The importance of being "consistent" in Cosmology

The study of primordial non-Gaussianity can be considered one of the best avenues to probe inflation, trying to resolve the large degeneracy among different models still present after analyzing Cosmic Microwave Background data. Indeed, on the one hand, the amount of non-Gaussianity in single-field inflation is very tiny and entirely fixed by the spontaneous breaking of time reparameterization; on the other hand, different symmetry breaking patterns, typical of more exotic theories, lead to quite different predictions. In this context, this Ph.D. thesis aims at analyzing the primordial imprints in two and three-point correlation functions relevant for future Large Scale Structure and gravitational waves detection, whose origins can be traced back to an inflationary phase of the Universe, discerning between single and multi-field models. Regarding single-field inflation, a controversial issue concerns the consistency relation describing the three-point correlation functions of the comoving curvature perturbation in the squeezed limit. The common lore is that the consistency relation, which implies a prediction for local non-Gaussianity with $f_{NL}=5, (n_s-1 )/12$, actually is not physical and can be gauged away by a constant rescaling of coordinates. This cancellation is the direct consequence of a gauge redundancy which becomes effective only if the squeezed long mode $k_L$ is precisely zero. Such a limit is not physically observable, and by taking into account a small but finite $k_L$, the cancellation of $f_{NL}$ does not take place. The same technique is often used in the literature to cancel any local $f_{NL}$ of quantities depending on the comoving curvature, as in matter density perturbations and related $f_{ NL}^{GR}=-5/3$, entering in the halo bias scale dependence. In the second part of this thesis, we study in detail a promising class of models that allows a systematic analysis of symmetric breaking pattern during inflation and the impact on primordial non-Gaussianity. By using an effective field theory description, the elementary excitations of the model can be interpreted as phonons in a supersolid. In supersolid inflation, the single-field consistency relation that relates the 3-point function in the squeezed limit to the square of the power spectrum does not hold in general. The lack of the consistency relation is deeply connected with the explicit violation of the Weinberg theorem hypothesis: ``adiabaticity'' (presence of one scalar degree of freedom) and ``isotropy'' (absence of anisotropic stress tensor on large scales). The first is violated by the particular symmetry breaking pattern that requires two propagating scalar degrees of freedom while the second is a consequence of the solid component. We have shown that one can significantly enhance the production of gravitational waves entering the window of LISA sensibility without affecting non-Gaussianity in the scalar sector constrained by PLANCK results.