Coexistence of strange quark stars and neutron stars: Metastability and nucleation in protoneutron stars

If strange quark matter (SQM) is absolutely stable, hadronic neutron stars (NSs) and strange quark stars (QSs) may coexist in the so-called two-families scenario. A key issue is explaining how hadronic NSs can survive as long-lived metastable objects, rather than promptly converting into stable QSs. Since all NSs are born as hot proto-neutron stars (PNSs), a necessary condition for their existence is that the conditions for conversion, triggered by the nucleation of the first critical SQM droplet, are not reached in every PNS. We investigate the thermal nucleation of critical SQM droplets under representative PNS conditions, accounting for flavor-composition fluctuations and finite-size effects on color superconductivity by suppressing pairing in droplets smaller than the diquark coherence length. We define the nucleation conditions as the thermodynamic state at which a critical SQM droplet is expected to nucleate statistically within the characteristic dynamical timescale of the system. Under these conditions, the reduced nucleation barrier is nearly universal, with $W_*/T\simeq167$ to within a few units. Exploring the SQM parameter space, we show that requiring a canonical $1.4~\rm{M}_{\odot}$ NS to survive the PNS stage sets a lower bound on the hadron-quark surface tension $σ\gtrsim 67$ MeVfm$^{-2}$.