The Nonlocal Temperature of Nonlocal Energy

Maximal symmetry of the conformal FLRW frame \( \bar{g}=a(t)^{-2}g \) only admits constant curvature solutions, where constant matter density \( \bar{\rho}_{\textrm{m}}=R_{0}^{-2} \) constrains cosmology to a hyperbolic de Sitter solution \( a(t)=\textrm{sinh}(2t/R_{0})^{\frac{1}{2}} \), including reparametrizations like the \( \Lambda\textrm{CDM} \) solution \( \hat{a}=a^{4/3} \). Equipartition of the nonlocal gravitational field energy at the horizon shows equilibrium of Ricci (matter) and Weyl(BAO) densities of \( 12R_{0}^{-2} \) each, or \( \bar{H}_{0}=\sqrt{24\bar{\rho}_{\textrm{m}}}\approx72.1\;\textrm{km/s/Mpc} \). The BAO half predicts a concordance \( \hat{H}_{0}=\frac{4}{3}\sqrt{12\bar{\rho}_{\textrm{m}}}\approx68.0\;\textrm{km/s/Mpc} \). Adaptation of the Stefan-Boltzmann law to symmetries and time-dilation at the horizon in \( \bar{g} \) relates the nonlocal field energy associated with \( \bar{\rho}_{\textrm{m}} \) to a constant CMB temperature \( \bar{T}_{\textrm{CMB}}=2.725\pm0.009\:\textrm{K} \), within FIRAS confidence limits. It also predicts a baryon/photon density ratio \( \frac{12^{4}}{23}-1=900.56.. \), within Planck 2018 confidence limits. The existence of \( \bar{g} \) by itself seems to make a strong case for a stationary universe, where one expects to find mature galaxies at high redshifts.