  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.
