Primordial trispectra from gauge fields in Inflation

Modern cosmology is remarkably successful in describing the evolution of the Universe. Nevertheless, the standard Hot Big Bang model alone cannot account for several fundamental features of the observed Universe. The inflationary paradigm was introduced to address these shortcomings and has since become the leading framework for explaining the origin of primordial density perturbations and the formation of cosmic structures. Among the many inflationary scenarios proposed in the literature, models involving axion fields coupled to gauge fields have attracted considerable interest because of their rich phenomenology and their ability to generate distinctive observational signatures, including parity violation and primordial non-Gaussianity. A particularly notable example is Chromo Natural Inflation, in which an axion field acts as the inflaton and interacts with a non-Abelian SU(2) gauge field through a Chern–Simons coupling. This interaction gives rise to the production of chiral gravitational waves and other characteristic signatures. However, despite its theoretical appeal, the original Chromo Natural Inflation scenario is unable to simultaneously satisfy current observational constraints on the scalar (i.e. density) and tensor (i.e. gravitational waves) perturbation spectra. To overcome these difficulties, the model can be extended to Spectator Chromo Natural Inflation, where Inflation is driven by an independent inflaton field, while the axion–gauge sector acts as a spectator sector. While the tensor perturbations of this framework have been extensively investigated, the scalar sector remains relatively unexplored (in particular with regard to higher-order correlation functions), despite possessing a remarkably rich interaction structure. The primary objective of this thesis is to investigate the scalar trispectrum (i.e. the Fourier counterpart of the 4-point correlation function) generated by axion fluctuations in Spectator Chromo Natural Inflation and to determine whether parity-violating signatures can arise purely from the scalar sector of the theory. To this end, the action governing scalar perturbations is expanded up to fourth order in fluctuations. After integrating out the non-dynamical degrees of freedom, the corresponding interaction Hamiltonian is derived and employed within the In-in formalism to compute the four-point correlation function. Particular attention is devoted to the contributions generated by scalar exchange and contact interaction diagrams, which constitute the sources of the scalar trispectrum considered in this work. The resulting trispectrum is then analyzed through its shape function, focusing on the equilateral momentum configuration. Numerical evaluations are performed to determine its kinematic dependence and to extract the associated non-linearity parameter $t_{NL}$, which characterizes the amplitude of the generated non-Gaussian signal in the regular tetrahedron limit. The analysis leads to two main conclusions. First, no parity-violating signatures are found in the purely scalar trispectrum, because the correlation functions computed in this work do not contain the pseudoscalar momentum structures required to generate parity-odd contributions. Second, the scalar trispectrum generated by the considered interactions is found to be suppressed in the regular tetrahedron limit. The results presented in this thesis contribute to the phenomenological characterization of Spectator Chromo Natural Inflation beyond its tensor predictions and provide new theoretical insights into the structure of scalar primordial non-Gaussianities in axion–gauge inflationary models.