• conformal anomaly
• cosmology
• axion
• slow-roll inflation
• adiabatic renormalization
• pseudo-scalar inflaton
• gauge fields
• backreaction
• second order perturbations
• chaotic inflation
• quasi-de Sitter space-time

In this master thesis we take in consideration a particular type of inflationary model called natural inflation, where the inflaton $\phi$ is a pseudo-scalar and has a potential given by
$$
V(\phi)=\Lambda^4 \qty[1+\cos{\qty(\frac{\phi}{f})}] \,.
$$
Furthermore we consider that the pseudo-scalar inflaton is coupled to some massless gauge fields through
the Lagrangian
$$
\mathcal{L}_{int}=-\frac{g\phi}{4}F^{\mu \nu}\tilde{F}_{\mu \nu} \,,
$$
where $g$ is the gauge coupling with dimension of an inverse of an energy. The investigated model is sometimes called axion-like inflation, being the axion a pseudo-scalar particle, hence we assume $g=\frac{\alpha}{f}$ with $\alpha$ is a dimensionless parameter and $f$ the axion decay constant.
\\
In this framework we examine a slow-roll scenario with the inflationary phase realized by a quasi-de Sitter expansion, generalizing previous works where the inflationary phase was approximated by a pure de Sitter stage.
Accordingly, in our computations we use the quasi-de Sitter scale factor $a(\tau)=-\frac{1}{H \tau(1-\epsilon)}$. Because of the ultraviolet divergences present in the model, we need to adopt a renormalization method: our choice is the adiabatic renormalization scheme. This renormalization approach eliminates the ultraviolet divergences but brings some unphysical infrared divergences in the model, such issue appears when the physical
scales are stretched over the Hubble horizon. In order to solve this puzzle,
we make use of a well justified comoving infrared cut-off $c = \beta a H$ for the adiabatic counterpart, with $\beta$ a further parameter inserted by the adiabatic renormalization procedure. However, this parameter can be univocally fixed to a finite value $\beta \simeq 0.359$ by requiring to have the conformal
anomaly in a particular limit. This anomaly has, in fact, to be obtained for the case of massless conformally coupled fields in a curved space-time, this set the $\beta$ parameter to the mentioned value.
Since they exhibit ultraviolet divergences, the physical quantities to which we apply this renormalization procedure are the vacuum expectation values of the energy density $\frac{1}{2}\langle \textbf{E}^2+\textbf{B}^2 \rangle$ and the helicity integral $\langle \textbf{E}\cdot \textbf{B} \rangle$ of the gauge fields, where $\textbf{E}$ and $\textbf{B}$ are the electric and magnetic fields related to the gauge field $A_{\mu}$.
\\
The final aim of the thesis is to evaluate the quantum backreaction of the gauge fields on the inflationary dynamic, trying to understand if the presence of the gauge fields can help in obtaining an enough long inflationary phase.
Therefore, we first calculate the renormalized averaged physical quantities mentioned above, to the leading order in the slow-roll parameter $\epsilon$, and then we compute the quantum backreaction of the gauge fields using a gauge invariant method. In order to do that in a consistent way one has to consider perturbations of the matter and metric fields up to second order. To evaluate such backreaction we finally consider, in the long wavelength limit and using the uniform curvature gauge, the case of a chaotic inflationary model with potential $V(\phi)=\frac{1}{2}m^2\phi^2$, where $m=\frac{\Lambda^2}{f}$. Indeed, this potential approximates very well the natural inflation one if $f \gg m_{Pl}$, which is the bound permitted by the cosmic microwave background anisotropy observations. As a final result of our gauge invariant approach, we obtain the correction to the effective expansion rate of the universe as a consequence of the quantum backreaction of the gauge fields on the background dynamics. Such correction goes in the right direction of helping to obtain more inflation, for a large region of the parameters space.