Inflation-produced primordial gravitational waves (PGWs) carry a percent-level imprint of the cosmic QCD transition: as the plasma cools through T≃155T\simeq155 MeV, the effective number of relativistic degrees of freedom drops sharply, modifying the tensor transfer function and producing a characteristic feature in the present-day spectrum ΩGW(f)\Omega_{\rm GW}(f) around 10−1010^{-10}-10−710^{-7} Hz. Hajkarim, Schaffner-Bielich, Wystub and Wygas (Phys. Rev. D 99, 103527, 2019) quantified this imprint with then-available lattice-QCD equations of state (up to ∼4%\sim4\% spread between EOS choices) and lepton asymmetry (22-10%10\% effects), but deferred the isospin-chemical-potential/pion-condensation sector and left the lattice systematics open. Later work (2021-2026) computed pion-condensation onset criteria and their ΩGW\Omega_{\rm GW} imprint in effective models, produced lattice equations of state at nonzero isospin density, and traced cosmic trajectories with a state-of-the-art (2+1+1)(2+1+1)-flavor lattice EOS, which favor a smooth crossover for allowed lepton asymmetries. What remains open is the full problem: a first-principles, uncertainty-controlled prediction of ΩGW(f)\Omega_{\rm GW}(f) through the QCD epoch along cosmologically allowed trajectories — including whether a pion-condensed phase or a first-order transition can occur for allowed lepton flavor asymmetries, and with the lattice, trajectory, anisotropic-stress, and neutrino-decoupling systematics all quantified at the percent level.

Background

Primordial gravitational waves (PGWs) are the tensor (spin-2) perturbations of the spacetime metric generated during cosmic inflation, the near-exponential early expansion. Their present energy-density spectrum per logarithmic frequency interval, ΩGW(f)\Omega_{\rm GW}(f), is shaped not only by the inflationary amplitude and tilt but by the entire subsequent thermal history: after a Fourier mode re-enters the Hubble horizon, its evolution is governed by the equation of state (EOS), i.e. the relation between the total energy density ρ(T)\rho(T) and pressure p(T)p(T) of the cosmic plasma as a function of temperature TT. Around T≃155T\simeq155 MeV the strong interaction (QCD, quantum chromodynamics) confines quarks and gluons into hadrons; at the small net baryon density of the early universe this QCD transition is believed to be a smooth crossover, in which the effective number of relativistic degrees of freedom drops sharply. This leaves a step-like, percent-level feature in ΩGW(f)\Omega_{\rm GW}(f), sharpest at frequencies ∼10−9\sim10^{-9}-10−810^{-8} Hz — the nano-Hertz band of pulsar timing arrays such as SKA and EPTA — within the broader affected band ∼10−10\sim10^{-10}-10−710^{-7} Hz, while higher frequencies probe the perturbative electroweak-era plasma targeted by LISA and DECIGO. The thermodynamic input comes from lattice QCD (Monte-Carlo simulation of the theory on a discrete spacetime grid), which yields ρ(T)\rho(T) and p(T)p(T) through the trace anomaly (the interaction measure ρ−3p\rho-3p); residual systematics — continuum extrapolation to zero grid spacing, flavor content (2+12+1 sea quarks vs 2+1+12+1+1 including the charm quark), and matching to high-temperature perturbation theory — propagate directly into ΩGW(f)\Omega_{\rm GW}(f). A further input is the lepton asymmetry: the plasma may carry an excess of leptons over antileptons, quantified per flavor by lα=(nα−nαˉ)/sl_\alpha=(n_{\alpha}-n_{\bar\alpha})/s with ss the entropy density. Electric-charge neutrality, fixed net baryon and lepton numbers, and approximate chemical equilibrium tie together the baryon, electric-charge (isospin), and lepton chemical potentials along the 'cosmic trajectory' through the QCD phase diagram. Sufficiently large flavor asymmetries push the trajectory to nonzero isospin chemical potential μQ\mu_Q (an up-down quark chemical-potential imbalance), where low-temperature QCD forms a Bose-Einstein condensate of charged pions — pion condensation — with a very different EOS, including a speed of sound exceeding the conformal value cs2=1/3c_s^2=1/3. In addition, free-streaming photons and neutrinos source an anisotropic stress that damps tensor modes at the lowest frequencies (≲5×10−11\lesssim5\times10^{-11} Hz), and the modeling of neutrino decoupling near T∼1T\sim1 MeV shifts the frequency-to-horizon-crossing-temperature map. Hajkarim et al. (2019) computed ΩGW(f)\Omega_{\rm GW}(f) with then-available lattice EOSs, finding up to ∼4%\sim4\% differences between EOS choices near 3×10−93\times10^{-9} Hz, ∼1.1%\sim1.1\% from lattice error bands, ∼2.6%\sim2.6\% from charm inclusion, and 22-10%10\% effects from nonzero lepton asymmetry at f≳fQCD≃3×10−9f\gtrsim f_{\rm QCD}\simeq3\times10^{-9} Hz — sizes comparable to the percent-level sensitivity projected for future detectors. They explicitly deferred the isospin/pion-condensation sector to future work and acknowledged the EOS systematics and the untreated subleading effects (full anisotropic stress, precise neutrino decoupling, higher-order perturbative QCD) as limitations.

Problem Statement

Determine, from first-principles QCD thermodynamics and with uncertainties controlled at the percent level, how the cosmic QCD transition imprints itself on the primordial gravitational-wave spectrum ΩGW(f)\Omega_{\rm GW}(f). This comprises two jointly linked parts. (1) Thermal history: compute the cosmic trajectory through the QCD phase diagram (temperature vs baryon, electric-charge/isospin, and lepton-flavor chemical potentials, from charge neutrality, conserved asymmetries, and chemical equilibrium) for lepton flavor asymmetries allowed by cosmic-microwave-background (CMB) and big-bang-nucleosynthesis (BBN) constraints, closed by a first-principles EOS; in particular establish whether any allowed trajectory enters a pion-condensed phase or a first-order transition region, and determine the EOS along such trajectories (trace anomaly, speed of sound, effective degrees of freedom) with quantified lattice and matching uncertainties; both the determination and the EOS are required, and if no allowed trajectory enters such a region the EOS requirement is met by that null result together with the EOS along the allowed crossover trajectories. The continuum-extrapolated lattice EOS at high isospin chemical potential near the transition is not yet available and is therefore itself part of the target to be obtained first-principles or, where that cannot be achieved, to be carried as an explicitly quantified systematic in the error budget, in which case its impact must be shown to lie below the percent-level target using the spread of the existing nonzero-isospin lattice equations of state (refs 3 and 4). (2) Forward prediction: propagate this thermal history into the present-day spectrum ΩGW(f)\Omega_{\rm GW}(f) across the QCD-sensitive band ∼10−10\sim10^{-10}-10−710^{-7} Hz — including the tensor transfer through the time-dependent EOS, the damping from anisotropic stress of free-streaming photons and neutrinos, which is effective at the lowest frequencies below the band's lower edge (≲5×10−11\lesssim5\times10^{-11} Hz), realistic neutrino-decoupling treatment, higher-order perturbative-QCD matching at high temperature, and any additional GW component or spectral distortion sourced by a first-order transition or a stiff (cs2>1/3c_s^2>1/3) epoch — of which the tensor transfer, the neutrino-decoupling treatment and the high-temperature perturbative-QCD matching are required within the band, while the anisotropic-stress damping (effective only below the band) and any component or distortion sourced by a first-order transition or a stiff epoch are required wherever they affect the band — with uncertainty bands whose total width remains at or below the percent level (about 1% of ΩGW(f)\Omega_{\rm GW}(f)) across the band and in which neither model choice nor lattice systematics dominates, i.e. neither contributes more than half of the total band, matching the percent-level target of SKA, EPTA, LISA and DECIGO. The common objective is a quantitative, uncertainty-controlled mapping between QCD-transition microphysics and the observable ΩGW(f)\Omega_{\rm GW}(f); the thermal history determined in part (1) is the input that part (2) propagates into the present-day spectrum.

References

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