Can topological defects be formed during preheating ?

We study the dynamics of a scalar field $\Phi$ with potential $g(|\Phi|^2-\eta^2)^2/2$ (g: self-coupling constant, $\eta$: symmetry breaking scale) after inflation and make clear whether topological defects can ever be formed during preheating. In particular, we pay attention to the GUT defects ($\eta \simeq 10^{15} GeV$), and find that non-thermal fluctuations of the scalar field produced by parametric resonant decay of its homogeneous part do not lead to defect formation. This fact shows that the positiveness of the effective mass square of the field and production of large fluctuations whose amplitude is as large as that of homogeneous mode are not enough conditions for the full symmetry restoration, which leads to the formation of the topological defects. We also study another mechanism for the production of topological defects. It is due to the facts that the effective potential of the field has two minima in the radial direction and the field has small fluctuations at the end of inflation. We find the condition for the topological defects to be formed: $\eta \lesssim 10^{14} GeV$. Therefore, the GUT defects cannot be produced.