Physical Foundations of Cosmology

4.6 Beyond the Standard Model 217

vacuum is isomorphic to a circleS^1 (the “bottom of the bottle” ). In the casen= 3 ,
the vacuum states form a two dimensional sphereS^2.
Topological defects are solitonic solutions of the classical equations for the
scalar (and gauge) fields. They can be formed during a phase transition and since
they interpolate between vacuum states they reflect the structure of the vacuum
manifold. One real scalar field with two degenerate vacua (n=1 andM=S^0 )
leads todomain walls. In the case of a complex scalar field (n= 2 ,M=S^1 )
cosmic stringscan be formed. If the symmetry is broken with a triplet of real scalar
fields (n= 3 ,M=S^2 ),the topological defects aremonopoles. Finally, in the
casen=4 (four scalar fields or, equivalently, a doublet of complex scalar fields)
the vacuum manifold is a 3-sphereS^3 and the corresponding defects aretextures.
Depending on whether we consider a theory with or without local gauge invariance,
the topological defects are calledlocalorglobalrespectively.

Domain walls Let us first consider one real scalar field which has the double-well
potential in (4.230). The statesφ=σandφ=−σcorrespond to two degenerate
minima of the potential and during symmetry breaking the fieldφacquires the
valuesσand−σwith equal probability. The important thing is that the phase
transition sets the maximum distance over which the scalar field is correlated. It is
obvious that in the early universe the correlation length cannot exceed the size of
the causally connected region. Let us consider two causally disconnected regionsA
andB,and assume that the fieldφin regionAwent to the minimum atσ. The field
in regionBdoes not “know” what happened in regionAand, with probability 1/2,
goes to the minimum at−σ.Since the scalar field changes continuously from−σ
toσ,it must vanish on some two-dimensional surface separating regionsAandB.
This surface, determined by the equationφ

(

x^1 ,x^2 ,x^3

)

= 0 ,is called the domain
wall (see Figure 4.23). The domain wall (Figure 4.24(a)) has a finite thicknessl,
which can be estimated with the following simple arguments. Let us assume, for
simplicity, that the domain wall is static and not curved. The energy density of the
scalar field is

ε=^12 (∂iφ)^2 +V; (4.231)

it is distributed as shown in Figure 4.24(b). The total energy per unit surface area
can be estimated as

E∼εl∼

(σ

l

) 2

l+λσ^4 l, (4.232)

where the first contribution comes from the gradient term. This energy is minimized
forl∼λ−^1 /^2 σ−^1 and is equal toEw∼λ^1 /^2 σ^3.