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 1.30.3em0exM_. We study the cooling of these objects in isolation for different values of the gravitational mass. Our equation of state (EoS) allows for two-flavor color superconductivity (2SC) quark matter with a large quark gap (~1000.3em0exMeV) 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 occurred for M<1.30.3em0exM_, as follows from our EoS, one could not appropriately describe the neutron star cooling data existing today. We discuss a ``2SC+X'' phase as a possibility for having all quarks paired in two-flavor quark matter under neutron star constraints, where the X gap is of the order of 100.3em0exkeV--10.3em0exMeV. Density-independent gaps do not allow us to fit the cooling data. Only the presence of an X gap that decreases with increasing density would allow us 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.