Implication of the Proton-Deuteron Radiative Capture for Big Bang Nucleosynthesis

The astrophysical $S$ factor for the radiative capture $d(p,\gamma ){}^3\mathrm{He}$ in the energy range of interest for big bang nucleosynthesis (BBN) is calculated using an ab initio approach. The nuclear Hamiltonian retains both two- and three-nucleon interactions—the Argonne $v_{18}$ and the Urbana IX, respectively. Both one- and many-body contributions to the nuclear current operator are included. The former retain for the first time, besides the $1/m$ leading order contribution ($m$ is the nucleon mass), also the next-to-leading order term, proportional to $1/m^3$. The many-body currents are constructed in order to satisfy the current conservation relation with the adopted Hamiltonian model. The hyperspherical harmonics technique is applied to solve the $A=3$ bound and scattering states. Particular attention is paid in this second case in order to obtain, in the energy range of BBN, an uncertainty on the astrophysical $S$ factor of the order or below $\sim 1\%$. Then, in this energy range, the $S$ factor is found to be $\sim 10\%$ larger than the currently adopted values. Part of this increase (1%–3%) is due to the $1/m^3$ one-body operator, while the remaining is due to the new more accurate scattering wave functions. We have studied the implication of this new determination for the $d(p,\gamma ){}^3\mathrm{He}$ $S$ factor on the deuterium primordial abundance. We find that the predicted theoretical value for ${}^2\mathrm{H}/\mathrm{H}$ is in excellent agreement with its experimental determination, using the most recent determination of the baryon density of the Planck experiment, and with a standard number of relativistic degrees of freedom $N_{\text{eff}}=3.046$ during primordial nucleosynthesis. This calls for a more accurate measurement of the astrophysical $S$ factor in order to confirm the present predictions.

Figure 1

The astrophysical $S$ factor obtained in the present work (red points) is plotted together with the available experimental data of Refs. [10, 11, 23, 24], the calculation of Ref. [14] (solid black line), and the quadratic best fit to the data of Ref. [7] (green band). The inset shows the astrophysical $S$ factor in the 0–300 keV energy range, since the relevant BBN energy range is 30–300 keV.

Figure 2

The likelihood contours (68, 95, and 99% C.L.) in the ${\mathrm{\Omega }}_b{h}^2-N_{\text{eff}}$ plane from ${}^2\mathrm{H}/\mathrm{H}$, with the Planck 2015 prior on ${\mathrm{\Omega }}_b{h}^2$, a free $N_{\text{eff}}$, and using the experimental result of Ref. [8]. The triangle is the best fit value of Planck 2015 results for these parameters, with corresponding 68% C.L. error bars [3].