Production of thermal axions across the electroweak phase transition

Theoretical frameworks for axion-like particles address outstanding questions in the physics of fundamental interactions. An astonishing experimental effort paves the way toward discoveries that could lead to remarkable advancements in the understanding of both particle interactions and the composition of our universe. Indeed the QCD axion provides an elegant solution to two well-known drawbacks of the standard model (SM) of particle physics: the strong CP problem and the observed dark matter abundance. The introductory part of this thesis will present the strong CP problem and how the Peccei-Quinn mechanism provides a solution to these two open questions. The core of this work will be to investigate a complementary manifestation of the axion on cosmological scales. In particular, we will study an additional dark radiation component in the form of relativistic axions generated at high temperatures in the Early Universe, specifically around the ElectroWeak Phase Transition (EWPT). Thermal production could lead to a potentially observable axion contribution to the cosmic energy budget, conventionally parameterized by an effective number of additional neutrinos. The main goal of this thesis is to predict the axion contribution to it. We will consider the relevant processes for the thermal production of axions and derive explicit expressions for cross-sections. We will compute them above and below the EWPT and will connect the results across this threshold. With these quantities then we feed the Boltzmann equations. The novel aspect of this research work involves investigating the thermal production of axion dark radiation in momentum space: we will apply a new formalism based entirely on phase space analysis to axion dark radiation and provide predictions. Current observations align with the SM prediction for the effective number of neutrinos, yet upcoming experiments promise to refine bounds on this parameter and potentially discover deviations from the SM. In this context, the implications of our findings could be significant as an accurate prediction of the axion dark radiation impact on the effective number of neutrinos. If detected, this contribution could help to validate the axion's role in the ongoing exploration of physics beyond the SM.