Observing evolution from steady state

Abstract

The time-translation symmetry of the conformal FLRW frame $\bar{g}=a^{-2}g$ allows reinterpretation of cosmological observation in the static space of a stationary universe, where constant matter density $\bar{\rho}_{\textrm{m}}=\rho_{\textrm{m0}}$ induces constant curvature $R_{0}^{-2}$. A hyperbolic de Sitter solution arises from equipartition of the kinetic energy of recessional and peculiar components of the gravitational field, corresponding to a total density $24R_{0}^{-2}$ of twice the scalar curvature. This predicts a matter density $\Omega_{\textrm{m}}=\frac{1}{24}$, or a Hubble constant $h=\sqrt{24\rho_{\textrm{m0}}}\approx0.72$, in agreement with distance-ladder estimates. Projecting the equilibrium state onto the $\Lambda\textrm{CDM}$ model returns $\hat{h}=\frac{4}{3}\sqrt{12\rho_{\textrm{m0}}}\approx0.68$ and exact densities $\hat{\Omega}_{\textrm{m}}=1-\hat{\Omega}_{\Lambda}=[\textrm{sinh}(\frac{4}{3}\textrm{asinh}(1))^{2}+1]^{-1}=0.3179...$, within confidence limits of Planck 2018 results.