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, based on available experimental information at carbon--carbon relative energies $E \gtrsim 2~\mathrm{MeV}$, including direct measurements, indirect Coulomb-renormalized Trojan Horse Method (THM) results, and recent inverse-kinematics data. The Bayesian inference is performed on the quantity $\log_{10}S^{*}(E)$ rather than on $S^{*}(E)$ itself, which naturally accommodates the wide dynamic range of the data and leads to approximately Gaussian uncertainties. The logarithm of the astrophysical factor is parametrized by a quadratic polynomial in energy, and the posterior distribution of the fit coefficients is determined using a weighted Bayesian regression. From this posterior, a global median $S^{*}(E)$ curve is constructed, and the associated covariance matrix is used to define a low/medium/high (LO/MED/HI) band corresponding to a $68\%$ credible interval. Particular emphasis is placed on the extrapolation below $E_{\mathrm{cm}}=2~\mathrm{MeV}$, where the fusion reaction rate is most relevant for stellar carbon burning. At $E_{\mathrm{cm}}=1.5~\mathrm{MeV}$, the posterior distribution yields $S_{\mathrm{global}}^{*}(1.5~\mathrm{MeV})= \left(2.13^{+0.01}_{-0.01}\right)\times10^{16}\,\mathrm{keV\,b},$ corresponding to a $68\%$ credible interval. The extracted result is consistent with recent inverse-kinematics measurements and with Coulomb-corrected Trojan Horse Method constraints, providing a tightly constrained estimate of the $^{12}\mathrm{C}+^{12}\mathrm{C}$ fusion $S$ factor in the energy region relevant for stellar carbon burning.