Evolutions 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 at beam energies from 150 to 1200 MeV/nucleon taken 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. 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.