Past exam of the particle Astrophysics course of the University of Cambridge / 2015 / Paper 1 / 3 / a / 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.^[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.