We compute the full cosmic trajectories of the early Universe across the QCD phase diagram as the plasma cools from $T\simeq 500\mathrm{MeV}$ to 30 MeV, assuming $\beta$-equilibrated matter. The trajectories are obtained by simultaneously solving baryon-number, electric-charge, and lepton-asymmetry conservation, closed by a state-of-the-art lattice-QCD equation of state: a fourth-order Taylor expansion in the chemical potentials that merges the latest ($2+1$)-flavor susceptibilities with charm-quark contributions, thus delivering a consistent $(2+1+1)$-flavor equation of state. Results are compared with an ideal quark-gluon plasma and with a hadron-resonance gas to highlight interaction effects. Two cases of primordial lepton asymmetries are analyzed: a symmetric configuration $({\ell }_e={\ell }_{\mu }={\ell }_{\tau }=\ell /3)$ and an asymmetric one $({\ell }_e=0,{\ell }_{\mu }=-{\ell }_{\tau })$. Increasing $|\ell |$ systematically drives the trajectories toward larger values of ${\mu }_B$ and more negative ${\mu }_Q$. In the asymmetric case, a nonmonotonic “bounce” develops when the $\tau$ chemical potential reaches $m_{\tau }$, generating a maximum in ${\mu }_B(T)$, the position of which depends on ${\ell }_{\tau }$. Assuming a modest ${\mu }_Q$ dependence of the lattice-QCD critical end point estimates (obtained at ${\mu }_Q=0$), the trajectories for all lepton asymmetries explored ($|\ell |\lesssim 0.1$) lie to their left, implying that in a standard cosmological scenario the QCD transition is almost certainly a smooth crossover. Nevertheless, we estimate the magnitude of baryon and lepton asymmetries needed to obtain a cosmic trajectory closer to the QCD critical point, providing inputs for future studies of the strong-interaction epoch.

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• Cosmological parameters
• Cosmology
• Evolution of the Universe
• QCD phase transitions
• Quark-gluon plasma