Toward the primordial black hole formation threshold in a time-dependent equation-of-state background

The study of the primordial black hole (PBH) gravitational collapse process requires determination a critical energy density perturbation threshold ${\ensuremath{\delta}}_{\mathrm{c}}$, which depends on equation state universe at time PBH formation. Up to now, majority analytical and numerical techniques calculate ${\ensuremath{\delta}}_{\mathrm{c}}$ by assuming constant equation-of-state (EoS) parameter $w$ In this work, after generalizing prescription [T. Harada et al., Phys. Rev. D 88, 084051 (2013); T. al.Phys. D89, 029903(E) (2014).] for computation commenting its limitations we give first estimate in case time-dependent background. particular, apply our formalism QCD phase transition, where EoS varies significantly with one expects an enhanced production due abrupt softening $w$. At end, compare results analytic approaches parameter.