Horizon-density monopole abundance bound (${\rho }_{M0}\propto {\eta }^4$)

For a radiation-era transition at $T_i\sim \eta$, use $H_i=1.66\sqrt{g_{\ast }}{\eta }^2/m_{\mathrm{pl}}$ and initial number density $n_i=C_n{H}_i^3$. Stable monopoles of mass $C_M\eta$ redshift as nonrelativistic matter. With comoving entropy conservation,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