At the quantum chromodynamics (QCD) epoch, about 10^-5 seconds after the Big Bang, the Universe's strong-interaction content passed through a smooth crossover rather than a first-order phase transition, so no latent-heat-driven fluid motion is expected. Mourão Roque and Lugones (2013) showed, using one-dimensional relativistic hydrodynamic simulations with a lattice-QCD equation of state, that if temperature and velocity fluctuations of earlier origin (amplitude Δv/c≳10−2\Delta v/c\gtrsim 10^{-2} or ΔT/Tc≳10−3\Delta T/T_c\gtrsim 10^{-3}, correlation scale ≲1\lesssim 1 m) survive to this epoch, the resulting freely decaying turbulent motion radiates a stochastic gravitational-wave background potentially detectable by an eLISA/LISA-class detector above ∼10−4\sim 10^{-4} Hz. Their prediction rests on unverified ingredients: one-dimensional cascades with angular extrapolation, ideal (inviscid) hydrodynamics despite lepton- and neutrino-induced viscosity possibly exceeding the quark-gluon-plasma value by several orders of magnitude, and ad hoc initial perturbations whose physical origin and survival are unestablished. The open problem is the completion of this prediction: compute the QCD-epoch gravitational-wave signal in two and three dimensions with the lattice-QCD equation of state, quantify realistic viscous damping of the small-scale cascade, establish which mechanisms can supply the assumed fluctuations and whether they can survive to the QCD epoch, and determine the resulting robust detectability and separability from inflationary and other stochastic backgrounds.

Background

Quantum chromodynamics (QCD), the theory of the strong interaction, predicts that as the early Universe cooled to a temperature of order 150–170 MeV (about 10−510^{-5} s after the Big Bang), quarks and gluons condensed into hadrons. Lattice QCD simulations (discrete-space-time calculations of the thermal QCD partition function) established that at the small baryon chemical potential of the early Universe this quark-hadron transition is a smooth crossover — an analytic change in the equation of state (EoS) with no latent heat and no bubble nucleation — not a first-order phase transition (Aoki et al. 2006). This matters for gravitational-wave (GW) cosmology: a first-order transition would stir the primordial plasma through bubble collisions and turbulence, producing a stochastic gravitational-wave background (SGWB, an incoherent superposition of waves characterized by its fractional energy-density spectrum ΩGW(f)\Omega_{\mathrm{GW}}(f)). Standard modeling of such backgrounds (reviewed in Binétruy et al. 2012) assumes Kolmogorov turbulence, i.e. a stationary cascade in which energy injected at a stirring scale LL by continuous driving is transferred to smaller eddies and dissipated by viscosity at a damping scale lDl_D set by Reynolds number Re=vL/ν∼1\mathrm{Re}=vL/\nu\sim 1, with ν\nu the kinematic shear viscosity. The QCD crossover provides no such continuous stirring. Mourão Roque and Lugones (2013) therefore considered the opposite limiting case: temperature and/or velocity fluctuations supposed to have been generated by some earlier event (for example at the electroweak epoch — the earlier hot-Big-Bang era at which the electromagnetic and weak interactions first separated — giving a correlation scale L≲1L\lesssim 1 m, the size at the QCD epoch of a region that lay within the horizon at the electroweak epoch) and conjectured to survive until the QCD epoch because the Reynolds number of the primordial fluid is enormous. Evolving one-dimensional special-relativistic, inviscid hydrodynamics with a lattice-QCD EoS supplemented by leptons and photons, they found a freely decaying cascade whose velocity spectrum strongly deviates from the Kolmogorov law: because the physical viscous damping scale (lD<10−7l_D<10^{-7} m even for the most viscous composition, with neutrinos raising the shear viscosity by up to eight orders of magnitude over the quark-gluon-plasma value) lies far below the grid cell (Δx=0.02\Delta x=0.02 m), kinetic energy piles up at the smallest resolved scales. From the simulated anisotropic stress they computed the GW spectrum and concluded that fluctuations with Δv/c≳10−2\Delta v/c\gtrsim 10^{-2} and/or ΔT/Tc≳10−3\Delta T/T_c\gtrsim 10^{-3} would be detectable by the proposed eLISA/NGO (now LISA) space interferometer at frequencies above ∼10−4\sim 10^{-4} Hz. The authors themselves flagged several of the open issues this prediction rests on: the one-dimensional treatment (emission is axially anisotropic and multi-dimensional cascade dynamics is absent, differences 'to be expected... which will be addressed in future work'), the unknown origin, spectrum, coherence length, and generation mechanism of the assumed initial fluctuations, and the unexamined modification ('reprocessing') of any pre-existing inflationary GW spectrum by QCD-epoch physics. Viscosity, by contrast, was not flagged as an open issue: the authors argued that 'the viscosity is still very small and the Reynolds number very large', so that ideal (inviscid) hydrodynamics is 'fully justified'. The open viscous-physics question is instead exposed by the paper's own numbers — the physical damping scale lies far below the grid cell, so the sensitivity of the predicted GW spectrum to realistic lepton, photon, and especially neutrino dissipation was never quantified. These issues must all be resolved for the QCD-epoch turbulence GW prediction to be physically and observationally robust.

Problem Statement

Complete the quantitative prediction of the stochastic gravitational-wave background produced by hydrodynamic motion at the QCD epoch in the Standard-Model crossover scenario, to the point of a robust detectability statement, as framed by Mourão Roque and Lugones (2013). Concretely, the problem comprises four jointly supporting parts. (1) Multi-dimensional dynamics: replace the one-dimensional, angularly extrapolated calculation by genuinely two- and three-dimensional relativistic hydrodynamic simulations with the lattice-QCD equation of state (plus leptons and photons), establishing how the freely decaying cascade — whose one-dimensional treatment piles kinetic energy up at the smallest resolved scales and produces a non-Kolmogorov velocity spectrum — and the emitted GW spectrum change with dimensionality. (2) Viscous physics: quantify how realistic shear viscosity, including lepton, photon, and especially neutrino contributions that can exceed the quark-gluon-plasma value by several orders of magnitude, modifies the small-scale cascade, the velocity spectrum, and the GW output, given that the physical damping scale (≲10−7\lesssim 10^{-7} m) lies many orders of magnitude below any feasible simulation grid and must be treated by subgrid or analytic means. (3) Initial conditions: determine which physical mechanisms (electroweak-era or earlier events, phase transitions, cosmic strings or other topological defects left over from early-universe symmetry breaking, inflationary-era processes) can generate temperature and velocity fluctuations of the required amplitude (ΔT/Tc∼10−3\Delta T/T_c\sim 10^{-3}–10−210^{-2} and/or Δv/c∼10−2\Delta v/c\sim 10^{-2}) at correlation scales ∼1\sim 1 m and smaller, and whether such fluctuations can actually survive until the QCD epoch — noting that freely decaying turbulence dissipates within a few eddy turnover times, the characteristic time L/vL/v for flow at speed vv to cross the correlation scale LL. (4) Detectability and separation: determine, robustly with respect to dimensionality, viscosity treatment, and initial-condition statistics, the smallest fluctuation amplitudes for which the QCD-epoch signal is detectable by a LISA-class detector at frequencies ≳10−4\gtrsim 10^{-4} Hz, and whether such a contribution can be separated from the inflationary stochastic background and other radiation-era components, including how QCD-epoch physics modifies any pre-existing inflationary spectrum. Parts (1)-(4) are jointly required for an affirmative result; a negative resolution may instead rest on part (3) alone. A complete answer fixes the predicted ΩGW(f)\Omega_{\mathrm{GW}}(f) and its uncertainties, or demonstrates that no admissible initial-condition class makes the signal physically realizable or detectable, in which case it states which assumption of the source scenario fails.

References

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