A geometric-mean CMB temperature law in an FRW RH = ct cosmology with modified redshift and comparison with Melia’s Rh = ct universe

We examine whether a recently suggested geometric-mean expression, \[ T_{\rm cmb}=\sqrt{T_{\max}T_{\min}} = \frac{\hbar c}{4\pi k_B\sqrt{2R_Hl_p}}, \] for the cosmic microwave background temperature can be incorporated into a Friedmann--Robertson--Walker cosmology satisfying $R_H=ct$. In Haug and Tatum's formulation, the CMB temperature is related to the geometric mean of two Hawking temperatures: a minimum temperature associated with the Hubble radius and a maximum temperature associated with a Planck-mass Schwarzschild black hole. Related work by Haug and Wojnow has demonstrated that the same temperature expression can be obtained from the Stefan--Boltzmann law, while Tatum, Seshavatharam and Lakshminarayana earlier arrived at a closely related expression by a more heuristic route. We also discuss Haug's recent derivation of an exact photon energy-density parameter and an associated CMB photon number-density formula in an $R_H=ct$ framework, as well as Haug and Tatum's reported fit to the PantheonPlusSH0ES supernova database using a modified redshift relation. We show that if the standard FRW metric redshift is used, the geometric-mean temperature scales as $T_{\rm cmb}\propto (1+z_{\rm FRW})^{1/2}$. However, if one adopts the modified redshift definition $1+z_{\rm H}=(R_{H0}/R_H)^{1/2}$, then the same temperature law gives the linear relation $T_{\rm cmb}=T_0(1+z_{\rm H})$. The resulting CMB energy-density fraction is constant and equal to $1/(5760\pi)$, while the blackbody photon number density may be written directly in terms of $H$ and $l_p$.