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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 cosmic string network averages the directional stress and energy density of many long cosmic strings. For an ideal smooth string whose tension equals its rest energy per unit length, a network frozen into the expansion has negative averaged pressure,A comoving string segment then has physical length proportional to $a$, while its enclosing volume grows as $a^3$, giving $\rho \propto a^{-2}$. This is the nonmoving, noninteracting limit, not the general evolution law of a mobile network.

In units $c=1$, let $v^2$ be the energy-weighted mean squared transverse string velocity. For an isotropic smooth-string network the averaged equation of state isThe kinetic contribution raises the averaged pressure; for example $v^2=1/2$ gives $w=0$. Motion, reconnection and production of smaller loops therefore matter even when the initial network is statistically isotropic.

The corresponding long-string balance, neglecting friction against matter, can be writtenwhere $\mathcal{Q}\ge 0$ is energy removed from the long-string population into loops or other radiation channels. For $\mathcal{Q}=0$ this agrees with $\dot{\rho }+3H(\rho +P)=0$ and proves the equation-of-state relation. If $v$ is constant and losses vanish, the solution is $\rho \propto a^{-2(1+v^2)}$. The primary velocity-dependent model defines $\rho =\mu /L^2$ and evolves both its characteristic length $L$ and its velocity.^[ref]

A horizon-scaling cosmic string network is consequently distinct from a frozen fluid with fixed $w=-1/3$. The ideal frozen equation of state can be used in a specified cosmological toy model, but it cannot be inferred from isotropy alone.

The Kibble mechanism explains how a spontaneous symmetry breaking transition can form topological defects. Regions that order without coordinating their choices can settle at different points of a vacuum manifold. Smooth relaxation removes many mismatches, but some cannot be removed while the field remains in that vacuum manifold everywhere.^[ref]

The possible defects depend on the homotopy groups of the vacuum manifold: disconnected components permit domain walls, noncontractible loops permit cosmic strings, and nontrivial surrounding two-spheres permit magnetic monopoles in three spatial dimensions. This is the homotopy classification of cosmological defects, rather than a prediction that every permitted defect must occur.^[ref]

For a direct winding example, an order parameter represented by a complex scalar field with fixed nonzero magnitude has a circular vacuum manifold. If its order-parameter phase changes by $2\pi$ around a closed spatial loop, it cannot extend continuously across a spanning disk while keeping its magnitude nonzero. A zero of the order parameter supplies the core; in three dimensions the corresponding defect can be a string.

A formation estimate needs a domain correlation length $\xi$ and a model for the chance of trapping a defect. For point defects, a dimensional estimate is $n_{\mathrm{defect}}\sim p_{\mathrm{defect}}/{\xi }^3$, with dimensionless formation efficiency $p_{\mathrm{defect}}$. Causality restricts how ordering can spread; taking a horizon-sized $\xi$ supplies a scale estimate, not by itself a guaranteed number of defects in every horizon volume. Subsequent annihilation or expansion can further change the abundance. Kibble's original discussion separately estimates the initial domain scale from fluctuations and follows later evolution.^[ref]

Outside the core, $f^{\prime \prime }=0$ and the metric is locally flat: set $\tilde{\varphi }=(1-\lambda )\varphi$. Static test particles have no Newtonian-like attraction because $g_{00}$ is constant and ${\Gamma }^r{}_{00}=0$. Local trajectories are straight lines in locally unwrapped coordinates, so a very thin dark core is hard to identify through ordinary orbital accelerations or local tidal forces.

The global angular range is nevertheless $2\pi (1-\lambda )$ rather than $2\pi$. The missing angle $\delta =2\pi \lambda$ can give gravitational lensing with two nearly identical, essentially undistorted images of a background galaxy, and multiple such pairs aligned along the projected object. For a small deficit, a straight string and favourable geometry, their angular separation is of order $\delta D_{ls}/D_s$, with an additional inclination factor for a tilted string. The matching spectra and common line of image pairs distinguish this from an ordinary compact lens.

A moving cosmic string can also induce a step in the cosmic microwave background temperature across its path. That signature needs motion in addition to the static geometry supplied here. These are consequences of the exterior conical geometry; interpreting the inferred deficit as a positive mass per length requires the regular-axis/core assumptions distinguished in part b.

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.

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.

Tom Kibble related spontaneous symmetry breaking in the early universe to the formation of topological defects. Different regions can choose different states on a vacuum manifold. Its homotopy groups determine whether boundaries, loops or surrounding surfaces can be continuously unwound, giving possible domain walls, cosmic strings or monopoles. His 1976 paper also examines their subsequent cosmological effects. The connection makes the global shape of a space of field states relevant to large-scale physical structures.^[ref]