Supercooled first-order phase transitions and gravitational waves in sub-GeV dark sectors

In the summer of 2023, various pulsar timing array (PTA) collaborations reported the first evidence for a stochastic gravitational wave background (SGWB) in the nHz frequency range. Among the possible explanations for this signal, a cosmological first-order phase transition (FOPT) is one of the most intriguing, as its realisation would open a new window into the early Universe to probe new physics. In this thesis, we present a novel semi-analytic, model-independent framework to study supercooled FOPTs in classically scale-invariant models. The goal is to establish the relation between the Lagrangian parameter space and the SGWB signal as clearly as possible. The main idea is to exploit collective couplings and the high-temperature expansion to write the effective potential as a polynomial, for which analytic results for the bounce action are known. We then derive a simplified equation for the computation of the percolation temperature, together with analytic expressions for the thermal parameters that enter into the expected SGWB spectrum. To validate and illustrate our proposal, we apply our framework to phenomenological sub-GeV dark abelian U(1) models, which are able to explain the PTA signal. We find that our approach not only reproduces full numerical analyses with high accuracy and significantly reduced computational time, but also provides valuable physical insights.