Cooling of Neutron Stars with Color Superconducting Quark Cores

We show that within a recently developed nonlocal chiral quark model the critical density for a phase transition to color superconducting quark matter under neutron star conditions can be low enough for these phases to occur in compact star configurations with masses below 1.3 M_solar. We study the cooling of these objects in isolation for different values of the gravitational mass. Our equation of state (EoS) allows for 2SC quark matter with a large quark gap \~100 MeV for u and d quarks of two colors that coexists with normal quark matter within a mixed phase in the hybrid star interior. We argue that, if the phases with unpaired quarks were allowed, the corresponding hybrid stars would cool too fast. If they occured for M < 1.3 M_solar, as it follows from our EoS, one could not appropriately describe the neutron star cooling data existing by today. We discuss a "2SC+X" phase, as a possibility to have all quarks paired in two-flavor quark matter under neutron star constraints, where the X-gap is of the order of 10 keV - 1 MeV. Density independent gaps do not allow to fit the cooling data. Only the presence of an X-gap that decreases with increase of the density could allow to appropriately fit the data in a similar compact star mass interval to that following from a purely hadronic model. This scenario is suggested as an alternative explanation of the cooling data in the framework of a hybrid star model.