Appendix C

Number Density Integrals

In this appendix, we give some integrals of equilibrium phase-space distributions used in sect. 2. As elsewhere, we take units such that ( is Boltzmann's constant)

We consider a uniform ideal gas of particles with mass in thermal equilibrium at temperature , and write

The number density of such particles in phase space is given by (see subsect. 2.1) (29)

where for particles obeying Fermi-Dirac (FD) (Bose-Einstein (BE)) statistics, and in the classical (distinguishable particles) approximation of Maxwell-Boltzmann (MB) statistics, used extensively above. gives the number of spin states for each particle. [Usually for massive particles and for However, because the interaction cross sections for the various spin states of a weakly interacting particle may differ, its full complement of spin states may not be in thermal equilibrium at a particular temperature. Thus, for example, the second spin state of a neutrino carrying a small but non-zero mass may not be in thermal equilibrium if only one of its spin states may participate directly in weak interactions.] The appearing in eq. (C.3) is a possible chemical potential which serves to constrain the total number of particles. When a species of particles are distributed in phase space according to eq. (C.3), they are said to be in kinetic equilibrium. They are in chemical equilibrium only if (unless they carry an absolutely conserved quantum number, such as electric charge, with respect to which the complete system is not neutral). Baryons in the early universe should quickly be brought into kinetic equilibrium by collisions; only much slower - and -violating scatterings can produce chemical equilibrium, with

The total particle number density may be obtained by integrating (C.3) over the available momentum states:

C.1. Number Densities with Maxwell-Boltzmann Statistics

In the approximation of Maxwell-Boltzmann statistics, the number density integral becomes

This integral may be expressed in terms of a modified Bessel function (using the notation of ref. [27])

For large x (corresponding to low temperatures) the asymptotic expansion of the Bessel function

gives the usual Boltzmann factor

At small (high temperatures) expansion of the Bessel function

gives

where is Euler's constant. Hence the ratio of the number density of a massive species of particles to the number density of photons at high temperatures is roughly

where, for consistency, we have approximated the photon number distribution by Maxwell-Boltzmann statistics.

C.2. Moments of the Maxwell-Boltzmann Distribution

The mean time dilation in the decay lifetime of Maxwell-Boltzmann particles in thermal equilibrium at a temperature is given by

At high temperatures (small ) the particles are relativistic, with energies , and

while in the non-relativistic limit (high ):

Next, we consider the mean energy of the particles in a Maxwell-Boltzmann gas. This may be found directly by integrating (C.1) with weight However, we shall here use a less direct method, since it introduces several useful results. In non-relativistic classical statistical mechanics the equipartition theorem states that each (quadratic) degree of freedom of a particle which is Boltzmann distributed has a mean thermal energy of To obtain the relativistic generalization of the equipartition theorem one must find a quantity whose mean value

depends only on and not on or A suitable such quantity is

where labels some component of the three-momentum . Inserting the choice (C.16) into (C.15) one finds on integrating by parts:

for all and In the non-relativistic limit, so that and summing over one regains the standard result that the mean kinetic energy of a particle in thermal equilibrium at a temperature is For relativistic particles, so that

regardless of the value of (or ). This result is therefore a relativistic generalization of the equipartition theorem. One may now use eqs. (C.12) and (C.18) to find

Eq. (C.6) then gives the energy density of the gas:

C.3. Number Densities with Quantum Statistics

For fermions (bosons) the number density integral (C.4) becomes

which cannot be expressed in terms of the usual special functions, but may formally be written as

since falls off exponentially for large this series converges rapidly, and is convenient for numerical evaluation. However, for small the series is not uniformly convergent. Nevertheless it is easy to find the high temperature behavior of eq. (C.21) when

where while for Maxwell-Boltzmann statistics, eq. (C.10) gives

At low temperatures, of course, the effects of quantum statistics (represented by the terms in (C.22) with ) are exponentially unimportant, and thus [from eq. (C.8)]

[Further terms in the asymptotic series are given in eq. (C.37)]. The ratio goes monotonically from to 1 as goes from to the largest changes occur around

C.4. Massless Particle Number Densities

For massless particles, it is simple to perform the integrals (C.4) retaining the chemical potential in terms of the polylogarithm functions [28]

using the formula

One finds (30)

where for the Bose-Einstein case. [If the number density of bosons in a system increases above its value given above when then Bose condensation must occur: many particles collect in the ground state and the approximation of a sum over discrete energy levels by the integral (C.4) is no longer satisfactory (so that finite volume effects become important).]

C.5. Energy Densities

The expansion rate of the early universe is determined by its energy density, which is conveniently parametrized in terms of the ``effective number of species in thermal equilibrium'', defined by

where is the energy density of a (genuine Bose-Einstein) photon gas

The contribution of a massive particle with to is given by eq. (C.25) as

which is usually negligibly small. Eq. (C.20) gives us complete form of for particles obeying Maxwell-Boltzmann statistics. At high temperatures, one finds

For zero-mass particles with non-zero chemical potential (31)

where the expansions are for The mean energies for zero-mass particles in thermal equilibrium with are given by

while the dispersions of the energy distributions about these means are

Since no known bosons carry absolutely conserved quantum numbers (other than electric charge) it seems unlikely that a high chemical potential for a boson species, leading to Bose condensation, could be enforced in the early universe. However, according to ``cold'' models for the early universe (discussed in sect. 4), degenerate Fermi gases existed at early times, having

C.6. Non-relativistic Limit of Quantum Statistics

In the non-relativistic limit , eq. (C.4) becomes

which may be written in terms of polylogarithm functions

where from eq. (C.26)

while

(Note that for has an infinite derivative at The energy densities of massive fermions and bosons in thermal equilibrium at low temperatures are given by

C.7. A Two-particle Integral

In subsect. 2.3, the two-particle integral

where appeared in connection with the rate for scattering of massless b via nearly on-shell exchange. Performing the integral over c.m. angles gives

where is a modified Bessel function. Making use of (C.6) and (C.12), may be written as [cf., (2.3.17)

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