On gauge dependence of gravitational waves from a first-order phase transition in classical scale-invariant $U(1{)}^{\mathrm{\prime }}$ models

We study gauge dependence of gravitational waves produced from a first-order phase transition in classical scale-invariant $U(1{)}^{\mathrm{\prime }}$ models. Accidental gauge independence of the one-loop effective potential in this class of models is spoiled by including thermal resummation. The gauge artifact in the resummed effective potential propagates to the gravitational wave spectrum and results in one order of magnitude uncertainties in the prediction under a specific gauge choice.