= Solution
A <topological defect> can remain after <spontaneous symmetry breaking> when the field cannot be continuously put into the vacuum everywhere. If the <vacuum manifold> is $\mathcal M=G/H$, its <topology> classifies possible obstructions. Causally separated regions can settle into different local order-parameter configurations; joining them during a <phase transition> can trap defects. This is a formation mechanism, not an assertion that every transition creates every possible defect.
In three spatial dimensions the <homotopy classification of cosmological defects> gives are
$$
\begin{array}{c|c|c}
\text{vacuum topology}&\text{configuration}&\text{spatial character}\\ \hline
\pi_0(\mathcal M)\ne0&\text{domain walls}&\text{surfaces}\\
\pi_1(\mathcal M)\ne0&\text{strings}&\text{lines}\\
\pi_2(\mathcal M)\ne0&\text{monopoles}&\text{points}\\
\pi_3(\mathcal M)\ne0&\text{textures}&\text{extended winding}
\end{array}
$$
Disconnected vacua can meet across a wall; winding around a line can obstruct filling its core; a sphere around a point can carry monopole charge. Textures differ from permanent localized cores: they can collapse and unwind when the field leaves the <vacuum manifold>. Local gauge and global symmetry breaking also differ in their long-range fields and <radiation>. The original cosmological analysis connects vacuum <topology> with domain and string formation and their subsequent evolution.<Topology of cosmic domains and strings by Thomas Walter Bannerman Kibble>[J. Phys. A 9, pp. 1387–1398 (1976), vacuum topology and domain/string evolution]{ref}
\Image[/past-exam-of-the-particle-astrophysics-course-of-the-university-of-cambridge/2016/paper-1-defect-topology.png]
{title=Disconnected, circular and spherical <vacuum manifolds> support different defects}
{description=Original schematic vacuum spaces, not pictures of astronomical objects. The main text specifies the physical-space loops or spheres whose mappings classify walls, strings and monopoles.}
The <energy> budget determines cosmological importance. Nonrelativistic monopoles dilute approximately as $a^{-3}$, so a large primordial abundance can become unacceptable as <radiation> falls faster. Inflation can dilute an earlier population, provided the same defect-producing transition is not repeated afterwards. A slowly evolving wall network dilutes especially slowly and can dominate the <energy> budget or create excessive anisotropy.
Strings can lose <energy> by intercommutation and loop formation, with loops radiating <gravitational waves> or field quanta. A <horizon-scaling cosmic string network> whose characteristic separation is of order the horizon can have density $\rho_{\rm string}\sim\mu/t^2$ in natural units, keeping a limited fraction of the total rather than accumulating indefinitely. This differs from an ideal frozen noninteracting network's simple $a^{-2}$ dilution law. Similarly, a <horizon-scaling domain wall network> has $\rho_{\rm wall}\sim\sigma/t$, while a frozen wall network follows $a^{-1}$. The evolutionary assumptions must accompany any quoted scaling.
Defects can seed density wakes and other inhomogeneities, perturb the microwave background and generate non-Gaussian signatures. A sufficiently straight ideal <cosmic string> has conical spacetime and can create doubled background images; a cosmic-string network can generate gravitational-wave bursts and a stochastic background. Textures can produce localized microwave-background <temperature> features while collapsing. These signatures constrain defect abundance and symmetry-breaking scale; an isolated suggestive feature is not a unique defect detection.
The predictions depend on network evolution, reconnection, loop sizes, <radiation> channels and the relation between symmetry breaking and inflation. \b[<topology> identifies which configurations are possible; formation and evolution determine whether they survive and whether their gravitational or radiative effects are observable.] A defect's unseen <energy> does not automatically make it the dominant <cold dark matter>.
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