Notes

(1) This relation is modified if some of the final-state particles are indistinguishable from those in an ambient gas. The necessary quantum statistics corrections are given in appendix A.

(2) This result holds if all particles obey Maxwell-Boltzmann statistics. The modified form with quantum statistics is given in Eq. (A.23).

(3) This result may essentially be obtained by Newtonian considerations. Taking the universe to contain a homogeneous gas with density the energy equation for a unit mass shell of the gas with radius becomes where is the total mass contained within the shell (relativistic considerations show that pressure terms do not contribute). At least in the early universe the total energy may be neglected with respect to the separate kinetic and potential energies and

(4) The temperature dependence of the expansion rate in eq. (2.1.14) obtains only in a homogeneous universe.

(5) We comment on the consistency of this assumption in sect. 4.

(6) This increase may be considered to result from the appearance of a bulk (or second) viscosity [24] which opposes the expansion, vanishes for ideal gases in the limits and but is positive at intermediate temperatures (and is typically proportional to the relaxation time).

(7) These equations are entirely classical. Identical particle corrections are discussed in subsect. 2.4. Genuine quantum-mechanical interference effects should be important only when the mean distance between successive collisions is shorter than the wavelengths of the participating particles. This circumstance may well occur at high temperatures, and a discussion of its consequences is given in sect. 4. The rough number of particles in causal contact at a temperature T is given (see subsect. 2.4) roughly by and is therefore sufficiently large at the temperatures we consider for statistical methods to be applicable.

(8) A model similar to this was proposed in ref. [6].

(9) We may take the two-body decay matrix element to be independent of any momenta. Even terms must vanish on averaging over initial and final spins.

(10) In addition to the finite intrinsic width of the resonance, there should be additional collision broadening at high temperatures; the resonance width should become of order the inverse time between collisions or (see sect. 4). However, most decay at temperatures where such effects are probably negligible. (Doppler broadening, familiar from the spectra of hot gases, is irrelevant here; it serves only to smear the energies of emitted in decay.)

(11) Since quarks have third-integer electric charges, but leptons integer ones, -violating bosons in grand unified gauge models typically carry electric charge, and therefore cannot mix with their antiparticles.

(12) In general

while

(13) If a massive species survives for a long time between decoupling from thermal equilibrium and decaying, the large deviations from equilibrium can occur, and large amounts of entropy may be generated [31].

(14) Much more complicated behavior, perhaps with can be obtained by introducing several coupled Higgs fields.

(15) In this equation, we have approximated the intermediate propagator by its zero-temperature form. As discussed in sect. 4, the effective mass at high temperatures is probably given by an inverse Debye screening length.

(16) When applied solely to the matter in the universe, but not to the gravitational field generated by it.

(17) For two quarks within a proton, GeV), while if the quarks were free in an ideal gas of number density

(18) Note that even within this specific model, only leading log contributions are included in deducing [16]; the subheading log terms have not yet been calculated using consistent prescriptions.

(19) Note, of course, that by construction so that models giving too small even when should be considered in disagreement with the standard cosmology.

(20) In fact, in the presence of ``long-range forces'' (acting over times longer than the collision time), the Boltzmann equation ceases to be applicable. The equation assumes that the momenta of particles are uncorrelated before each scattering, but ignores processes involving more than two initial particles (e.g., two sequential two-body interactions). When long-range forces exist, the effects neglected in this way become important, and one must formally resort to more complicated equations [19]. So long as kinetic equilibrium prevails, however, consideration of an effective screened cross section should be adequate.

(21) In electron-ion plasmas, the effective Coulomb cross section involves a log, rather than a power of in that case, the change only if scatterings deflect particles into a different momentum state, so that for the relevant effective total cross section, the differential cross section is weighted by the change in momentum. Here, the may change due to changes in quantum numbers alone, with no change in momentum.

(22) Bounds on Higgs couplings [20] based on the structure of the effective potential (vacuum energy density as a function.of Higgs field strength) responsible for spontaneous symmetry breakdown need only apply at low temperatures at higher temperatures, large thermal fluctuations in the fields restores the broken symmetry (see subsect. 2.4) and render the effective potential irrelevant.

(23) Recall that asymptotically free coupling constants behave as and diverge in the small infrared region, while asymptotically strong couplings behave according to For QED,

(24) Which represent (in ladder approximation) only spin-0 exchanges, yielding universally attractive forces.

(25) At high temperatures, the Boltzmann distribution of the screening particles is reflected in the exponential form for the screened potential. In a degenerate Fermi gas, the potential again reflects the particle distribution, and .

(26) One might expect that the equilibrium condition (A.8) could be deduced from for a single state However, this implies only

individual terms could be non-zero but cancel in the complete sum. Consideration of yields thus providing an alternative derivation of (A.8).

(27) The proof given here finally dispels doubts raised e.g., in refs. [25,30].

(28) The factor arises because a phase-space cell containing a single mode of the field has volume rather than .

(29) For these expansions, we used the relations (valid for

The last result may trivially be derived from the series expansion in (C.26). It is also convenient to apply the relation

to obtain (for

(30) For these expansions we used the result

where the second term is absent for

previous