Upper Bound on the Cosmic Baryon Chemical Potential from Lepton-Flavor Asymmetry

We study the early Universe trajectory around the QCD transition in lepton-flavor-asymmetric cases with small total lepton asymmetry ($|\ell|\lesssim 10^{-2}$), while allowing large individual lepton asymmetries. For each temperature, we find an upper bound on the baryon chemical potential $\mu_{\mathrm B}(T)$: $\tau$--$\mu$ asymmetric cases exhibit a local maximum, whereas $\mu$--$e$ cases approach a limiting curve. Thus, even extreme lepton-flavor asymmetry alone cannot reach a first-order region, unless the critical point is moved to a substantially lower $\mu_\mathrm{B}/T$ because of the nonzero $\mu_\mathrm{Q}$. Therefore, we constrain the QCD-era relic to the standard scenario of a chiral crossover transition.