Physical Foundations of Cosmology

4.6 Beyond the Standard Model 221

Here we take into account thatθchanges around the contour by the integerm
multiplied by 2π. Thus, in the gaugedU( 1 )theory, strings carry a magnetic flux
inversely proportional to the electric chargee, and this flux is quantized.
After symmetry breaking the vector and scalar fields acquire massesmA=eσ
andmχ=

√

2 λσrespectively. These masses determine the “thickness” of the string.
Outside the string core both fields tend to their true vacuum values exponentially
quickly. This is not surprising because in the brokenU( 1 )gauge theory no massless
fields remain after symmetry breaking (compare this to the global string, where one
masslessscalar field “survives”). The thickness of the string core is determined by
the Compton wavelengthsδχ∼m−χ^1 andδA∼m−A^1 .Formχ>mA,the size of the
magnetic core(∼δA)exceeds the size of the false vacuum tube

(

∼δχ

)

. In this case,
the scalar and magnetic fields give about the same contribution to the energy per
unit length:

μ(χ)∼λσ^4 δχ^2 ∼σ^2 and μ(A)∼B^2 δ^2 A∼

(

eδ^2 A

)− 2

δ^2 A∼σ^2.

The total mass of a string with length of order the present horizon is about

σ^2 t 0 ∼ 1048

(

σ/ 1015 GeV

) 2

g.

Hence, even if symmetry breaking occurred at Grand Unified Theory scales, any
strings produced would not be in immediate conflict with observations. Moreover,
such Grand Unified Theory strings could, in principle, serve as the seeds for galaxy
formation. CMB measurements, however, rule out this possibility; cosmic strings
cannot play a dominant role in structure formation. Nevertheless this does not mean
that they were not produced in the early universe. If they were detected, cosmic
strings would reveal important features of the theory beyond the Standard Model.

Monopoles Monopoles arise if the vacuum manifold has the topology of a two-
dimensional sphere; for example, in theories where the symmetry is broken with
three real scalar fields. In this case the vacuum manifold described by
(
φ^1

) 2

+

(

φ^2

) 2

+

(

φ^3

) 2

=σ^2 (4.236)

is obviouslyS^2 .Again considering two causally disconnected regionsAandB,
we find that with probability of order unity one can obtain after symmetry break-
ingφiA>0 andφiB<0, wherei= 1 , 2 ,3 and the fieldsφisatisfy (4.236). Three
two-dimensional hypersurfaces, determined by the equationsφi

(

x^1 ,x^2 ,x^3

)

= 0 ,

generically cross each other at a point. This is the point of false vacuum because
there all three fieldsφivanish. Thus a zero-dimensional topological defect−a
monopole−is formed (Figure 4.23). Solving the equations for the scalar field,
one can find the classical spherically symmetric scalar field configuration which