How constant is the Fermi coupling constant?

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