Cosmological Implications of Gravitational Wave Observations

GWs are a very powerful probe to understand the Universe and provide a unique way to study its evolution. First, the direct detection of gravitational waves by the LVK collaboration, and more recently the claim from Pulsar Timing Array collaborations, have further confirmed the potential of these signals. In this thesis we explore the cosmological implications associated to GW detections focusing mainly on primordial NG and on the Hubble parameter $H_0$. After introducing GWs and primordial NG, we study the imprints of local NG on the SIGW spectrum. Hence, we forecast the capability of LISA to detect NG, finding that it could be constrained up to the percent level. In the context of SIGW, we assume that the PTA signal can be attributed to SIGWs, while considering various possible shapes for the scalar power spectrum. We find upper bounds on the amplitude of the scalar power spectrum that we use to constrain the abundance of PBHs in the stellar mass range. We then study the impact on the abundance due to the presence of primordial local NG, finding that it could also exclude PBHs. Then, we consider different imprints of primordial NG, considering its signatures on the tracer bias, with the tracers being GW sources contributing to an astrophysical background of GWs. After characterizing the anisotropies of the signal, we evaluate their cross-correlation with the Cosmic Microwave Background ones to forecast the capability of future interferometers to detect the imprint of NG on the largest scales. Finally, we focus on resolved GWs, in the context of dark sirens combined with a galaxy catalog, to constrain the Hubble parameter. We study the presence of possible biases due to a mismodelling of the galaxy host probability, finding that mismatching the redshift or the luminosity dependence could induce different biases towards higher or lower values of $H_0$, spoiling the inference. Thus, in this thesis, we show with different examples that GWs, whether originating in the early Universe or in more recent times and both of astrophysical and cosmological origin, represent an important tool to probe the Universe.