Electroweak and Left-Right Phase Transitions in $SO(5)\times U(1)\times SU(3)$ Gauge-Higgs Unification

The electroweak phase transition in GUT inspired $SO(5)\times U(1) \times SU(3)$gauge-Higgsunificationisshowntobeweaklyfirstorderandoccursat$T = T_c^{\rm EW} \sim 163,$GeV, which is very similar to the behavior in the standard model in perturbation theory. A new phase appears at higher temperatures. $SU(2{)}_L\times U(1{)}_Y$ (${\theta }_H=0$) and $SU(2)R \times U(1){Y’}$($\theta_H=\pi$)phasesbecomealmostdegenerateabove$T \sim m_{\rm KK}$where$m_{\rm KK}$ is the Kaluza-Klein mass scale typically around 13$$TeV and ${\theta }_H$ is the Aharonov-Bohm phase along the fifth dimension. The two phases become degenerate at $T=T_c^{\mathrm{L}\mathrm{R}}\sim m_{\mathrm{K}\mathrm{K}}$. As the temperature drops in the evolution of the early Universe the $SU(2)R \times U(1){Y’}$phasebecomesunstable\mathrm{.}Thetunnelingratefromthe$SU(2)R \times U(1){Y’}$phasetothe$SU(2)_L \times U(1)_Y$ phase becomes sizable and a first-order phase transition takes place at $T=2.5\sim 2.6$TeV. The amount of gravitational waves produced in this left-right phase transition is small, far below the reach of the sensitivity of Laser Interferometer Space Antenna (LISA). A detailed analysis of the $SU(2{)}_R\times U(1{)}_{Y^{\mathrm{\prime }}}$ phase is also given. It is shown that the $W$ boson, $Z$ boson and photon, with ${\theta }_H$ varying from 0 to $\pi$, are transformed to gauge bosons in the $SU(2)R \times U(1){Y’}$ phase. Gauge couplings and wave functions of quarks, leptons and dark fermions in the $SU(2{)}_R\times U(1{)}_{Y^{\mathrm{\prime }}}$ phase are determined.