Bayesian Smooth-Fit Extrapolation of the ¹²C+¹²C Astrophysical S Factor

A Bayesian analysis of the astrophysical S factor for the ¹²C+¹²C fusion reaction is presented, based on available experimental information at carbon--carbon relative energies E 2~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 log₁₀S*(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 Ecm=2~MeV, where the fusion reaction rate is most relevant for stellar carbon burning. At Ecm=1.5~MeV, the posterior distribution yields Sglobal*(1.5~MeV)= (2.13+0.01-0.01)×10¹⁶\,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 ¹²C+¹²C fusion S factor in the energy region relevant for stellar carbon burning.