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I will discuss the derivation of a microscopic equation of state of neutron star matter suitable for application to neutron stars merging and core collapse supernovae. I will focus on the various possibilities concerning the composition of neutron star matter and on the theoretical problems about the modeling of the different scenarios. In particular, I will consider both the mechanism that may lead to the formation of hyperons in the core of neutron stars, as well as the possibility that hadronic matter undergoes a phase transition to deconfined quark matter. Astrophysical consequences and possible signatures in merging neutron
stars will be discussed.