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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.^[ref]

The relevant homotopy classification of cosmological defects uses the vacuum map on a surface surrounding a proposed defect: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. 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.

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 areDisconnected 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.^[ref]

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 ${\rho }_{\mathrm{string}}\sim \mu /t^2$ 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 ${\rho }_{\mathrm{wall}}\sim \sigma /t$, 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. 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.