Numerical approach to the onset of the electroweak phase transition

We investigate whether the universe was homogeneously in the false vacuum state at the critical temperature of a weakly first-order phase transition such as the electroweak phase transition in terms of a series of numerical simulations of a phenomenological Langevin equation, whose noise term is derived from the effective action but the dissipative term is set so that the fluctuation-dissipation relation is met. The correlation function of the noise terms given by a non-equilibrium field theory has a distinct feature if it originates from interactions with a boson or with a fermion. The spatial correlation function of noises from a massless boson damps with a power-law, while the fermionic noises always damp exponentially above the inverse-temperature scale. In the simulation with one-loop effective potential of the Higgs field, the latter turns out to be more effective to disturb the homogeneous field configuration. Since noises of the both types are present in the electroweak phase transition, our results suggest that conventional picture of a phase transition, namely, nucleation of critical bubbles in a homogeneous background does not apply or the one-loop approximation breaks down in the standard model.