Physical Foundations of Cosmology

222 The very early universe

corresponds to the false vacuum in the center and approaches the true vacuum
asr→∞. Without going into the detailed structure of the exact solutions, we
can analyze the properties of monopoles using dimensional arguments. In the the-
ory without gauge fields, two massless bosons survive after symmetry breaking.
Therefore the fieldsφi,smoothly changing from zero in the core of the monopole,
approach their true vacuum configuration only as some power of distancer.On
dimensional grounds,∂φ∼σ/r(here and in the following formulae we skip all
indices) and the mass of the global monopole

M∼σ^2

∫

1

r^2

d^3 x (4.237)

diverges linearly. The cut-off scale should be taken to be of order the correlation
length; this can never exceed the horizon scale.
Localmonopoles have quite different properties. As an example, let us consider
SO( 3 )SU( 2 )local gauge theory with the triplet of real scalar fields. After sym-
metry breaking, two of the three gauge fields acquire massmW=eσ. One gauge
field,A, remains massless andU( 1 )gauge invariance survives. The massive vector
fieldsWdecay exponentially quickly beyond their interaction distance, determined
by the Compton wavelengthδW∼m−W^1. In the monopole solution the massless
U( 1 )gauge fieldAcompensates the gradient term in the covariant derivative and
Dφ=∂φ+eAφdecays exponentially asr→∞.Therefore at large distancesr
we have

A∼

1

e

∂φ

φ

∼

( 1 /e)

r

(4.238)

and the corresponding magnetic field is of order

B=∇×A∼

( 1 /e)

r^2

. (4.239)

Thus the local monopole has a magnetic chargeg∼ 1 /e.The exact calculation
givesg= 2 π/ein agreement with the result for the Dirac monopole. We have to
stress, however, that in contrast with the Dirac monopole, which is a fundamental
point-likemagnetic charge, monopoles in gauge theories areextendedobjects. They
are classical solutions of the field equations. As with the local string, the monopole
has two cores−scalar and magnetic−with radii

δs∼m−s^1 =( 2 λ)−^1 /^2 σ−^1 and δW∼m−W^1 =e−^1 σ−^1

respectively. Let us assume that the scalar core is smaller than the magnetic one,
that is,δW>δs.It is easy to verify that in this case the dominant contribution to