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 $E<3.5\mathrm{MeV}$, 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 $S^{*}\left(E\right)$ using the direct and inverse-kinematics constraints. A conservative uncertainty model is adopted in order to combine heterogeneous datasets with different systematic uncertainties and normalizations. The inference is performed for $y\left(E\right)={log}_{10}{S}^{*}\left(E\right)$, which reduces the dynamic range of the data and leads to approximately Gaussian uncertainties. The function $y\left(E\right)$ is represented by a quadratic polynomial, and the posterior distribution of its coefficients is determined. Propagation of this posterior gives the median smooth curve and the corresponding $68\%$ posterior interval for $S^{*}\left(E\right)$. The resulting posterior $S^{*}\left(E\right)$ lies systematically below the traditional FCZ75 reference normalization over the energy interval considered. For example, at $E=1.5\mathrm{MeV}$ the median posterior value is $S^{*}\left(1.5\mathrm{MeV}\right)=1.20\times {10}^{16}\mathrm{keV}\mathrm{b}$, whereas the Fowler–Caughlan–Zimmerman reference value is $S_{\mathrm{FCZ}75}^{*}=3.0\times {10}^{16}\mathrm{keV}\mathrm{b}$ [W. A. Fowler, G. R. Caughlan, and B. A. Zimmerman, Annu. Rev. Astron. Astrophys. 13, 69 (1975)]. Thus, the updated posterior favors a lower smooth low-energy trend and does not support a sharp increase of $S^{*}\left(E\right)$ as the energy decreases. The astrophysical consequence of the analysis is assessed through the thermonuclear reaction rate $N_A\langle \sigma v\rangle$. The resulting median rate is lower than the CF88 analytic rate [G. R. Caughlan and W. A. Fowler, At. Data Nucl. Data Tables 40, 283 (1988)] over the temperature interval considered, with $R_{\mathrm{present}}/R_{\mathrm{CF}88}\simeq 0.33–0.46$ 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.

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• Low & intermediate energy heavy-ion reactions
• Nuclear physics of explosive environments
• S-factor
• Bayesian methods