Exact determination of asymptotic CMB temperature-redshift relation

Abstract

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 ${\nu }_{\mathrm{CMB}}$ in the high-temperature ($T$) -- redshift ($z$) relation, $T/T_0={\nu }_{\mathrm{CMB}}(z+1)$, 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 ${\nu }_{\mathrm{CMB}}={\left(1/4\right)}^{1/3}=0.629960(5)$, representing a topological invariant. Interestingly, the relative deviation of ${\nu }_{\mathrm{CMB}}$ from the critical exponent associated with the correlation length $l$ of the 3D Ising model, ${\nu }_{\mathrm{Ising}}=0.629971(4)$, is less than $2\times 10^{-5}$. We are not yet in a position to establish a rigorous theoretical link between ${\nu }_{\mathrm{CMB}}$ and ${\nu }_{\mathrm{Ising}}$ as suggested by the topological nature of ${\nu }_{\mathrm{CMB}}$ 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.

v2: 4 pp, 1 fig, discussion added

References in corpus (2)

• Precision Islands in the Ising and $O(N)$ Models
• Relic photon temperature versus redshift and the cosmic neutrino background

Cited by in corpus (5)

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• An SU(2) gauge principle for the Cosmic Microwave Background: Perspectives on the Dark Sector of the Cosmological Model
• Frequency-redshift relation of the Cosmic Microwave Background
• Cosmological parameters from Planck data in SU(2)${}_{\mathrm{CMB}}$, their local $\Lambda$CDM values, and the modified photon Boltzmann equation
• SU(2) Quantum Yang-Mills Thermodynamics: Some Theory and Some Applications