Dynamics of symmetry breaking in FRW cosmologies: Emergence of scaling

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 ξphys=dH(t) with dH(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.