Stochastic evolution of scalar fields with continuous symmetries during inflation

During inflation, scalar fields with masses less than the Hubble scale acquire vacuum expectation values (VEVs) via stochastic processes driven by quantum fluctuations. For nearly massless spectator scalars transforming nontrivially under a continuous symmetry group, we demonstrate that the evolution of the VEV depends on the dimensionality of the scalar field space. Fields in larger representations both attain larger vacuum expectation values and converge more rapidly to equilibrium. We present an argument demonstrating how this higher-dimensional evolution can be obtained in unitary gauge for fields transforming under local symmetries with a mass gap that is small compared to the Hubble scale. Finally, we show that accounting for the full number of degrees of freedom in the Standard Model Higgs multiplet tightens Higgs stability constraints on the inflationary scale at the percent level and has more dramatic consequences for both the VEV and the energy stored in the Higgs field after inflation.