The Bubble Wall Velocity in Local Thermal Equilibrium and Energy Budget with Full Effective Potential

We develop a framework based on the full one-loop finite-temperature effective potential model, within which the bubble wall velocity is calculated using the local thermal equilibrium (LTE) approximation, and the kinetic energy fraction $K$ is computed directly. In cosmological phase transitions, these quantities play a critical role in determining the resulting gravitational wave signals. Using the xSM as a benchmark model, we compute the peak gravitational wave spectra under different methods for determining the wall velocity and the kinetic energy fraction $K$, and compare these results to those obtained using the commonly employed bag model. Within the scanned parameter space, we find: (1) Deflagration is the most prevalent mode of fluid motion.(2) Gravitational wave spectra based on the full effective potential with LTE-derived wall velocity and integrated $K$ can differ significantly from those using the bag model with fitted $K$. In the deflagration regime, discrepancies reach up to 48\% in peak frequency and 90\% in amplitude.(3) The bag model provides a good approximation to the full equation of state in many cases. Notably, in deflagration scenarios with input wall velocity, the gravitational wave spectra obtained from the bag model more closely resemble the LTE-based results than those derived using the full potential with this input wall velocity.