Topological electroweak phase transition

In this work we shall consider the effects of a non-trivial topology on the effective potential of the Standard Model. Specifically we shall assume that the spacetime topology is $S^1\times R^3$ and we shall calculate the Standard Model effective potential for such a topological spacetime. As we demonstrate, for small values of the compact dimension radius, the electroweak symmetry is unbroken, but for a critical length and beyond, the electroweak symmetry is broken, since the configuration space of the Higgs field contains an additional energetically favorable minimum, compared to the minimum at the origin. The two minima are separated by a barrier, thus a phase transition can occur, via quantum tunneling, which mimics a first order phase transition. This is a non-thermal phase transition, similar possibly to quantum Hall topological phase transitions, hence in the context of this scenario, the electroweak symmetry breaking does not require a high temperature to occur. The present scenario does not rely on the occurrence of the inflationary era, only on the expansion of the Universe, however we briefly discuss the freezing of the superhorizon terms in $S^1\times R^3$ spacetime, if inflation occurred. We also investigate the large scale differences of the gravitational potential due to the non-trivial topology. Finally, we briefly mention the distinct inequivalent topological field configurations that can exist due to the non-trivial topology, which are classified by the first Stieffel class which in the case at hand is $H^1(S^1{\times R}^3,Z_{\tilde{2}})=Z_2$, so even and odd elements can exist. We also briefly qualitatively discuss how a topologically induced electroweak phase transition can yield primordial gravitational waves.