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The study of the primordial black hole (PBH) gravitational collapse processrequires the determination of a critical energy density perturbation threshold$\delta_\mathrm{c}$, which depends on the equation of state of the universe atthe time of PBH formation. Up to now, the majority of analytical and numericaltechniques calculate $\delta_\mathrm{c}$ by assuming a constantequation-of-state (EoS) parameter $w$ at the time of PBH formation. In thiswork, after generalizing the constant $w$ prescription of [1] for thecomputation of $\delta_\mathrm{c}$ and commenting its limitations we give afirst estimate for the PBH threshold in the case of a time-dependent $w$background. In particular, we apply our formalism in the case of the QCD phasetransition, where the EoS parameter varies significantly with time and oneexpects an enhanced PBH production due to the abrupt softening of $w$. At theend, we compare our results with analytic and numerical approaches for thedetermination of $\delta_\mathrm{c}$ assuming a constant EoS parameter.

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