The masses of weakly-coupled scalar fields in the early Universe

We consider the effective mass-squared in the early Universe, of a scalar field which has only Planck-suppressed couplings with light fields and whose true mass is less than the Hubble parameter $H$. A detailed investigation shows that the effective mass-squared generically is of order $\pm H^2$ during inflation and matter-domination, but much smaller during radiation domination. We consider the special circumstances under which the mass-squared may be much bigger or much smaller than the generic value.