A Comprehensive Framework for Electroweak Phase Transitions: Thermal History and Dynamics from Bubble Nucleation to Percolation

The electroweak phase transition (EWPT) is crucial for cosmology and particle physics, with a profound impact on electroweak baryogenesis, symmetry breaking, and gravitational wave (GW) signals. However, many studies overlook key aspects of EWPT dynamics, leading to misidentified patterns and overestimated GW signals. To address these gaps, we present a comprehensive framework for analyzing EWPTs, focusing on the vacuum's thermal history and dynamics from bubble nucleation to percolation. Using the $\mathbb{Z}_2$-odd real scalar singlet model, we demonstrate the occurrence of spontaneous $\mathbb{Z}_2$ symmetry breaking in the high-temperature vacuum, leading to diverse EWPT processes, including multi-step transitions and inverse symmetry breaking. We identify four distinct EWPT patterns, each characterized by unique symmetry-breaking mechanisms and associated with bubbles exhibiting distinct field configurations, which can be analyzed using a formalism based on energy density distributions developed here. A key finding is that bubble nucleation fails in extremely strong phase transitions (PTs) with low nucleation rates, or in ultra-fast PTs involving inverse $s$-bubbles that collapse instantly upon formation, both of which lead to false vacuum trapping and the absence of observable GW signals. In first-order PTs where nucleation succeeds, stronger transitions occur later in the universe's evolution, while weaker transitions proceed more rapidly. Multi-step transitions involving (inverse) $\mathbb{Z}_2$ symmetry breaking give rise to complex transition sequences and exotic bubble dynamics, such as sequential nucleation or the coexistence of bubbles from different vacua -- phenomena with significant implications for GW spectra, dark matter, and baryogenesis. This work advances our understanding of EWPT dynamics and lays the groundwork for future studies of EWPTs in BSM physics.