happens if $\epsilon_1 = 0$ and $\epsilon_2 \neq 0$. Two regions of space which correspond to the two degenerate vacua $\sigma = f_\pi$ and $\sigma = - f_\pi$, respectively, (see eq. (14)), will be separated by a domain wall where the chiral field smoothly interpolates between the two vacua. In the context of the early Universe, stable domain walls are almost always disastrous unless the phase transition happens at extremely late times. This already suggests that the parameter $\epsilon_1$ cannot be identically zero.

By expanding $\sigma$ and , as in eqs. (9-10), we can determine the effective Lagrangian for $\theta$ from eqs. (1) and (20) to be $${\cal L}_\theta = {v^2 \over 2} (\partial_\mu \theta)^2 + \epsilon_1 v
\cos\theta + {\epsilon_2 \over 2} v^2 \cos^2\theta \, .$$ Here $v$ minimizes the effective potential at fixed $\theta$ and $T$; that is, it traces the bottom of the valley of the potential. Since this valley has an almost constant radius at fixed $T$, we can neglect its very weak dependence on $\theta$. Hence $v = v(T)$.

As we will show in the following, cosmology places a lower limit on $\epsilon_1$ which is very small. Thus, as far as cosmological considerations are concerned, 