Bayesian Smooth-Fit Extrapolation of the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ Astrophysical $S$ Factor

A Bayesian analysis of the astrophysical $S$ factor for the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion reaction is presented using available low-energy information at carbon–carbon relative energies $E<3.5\mathrm{M}\mathrm{e}\mathrm{V}$, including direct measurements and recent inverse-kinematics data. The goal of the global Bayesian fit is not to reproduce the local resonance-by-resonance structure of the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ system, but rather to constrain the smooth global component of $S^{\ast }(E)$ using the direct and inverse-kinematics constraints. The resulting posterior $S^{\ast }(E)$ lies systematically below the traditional FCZ75 reference normalization over the energy interval considered. For example, at $E=1.5\mathrm{M}\mathrm{e}\mathrm{V}$ the median posterior value is $S^{\ast }(1.5\mathrm{M}\mathrm{e}\mathrm{V})=1.20\times {10}^{16}\mathrm{k}\mathrm{e}\mathrm{V}\mathrm{b}$, whereas the Fowler–Caughlan–Zimmerman reference value is $S_{\mathrm{F}\mathrm{C}\mathrm{Z}75}^{\ast }=3.0\times {10}^{16}\mathrm{k}\mathrm{e}\mathrm{V}\mathrm{b}$ [W.~A. Fowler, G.~R. Caughlan, and B.~A. Zimmerman, Annu. Rev. Astron. Astrophys. $\mathbf{\text{13}}$, 69 (1975)]. Thus, the updated posterior favors a lower smooth low-energy trend and does not support a sharp increase of $S^{\ast }(E)$ as the energy decreases. The astrophysical consequence of the analysis is assessed through the thermonuclear reaction rate $N_A⟨\sigma v⟩$. The resulting median rate is lower than the CF88 analytic rate [G.~R. Caughlan and W.~A. Fowler, At. Data Nucl. Data Tables $\mathbf{\text{40}}$, 283 (1988)] over the temperature interval considered, with $R_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}}\mathrm{/}{R}_{\mathrm{C}\mathrm{F}88}\simeq 0.33$–$0.46$ for $0.3\le T_9\le 1.2$. At higher temperatures, where the Gamow window extends above the upper end of the adopted integration interval, the calculated rate should be regarded as a truncated rate rather than a complete stellar rate.