If ₀() 1, the manifold is disconnected. (A zero-sphere is the set of two points a fixed distance from the origin in R¹. The set of topologically equivalent maps with fixed base point from such such a sphere into a manifold is simply the set of disconnected pieces into which the manifold falls.) Let's assume for simplicity that the vacuum manifold consists of just two disconnected components ₁ and ₂ and restrict ourselves to one spatial dimension. Then, if we apply boundary conditions that the vacuum at - lies in ₁ and that at + lies in ₂, by continuity there must be a point somewhere where the order parameter does not lie in the vacuum manifold as the field interpolates between the two vacua. The region in which the field is out of the vacuum is known as a domain wall.

Similarly, suppose ₂() 1. This implies that the vacuum manifold is not simply-connected: there are non-contractible loops in the manifold. The order parameter for such a situation may be considered complex and if, when traveling around a closed curve in space, the phase of the order parameter changes by a non-zero multiple of 2, by continuity there is a point within the curve where the field is out of the vacuum manifold. Continuity in the direction perpendicular to the plane of the curve implies that there exists a line of such points. This is an example of a cosmic string or line defect. Since we shall mostly concentrate on these defects let us now give a detailed analysis of the simplest example, the Nielsen-Olesen vortex in the Abelian Higgs model.

Consider a complex scalar field theory based on the Abelian gauge group U(1). The Lagrangian density for this model is

(136)

where (x) is the complex scalar field, the covariant derivative, D_µ, and field strength tensor F_µ are defined in terms of the Abelian gauge field A_µ as

	(137)
	(138)

and the symmetry breaking (or "Mexican hat") potential is

(139)

Here e is the gauge coupling constant and is a parameter that represents the scale of the symmetry breaking. In preparation for our discussion of the more general cosmological situation, let us note that at high temperatures the Lagrangian contains temperature-dependent corrections given by

(140)

where C is a constant. For T > T_c, where T_c is defined by

(141)

we see that

(142)

is minimized by <> = 0. However, for T < T_c, minimizing the above expression with respect to yields a minimum at

(143)

Thus, for T > T_c the full symmetry group G = U(1) is restored and the vacuum expectation value of the -field is zero. As the system cools, there is a phase transition at the critical temperature T_c and the vacuum symmetry is spontaneously broken, in this case entirely:

(144)

The vacuum manifold is

(145)

The group U(1) is topologically a circle, so

(146)

The set of topologically equivalent ways to map one circle to another circle is given by the integers. (We can wrap it any number of times in either sense.) The first homotopy group is therefore the integers. Calculating homotopy groups in general is difficult, but for S¹ all of the other groups are trivial. We therefore have

(147) (148) (149) (150)

The Abelian Higgs model therefore allows for cosmic strings.

To see how strings form in this model, consider what happens as T decreases through T_c. The symmetry breaks and the field acquires a VEV given by

(151)

where may be chosen differently in different regions of space, as implied by causality (we shall discuss this shortly). The requirement that <> be single valued implies that around any closed curve in space the change in must satisfy

(152)

If for a given loop we have n 0, then we can see that any 2-surface bounded by the loop must contain a singular point, for if not then we can continuously contract the loop to a point, implying that n = 0 which is a contradiction. At this singular point the phase, , is undefined and <> = 0. Further, <> must be zero all along an infinite or closed curve, since otherwise we can contract our loop without encountering a singularity. We identify this infinite or closed curve of false vacuum points as the core of our string. We shall restrict our attention to the case where n = 1, since this is the most likely configuration.

The first discussion of string solutions to the Abelian Higgs model is due to Nielsen and Olesen. Assume the string is straight and that the core is aligned with the z-axis. In cylindrical polar coordinates, (r, ), let us make the ansatz

(153)

The Lagrangian then yields the simplified equations of motion

(154)

where a prime denotes differentiation with respect to r. It is not possible to solve these equations analytically, but asymptotically we have

(155)

This solution corresponds to a string centered on the z-axis, with a central magnetic core of width ~ (e)^-1 carrying a total magnetic flux

(156)

The core region over which the Higgs fields are appreciably non-zero has width ~ (^1/2 )^-1. Note that these properties depend crucially on the fact that we started with a gauged symmetry. A spontaneously broken global symmetry with nontrivial _i() would still produce cosmic strings, but they would be much less localized, since there would be no gauge fields to cancel the scalar gradients at large distances.

Strings are characterized by their tension, which is the energy per unit length. For the Nielsen-Olesen solution just discussed, the tension is approximately

(157)

Thus, the energy of the string is set by the expectation value of the order parameter responsible for the symmetry breaking. This behavior is similarly characteristic of other kinds of topological defects. Again, global defects are quite different; the tension of a global string actually diverges, due to the slow fall-off of the energy density as we move away from the string core. It is often convenient to parameterize the tension by the dimensionless quantity Gµ, so that a Planck-scale string would have Gµ ~ 1.

From the above discussion we can see that cosmology provides us with a unique opportunity to explore the rich and complex structure of particle physics theories. Although the topological solutions discussed above exist in the theory at zero temperature, there is no mechanism within the theory to produce these objects. The topological structures contribute a set of zero measure in the phase space of possible solutions to the theory and hence the probability of production in particle processes is exponentially suppressed. What the cosmological evolution of the vacuum supplies is a concrete causal mechanism for producing these long lived exotic solutions.

At this point it is appropriate to discuss how many of these defects we expect to be produced at a cosmological phase transition. In the cosmological context, the mechanism for the production of defects is known as the Kibble mechanism. The guiding principle here is causality. As the phase transition takes place, the maximum causal distance imposed on the theory by cosmology is simply the Hubble distance - the distance which light can have traveled since the big bang.

As we remarked above, as the temperature of the universe falls well below the critical temperature of the phase transition, T_c, the expectation value of the order parameter takes on a definite value (~ eⁱ in the Abelian Higgs model) in each region of space. However, at temperatures around the critical temperature we expect that thermal fluctuations in <> will be large so that as the universe cools it will split into domains with different values of in different domains. This is the crucial role played by the cosmological evolution.

The Kibble mechanism provides us with an order of magnitude upper bound for the size of such a region as the causal horizon size at the time of the phase transition. The boundaries between the domains will be regions where the phase of <> changes smoothly. If the phase changes by 2n for some n 0 when traversing a loop in space, then any surface bounded by that loop intersects a cosmic string. These strings must be horizon-sized or closed loops. Numerical simulations of cosmic string formation indicate that the initial distribution of strings consists of 80% horizon-sized and 20% loops by mass. If we consider the temperature at which there is insufficient thermal energy to excite a correlation volume back into the unbroken state, the Ginsburg temperature, T_G, then, given the assumption of thermal equilibrium above the phase transition, a much improved estimate for the initial separation of the defects can be derived and is given by

where is the self coupling of the order parameter. This separation is microscopic.