Chapter 2Quark-Gluon Plasma and the EarlyUniverseThere is now considerable evidence that the universe began as a fireball, the so called “Big-Bang”,with extremely high temperature and high energy density. At early enough times, the temperaturewas certainly high enough (T > 100 GeV) that all the known particles (including quarks, leptons,gluons, photons, Higgs bosons, W and Z) were extremely relativistic. Even the “strongly inter-acting” p articles, quarks and gluons, would interact fairly weakly due to asymptotic freedom andperturbation theory should be s ufficient to describe them. Thus this was a system of hot, weaklyinteracting color-charged particles, a quark-gluon plasma (QGP), in equilibrium with the otherspecies.Due to asymptotic freedom, at sufficiently high temperature the quark-gluon plasma can bewell-describ ed using statistical m echanics as a free relativistic parton gas. In this Chapter, weexplore the physics of QGP, perhaps the simplest system of strong-interaction particles that existsin the context of QC D. As the universe cooled during the subsequent expansion phase, the quarks,antiquarks, and gluons combined to form hadrons resulting in the baryonic matter that we observetoday. The transition from quarks and gluons to baryons is a fascinating subject that has beendifficult to address quantitatively. However, we will discuss this transition by considering the basicphysics issu es without treating the quantitative details. At p resent there is a substantial effort intheoretical physics to address this transition by using high-level computational methods known aslattice gauge theory. This s ubject is somewhat technical and we will discuss it only very briefly.However, the general features that h ave emerged from lattice studies to date are rather robust andcan be discussed in some detail.The relatively cold matter that presently comprises everything around us is actually a residu eof the annihilation of matter and anti-matter in the early universe. The origin of the matter-antimatter asymmetry wh ich is critical for generating the small amount of residual matter is stilla major subject of study, and we discuss this topic at the end of this Chapter.Another major th rust associated with the transition between the QGP and baryonic matter isthe experimental program underway to study observable phenomena associated with the dynamicsof this interface. This experimental program involves th e collision of relativistic heavy ions thatshould produce (relatively) small drops of QGP. Large particle detector systems then enable studiesof the products of these collisions, which can (in principle) yield information on the transition tothe baryonic phase and the QGP itself. The program of experiments and the present state of the202.1. THERMODYNAMICS OF A HOT RELATIVISTIC GAS 21experimental data will be discussed in Chapter XX.2.1 Thermodynamics of A Hot Relativistic GasAt very high-temperature such that the particles h ave energy much larger than their rest mass,we m ay describe them using relativistic kinematics and ignore their masses. Thus these energeticweakly interacting particles form a system that is, to an excellent approximation, a hot relativisticfree gas. Since particles and antiparticles can be created and annihilated easily in such an environ-ment, their densities are much higher than their differences. Therefore the chemical potential µ canbe neglected. The number densities of the p artons (species i) are then described by the quantumdistribution functionsni=Zd3pi(2π)31eβEi± 1, (2.1)where β = 1/kBT and the − sign is for bosons and the + is for fermions. For relativistic particles,pi= Ei. For Eiβ < 1, the exponential factor is small and there is a large difference betweenfermions and bosons. For Eiβ ≥ 1 the ±1 becomes increasingly unimportant, and the distributionsbecome similar. Integrating over the phase space, one fin ds,ni=(ζ(3)/π2T3(boson)(3/4)ζ(3)/π2T3(fermion)(2.2)where ζ(3) = 1.20206... is a Riemann zeta fu nction. The T3-dependence follows simply fromdimensional analysis (the Boltzmann constant kBcan also be taken to be 1).The energy density for a free gas can be computed from the same quantum distribution func-tions:ǫi=Zd3pi(2π)3EieβEi± 1=(π2/30T4(boson)(7/8)(π2/30)T4(fermion)(2.3)where th e fermion energy density is 7/8 of that of boson.These expressions are valid for each s pin/flavor/charge/color state of each particle. For a systemof fermions and bosons, we need to include separate degeneracy factors for the various p articles:ǫ =Xigiǫi= g∗π230(kBT )4, (2.4)where g∗=gb+78gfwith gband gfare the degeneracy factors for bosons and fermions, respec-tively. Each of these degeneracy factors counts the total number of degrees of freedom, summedover the spins, flavors, charge (particle-antiparticle) and colors of particles. When some species arethermally decoupled from others due to the absence of interactions (such as neutrinos at presentepoch), they no longer contribute to the degeneracy factor. For example, at temperature above 100GeV, all particles of the standard model are present. At lower temperatures, the W an d Z bosons,22 CHAPTER 2. QUARK-GLUON PLASMA AND THE EARLY UNIVERSEtop, bottom, and charm quarks freeze out and g∗decreases. Therefore g∗is generally a decreasingfunction of temperature.We can now calculate the contribution to the energy d en sity from the quark-gluon plasma as arelativistic free parton gas. For a gluon, there are 2 helicity states and 8 choices of color so we havea total degeneracy of gb= 16. For each quark flavor, there are 3 colors, 2 spin states, and 2 chargestates (corresponding to quarks and antiquarks). At temperatures below kBT ∼ 1 GeV, there are3 active flavors (up, down and strange) so we expect the fermion degeneracy to be a large numberlike gf≃ 36 in this case. Thus we expect for the QGP:ǫQGP≃ 47.5π230(kBT )4. (2.5)With two quark flavors, the pr efactor is g∗= 37. (For reference, if one takes into account allstandard model particles, g∗= 106.75.)The pressure of the free gas can be calculated just like the case of black-body radiation. Forrelativistic species,p =13ǫ , (2.6)which is the equation of state.To calculate the entropy of the relativistic gas, we consider the thermodynamics relation, dE =T dS − pdV . At constant volume we would have just dE = T dS, or dǫ = T ds where ǫ (s) is theenergy (entropy) per unit volume. Since ǫ ∝ T4, we can easily find