TITLE:
An Exact CMB Photon Radiation Density Ω γ of the Universe Derived from R H t =ct Cosmology
AUTHORS:
Espen Gaarder Haug
KEYWORDS:
Photon Energy Density Parameter, Friedmann Equation, Critical Density, Number Density CMB Photons, Thermodynamical Friedmann Equation
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.1,
January
30,
2026
ABSTRACT: We will demonstrate that the photon energy density parameter
Ω
γ
of the universe can be derived exactly within the
R
H
t
=ct
cosmology. We find that it must be precisely:
Ω
γ
=
1
5760π
≈5.526×
10
−5
, which lies well within the 95% confidence interval for the photon radiation density reported by the Particle Data Group (PDG). This exact result implies that there is no uncertainty in the radiation density—at least within certain subclasses of the
R
H
t
=ct
cosmology—and that such a model is consistent with observations. Furthermore, in the standard model, the number density of CMB photons cannot be predicted from the Hubble parameter, but only from the CMB temperature. For the first time, we derive a new equation for the number density of CMB photons that allows prediction based solely on the Hubble parameter and the Planck length. This leads to a predicted number density of CMB photons equal to
n
γ
=410.71±0.26
photons per cm3, which is remarkably close to the value reported by the PDG when we use the
H
0
value recently proposed by Haug and Tatum,
H
0
=66.8943±0.0287
km/s/Mpc. However, when using the value
H
0
=73.30±1.04
km/s/Mpc predicted by Riess et al., the photon number density prediction falls far outside even the five-sigma confidence interval reported by the PDG. As we briefly discussed, this discrepancy is related to the Hubble tension observed in the ΛCDM model.