Order of the cosmological QCD confinement transition, stability of strange quark matter, and survival of quark-nugget relics: the open questions of Witten's 'Cosmic separation of phases'

In 'Cosmic separation of phases', Witten (1984) argued that a first-order QCD phase transition in the hot early universe would segregate the tiny baryon asymmetry into dense quark nuggets that could constitute the dark matter, and posed seven linked open questions: whether real QCD makes the cosmological transition first order; what nucleates it and at what mean spacing; whether shrinking high-temperature regions cool mainly by neutrino emission (which traps baryons) or by surface evaporation (which ejects them); whether strange quark matter is stable; whether nascent nuggets survive evaporation before neutrino decoupling; and whether the QCD epoch produces detectable gravitational waves or primordial black holes. Lattice QCD has since answered the first question within the Standard Model: at the small baryon chemical potential of the early universe the transition is a smooth crossover, so no latent heat, bubble nucleation, or baryon segregation occurs in the minimal theory. The surviving open core is precise and nonempty: whether strange quark matter is stable (the Witten hypothesis); whether non-minimal but cosmologically admissible conditions (large baryon density, large lepton asymmetry, primordial magnetic fields, or beyond-Standard-Model sectors) make the transition first order with relic quark nuggets, gravitational waves, or primordial black holes; and whether any such relics are observable today.

Background

Quantum chromodynamics (QCD) is the $SU(3)$ gauge theory of the strong interaction. At temperatures of order $150\ \mathrm{MeV}$ (about $10^{12}\ \mathrm{K}$), hadronic matter deconfines into a quark-gluon plasma, a phase of quasifree quarks and gluons. The hot Big Bang passed through this transition roughly ten microseconds after the beginning. If the transition were first order — like boiling water, with a latent heat and a metastable supercooled phase — the universe would briefly have been a mixture of expanding hadronic bubbles and shrinking droplets of quark-gluon plasma. In 1984 Witten analyzed this scenario and found a surprising consequence: because the primordial baryon asymmetry (the tiny excess $\eta\sim10^{-10}$ of baryons over antibaryons) is carried by nearly massless quarks in the high-temperature phase but by baryons of mass $938\ \mathrm{MeV}$ in the low-temperature phase, thermal equilibrium expels baryon number from the hadronic bubbles into the quark phase. If the shrinking droplets then lose heat mainly by emitting neutrinos — which escape from within one neutrino mean-free path ($\sim10\ \mathrm{cm}$) of the surface and carry heat but no baryon number — their baryon excess becomes trapped and concentrated, ending in dense quark nuggets with masses of order $10^{9}$–$10^{18}$ grams (baryon numbers of order $10^{32}$–$10^{42}$) that could survive as an invisible dark-matter component made of QCD matter alone.

Witten was explicit that this chain was conjectural. The two competing heat-loss channels — neutrino emission versus surface evaporation, which removes baryons along with energy — appear comparable, and only neutrino-dominated cooling traps baryons. The characteristic bubble spacing $R_0$ set by nucleation was unknown: thermal-fluctuation nucleation theory (Hogan) gives a fraction $\epsilon=R_0/R_H\lesssim10^{-2}$ of the horizon radius $R_H$ (the scale beyond which causal processes cannot act), while nonthermal agents — impurities, magnetic monopoles, cosmic strings (hypothetical one-dimensional topological defects), or pre-existing turbulence — could nucleate at much larger separations. The whole scenario moreover requires strange quark matter — deconfined $u,d,s$ matter, studied contemporaneously by Farhi and Jaffe with the bag model, which treats quarks as a perturbative Fermi gas confined by a vacuum-energy 'bag' — to have an energy per baryon below $938\ \mathrm{MeV}$; contemporary estimates carried uncertainties of order $100\ \mathrm{MeV}$ per baryon, leaving four possibilities open: absolutely stable strange matter, metastability only in finite droplets, instability only through weak interactions, or instability to strong decay. Finally, Witten noted in appendices that horizon-scale nucleation could produce a stochastic gravitational-wave background of primordial amplitude $h_0\sim\epsilon$ (detectable by pulsar timing with then-available technology if $\epsilon\gtrsim1/10$) and that in a fraction $\sim10^{-4}$ of horizon volumes colliding bubble walls could collapse into primordial black holes (black holes formed in the early universe rather than by stellar collapse).

The modern tools relevant to these questions are: lattice QCD — Monte Carlo simulation of the theory on a discretized space-time grid, which fixes the equilibrium thermodynamics at physical quark masses; at nonzero baryon chemical potential $\mu_B$ the fermion determinant becomes complex (the 'sign problem'), so the phase boundary is probed indirectly by Taylor expansion, reweighting, analytic continuation from imaginary $\mu_B$, or functional (Dyson–Schwinger) methods, which solve the nonperturbative continuum equations for the quark and gluon propagators; pulsar timing arrays, which now measure nanohertz gravitational-wave strains; strangelet searches in cosmic rays, lunar soil, and heavy-ion collisions; and microlensing, merger, and evaporation constraints on primordial black holes.

Problem Statement

The problem is the question Witten (1984) left open about the epoch, roughly ten microseconds after the Big Bang, when the universe cooled through the QCD confinement temperature $T_c$:

(1) Order. Determine, using the actual QCD parameters and under non-minimal but cosmologically admissible conditions (a large baryon density, a large lepton asymmetry, primordial magnetic fields, or beyond-Standard-Model sectors), whether the confinement transition is first order — with latent heat, supercooling, and nucleation of low-temperature-phase bubbles at mean spacing $R_0$, where $\epsilon=R_0/R_H$ is the ratio to the horizon radius — or an analytic crossover; in the first-order case, determine the nucleation mechanism (thermal fluctuations versus nonthermal agents such as impurities, magnetic monopoles, cosmic strings, or pre-existing turbulence) and the resulting value of $\epsilon$.

(2) Fate of baryon number. In a first-order transition, determine the dominant heat-loss channel of the shrinking high-temperature regions — neutrino emission from within one neutrino mean-free path of the surface, which traps and concentrates the baryon excess, versus surface evaporation, which ejects baryons — and hence the survival probability and mass spectrum of the quark lumps formed, over the temperature range from $T_c$ down to neutrino decoupling (the point, at $T\sim1$–$2\ \mathrm{MeV}$, where neutrinos cease interacting with the cosmic plasma).

(3) Stability of quark matter. Determine whether strange quark matter — deconfined $u,d,s$ matter with energy per baryon below $938\ \mathrm{MeV}$ — is absolutely stable, metastable only in finite droplets, unstable only through weak interactions, or unstable to strong decay.

(4) Relics. Determine whether any relics of the QCD epoch survive to the present — quark nuggets constituting (part of) the dark matter, primordial black holes from the rare tail of horizon-volume collapses, or a stochastic gravitational-wave background of primordial amplitude $h_0\sim\epsilon$ — and predict their abundance, mass or frequency spectrum, and detectability.

Component (1) is already settled for the minimal Standard Model at the negligible baryon chemical potential of the early universe: the transition is an analytic crossover there (Aoki et al. 2006; Bazavov et al. 2012), so that subcase belongs to the answer rather than the open question; the open content of (1) is whether non-minimal but cosmologically admissible conditions — a large lepton or baryon asymmetry, primordial magnetic fields, or beyond-Standard-Model sectors — make the transition first order.

These four components form one scientific objective: to establish, from QCD and cosmology, what the confinement epoch did to baryon number and whether it left observable relics. Each component is posed at the level of the real QCD parameters and the stated cosmological conditions, not only within an assumed first-order toy model. Cited results that analyze the transition or its relics within a specific non-minimal scenario — a large lepton asymmetry, a strong primordial magnetic field, or a large baryon density — are conditional on that scenario's cosmological admissibility.

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