Dynamical properties of nuclear and stellar matter and the symmetry energy

The effects of density dependence of the symmetry energy on the collective modes and dynamical instabilities of cold and warm nuclear and stellar matter are studied in the framework of relativistic mean-field hadron models. The existence of the collective isovector and possibly an isoscalar collective mode above saturation density is discussed. It is shown that soft equations of state do not allow for a high density isoscalar collective mode, however, if the symmetry energy is hard enough an isovector mode will not disappear at high densities. The crust-core transition density and pressure are obtained as a function of temperature for $\beta$-equilibrium matter with and without neutrino trapping. An estimation of the size of the clusters formed in the non-homogeneous phase as well as the corresponding growth rates and distillation effect is made. It is shown that cluster sizes increase with temperature, that the distillation effect close to the inner edge of the crust-core transition is very sensitive to the symmetry energy, and that, within a dynamical instability calculation, the pasta phase exists in warm compact stars up to 10 - 12 MeV.