^{nd}$-generation interferometers on Earth, such as Kagra, Ligo, Virgo with their planned upgrades ``Advanced Ligo” and ``Advanced Virgo” and upcoming $3^{rd}$-generation ground-based detectors, such as Einstein Telescope (ET) and Cosmic Explorer (CE). The key quantity we look for is the gravitational wave background energy density for each polarization mode. In order to distinguish tensor, vector and scalar contributions to the SGWB energy density at least three detectors are needed. Since no ultimate location for ET and CE has been officially decided yet, we investigate different network configurations to show which may be the optimal ones. We find that the Einstein Telescope alone in the proposed triangular configuration cannot separate tensor, vector and scalar contributions to the SGWB exploiting its three detectors. We find that using ET and CE greatly improves sensitivity to extra polarizations and that all networks show almost the same sensitivity to tensor, vector and scalar modes, thus returning similar values for each polarization SGWB energy density contribution. While considering GW frequencies lower than a characteristic value depending on the detector geometry, ground-based interferometers show degenerate responses to the two allowed scalar modes, which then result undistinguishable. We show a possible way to break this degeneracy with $3^{rd}$-generation interferometers, which are expected to be more sensitive to GWs of higher frequencies. Finally, we consider Earth rotation to investigate and obtain maps of the response of the detectors to different polarizations.