Exact determination of asymptotic CMB temperature-redshift relation - Hahn, Steffen; Hofmann, Ralf (Modern physics letters, 2018)
Based on energy conservation in a Friedmann–Lemaître–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 νCMB in the high-temperature (T) — redshift (z) relation, T/T0=νCMB(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 νCMB=(1/4)1/3=0.629960(5), representing a topological invariant. Interestingly, the relative deviation of νCMB from the critical exponent associated with the correlation length l of the 3D Ising model, νIsing=0.629971(4), is less than 2×10−5. We are not in a position to establish a direct theoretical link between νCMB and νIsing as suggested by the topological nature of νCMB and the fact that both theories are members of the same universality class. We do, however, spell out a somewhat speculative, strictly monotonic map from the physical Ising temperature θ to a fictitious SU(2) Yang–Mills temperature T, the latter continuing the asymptotic dependence of the scale factor a on T/T0 for T/T0≫1 down to T=0, and we identify an exponential map from a to l to reproduce critical Ising behavior.