The core-collapse supernova (CCSN) phenomenon, one of the most explosive events in the Universe, presents a challenge to theoretical astrophysics. Stellar matter in supernovae, experiencing most extreme pressure and temperature, undergoes transformations that cannot be simulated in terrestrial laboratories. Construction of astrophysical models is the only way towards comprehension of CCSN. The key microscopic input into CCSN models is the Equation of State (EoS), connecting the pressure of stellar matter to the energy density and temperature, dependent upon its composition. Of the large variety of forms of CCSN matter, we focus on the transitional region between homogeneous and inhomogeneous phases. Here the nuclear structures undergo a series of changes in shape from spherical to exotic deformed forms: rods, slabs, cylindrical holes and bubbles, termed “nuclear pasta”. We perform a three-dimensional, finite temperature Skyrme-Hartree-Fock + BCS (3D-SHF) study of the inhomogeneous nuclear matter, where we calculate self-consistently the nuclear pasta phase and determine the phase transition between pasta and uniform matter and its character. As the nuclear matter properties depend on the effective nucleon-nucleon interaction in the 3D-SHF model, we employ four different parametrizations of the Skyrme interaction, SkM*, SLy4, NRAPR and SQMC700. For each of these interactions we calculate free energy, pressure, entropy and chemical potentials in the space of particle number densities, temperatures and proton fractions, expected to cover the pasta region. The available data analysed are for particle number densities 0.02 - 0.12 fm−3 [reciprocal of cubic fermi], temperatures 0 - 10 MeV and a proton fraction equal to 0.3. The data indicate a distinct discontinuity in the first derivatives of the free energy, which can be interpreted as a fingerprint of the first order transition between inhomogeneous and homogeneous supernova matter. This transition occurs naturally in our model, without a need for thermodynamic constructions. However, the transitions between distinct pasta formations are much less pronounced and hard to detect with certainty
