• Adiabatic Renormalization
• Axion Gauge Filed Inflation
• Early Universe
• Pre-Big-Bang Scenario
• Primordial Black Holes
• String Cosmology

The standard cosmological model describes beautifully many features of our universe, however, several problems remain open. This thesis is motivated by this challenging context, trying to address several aspects from different prospects. On one hand, we try to obtain a robust description of the beginning of our Universe, via the Pre-Big Bang scenario, looking for non-singular solutions, in which we try also to explain the observed dark matter via primordial black holes. On the other hand, we develop a new formulation for the adiabatic renormalization, an important tool to study the quantum field in curved space-time, fundamental to understanding properly the physics of the inflationary epoch. In particular, we apply this new approach to study an inflationary model described by a pseudo-scalar field coupled to gauge bosons. The thesis is organized into three parts, as follows: In part I, we introduce the standard cosmological model, inflation accelerated, cosmological perturbation theory, and primordial black holes. Part II is focused on string cosmology, introducing the basics of string theory and the Pre-Big-Bang scenario. In Ch. 7 it is described how string correction can produce primordial black holes. Ch. 8 is related to the possibility of characterizing the final stage of the pre-big bang scenario by a gas of string holes, discussing the stability of such a possibility. Ch. 9 It is focused on obtaining smooth solutions connecting smooth solutions connecting the pre-big bang and post-big bang evolutions, considering higher order correction in higher derivatives (the so-called α ′ -corrections), and loop correction by an effective dilaton potential. Part III is devoted to quantum fields in curved space-time. How to extend adiabatic renormalization to obtain well-defined renormalized quantities in curved space-time. We consider, in particular, a peculiar model of a pseudo-scalar inflaton field coupled with an abelian gauge field. In such a case, the standard adiabatic procedure, needed to evaluate properly the backreaction of gauge fields on the background, fails to introduce fictitious divergences in the infrared domain.