Biased discrete symmetry and domain wall problem

We reconsider the cosmological evolution of domain walls produced by the spontaneous breaking of an approximate discrete symmetry. We show that domain walls may never collapse, even if the standard bound on the vacuum energy asymmetry is satisfied. Instead of disappearing, these defects may form stable "bound states"-double wall systems. Possible stability of such a wall is a dynamical question and consequently restricts the allowed range of parameters. In particular, in the two Higgs doublet standard model with an anomalous Z₂ symmetry, the above restriction suggests the mass of the pseudoscalar Higgs (would be axion) being close to the mass of the scalar one.