Exact determination of asymptotic CMB temperature-redshift relation

Based on energy conservation in a Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe, on the Legendre transformation between energy density and pressure, and on nonperturbative asymptotic freedom at high temperatures we derive the coefficient $\nu_{\rm CMB}$ in the high-temperature ($T$) -- redshift ($z$) relation, $T/T_0=\nu_{\rm CMB}(z+1)$, of the Cosmic Microwave Background (CMB). Theoretically, our calculation relies on a deconfining SU(2) rather than a U(1) photon gas. We prove that $\nu_{\rm CMB}=\left(1/4\right)^{1/3}=0.629960(5)$, representing a topological invariant. Interestingly, the relative deviation of $\nu_{\rm CMB}$ from the critical exponent associated with the correlation length $l$ of the 3D Ising model, $\nu_{\rm Ising}=0.629971(4)$, is less than $2\times 10^{-5}$. We are not yet in a position to establish a rigorous theoretical link between $\nu_{\rm CMB}$ and $\nu_{\rm Ising}$ as suggested by the topological nature of $\nu_{\rm CMB}$ and the fact that both theories share a universality class. We do, however, line out a somewhat speculative map from the physical Ising temperature $\theta$ to a fictitious SU(2) Yang-Mills temperature $T$, the latter continuing the asymptotic behavior of the scale factor $a$ on $T/T_0$ for $T/T_0\gg 1$ down to $T=0$, and an exponential map from $a$ to $l$ to reproduce critical Ising behavior.