Effective QCD models predict that a large baryon chemical potential at the QCD epoch makes the confinement/chiral transition first order with a strongly suppressed bubble-nucleation rate, opening two provocative possibilities: a stochastic nanohertz gravitational-wave background detectable by pulsar timing arrays, and survival of supercooled quark-matter false-vacuum domains as primordial quark nuggets that could seed early compact stars and high-redshift galaxies. The source work (Shao, Mao, Huang, Chin. Phys. C 49, 065103 (2025)) computed the transition rate β/H(μ)\beta/H(\mu) and the resulting gravitational-wave spectra in the Friedberg-Lee and quark-meson models but left the scenario's load-bearing physics uncomputed: the effect of color-superconducting (diquark-condensed) phases and of primordial magnetic fields on the transition line and rate, which the source explicitly flags; the quantum-tunneling decay and cosmological survival of the nuggets against nucleosynthesis and microwave-background constraints; and a quantitative confrontation of the predicted spectra with pulsar timing array data. The source treats the cosmological history that generates the required large chemical potential — either a homogeneous Affleck-Dine-generated chemical potential subsequently diluted by little inflation, or inhomogeneously distributed high-baryon-density clumps — as an established premise rather than deriving it, and the cosmological consistency of the resulting scenario remains part of the viability question. The problem is to determine, quantitatively and self-consistently, whether this scenario is cosmologically viable and observationally testable — that is, to supply these missing pieces or to exclude the scenario within the source's effective-model setting (the Friedberg-Lee and quark-meson high-μ\mu family together with its color-superconducting and magnetized extensions) under one of the source-allowed cosmological histories for generating μ\mu.

Background

At zero baryon chemical potential, lattice QCD (Monte Carlo simulation of quantum chromodynamics on a discrete spacetime grid) establishes that the cosmic quark–hadron transition is a smooth crossover, but at large baryon density the question reopens. In effective models — the Friedberg–Lee (FL) soliton model and the quark-meson (QM) model, in which chiral symmetry breaking is described by a sigma field interacting with quarks — a first-order transition line extends down in temperature from a critical end point (CEP), the endpoint of the first-order line where the transition becomes continuous. Shao, Mao, and Huang studied this transition in the early universe with a baryon chemical potential μ\mu generated locally by Affleck–Dine baryogenesis (a mechanism in which scalar-field dynamics of the early universe produce a large baryon asymmetry, potentially inhomogeneously). Their key quantities: the bubble nucleation rate, encoded in the inverse transition timescale β/H\beta/H (β\beta is the time-derivative of the nucleation rate at percolation and HH the Hubble rate); the transition strength α\alpha (latent heat over radiation density); the critical nucleation point (CNP), the chemical potential beyond which the thermal nucleation rate vanishes entirely and the high-density quark phase is trapped as a false vacuum (a metastable state separated from the true vacuum by a potential barrier). They found β/H\beta/H dropping from ∼104\sim 10^4 at μ=0\mu = 0 to ∼101\sim 10^1 in a narrow high-μ\mu window — pushing the gravitational-wave peak into the nanohertz band where pulsar timing arrays (PTAs, collaborations timing millisecond pulsars to detect nanohertz gravitational waves) reported a stochastic common-spectrum signal in 2023 — and to zero at the CNP, where the false vacuum survives cosmologically, realizing Witten's 1984 primordial-quark-nugget (PQN) proposal in which compact remnants of the cosmic quark–hadron transition persist as stable quark-matter objects. The scenario's load-bearing physics is, however, uncomputed. At the relevant densities, QCD may instead favor color superconductivity — pairing of quarks into diquark condensates (two-flavor 2SC or three-flavor CFL phases), which would restructure the first-order line, the CEP/CNP locations, and β/H(μ)\beta/H(\mu). Primordial magnetic fields modify the effective potential and bubble dynamics. Below the temperature where thermal nucleation ceases, the false vacuum can still decay by quantum tunneling (nucleation of a critical bubble described by a bounce action), at a rate the source did not compute, so the survival of nuggets against boiling, evaporation, and constraints from Big Bang nucleosynthesis (BBN, the cosmic production of light nuclei) and the cosmic microwave background (CMB) is undetermined. Finally, the source assumes rather than derives the cosmological history generating the required large chemical potential — homogeneous Affleck-Dine baryogenesis subsequently diluted by little inflation, or inhomogeneously distributed high-baryon-density clumps — and the predicted spectra have not been fit to PTA data in this model family. Independent lines make the question timely rather than settled: functional-QCD calculations (continuum studies of QCD via the Dyson–Schwinger and functional-renormalization-group equations) find the high-μ\mu chiral transition too weak and fast to source the PTA signal, Bayesian fits in holographic models (effective strongly-coupled-QCD descriptions derived from gauge/gravity duality) both support and constrain QCD-transition interpretations of the PTA data, and analyses of supercooled QCD scenarios — in which the transition is delayed far below its critical temperature before bubbles nucleate — find the required supercooling difficult to achieve homogeneously.

Problem Statement

Determine, quantitatively and self-consistently, whether a first-order QCD phase transition at large baryon chemical potential in the early universe can (i) produce a stochastic nanohertz gravitational-wave background consistent with pulsar timing array data and (ii) leave behind long-lived primordial quark nuggets that satisfy all cosmological constraints. These two phenomenological endpoints rest on one common object — the high-μ\mu transition dynamics — so the following pieces missing from the source work bear on the question within a single consistent treatment: (a) the modification of the first-order transition line, of the nucleation rate β/H(μ)\beta/H(\mu), and of the critical-end-point and critical-nucleation-point locations when color-superconducting (diquark-condensed) phases are included; (b) the combined effect of finite chemical potential and primordial magnetic fields on the effective potential, bubble dynamics, and the gravitational-wave spectrum; (c) the quantum-tunneling decay rate of supercooled quark-matter false-vacuum domains at low temperature and the survival probability of the nuggets through Big Bang nucleosynthesis and microwave-background constraints; (d) the cosmological history for the generation of the required large baryon chemical potential — the source allows either a homogeneous Affleck-Dine-generated chemical potential subsequently diluted by little inflation, or inhomogeneously distributed high-baryon-density clumps — stated explicitly for the treatment, together with an accounting of any dilution or volume-fraction suppression of the gravitational-wave amplitude implied by the history adopted, consistently with the observed mean cosmic baryon density; and (e) a confrontation of the resulting gravitational-wave spectra with pulsar timing array observations, including whether any parameter region of this model setting is both cosmologically allowed and observationally compatible. A single consistent treatment means that endpoints (i) and (ii) are derived from the same effective-model treatment of the high-μ\mu transition and from the same stated cosmological history for the generation of μ\mu — they need not occur at one common (T,μ)(T,\mu) point, since the source places the nanohertz window just below the critical nucleation point and nugget survival where β/H→0\beta/H \to 0 — and the gravitational-wave amplitude invoked for (i) must be consistent with that history. The model setting is the Friedberg–Lee and quark-meson high-μ\mu family of the source work together with its color-superconducting (diquark-condensed) and magnetized extensions. The question is resolved by a quantitative account that either establishes a cosmologically consistent, testable realization of the scenario within this setting under one of the source-allowed cosmological histories for the generation of μ\mu, or excludes it there by showing that no parameter region of this model setting is both cosmologically allowed and observationally compatible under either of those histories. The establish-or-exclude verdict requires (a)-(e) within the single consistent treatment.

References

  1. Jingdong Shao, Hong Mao, and Mei Huang, Nanohertz gravitational waves and primordial quark nuggets from dense QCD matter in the early Universe, Chin. Phys. C 49, 065103 (2025), DOI 10.1088/1674-1137/adbeeb, arXiv:2410.00874, https://arxiv.org/abs/2410.00874
  2. Jingdong Shao, Hong Mao, and Mei Huang, Transition rate and gravitational wave spectrum from first-order QCD phase transitions, Phys. Rev. D 111, 023052 (2025), DOI 10.1103/PhysRevD.111.023052, arXiv:2410.06780, https://arxiv.org/abs/2410.06780
  3. Jingdong Shao and Mei Huang, Primordial Quark Stars Made of Long-lived False Vacuum (2025), arXiv:2510.06065, https://arxiv.org/abs/2510.06065
  4. Bikai Gao, Jingdong Shao, and Hong Mao, Gravitational wave spectrum from first-order QCD phase transitions based on a parity doublet model (2026), arXiv:2604.00465, https://arxiv.org/abs/2604.00465
  5. Tae Hyun Jung, Seyong Kim, Jong-Wan Lee, Chang Sub Shin, and Hee Beom Yang, Revisiting QCD-induced little inflation with chiral density wave state and its implications on pulsar timing array gravitational-wave signals (2026), DOI 10.1103/px47-qkxm, arXiv:2603.29772, https://arxiv.org/abs/2603.29772
  6. Francesco Di Clemente, Alessandro Drago, Lorenzo Formaggio, Claudia Ratti, Volodymyr Vovchenko, and Geetika Yadav, Upper Bound on the Cosmic Baryon Chemical Potential from Lepton-Flavor Asymmetry (2025), arXiv:2511.11995, https://arxiv.org/abs/2511.11995
  7. Hui-wen Zheng, Fei Gao, Ligong Bian, Si-xue Qin, and Yu-xin Liu, Quantitative analysis of the gravitational wave spectrum sourced from a first-order chiral phase transition of QCD, Phys. Rev. D 111, L021303 (2025), DOI 10.1103/PhysRevD.111.L021303, arXiv:2407.03795, https://arxiv.org/abs/2407.03795
  8. Song He, Li Li, Sai Wang, and Shao-Jiang Wang, Constraints on holographic QCD phase transitions from PTA observations, Sci. China Phys. Mech. Astron. 68, 210411 (2025), DOI 10.1007/s11433-024-2468-x, arXiv:2308.07257, https://arxiv.org/abs/2308.07257
  9. Zu-Cheng Chen, Shou-Long Li, Puxun Wu, and Hongwei Yu, NANOGrav hints for first-order confinement-deconfinement phase transition in different QCD-matter scenarios, Phys. Rev. D 109, 043022 (2024), DOI 10.1103/PhysRevD.109.043022, arXiv:2312.01824, https://arxiv.org/abs/2312.01824
  10. Man-Man Sun, Man-Li Tian, and Zhou-Run Zhu, Gravitational waves from holographic first-order QCD phase transition with magnetic field (2026), arXiv:2603.21960, https://arxiv.org/abs/2603.21960
  11. G. Lugones and J. E. Horvath, Primordial nuggets survival and QCD pairing, Phys. Rev. D 69, 063509 (2004), DOI 10.1103/PhysRevD.69.063509, arXiv:astro-ph/0310082, https://arxiv.org/abs/astro-ph/0310082
  12. Kiyoharu Kawana, Philip Lu, and Ke-Pan Xie, First-order phase transition and fate of false vacuum remnants, J. Cosmol. Astropart. Phys. 10 (2022) 030, DOI 10.1088/1475-7516/2022/10/030, arXiv:2206.09923, https://arxiv.org/abs/2206.09923
  13. Yang Bai and Mrunal Korwar, Radioactivity of quark nuggets, J. High Energy Phys. 06 (2025) 059, DOI 10.1007/JHEP06(2025)059, arXiv:2409.16487, https://arxiv.org/abs/2409.16487
  14. Edward Witten, Cosmic separation of phases, Phys. Rev. D 30, 272 (1984), DOI 10.1103/PhysRevD.30.272, https://doi.org/10.1103/PhysRevD.30.272