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The theory of General Relativity was established on a spacetime manifoldequipped with a metric tensor, $(\mathcal{M}_4,\text{g})$, and the connectionon $\mathcal{M}_4$ identified with the Levi-Civita one. Even though there arevalid reasons to assume a torsionless manifold that preserves the metric, itwas shown that dealing away with these assumptions the Levi-Civita conditioncan be reproduced at the level of equations of motion of GR. It was not longbefore the equivalence of General Relativity between the two descriptions,known as the Palatini or first-order formalism in which the connection isindependent of the metric, and the conventional metric or second-orderformalism, was broken for more complicated actions involving higher-ordercurvature invariants and/or nonminimal couplings between the gravitational andmatter sector. Nowadays these types of theories are prominent in modelinginflation where they have found major success. Since the paradigm of inflationis fused with the gravitational degrees of freedom and thus theirparametrisation, it is interesting to understand how the predictions of thesemodels differ between the two formulations. One of the outstanding models ofinflation is the Starobinsky or quadratic gravity model. However in thePalatini formalism the scalar degree of freedom sourced by the $R^2$ term isnondynamical and is unable to drive an inflationary phase. In order forinflation to be realised in the first-order formalism the Starobinsky model hasto be coupled with a fundamental scalar field that will assume the role of theinflaton. In this thesis we investigate different inflationary scenarios,starting with previously ruled-out models, where we find that the $R^2$ termhas a significant role in flattening the Einstein-frame inflaton potential andthus giving the opportunity for these models to come in contact withobservations.

Inflationary phenomenology of quadratic gravity in the Palatini formulation