Introduction to Cosmology

156 Cosmic Inflation

the domains would have to be separated bydomain walls,whicharetopological defects

of space-time. Such domain walls would contain enormous amounts of energy and,

in isolation, they would be indestructible. Intersecting domain walls would produce

other types of topological defects such asloopsorcosmic stringswiggling their way

through the Universe. No evidence for topological defects has been found, perhaps

fortunately for us, but they may still lurk outside our horizon.

Magnetic Monopoles. A particular kind of topological defect is a magnetic
monopole. Ordinarily we do not expect to be able to separate the north and south
poles of a bar magnet into two independent particles. As is well known, cutting a
bar magnet into two produces two dipole bar magnets. Maxwell’s equations account
for this by treating electricity and magnetism differently: there is an electric source
term containing the charge푒, but there is no magnetic source term. Thus free elec-
tric charges exist, but free magnetic charges do not. Stellar bodies may have large
magnetic fields, but no electric fields.
Paul A. M. Dirac(1902–1984) suggested in 1931 that the quantization of the elec-
tron charge might be the consequence of the existence of at least one free magnetic
monopole with magnetic charge

푔M=

1

2

ℏcn

푒

≃ 68. 5 en, (7.14)

where푒is the charge of the electron and푛is an unspecified integer. This would then

modify Maxwell’s equations, rendering them symmetric with respect to electric and

magnetic source terms. Free magnetic monopoles would have drastic consequences,

for instance destroying stellar magnetic fields.

Without going into detail about how frequently monopoles might arise during the

GUT phase transition, we assume that there could arise one monopole per ten horizon

volumes

푁M(푡GUT)= 0. 1 ×( 2 × 10 −^29 m)−^3 ,

and the linear scale has grown by a factor 4. 4 × 1027. Nothing could have destroyed

them except monopole–anti-monopole annihilation, so the monopole density today

should be

푁M(푡 0 )≃ 0. 1 ×( 4. 4 × 0 .02 m)−^3 ≃150 m−^3. (7.15)

This is quite a substantial number compared with the proton density which is at

most 0.17m−^3. Monopoles circulating in the Galaxy would take their energy from

the galactic magnetic field. Since the field survives, this sets a very low limit to the

monopole flux called theParker bound. Experimental searches for monopoles have

not yet become sensitive enough to test the Parker bound, but they are certainly in

gross conflict with the above value of푁M; the present experimental upper limit to푁M

is 25 orders of magnitude smaller than푁훾.

Monopoles are expected to be superheavy,

푚M≳

푚X

훼GUT

≃ 1016 GeV≃ 2 × 10 −^11 kg. (7.16)