  We study the onset of symmetry breaking after hybrid inflation in a model
having the field content of the SU(2) gauge-scalar sector of the standard
model, coupled to a singlet inflaton. This process is studied in
(3+1)-dimensions in a fully non-perturbative way with the help of lattice
techniques within the classical approximation. We focus on the role played by
gauge fields and, in particular, on the generation of Chern-Simons number. Our
results are shown to be insensitive to the various cut-offs introduced in our
numerical approach. The spectra preserves a large hierarchy between long and
short-wavelength modes during the whole period of symmetry breaking and
Chern-Simons generation, confirming that the dynamics is driven by the low
momentum sector of the theory. We establish that the Chern-Simons production
mechanism is associated with local sphaleron-like structures. The corresponding
sphaleron rates are of order 10^{-5} m^4, which, within certain scenarios of
electroweak baryogenesis and a (not unnaturally large) additional source of CP
violation, could explain the present baryon asymmetry of the universe.
