= Solution
When a cosmological <phase transition> breaks a <symmetry> $G$ to a <subgroup> $H$, the degenerate vacuum configurations form a <vacuum manifold> $\mathcal M=G/H$. Causally separated regions need not choose the same vacuum orientation. The <Kibble mechanism> explains how joining those choices can leave a field configuration that cannot be continuously deformed to a uniform vacuum without leaving $\mathcal M$. The field then stores <energy> in a defect core, gradients or <gauge fields>.<Topology of cosmic domains and strings by Thomas Walter Bannerman Kibble>[Sections on vacuum topology, domains and strings]{ref}
The relevant <homotopy classification of cosmological defects> uses the vacuum map on a surface surrounding a proposed defect:
$$
\begin{array}{c|c|c}
\text{nontrivial topology}&\text{defect in three spatial dimensions}&\text{example}\\
\hline
\pi_0(\mathcal M)&\text{domain wall}&\mathcal M=\{+v,-v\}\\
\pi_1(\mathcal M)&\text{string}&\mathcal M=S^1\\
\pi_2(\mathcal M)&\text{monopole}&\mathcal M=S^2
\end{array}
$$
A <domain wall> separates different disconnected vacua. A <cosmic string> has nonzero <winding number> on a loop around its core. A <magnetic monopole> has a nontrivial vacuum map on an enclosing sphere. Nontrivial $\pi_3(\mathcal M)$ can instead support <cosmological texture> configurations extending over a volume; <cosmological textures> can unwind dynamically by leaving the <vacuum manifold> and need not possess a permanently localized stable core. Nontrivial topology permits a defect; formation efficiency, interactions and the cosmological history still determine its abundance.
For example, a complex scalar with a circular set of minima can wind by $2\pi n$ around a <cosmic string>. Continuity forces the field away from its vacuum value near the core. A local <gauge symmetry> allows <gauge fields> to screen much of the exterior gradient and trap flux, whereas a global <symmetry> leaves long-range Goldstone-field gradients. Thus local and global <cosmic strings> have different energy-loss channels; they should not be treated as identical networks.
Relic defects can be tightly constrained because their <energy> dilutes differently from <radiation>. Nonrelativistic <magnetic monopoles> approximately dilute as $a^{-3}$, so excessive production causes the <cosmological monopole problem>. A roughly frozen wall network has <energy density> proportional to $a^{-1}$, even slower dilution. An evolving <cosmic string network> can approach a scaling regime with characteristic separation proportional to <cosmic time> and density of order $\mu/t^2$ in $c=1$ units, losing <energy> into loops and <radiation>. That is a dynamical network statement, not the dilution law of a fixed collection of <cosmic strings>.
<cosmic inflation> can dilute defects made before it; <reheating> and later <symmetry> breaking must not regenerate an unacceptable abundance. Observational tests include microwave-background anisotropy, <gravitational lensing> by <cosmic strings>, <gravitational radiation> from string loops, and searches for <magnetic monopoles>. Each is sensitive to model-dependent tensions, couplings and evolution. \b[Defects are possible relics of <symmetry> breaking, not an inevitable dominant dark-matter population.] Their absence constrains a transition and its subsequent history rather than every theory possessing a nontrivial vacuum topology.
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