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A cosmic string is a hypothetical one-dimensional topological defect formed by symmetry breaking in the early universe. A cosmic string network consists of many such defects and requires additional assumptions to infer an averaged equation of state.

An isotropic network of long cosmic strings has average equation-of-state parameter $w=-1/3$ and energy density proportional to $a^{-2}$.

The equation of state averages the stress of a cosmic string over an isotropic network; it is not the definition of the individual defect.

A spontaneous symmetry breaking chooses order-parameter directions independently in regions that have not been in causal contact. Mismatches around disconnected components, loops or enclosing spheres can trap domain walls, cosmic strings or magnetic monopoles. The correlation length cannot exceed the causal horizon, so one monopole per horizon volume estimates the smallest causal initial abundance up to geometrical and efficiency factors.

// Target: classical-field-theory-soliton.bigb

Finite-energy configurations approach the vacuum manifold outside defect cores. The surrounding boundary in the transverse directions is a sphere, and noncontractible boundary data prevent a smooth deformation to the uniform vacuum.

Nontrivial ${\pi }_0(M)$ means disconnected vacuum components. Choosing different components on the two sides of an interface produces domain walls. Here ${\pi }_0$ labels components and need not itself carry a group structure. Nontrivial ${\pi }_1(M)$ means noncontractible loops: a circle surrounding a line can wind around the vacuum manifold, producing cosmic strings. Nontrivial ${\pi }_2(M)$ means noncontractible maps from an enclosing two-sphere, producing pointlike magnetic monopoles in three spatial dimensions.During the transition, initially uncorrelated vacuum choices in causally separated regions can generate these boundary classes through the Kibble mechanism. Nontrivial topology allows stable defects; it does not specify their exact formation efficiency or abundance.