Expanding Bubbles in a Thermal Background

The behavior of fields during weak first order phase transitions has been a source of great interest. Several studies have shown either a need for changes to the current theory of thermal phase transitions or the inclusion of carefully calculated corollaries to the present models [1–3]. Sub-critical bubbles and oscillon solutions may give rise to phase mixing in which these fluctuations in the smooth homogeneous background can have significant dynamical effects. Thus in the limit of weak first order phase transitions, the assumption of small amplitude fluctuations may need revision. For stronger transitions, the fluctuations are expected to diminish in importance validating the use of standard homogeneous nucleation theory. Yet, in the realm of the expanding bubble, precious few analytic and numerical studies have been performed to determine the effect of the thermal bath on the expansion of the bubbles [4,5]. With this in mind, we numerically examine expanding bubbles in 3 + 1 dimensions using a Langevin equation in hopes of gaining either qualitative or quantitative results that describe their behavior. Symmetry restoration at various stages in the Universe’s life has become an important and powerful concept in modern cosmology. Several examples of the relevance of phase transitions are given by inflation [6], the electroweak phase transition [7], and the quark hadron phase transition [8]. In particular, the phase transition that occured when the universe cooled to a critical temperature of about 300 GeV broke the symmetry of the weak and electromagnetic interactions. The breakdown of SU(2) × U(1) which may have created the baryon asymmetry we see today is expected to be of first order [9]. This satisfies the third of Sakharov’s conditions for baryogenesis, namely that no thermal equilibrium exist [10]. At the electroweak scale, the rate of expansion of the universe is such that thermal equilibrium is maintained [11]. To create an out of equilibrium process, a bubble of true vacuum appears within the false vacuum state. As the bubble expands, baryogenesis takes place in the neighborhood of the boundary of the bubble, the bubble wall. Thus, the detailed behavior of the bubble wall is of great importance in the electroweak phase transition. The nucleation of bubbles in the context of field theory has by now a long history. Coleman has shown that at zero temperature, the transition to the true vacuum takes place via the quantum nucleation of bubbles [12]. This process corresponds to a generalization of barrier penetration in quantum mechanics. The probability for bubble nucleation at zero temperature is found by determining the “bounce” configuration from the Euclidean equation of motion subject to the boundary conditions of restricting the field to the metastable minimum at tE = −∞, the global minimum at tE = 0, and the metastable minimum at tE = +∞. The bounce solution is used to compute the Euclidean action, from which bubble nucleation probability per unit volume is determined, Γ = A exp (−SE). The dominant contribution to