218 The very early universe
domain wall string monopole
φ= 0
φ^1 = 0φ^2 = 0 φ(^2) = 0
φ^1 = 0
φ^3 = 0
Fig. 4.23.
Problem 4.33Consider an infinite wall in thex–yplane and verify thatφ(z)=
σtanh(z/l),wherel=(λ/ 2 )−^1 /^2 σ−^1 ,is the solution of the scalar field equation
with the potential in (4.230).
Domain walls are nonperturbative solutions of the field equations and they are
stable with respect to small perturbations. To remove the wall described by the
solution in Problem 4.33, one has to “lift” the scalar field over the potential barrier
fromφ=σtoφ=−σin infinite space. This costs an infinite amount of energy.
It is clear from the previous discussion that on average at least one domain
wall per horizon volume is formed during the cosmological phase transition. The
subsequent evolution of the domain wall network is rather complicated and has
been investigated numerically. The result is that one expects at least one domain
l
φ
σ
−σ
(a) (b)
Fig. 4.24.