= Horizon-density monopole abundance bound
{title2=$\rho_{M0}\propto\eta^4$}
For a radiation-era transition at $T_i\sim\eta$, use $H_i=1.66\sqrt{g_*}\eta^2/m_{\rm pl}$ and initial number density $n_i=C_nH_i^3$. Stable monopoles of mass $C_M\eta$ redshift as nonrelativistic matter. With <comoving entropy conservation>,
$$
\rho_{M0}\sim C_nC_M(1.66)^3g_*^{3/2}\frac{g_{*s,0}}{g_{*s,i}}\frac{\eta^4T_0^3}{m_{\rm pl}^3}.
$$
The fourth-power scale dependence makes even a small causal initial abundance incompatible with conventional high-scale grand unification unless inflation, later <entropy> production or efficient destruction reduces the relic population.
// Target: cosmology.bigb
Back to article page
