Chiral symmetry breaking and pion condensation in the early universe

TL;DR

This work uses the two-flavor quark-meson model to map how large primordial lepton asymmetries can drive chiral symmetry breaking and pion condensation in the early Universe's QCD epoch. By solving conservation constraints for $B$, $Q$, and $L_\alpha$ while minimizing the QM free energy, the authors derive cosmic trajectories through the $T$-$\mu_Q$ phase diagram, revealing a tricritical point and a possible first-order entry into a pion-condensed phase for $|l_e+l_\mu|\gtrsim 0.1$, followed by a second-order exit. Such a first-order transition could source primordial gravitational waves in the nanohertz range, with the signal's strength tied to the latent heat and lepton asymmetry, and thus potentially detectable by Pulsar Timing Arrays like SKA or informed by NANOGrav data. They also compare their results to lattice QCD and other effective models, discuss BBN/CMB consistency, and propose extensions (three flavors, Polyakov loop, vacuum corrections) to sharpen predictions and explore broader astrophysical implications.

Abstract

We determine the possible trajectories the universe may have followed in the QCD phase diagram during the QCD epoch. We focus on the roles of chiral symmetry breaking and pion condensation under high imbalances in lepton asymmetry. Adopting the quark-meson model as an effective description of QCD at finite temperature, charge and baryon chemical potentials we show that, for sufficiently large but physically motivated asymmetries, the universe may have entered the pion condensation phase through a first-order phase transition, followed by a second-order phase transition when exiting it. Such a first-order phase transition represents a new possible source of primordial gravitational waves during the QCD epoch.

Figures (4)

• Figure 1: Pion condensation in the early universe. When baryons appear, charge neutrality can still be maintained at high $\mu_Q$. The universe can then reach phases such as pion condensation, depending on the imbalance in the individual lepton asymmetries quantified by $l_e+l_\mu$ (with $l_e + l_\mu + l_\tau =0$). The dark gray line is the standard trajectory that assumes no imbalance in the asymmetries and a total asymmetry of the order of $b$. In the figure three possibilities are illustrated: A) at small $l_e+l_\mu$ the trajectory does not reach the pion condensate; B) at intermediate values of $l_e+l_\mu$ the universe enters and leaves the pion condensate through second-order PTs; C) at high $l_e+l_\mu$ the universe enters the pion condensed phase through a first-order PT and exits it through a second order PT.
• Figure 2: Left: Cosmic trajectories in the $T$ vs $\mu_{Q}$ plane for different values of $|l_e + l_\mu|$. For illustration, the background shows the phase diagram of the QM model, at fixed $\mu_B=100$ MeV. The blue shaded region corresponds to the pion condensation phase. The dashed segments indicates that the trajectories have entered the pion condensation phase. Notice that $\mu_B$ varies along the cosmic trajectories. Right: Region in the $T$ vs $l_e + l_\mu$ phase diagram in which the pion condensation can be formed.
• Figure 3: Condensate values along cosmic trajectories as a function of temperature. Pion condensate (left) and chiral condensate (right). The first-order phase transitions along the cosmic trajectories cause discontinuities in the condensates.
• Figure 4: Trace anomaly (left) and speed of sound (right) as functions of the temperature for different lepton asymmetries. Solid lines correspond to the total trace anomaly, and dashed lines correspond to the QCD sector only. For high values of $|l_e + l_\mu|$, the trace anomaly reaches negative values inside the pion condensation phase and the speed of sound exceeds the conformal value.