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

Abstract

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 $μ_{\mathrm B}(T)$: $τ$--$μ$ asymmetric cases exhibit a local maximum, whereas $μ$--$e$ 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 $μ_\mathrm{B}/T$ because of the nonzero $μ_\mathrm{Q}$. Therefore, we constrain the QCD-era relic to the standard scenario of a chiral crossover transition.

Figures (4)

• Figure 1: Cosmic trajectories in the $T$--$\mu_{\mathrm B}$ plane for $\ell_e \simeq 0$ with $\ell_\mu=-\ell_\tau$ ($\ell_\tau>0$), shown for several values of $\ell_\tau$. Top: free QGP EoS. Middle: lattice QCD-based EoS. Bottom: pure HRG behavior calculated with the Thermal-FIST package Vovchenko:2019pjl. The dashed curve on the top and middle panels connects the points where $\mu_\tau = m_\tau$ along each trajectory.
• Figure 2: Top: absolute net leptonic charge density $|Q_{\mathrm{lep}}(T)|$ for several $\ell_\tau$ (with $\ell_\mu=-\ell_\tau$) for the lattice QCD-based EoS, where $Q_{\mathrm{lep}}(T)\equiv n_{\tau^-}(T)-n_{\mu^+}(T)$. Bottom: relative charged-lepton imbalance $[n_{\tau^-}(T)-n_{\mu^+}(T)]/n_{\tau^-}(T)$ (normalized to the $\tau^-$ density). Using the absolute (net) difference, the curves cross; with the relative one, they do not cross and are ordered, indicating that the relative charge imbalance sets the limiting criterion for the trajectories.
• Figure 3: Cosmic trajectories in the $T$--$\mu_{\mathrm B}$ plane for $\ell_\tau \simeq 0$ with $\ell_e=-\ell_\mu$ ($\ell_\mu>0$) , shown for several values of $\ell_\mu$. Top: free QGP EoS. Bottom: lattice QCD-based EoS. Unlike the $\ell_e\simeq 0$ case, there is no sharp rebound: the curves approach an upper limiting trajectory that caps $\mu_\mathrm{B}(T)$ and increasing $|\ell_\mu|$ beyond this does not raise $\mu_\mathrm{B}$ further.
• Figure 4: Top: absolute net leptonic charge density $|Q_{\mathrm{lep}}(T)|$ for several $\ell_\mu$ (with $\ell_e=-\ell_\mu$) using the lattice QCD-based EoS, where $Q_{\mathrm{lep}}(T)\equiv n_{\mu^-}(T)-n_{e^+}(T)$. Bottom: relative charged-lepton imbalance $[n_{\mu^-}(T)-n_{e^+}(T)]/n_{\mu^-}(T)$ (normalized to the $\mu^-$ density). With the absolute (net) difference, the curves reproduce the ordering seen in the $(T,\mu_{\mathrm B})$ trajectories; with the relative difference, the curves are strictly ordered indicating that the relative charge imbalance sets the limiting criterion for the trajectories in this case as well.