Neutron stars with large quark cores

We describe charge-neutral neutron star matter in $\beta-$equilibrium using hybrid equations of state, where a first-order phase transition from hadronic to quark matter is realized. The hadronic matter is described in a model-independent way by a Taylor expansion around saturation density $n_0$, while the three-flavor NJL model is used for the quark matter. Exploring the present uncertainty on the empirical parameters of nuclear matter and the parameter space of the NJL model, we construct two datasets of thermodynamically consistent and causal hybrid EoSs, compatible with astrophysical observations. We conclude that, to sustain a considerable quark core size, the intensity of the phase transition from hadron to quark matter cannot be strong, having a energy density gap below $200$ MeV/fm$^3$, and must occur at baryon densities not above four times the saturation density. A non zero but not too strong quark vector-isoscalar term and a weak vector isovector quark term are required. Large quark cores carrying almost half of the star mass are possible inside neutron stars with a maximum mass $\approx 2.2 M_\odot$. To get a considerable number of hybrid EoS predicting quark matter already inside neutron stars with a mass $\sim 1.4 M_\odot$, we require that the onset of quarks occurs in the range $1.3n_0$ and $2.5n_0$. Neutron stars with large quark cores corresponding to more than one fourth of the total star mass, are possible if the energy density gap and the pressure at transition are below 100 MeV/fm$^3$. However, under these constraints, the maximum neutron star mass is limited to $ \lesssim 2.06 M_{\odot}$. No strong signatures from quark matter were found on the radius and the tidal deformability for neutron star masses below $1.8\, M_\odot$.