4.6 Beyond the Standard Model 219wall per present horizon scale∼t 0. Its mass can be estimated as
Mwall∼Ewt 02 ∼ 1065 λ^1 /^2 (σ/100 GeV)^3 g.For realistic values ofλandσ,the mass of the domain wall exceeds the mass of
matter within the present horizon by many orders of magnitude. Such a wall would
lead to unacceptably large CMB fluctuations. Therefore, domain walls are cosmo-
logically admissible only if the coupling constantλand the symmetry breaking
scaleσare unjustifiably small.
Homotopy groups give us a useful unifying description of topological defects.
Maps of then-dimensional sphereSninto a vacuum manifoldMare classified
by the homotopy groupπn(M).This group counts the number of topologically
inequivalent maps fromSnintoMthat cannot be continuously deformed into
each other. In the language of homotopy groups, domain walls correspond to the
groupπ 0
(
M=S^0
)
, which describes the maps of a zero-dimensional sphereS^0 =
{− 1 ,+ 1 }to itself.This group is nontrivial and is isomorphic to the group of integers
under addition modulo 2,that is,π 0
(
S^0
)
=Z 2.
Cosmic strings If the symmetry breaking occurs via aU( 1 )complex scalar field
φ=φ^1 +iφ^2 or, equivalently, with two real scalar fieldsφ^1 ,φ^2 ,cosmic strings are
formed. In this case the vacuum manifold described by
(
φ^1) 2
+
(
φ^2) 2
=σ^2 (4.233)is obviously a circleS^1.
Let us again consider two causally disconnected regionsAandBand assume that
inside regionAthe scalar fields acquired the expectation valuesφ^1 A>0 andφ^2 A> 0
satisfying (4.233). Because of the absence of communication, the expectation values
of the fields in regionBare not correlated with those in regionAand both can take
negative values:φ^1 B<0 andφ^2 B<0 (the only restriction is that they have to satisfy
(4.233)). The probability of this happening is 1/4. The fields are continuous and, in
changing from negative to positive values, must vanish somewhere between regions
AandB.Namely, fieldφ^1 vanishes on the two-dimensional surface determined
by the equationφ^1
(
x^1 ,x^2 ,x^3)
= 0 ,while fieldφ^2 is equal to zero on the surface
described byφ^2
(
x^1 ,x^2 ,x^3)
= 0 .These two surfaces generically cross each other
along a curve which is either infinite or closed. On this curve,φ^1 =φ^2 = 0 ,and
we have a false vacuum. Thus, as a result of symmetry breaking, one-dimensional
topological objects−cosmic strings−are formed (see Figure 4.23). It is clear that
they are produced with an abundanceat leastof order one per horizon volume.
As with domain walls, strings have a finite thickness. The fieldφsmoothly
changes from zero in the core of the string and approaches the true vacuum,|φ|=σ,