174 The very early universe
is separated from ̄χ=0 by a potential barrier of height,
Vc=e^12 Tc^4
4 ( 4 π)^4 λ^3 Tc, (4.128)
with the maximum at ̄χ=χ ̄c/ 2 .Note that the coupling constantλTcannot be taken
as arbitrarily small, because for largeTthe temperature corrections to it exceed
e^4. Hence, at the critical temperature the second minimum, located at ̄χc<T/e,
is always within the region of applicability of the high-temperature expansion.
As the temperature drops belowT 0 ,m^2 T becomes negative and the minimum at
χ ̄=0 disappears. Finally, at very low temperatures (see Problem 4.20) the potential
converges to (4.117).
4.4.4 Phase transitions
As the temperature drops below the critical valueTc,the minimum at ̄χm= 0
becomes energetically favorable. Therefore, the field ̄χcan change its value and
evolve to this second minimum. If at this time the two minima are separated by a
potential barrier, the transition occurs with bubble nucleation. Inside the bubbles
the scalar field acquires a nonvanishing expectation value. If the bubble nucleation
rate exceeds the universe’s expansion rate, the bubbles collide and eventually fill all
space. As a result, gauge bosons and fermions become massive. Such a transition
is called a first order phase transition. It is very violent and one can expect large
deviations from thermal equilibrium.
The other possible scenario takes place if ̄χ=0 and ̄χm=0 are never separated
by a potential barrier. In this case, the field ̄χgradually changes its value and the
transition is smooth. It can be either a second order phase transition or simply a
cross-over. As we have pointed out a second order phase transition is usually char-
acterized by a continuous change of some symmetry. Because the gauge symmetry
is never broken by the Higgs mechanism, we expect that in gauge theories a smooth
transition is a cross-over. From the point of view of cosmological scenarios, a cross-
over is not very different from a second order phase transition and for our purposes
we simply have to distinguish a violent from a smooth transition.
Let us now discuss what kind of transition one could expect inU( 1 )theory. To
answer this question we consider the high-temperature expansion ofVeff,given in
(4.122). First of all we note that the barrier is due entirely to the ̄χ^3 term, which
in turn appears because of the nonanalytic term∝
√
(mG/T)^2 in (3.44) forJ−(^1 ).
If the massmGwere zero, this term would be absent. Therefore, to establish the
character of the transition, we need to know when we can trust our calculation of
the ̄χ^3 contribution toVeff. As follows from (4.109), the temperature fluctuations