A Poisson-Shot Interpretation of the CMB Temperature in R_H = ct Cosmology

A recently proposed CMB temperature relation, obtained from applying the Stefan Boltzmann law to the Hubble sphere and from related Hawking–Planck–Hubble scale arguments, may be written in the compact form T_CMB(t) = T_P/(8π √(N_P(t))). Here N_P is the effective Poisson-shot count. In an R_H = ct cosmology, the normalization consistent with the Stefan–Boltzmann radiation density is N_P(t) = (R_H(t))/2l_P = t/2t_P = (M_c(t))/m_P, where M_c(t) = c²R_H(t)/(2G) is the critical Hubble mass. If instead one defines the doubled Hubble-sphere mass M_u(t) = c²R_H(t)/G, then M_u/m_P = 2N_P. The formula has the mathematical structure of a Poisson relative-fluctuation law, since σ_N/N = 1/√N for a Poisson count, and may equivalently be written T_CMB(t) = T_P/8π σ_NP/N_P.We call this the Poisson-shot CMB formula. Substitution back into the Stefan–Boltzmann law gives u_γ ∝ T_CMB⁴ ∝ R_H^-2, matching the critical-density scaling in R_H = ct and yielding a constant photon radiation density parameter. This provides additional blackbody support for the formula and connects it to the observed near-perfect blackbody spectrum of the CMB. By contrast, in the standard ΛCDM framework the present CMB temperature is normally an observational input: the model predicts the redshift scaling T(z) = T₀(1 + z) once T₀ is supplied, but it does not derive the absolute present value T₀ from the Planck scale and the Hubble scale.