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

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.