Big bang nucleosynthesis and cosmic microwave background constraints on the time variation of the Higgs vacuum expectation value

We derive constraints on the time variation of the Higgs vacuum expectation value $〈\phi 〉$ through the effects on big bang nucleosynthesis (BBN) and the cosmic microwave background (CMB). In the former case, we include the (previously neglected) effect of the change in the deuteron binding energy, which alters both the ${}^4\mathrm{He}$ and deuterium abundances significantly. We find that the current BBN limits on the relative change in $〈\phi 〉$ are $-(0.6\mathrm{}-0.7)\times {10}^{-2}<\Delta 〈\phi 〉/〈\phi 〉<(1.5\mathrm{}-2.0)\times {10}^{-2},$ where the exact limits depend on the model we choose for the dependence of the deuteron binding energy on $〈\phi 〉.$ The limits from the current CMB data are much weaker.