Bayesian Estimation of the D(p,$\gamma$)$^3$He Thermonuclear Reaction
  Rate
  Big bang nucleosynthesis (BBN) is the standard model theory for the
production of the light nuclides during the early stages of the universe,
taking place for a period of about 20 minutes after the big bang. Deuterium
production, in particular, is highly sensitive to the primordial baryon density
and the number of neutrino species, and its abundance serves as a sensitive
test for the conditions in the early universe. The comparison of observed
deuterium abundances with predicted ones requires reliable knowledge of the
relevant thermonuclear reaction rates, and their corresponding uncertainties.
Recent observations reported the primordial deuterium abundance with percent
accuracy, but some theoretical predictions based on BBN are at tension with the
measured values because of uncertainties in the cross section of the
deuterium-burning reactions. In this work, we analyze the S-factor of the
D(p,$\gamma$)$^3$He reaction using a hierarchical Bayesian model. We take into
account the results of eleven experiments, spanning the period of 1955--2021;
more than any other study. We also present results for two different fitting
functions, a two-parameter function based on microscopic nuclear theory and a
four-parameter polynomial. Our recommended reaction rates have a 2.2\%
uncertainty at $0.8$~GK, which is the temperature most important for deuterium
BBN. Differences between our rates and previous results are discussed.
