2.2. Are defects inflated away?
It is important to realize the relevance that the Kibble's mechanism has for cosmology; nearly every sensible grand unified theory (with its own symmetry breaking pattern) predicts the existence of defects. We know that an early era of inflation helps in getting rid of the unwanted relics. One could well wonder if the very same Higgs field responsible for breaking the symmetry would not be the same one responsible for driving an era of inflation, thereby diluting the density of the relic defects. This would get rid not only of (the unwanted) monopoles and domain walls but also of any other (cosmologically appealing) defect. Let us follow [Brandenberger, 1993] and sketch why this actually does not occur. Take first the symmetry breaking potential of Eq. (1.2) at zero temperature and add to it a harmless -independent term 3 m4 / (2). This will not affect the dynamics at all. Then we are led to
with = ( 6 m2 / )1/2 the symmetry breaking energy scale, and where for the present heuristic digression we just took a real Higgs field. Consider now the equation of motion for ,
for << very near the false vacuum of the effective Mexican hat potential and where, for simplicity, the expansion of the universe and possible interactions of with other fields were neglected. The typical time scale of the solution is t m-1. For an inflationary epoch to be effective we need t >> H-1, i.e., a sufficiently large number of e-folds of slow-rolling solution. Note, however, that after some e-folds of exponential expansion the curvature term in the Friedmann equation becomes subdominant and we have H2 8G V(0) / 3 (2m2 / 3 )( / mP)2. So, unless > mP, which seems unlikely for a GUT phase transition, we are led to << H-1 and therefore the amount of inflation is not enough for getting rid of the defects generated during the transition by hiding them well beyond our present horizon.
Recently, there has been a large amount of work in getting defects, particularly cosmic strings, after post-inflationary preheating. Reaching the latest stages of the inflationary phase, the inflaton field oscillates about the minimum of its potential. In doing so, parametric resonance may transfer a huge amount of energy to other fields leading to cosmologically interesting nonthermal phase transitions. Just like thermal fluctuations can restore broken symmetries, here also, these large fluctuations may lead to the whole process of defect formation again. Numerical simulations employing potentials similar to that of Eq. (1.5) have shown that strings indeed arise for values ~ 1016 GeV [Tkachev et al., 1998, Kasuya & Kawasaki, 1998]. Hence, preheating after inflation helps in generating cosmic defects.