Predicting the CMB temperature.

Big-bang nucleosynthesis (BBN), a pillar of modern cosmology, begins with the trailblazing 1948 paper of Alpher, Bethe and Gamow \cite{abc} which proposed nucleosynthesis during an early ($t \sim 1 - 1000\,$sec), radiation-dominated phase of the Universe to explain the abundances of the chemical elements. While their model was flawed, they called attention to the importance of a hot beginning and the radiation now known as the cosmic microwave background (CMB). In subsequent papers they made estimates for the temperature of the CMB, from 5\,K to 50\,K, all based upon wrong physics. To illustrate the prediction {\it that could have been made} and to elucidate the key physics underpinning BBN, I show that in the absence of a detailed model of BBN and the desired abundances, at best one could have estimated an {\it upper} limit to the CMB temperature of between $10\,$K and $60\,$K, predicated upon the assumption of some nucleosynthesis.