Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 70 / 3

Choose a vacuum order parameter ${\Phi }_0$ and let $H$ be its stabilizer in $G$. Acting with $G$ produces all symmetry-related minima; two elements produce the same minimum precisely when they differ by an element of $H$. The vacuum manifold is thereforeThis assumes a single transitive orbit of minima; additional independent vacuum sectors would require more than one orbit. For a local gauge symmetry, $G/H$ describes the vacuum directions used in finite-energy boundary conditions and defect classification, rather than distinct observable vacua after quotienting by every gauge redundancy.

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.

The Abelian Higgs model has local $G=U(1)$, and a nonzero minimally charged scalar expectation value has trivial stabilizer $H=\{1\}$. Thus $M\simeq U(1)\simeq S^1$ and ${\pi }_1(M)=\mathbb{Z}$, while ${\pi }_0$ and ${\pi }_2$ are trivial. The defects are local strings, or Nielsen-Olesen vortices, whose winding fixes flux $2\pi n/e$.

Put $R=\eta +\varphi /\sqrt{2}$ and $\vartheta =\psi /(\eta \sqrt{2})$. The specified gauge-field shift is $A=A^{\prime }+\partial \vartheta /e$, soExpanding the potential gives $V=\lambda {\eta }^2{\varphi }^2/2+O({\varphi }^3)$. Hence the broken-phase spectrum of the Abelian Higgs model follows fromThe phase is not an additional physical massless scalar: the Higgs mechanism supplies the vector's longitudinal polarization. In the symmetric phase the complex scalar has two real degrees of freedom and the massless vector has two polarizations, a total of four. In the broken phase there are one scalar and three massive-vector polarizations, again four. The symmetric phase is the thermally restored phase; the origin of the given zero-temperature potential is unstable, not a separate stable particle vacuum. The local polar change of variables cannot remove global vortex winding everywhere, so the massive spectrum is consistent with string defects.

The SU(5) grand unified theory has simply connected space $G=SU(5)$ and a connected Standard Model subgroup. More precisely that subgroup is $(SU(3)\times SU(2)\times U(1))/{\mathbb{Z}}_6$; the product notation suppresses a finite central quotient. The long exact sequence of homotopy groups of a fibration $H\to G\to G/H$ givesThus stable magnetic monopoles form, whereas this transition does not require walls or strings. The monopole core and mass scales in an SU(5) transition are set by the massive gauge and scalar fields:For order-one couplings, the core size is of order ${\eta }^{-1}$ and the mass is of order $\eta$ up to the often substantial factor $4\pi /g$. These heavy objects are nonrelativistic soon after formation and can survive as stable relics.

Take the transition temperature $T_i\sim \eta$ and radiation-dominated universe. With $g_{\ast }\simeq 100$ relativistic degrees of freedom, ${\rho }_{r,i}=({\pi }^2/30)g_{\ast }{T}_i^4$ and the Friedmann equation givesA saturated causal correlation length is of order the radiation-era horizon $H_i^{-1}=2t_i$. Write $C_n\sim 1$ for the geometric/formation efficiency and $C_M=M_M/\eta$. One monopole per horizon volume gives the horizon-density monopole abundance bound estimateThe first quantity is number density and the second is defect energy density. Relative to the initial radiation critical density,This can initially be small even when the surviving monopole abundance is unacceptable today. Causality supplies an order-of-magnitude lower abundance estimate under the stipulated saturation; it is not a precise statistical count of defects.

Without substantial annihilation or decay, monopole number in a comoving volume is conserved. Hence $n_M\propto a^{-3}$ and, for constant nonrelativistic mass, ${\rho }_M\propto a^{-3}$. Radiation redshifts as $a^{-4}$, so the monopole-to-radiation ratio grows with $a$.

Use comoving entropy conservation rather than assuming the same relativistic content at every epoch:With $g_{\ast s,i}\simeq 100$, $g_{\ast s,0}\simeq 3.91$ and $T_0\simeq 2.35\times 10^{-13}\mathrm{GeV}$, the surviving energy density isTaking $m_{\mathrm{pl}}\simeq 1.22\times 10^{19}\mathrm{GeV}$ and $H_0\simeq 70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ gives ${\rho }_{\mathrm{crit},0}\simeq 4.0\times 10^{-47}{\mathrm{GeV}}^4$. Requiring ${\rho }_{M0}<0.01{\rho }_{\mathrm{crit},0}$ yieldsUnknown geometric and mass prefactors change this estimate only through a fourth root. Conventional grand-unification scales near $10^{15}$–$10^{16}\mathrm{GeV}$ are far above the bound: this is the cosmological monopole problem, not a universal prohibition on unification. Inflation after monopole formation can dilute the relics; reheating must then stay below their regeneration temperature. Later entropy production, efficient annihilation, or a symmetry-breaking history avoiding stable monopoles can also weaken the bound. These mechanisms change the assumed conserved initial relic abundance.