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Very soon after the Big Bang, the universe enjoyed a brief phase where quarks and gluons roamed freely, not yet joined up into hadrons such as protons, neutrons and mesons. This state, called a , existed for a brief time until the temperature dropped to about 20 trillion Kelvin, after which this "hadronization" took place.

Now a research group from Italy has presented new calculations of the plasma's equation of state that show how important the strong force was before the hadrons formed. Their work is in ÌÇÐÄÊÓƵical Review Letters.

The equation of state of quantum chromodynamics (QCD) represents the collective behavior of particles that experience the strong force—a gas of strongly interacting particles at equilibrium, with its numbers and net energy unchanging. It's analogous to the well-known, simple equation of state of atoms in a gas, PV=nRT, but can't be so simply summarized.

But akin to a classical gas, a collection of QCD particles at equilibrium a temperature, pressure, energy density and entropy density, and can undergo phase transitions.

For the purposes here, the first important phase transitions are:

1. The electroweak phase transition occurring about 10^-12 seconds after the Big Bang at a temperature around 10¹⁵ Kelvin (K), where the electromagnetic and weak interactions split and particles gain mass via the . Multiplied by Boltzmann's constant, that temperature is an energy scale of about 100 giga electronvolts (GeV).
2. The QCD phase transition is the point of hadronization, where quarks and gluons in the quark gluon plasma begin to separate into protons, neutrons (each comprised of three quarks) and mesons (usually two quarks), about a microsecond (10^-6) after the Big Bang where the universe's temperature is 10¹² K, or an energy scale of 150 million electronvolts (MeV). In between these phase transitions is the quark gluon plasma phase, lasting about a microsecond, by the SPS Heavy Ion Program at CERN in 2000.

However, straightforward QCD does not explain its equation of state. Perturbation theory, a major weapon in a physicists' arsenal where Feynman diagram terms are calculated via powers of the coupling constant, does not work for the strong interaction as it does for the electromagnetic interaction and quantum electrodynamics (QED), where the coupling constant is the small, fine structure constant of approximately 1/137.

There, powers of the coupling constant—its square, its cube, etc.—quickly get smaller and smaller. QCD is much more complicated because the coupling constant is not small. The theory is non-abelian (unlike QED's photon that carries no electromagnetic change, QCD's force carriers, the gluons, do hold a color charge (two of them, in fact—a color and an anti-color).

Furthermore, QCD's coupling constant varies with the energy of the interaction—at small distances the interaction is small, but at large distances the force is huge, manifesting in .

So physicists have turned to lattice QCD to compute the equation of state.

In lattice QCD, spacetime is divided up into discrete points on a four-dimensional cube, and properties of the spacetime of QCD interactions calculated pointwise in a nonperturbative fashion, not utilizing Feynman diagrams.

In the end, the distance between the spacetime points is made smaller and smaller, but it still takes supercomputers to run most lattice QCD calculations.

Using lattice QCD, researchers at the University of Milano-Bicocca and the National Institute for Nuclear ÌÇÐÄÊÓƵics (INFN) in Italy set out to determine QCD's equation of state from a temperature of 3 GeV to the electroweak transition.

They focused on a strongly interacting system of massless particles of quarks, where most of their mass is tied up in the gluon fields surrounding them and are less than 500 MeV/c² at this temperature scale (so approximately zero relative to the energy of the plasma.)

The researchers say "the computational strategy is entirely new, and we focus on the theory with three flavors of massless quarks," by three of the co-authors in 2022. While highly technical, in essence the new strategy uses to study lattice QCD from low to very high temperatures from first principles, obtaining numerical results from random sampling.

After computing, the group obtained the equation of state for the entropy density of a quark gluon plasma from temperatures of 3 GeV to 165 GeV for three quark flavors, up to the temperature of the electroweak transition, expressing it numerically as a seventh-order polynomial (sum of powers) of a strong force coupling constant that is itself a function of temperature.

They took a limit numerically to in effect reduce the lattice spacing to zero, so their results apply to the real world continuum.

"Lattice artifacts turn out to be rather mild," they conclude. This is a large improvement on previous quark gluon plasma simulations, which were limited to temperatures below 1 GeV.

From the entropy density, the pressure and energy density were calculated by . They also determined that the pressures they calculated could not be accurately described by a model of weakly interacting quarks and gluons, indicating the strong force was influential in the early universe sooner after the Big Bang than previously thought.

To go further, they say they need faster computers or more computer time: "the numerical results presented here can indeed be systematically improved in the future by investing more computational resources."