Comparison of different computations of backreaction in axion inflation

Axion inflation coupled to a gauge field can give rise to a rich phenomenology: the motion of the axion can strongly amplify the gauge field, which in turn can produce observable density perturbations and gravitational waves. In the observationally interesting cases, the amplification of the gauge field is so strong that it significantly backreacts on the motion of the axion. Obtaining the precise evaluation of this coupled system, which is necessary for accurate phenomenological predictions, is still an open problem, which has been addressed with different analytical and numerical techniques in the literature. Two widely used schemes are: (1) a numerical computation of the evolution of the Fourier modes of the gauge field, coupled to an axion field approximated to be exactly homogeneous axion, and (2) the reformulation of the equations of motion in position space in terms of a tower of equations for two-point correlators of the gauge field with an increasing number of spatial derivatives acting on them. Although these two methods are formally equivalent, their implementations are based on different approximations and truncations. Works that have employed these two methods appear to be in good qualitative agreement, but a detailed quantitative comparison between them is still lacking. The thesis has the goal to perform such a comparison, to assess the accuracy and the performance of the two different implementations.