Cosmic Entities from the Geometric Mean of Maximum and Minimum Unruh Temperatures

We develop the phenomenological proposal that the present cosmic microwave background (CMB) temperature is the geometric mean of minimum and maximum Unruh temperatures associated, respectively, with the Hubble and Planck scales. The construction assumes from the outset a linearly related Hubble scale, $\RH=ct=c/H$, and the particular limiting accelerations $a_{\min}=c^2/(2\RH)$ and $a_{\max}=c^2/(4\lp)$. The observed monopole temperature $\Tcmb=\SI{2.72548(57)}{K}$ then implies $H_0=\SI{66.8944(280)}{km.s^{-1}.Mpc^{-1}}$. Standard blackbody thermodynamics supplies the corresponding CMB photon number density, energy density, two conventional spectral peak wavelengths, photon-number-weighted mean wavelength, and mean photon energy. Direct formulas in the geometric-mean temperature supply the Hubble parameter, time, radius, critical Friedmann density, and mass; $H_0$ is an output rather than an independent input. All numerical results are collected in two tables. The algebra is exact given the stated postulates, but the postulates are model assumptions; the calculation is therefore a consistency construction rather than evidence that the observed CMB is physically Unruh radiation.