• cosmology
• inflation
• axion
• U(1) gauge field
• backreaction
• renormalization
• natural inflation
• adiabatic renormalization

In this thesis, we consider an inflationary model known as axion inflation coupled to U(1) gauge fields. In this model, the inflaton is an axion-like particle and two essential quantities in such context are the energy density associated with the gauge fields and the helicity integral, which is defined as the vacuum expectation value of the scalar product between the electric and magnetic fields. These two observables quantify the backreaction of the gauge field on the inflationary dynamics and can influence the duration of inflation. It is important to note that these quantities are affected by divergences, a common feature for correlation functions of quantum fields in curved spacetime. To treat these divergences, one needs a renormalization scheme, in particular, we focus on the so-called adiabatic renormalization, using a procedure which has been recently developed to univocally fix the renormalization scheme. On the other hand, Starobinsky introduced the so-called stochastic formalism which is a non-perturbative approach that allows to develop an effective theory for the long-wavelength modes of a quantum field.
In the stochastic approach, one is able to derive a Langevin equation describing the super-horizon field as a classical stochastic variable subjected to a noise term given by the quantum fluctuations of the sub-horizon modes. The application of the stochastic formalism to the inflaton or to test scalar fields in inflationary spacetimes has been largely studied, yielding results in agreement with the quantum field theory approach. In this work, we consider the application of the stochastic approach within the considered model of axion inflation with a coupling to gauge fields, comparing the results obtained for the energy density and the helicity integral with those obtained through renormalization. We find that the stochastic formalism is able to capture the qualitative behaviour of the energy density and the helicity integral.
However, the results are not precise enough to use the stochastic approach to well quantify the backreaction.
Furthermore, in the region in which we find the best agreement between the two approaches, the stochastic equations start to lose validity as the approximations done in order to obtain them are no longer valid.
The reason for this behaviour can be found in the strong scale dependence of the gauge fields on super-Hubble regions.
Unlike the inflaton, gauge fields are not "frozen" on super-Hubble scales. Consequently, when we define the dynamics of the long-wavelength modes that we aim to describe effectively, the peaks present in the power spectra are not accurately considered. This leads to the discrepancy with the adiabatic results.
As a possible solution to improve the stochastic formalism applied to gauge fields, we propose the introduction of new variables, obtained through a transformation of the physical fields, which freeze on super-Hubble scales and on which the stochastic approach could be applied more reliably. After obtaining the stochastic results, one could go back to the original physical fields in order to obtain the backreaction terms.