Exact determination of asymptotic CMB temperature-redshift relation

Abstract

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 [Formula: see text] in the high-temperature [Formula: see text] — redshift [Formula: see text] relation, [Formula: see text], 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 [Formula: see text], representing a topological invariant. Interestingly, the relative deviation of [Formula: see text] from the critical exponent associated with the correlation length [Formula: see text] of the 3D Ising model, [Formula: see text], is less than [Formula: see text]. We are not in a position to establish a direct theoretical link between [Formula: see text] and [Formula: see text] as suggested by the topological nature of [Formula: see text] 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 [Formula: see text] to a fictitious SU(2) Yang–Mills temperature [Formula: see text], the latter continuing the asymptotic dependence of the scale factor [Formula: see text] on [Formula: see text] for [Formula: see text] down to [Formula: see text], and we identify an exponential map from [Formula: see text] to [Formula: see text] to reproduce critical Ising behavior.