Quark-hadron phase transition

3.4 Quark-hadron phase transition

The standard picture of cosmology assumes that a phase transition (associated with chiral symmetry breaking following the electroweak transition) occurred at approximately 10^–5 s after the Big Bang to convert a plasma of free quarks and gluons into hadrons. Although this transition can be of significant cosmological importance, it is not known with certainty whether it is of first order or higher, and what the astrophysical consequences might be on the subsequent state of the Universe. For example, the transition may play a potentially observable role in the generation of primordial magnetic fields. The QCD transition may also give rise to important baryon number inhomogeneities which can affect the distribution of light element abundances from primordial Big Bang nucleosynthesis. The distribution of baryons may be influenced hydrodynamically by the competing effects of phase mixing and phase separation, which arise from any inherent instability of the interface surfaces separating regions of different phase. Unstable modes, if they exist, will distort phase boundaries and induce mixing and diffusive homogenization through hydrodynamic turbulence [102

, 112 , 95 , 4 , 137 ].

In an effort to support and expand theoretical studies, a number of one-dimensional numerical simulations have been carried out to explore the behavior of growing hadron bubbles and decaying quark droplets in simplified and isolated geometries. For example, Rezolla et al. [138 ] considered a first order phase transition and the nucleation of hadronic bubbles in a supercooled quark-gluon plasma, solving the relativistic Lagrangian equations for disconnected and evaporating quark regions during the final stages of the phase transition. They investigated numerically a single isolated quark drop with an initial radius large enough so that surface effects can be neglected. The droplet evolves as a self-similar solution until it evaporates to a sufficiently small radius that surface effects break the similarity solution and increase the evaporation rate. Their simulations indicate that, in neglecting long-range energy and momentum transfer (by electromagnetically interacting particles) and assuming that baryon number is transported with the hydrodynamical flux, the baryon number concentration is similar to what is predicted by chemical equilibrium calculations.

Kurki-Suonio and Laine [108 ] studied the growth of bubbles and the decay of droplets using a one-dimensional spherically symmetric code that accounts for a phenomenological model of the microscopic entropy generated at the phase transition surface. Incorporating the small scale effects of finite wall width and surface tension, but neglecting entropy and baryon flow through the droplet wall, they simulate the process by which nucleating bubbles grow and evolve to a similarity solution. They also compute the evaporation of quark droplets as they deviate from similarity solutions at late times due to surface tension and wall effects. Update

Ignatius et al. [96 ] carried out parameter studies of bubble growth for both the QCD and electroweak transitions in planar symmetry, demonstrating that hadron bubbles reach a stationary similarity state after a short time when bubbles grow at constant velocity. They investigated the stationary state using numerical and analytic methods, accounting also for preheating caused by shock fronts.

Figure 5: Image sequence of the scalar field from a 2D calculation showing the interaction of two deflagration systems (one planar wall propagating from the right side, and one spherical bubble nucleating from the center). The physical size of the grid is set to 1000 × 1000 fm and resolved by 512 × 512 zones. The run time of the simulation is about two sound crossing times, where the sound speed is $c∕ √3-$ , so the shock fronts leading the condensing phase fronts travel across the grid twice. The hot quark (cold hadron) phases have smaller (larger) scalar field values and are represented by black (color) in the colormap.

Figure 6: Image sequence of the scalar field from a 2D calculation showing the interaction of two detonation systems (one planar wall propagating from the right side, and one spherical bubble nucleating from the center). The physical size of the grid is set to 1000 × 1000 fm and resolved by 1024 × 1024 zones. The run time of the simulation is about two sound crossing times.

Figure 7: Image sequence of the scalar field from a 2D calculation showing the interaction of shock and rarefaction waves with a deflagration wall (initiated at the left side) and a detonation wall (starting from the right). A shock and rarefaction wave travel to the right and left, respectively, from the temperature discontinuity located initially at the grid center (the right half of the grid is at a higher temperature). The physical size of the domain is set to 1806.1 × 451.53 fm and resolved by 2048 × 512 zones. The run time of the simulation is about two sound crossing times.

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Fragile and Anninos [76 ] performed two-dimensional simulations of first order QCD transitions to explore the nature of interface boundaries beyond linear stability analysis, and determine if they are stable when the full nonlinearities of the relativistic scalar field and hydrodynamic system of equations are accounted for. They used results from linear perturbation theory to define initial fluctuations on either side of the phase fronts and evolved the data numerically in time for both deflagration and detonation configurations. No evidence of mixing instabilities or hydrodynamic turbulence was found in any of the cases they considered, despite the fact that they investigated the parameter space predicted to be potentially unstable according to linear analysis. They also investigated whether phase mixing can occur through a turbulence-type mechanism triggered by shock proximity or disruption of phase fronts. They considered three basic cases (see image sequences in Figures 5 , 6 , and 7 below): interactions between planar and spherical deflagration bubbles, collisions between planar and spherical detonation bubbles, and a third case simulating the interaction between both deflagration and detonation systems initially at two different thermal states. Their results are consistent with the standard picture of cosmological phase transitions in which hadron bubbles expand as spherical condensation fronts, undergoing regular (non-turbulent) coalescence, and eventually leading to collapsing spherical quark droplets in a medium of hadrons. This is generally true even in the detonation cases which are complicated by greater entropy heating from shock interactions contributing to the irregular destruction of hadrons and the creation of quark nuggets.

However, Fragile and Anninos also note a deflagration ‘instability’ or acceleration mechanism evident in their third case for which they assume an initial thermal discontinuity in space separating different regions of nucleating hadron bubbles. The passage of a rarefaction wave (generated at the thermal discontinuity) through a slowly propagating deflagration can significantly accelerate the condensation process, suggesting that the dominant modes of condensation in an early Universe which super-cools at different rates within causally connected domains may be through supersonic detonations or fast moving (nearly sonic) deflagrations. A similar speculation was made by Kamionkowski and Freese [102 ] who suggested that deflagrations become unstable to perturbations and are converted to detonations by turbulent surface distortion effects. However, in the simulations, deflagrations are accelerated not from turbulent mixing and surface distortion, but from enhanced super-cooling by rarefaction waves. In multi-dimensions, the acceleration mechanism can be exaggerated further by upwind phase mergers due to transverse flow, surface distortion, and mode dissipation effects, a combination that may result in supersonic front propagation speeds, even if the nucleation process began as a slowly propagating deflagration.