How constant is the Fermi coupling constant?

We discuss various astrophysical limits on the spatial and time variation of the Fermi coupling constant $G_F$. We consider two cases: (a) $G_F$ and the fermion masses vary through a change in the vacuum expectation value of the Higgs field; (b) $G_F$ varies while the fermion masses are held constant. In the former case, redshift measurements probe both the spatial and time variation of $G_F$ through changes in the electron mass: the agreement between measurements of hyperfine and optical lines in distant galaxies and quasars indicates that $G_F$ varies by less than 0.04% on cosmological length scales. Such measurements also show that $G_F$ varies by less than 0.2% back to a redshift of $z=3.4\mathrm{}$. If $G_F$ varies without any change in the fermion masses, the best constraints on spatial variations in $G_F$ come from supernova light curves, whose slopes depend upon the lifetime of ${}_{}{}^{56}{}_{}{}^{}\mathrm{Co}$. The similarities between light curves argue that the Fermi coupling constant $G_F$ varies by less than 5% on cosmological scales. Big bang nucleosynthesis indicates that the Fermi coupling constant at $t\sim 1$ sec differed by less than ∼ 10-20% from the contemporary terrestrial value, with the exact limits depending on which model we choose for the variation in $G_F$. Variation in $G_F$ would allow big bang nucleosynthesis to produce a lower ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ abundance without changing significantly any of the other element abundances.