Bubble-Nucleation Rates for Cosmological Phase Transitions

We estimate bubble-nucleation rates for cosmological phase transitions. We concentrate on the evaluation of the pre-exponential factor, for which we give approximate analytical expressions. Our approach relies on the use of a real coarse-grained potential. The consistency of the calculation implies that the coarse-graining scale must be larger than the typical scale of the critical bubble. We show how this scale can be determined in the studies of high-temperature phase transitions. We discuss the metastability bound on the Higgs-boson mass and the electroweak phase transition. We find that the saddle-point approximation is reliable in the first case and breaks down in the second case. Introduction: The estimates of bubble-nucleation rates for cosmological first-order phase transitions are carried out within Langer’s theory of homogeneous nucleation [1], applied to relativistic field theory in refs. [2]. The nucleation rate is exponentially suppressed by the action (free energy rescaled by the temperature) of the critical bubble, a saddle point of the free energy of the system. Significant contributions to the nucleation rate may arise from higher orders in a systematic expansion around this saddle point. The first correction has the form of a pre-exponential factor that involves fluctuation determinants around the saddle-point configuration and the false vacuum. The evaluation of this factor is a difficult problem at the conceptual and technical level, as crucial issues associated with the convexity of the potential, the divergences of the fluctuation determinants and the double-counting of the effect of fluctuations must be resolved. Several approaches have been proposed in order to address these issues [3, 4]. In a series of recent works [5]–[8], following the proposal of refs. [9], we developed a consistent approach, based on the effective average action Γk [10] that can be interpreted as a coarse-grained free energy. Fluctuations with characteristic momenta larger than a coarsegraining scale (q >∼ k) are integrated out and their effect is incorporated in Γk. In the limit k → 0, Γk becomes equal to the effective action. The k dependence of Γk is described by an exact flow equation [11]. This flow equation can be translated into evolution equations for functions appearing in a derivative expansion of the action [12]. An approximation that is sufficient in most cases takes into account the effective average potential Uk and a standard kinetic term and neglects higher derivative terms in the action. The bare theory is defined at some high scale Λ that can be identified with the ultraviolet cutoff. It is, however, more convenient to choose a starting scale k0 below the temperature T , where the effective average action of a (3+1)-dimensional theory at non-zero temperature can be described in terms of an effective three-dimensional action at zero temperature [13, 14]. In ref. [5] we computed the form of Uk at scales