Evolution of in-medium baryon-baryon scattering cross sections and stiffness of dense nuclear matter from Bayesian analyses of FOPI proton-flow excitation functions
Within a Bayesian statistical framework using a Gaussian process (GP) emulator for an isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model simulator of heavy-ion reactions with momentum-independent Skyrme interactions, we infer from the proton directed and elliptical flow in mid-central Au + Au reactions, measured at beam energies from 150 to 1200 MeV/nucleon by the FOPI Collaboration, the posterior probability distribution functions (PDFs) of the in-medium baryon-baryon scattering cross section (BBSCS) modification factor 𝑋 (with respect to their free-space values) and the stiffness parameter 𝐾 of dense nuclear matter. Here, we find that the most probable value of 𝑋 evolves from around 0.7 to 1.0 as the beam energy 𝐸 beam /𝐴 increases. On the other hand, the posterior PDF(𝐾) may have dual peaks having roughly the same height or extended shoulders at high 𝐾 values. More quantitatively, the posterior PDF(𝐾) changes from having a major peak around 220 MeV, characterizing a soft EOS in the reaction at 𝐸 beam /𝐴 = 150 MeV, to one that peaks around 320 MeV, indicating a stiff EOS in the reactions at 𝐸 beam /𝐴 higher than about 600 MeV. The transition from soft to stiff happens in mid-central Au + Au reactions at beam energies around 250 MeV/nucleon in which 𝐾 = 220 MeV and 𝐾 = 320 MeV are approximately equally probable. Altogether, the FOPI proton-flow excitation function data indicate a gradual hardening of hot and dense nuclear matter as its density and temperature increase in reactions with higher beam energies.