Origin of cosmological temperature

Abstract

A classical solution of the Standard Model + General Relativity is given by an elliptic function whose periodicity in imaginary time is the origin of cosmological temperature. Nothing beyond the Standard Model is assumed. The solution is a $\mathrm{Spin}(4)$-symmetric universe expanding prior to the electroweak transition. A rapidly oscillating $\mathrm{SU}(2)$ gauge field holds the Higgs field to $0$ with strength inversely proportional to the scale factor $a$. When $a$ reaches $a_{\scriptscriptstyle\mathrm{EW}}$ the solution becomes unstable and the electoweak transition begins. $a_{\scriptscriptstyle\mathrm{EW}}$ is the only free parameter in the solution. The temperature at $a_{\scriptscriptstyle\mathrm{EW}}$ is $m_{H}/{(6\pi)^{1/2}} = 28.8\,\text{GeV} = 3.34 \times 10^{14}\, \text{K}$ whatever the value of $a_{\scriptscriptstyle\mathrm{EW}}$.