Dynamics of Symmetry Breaking in FRW Cosmologies: Emergence of Scaling

Abstract

The dynamics of a symmetry breaking phase transition is studied in a radiation and matter dominated spatially flat FRW cosmology in the large N limit of a scalar field theory.The quantum density matrix is evolved from an initial state of quasiparticles in thermal equilibrium at a temperature higher than the critical. The cosmological expansion decreases the temperature and triggers the phase transition. We identify three different time scales: an early regime dominated by linear instabilities and the exponential growth of long-wavelength fluctuations,an intermediate scale when the field fluctuations probe the broken symmetry states and an asymptotic scale wherein a scaling regime emerges for modes of wavelength comparable to or larger than the horizon.The scaling regime is characterized by a dynamical physical correlation length xi_{phys} = d_H(t) with d_H(t) the size of the causal horizon, thus there is one correlated region per causal horizon. Inside these correlated regions the field fluctuations sample the broken symmetry states. The amplitude of the long-wavelength fluctuations becomes non-perturbatively large due to the early times instabilities and a semiclassical but stochastic description emerges in the asymptotic regime. In the scaling regime, the power spectrum is peaked at zero momentum revealing the onset of a Bose-Einstein condensate.The scaling solution results in that the equation of state of the scalar fields is the same as that of the background fluid. This implies a Harrison-Zeldovich spectrum of scalar density perturbations for long-wavelengths. We discuss the corrections to scaling as well as the universality of the scaling solution and the differences and similarities with the classical non-linear sigma model.