Global cosmic strings
69
In the present case, at high temperatures we expect (t/» to be zero and,
after the phase transition, some regions will have (t/» = 11 and some will have
{t/>} = -". In this way topological defects can form. Although a domain wall can
change its shape, topology prevents it from disappearing once formed because its
ends are at discrete minima of V. In general, there will be curved domain walls as
well as flat domain walls, with the curved walls enclosing a region of space inside
which the VEV of t/> differs from the VEV of t/> outside.
Many domain walls will form constituting a random network whose
evolution with time may be studied. The result for non-relativistic domain walls
is that the energy density (i.e. the energy per unit volume) of the domain walls
scales as R- 1 , where R(t) is the scale factor of the universe. This should be
compared with the energy density due to radiation which scales as R- 4 and
that due to matter which scales as R- 3 (as in section 1.3). Consequently, as R
increases with time. the energy density of the universe comes to be dominated
by domain walls. The total energy associated with a plane domain wall with area
H02, where Ho is the present day Hubble constant, is far larger than the estimated
total energy due to matter within the Hubble radius. For example, for J.. not too
much different from I and" '" lOO GeV, the former is larger by 12 orders of
magnitude. A larger expectation value for the scalar field makes things worse
(exercise 2). Thus, domain walls appear undesirable. This suggests that a theory
is needed which does not have disconnected vacuum states, such as (t/» = ±" in
the present model, to avoid the existence of domain walls. Alternatively, a period
of inflation (see chapter 7) is needed to dilute the domain wall density.
3.3 Global cosmic strings
One-dimensional topological defects (cosmic strings) can also be produced by
phase transitions in the early universe. The simplest example of a cosmic string
[3] may be derived from the Lagrangian density for a complex scalar field t/>:
£. = a",t/J*a"'t/J - V(t/J)
(3.11 )
with
V(t/J) = ~(tP·t/J - rh 2
(3.12)
2
and J.. and " are real constants. This Lagrangian possesses a global U (I)
symmetry under
t/J ~ eiat/J
(3.13)
where a is an arbitrary constant real number. The potential V of (3.12) has a
maximum at t/J = 0 and minima with V = 0 when
(3.14)
t/J = "e i /3
where f3 is an arbitrary real number. The vacuum VEV (3.14) breaks the global
U (1) symmetry because it is not invariant under the transformation (3.13).
69
In the present case, at high temperatures we expect (t/» to be zero and,
after the phase transition, some regions will have (t/» = 11 and some will have
{t/>} = -". In this way topological defects can form. Although a domain wall can
change its shape, topology prevents it from disappearing once formed because its
ends are at discrete minima of V. In general, there will be curved domain walls as
well as flat domain walls, with the curved walls enclosing a region of space inside
which the VEV of t/> differs from the VEV of t/> outside.
Many domain walls will form constituting a random network whose
evolution with time may be studied. The result for non-relativistic domain walls
is that the energy density (i.e. the energy per unit volume) of the domain walls
scales as R- 1 , where R(t) is the scale factor of the universe. This should be
compared with the energy density due to radiation which scales as R- 4 and
that due to matter which scales as R- 3 (as in section 1.3). Consequently, as R
increases with time. the energy density of the universe comes to be dominated
by domain walls. The total energy associated with a plane domain wall with area
H02, where Ho is the present day Hubble constant, is far larger than the estimated
total energy due to matter within the Hubble radius. For example, for J.. not too
much different from I and" '" lOO GeV, the former is larger by 12 orders of
magnitude. A larger expectation value for the scalar field makes things worse
(exercise 2). Thus, domain walls appear undesirable. This suggests that a theory
is needed which does not have disconnected vacuum states, such as (t/» = ±" in
the present model, to avoid the existence of domain walls. Alternatively, a period
of inflation (see chapter 7) is needed to dilute the domain wall density.
3.3 Global cosmic strings
One-dimensional topological defects (cosmic strings) can also be produced by
phase transitions in the early universe. The simplest example of a cosmic string
[3] may be derived from the Lagrangian density for a complex scalar field t/>:
£. = a",t/J*a"'t/J - V(t/J)
(3.11 )
with
V(t/J) = ~(tP·t/J - rh 2
(3.12)
2
and J.. and " are real constants. This Lagrangian possesses a global U (I)
symmetry under
t/J ~ eiat/J
(3.13)
where a is an arbitrary constant real number. The potential V of (3.12) has a
maximum at t/J = 0 and minima with V = 0 when
(3.14)
t/J = "e i /3
where f3 is an arbitrary real number. The vacuum VEV (3.14) breaks the global
U (1) symmetry because it is not invariant under the transformation (3.13).