4. Statistical Mechanics of Cosmological Baryon Number Generation

4.1. Introduction

Having outlined above the possible forms of -, -violating interactions, we now consider how these may act in the early universe to generate an excess of baryons over antibaryons from an initially symmetric state [3,4,6,7,24,25,26,27,28,29,30] (the destruction of a possible initial baryon number was described in sect. 4 of I, and will not be discussed here). If all particles in the universe remained in thermal equilibrium, then no direction for time would be defined, and invariance would render violation in the interactions irrelevant, and prevent the appearance of any baryon excess [see (I.4.1) and appendix A of I]. We shall assume that the early universe is homogeneous and isotropic (Friedmann model). Deviations from thermal equilibrium cannot occur in a homogeneous isotropic universe containing only massless particles: if massive particles are present, such deviations may occur when .

We shall assume here that all particles obey Maxwell-Boltzmann statistics (the negligible corrections from proper use of Fermi-Dirac or Bose-Einstein statistics were discussed in subsect. 2.4 of I). In this section, we make the simplification that all particles have only one spin/color state; the correct counting of states will be included in sects. 6 and 7. In thermal equilibrium, therefore, the density of a species of particles (with mass ) in phase space is given by (5)

where is the temperature of the species, and is a possible chemical potential. Note that, in keeping with the standard and simplest cosmological model, we assume throughout that the universe may be treated as homogeneous and isotropic. The total number density in equilibrium of the species is given from (4.1.1) by

where is a modified Bessel function (see appendix C of I). For , (4.1.2) becomes

while for ,

At early times, most of the energy density of the universe was presumably contained in essentially massless particles. Their chemical potentials were probably very small, since the universe appears to carry zero or very small net quantum numbers. Hence the energy density contributed by each species was

we shall denote such species generically as , and take them to be in number (for , typical grand unified gauge models imply ).

The Robertson-Walker scale factor for a homogeneous universe of total energy density expands at a rate

where is the Planck mass. In this expansion, the momenta of all particles are redshifted according to . The phase space distributions for massless particles (in thermal equilibrium, and with ) remain self-similar under this resealing, with a temperature . Using (4.1.5), eq. (4.1.6) then implies (6)

where is the number of (Maxwell-Boltzmann) particle species. The equilibrium phase-space densities of massive particles are not, however, self-similar in the expansion : their forms change when becomes smaller than . Thus the expansion of the universe may cause these phase-space densities to deviate from their equilibrium form: collisions with light particles will nevertheless restore the equilibrium form, but only after a finite relaxation time. During this temporary deviation from equilibrium -, -violating processes may generate a baryon asymmetry: under certain conditions this asymmetry may survive even after the massive particles have disappeared (through decays); the time necessary to relax the asymmetry may increase faster than the age of the universe.

The Boltzmann equation will be used below to describe the effect of reactions on the densities of particle species or quantum numbers. The validity of this approach is discussed in subsect. 4.5.

4.2. Evolution of a Single Massive Particle Species

Consider a particle of mass , which decays with a width into two massless particles . Then the number density of evolves with time in the early universe according to

(for the derivation of this result from the complete Boltzmann transport equation for the phase-space density see subsect. 2.3 of I). The expansion of the universe appears only implicitly in (4.2.1) through the dependence of the temperature in on the time : the explicit dependence has been removed by consideration of the scaled quantity . The first term in (4.2.1) accounts for the decays , and has the usual radioactive decay form. The second term in (4.2.1) accounts for inverse decay processes , which occur when the invariant mass of the initial system lies within the `` resonance curve.'' Clearly such processes can occur only if is sufficiently high that the energies reach : when , the inverse processes become exponentially improbable, as exhibited by the behavior (4.1.4) of in this region. The time appearing in (4.2.1) is measured in the c.m.s. of the complete universe: in that frame, the lifetimes of are dilated by factors , and hence the effective width in (4.2.1) must be averaged over the relevant energy spectrum. Typically, in the rest frame, , where is a coupling constant. We assume, in keeping with the simplest big bang cosmology, that at sufficiently early time, all species of particles are in thermal equilibrium, so that at , in eq. (4.2.1).

Fig. 2 shows numerical solutions to eq. (4.2.1) with equilibrium initial conditions. Deviations from equilibrium are initiated by the expansion of the universe. They are relaxed at a rate . Their extent depends on the relative magnitude of this relaxation rate, and the expansion rate of the universe, at a temperature . The larger is, the smaller the maximal deviations from thermal equilibrium are. Notice that, in fig. 2, curves with adjusted so as to give the same still differ slightly, because the averaging over time-dilation factors in depends on . In the high and low temperature limits, eq. (4.2.1) admits of simple approximate solutions. Defining , eq. (4.2.1) becomes

For the averaged width we approximate

At high temperatures, therefore, eq. (4.2.2) becomes [using eq. (4.1.3)]

The solution to this equation is

so that

It is clear that the parameter governs the magnitude of deviations from equilibrium. The numerical results of fig. 2 indicate that terms kept in eq. (4.2.7) are adequate until close to the maximum in at . Keeping only these terms suggests that is obtained as the real root of the equation

or roughly

At low temperatures, (4.2.2) becomes [using eq. (4.1.4)]

In the limit the expansion rate of the universe becomes negligible with respect to the rate of reactions, so that . For large , (4.2.10) may be solved with this boundary condition to give

Again, fig. 2 shows that this approximation is numerically accurate. Note that the presence of inverse decays remains crucial in determining deviations from equilibrium even at large times. If the decayed freely, with no back reactions, then the second term in eq. (4.2.2) would be absent, so that at low temperatures

On the other hand, the rate for inverse decays is proportional to , which falls much more slowly than eq. (4.2.13). The behavior of in (4.2.13) as a function of depends on the expansion rate (4.1.7) of the universe. If the temperature of the universe decreased faster than , then the equilibrium number density would fall more rapidly than the free decay probability , and at low temperatures (large times) would approach (4.2.13) (7).

[ Figure 2 ] Deviation of massive particle number density from equilibrium as a function of temperature in the early universe, as calculated from eq. (4.2.1). The particle is taken to have mass , and decay width . , is the scaled number density for the particle. The expansion rate of the universe is governed by an effective Planck mass . Unless otherwise indicated, and .

4.3. Evolution of Light Particle Species

In the previous section, we have assumed that neither the massive particle , nor its massless decay products carry any quantum number, so that both are charge conjugation eigenstates. This assumption is clearly inappropriate for particles which, for example, carry baryon number. The generation of a net baryon number relies on the development of a difference between the number densities of a particle and its antiparticle. Consider a massless particle species c which carries an absolutely conserved quantum number . The antiparticle carries . In a gas of c, with net , the equilibrium distribution of c, in phase space is . However, if the gas has net , then the equilibrium distributions will involve a chemical potential , and become

The chemical potentials of c and are forced to be opposite by the presence of processes such as , where has . The distributions (4.3.1) lead to a net density of the quantum number

where the final approximation holds so long as . The characteristic rate at which a gas of c, with an arbitrary initial configuration will relax into the equilibrium distributions (4.3.1) is governed by the total rate for interactions of the c, . If, however, some rare interactions of c violate the quantum number , then the effective chemical potential may change at a rate governed by the rate for these rare interactions. In the absence of external influences (such as the expansion of the universe), a gas of c, in the presence of -violating interactions will eventually relax into a state of ``chemical equilibrium'' in which . However, the rate of this relaxation will be much smaller than the rate at which ``kinetic equilibrium'' [leading to the phase-space distributions (4.3.1)] will be established. It is therefore sufficient to approximate the c, phase-space distributions by (4.3.1) in investigating their approach to ``chemical equilibrium.''

Most particles which undergo -violating interactions also participate in -conserving interactions, some of which are mediated by light bosons (, G, , ). As discussed above, baryon number generation requires deviations from ``kinetic equilibrium'' which occur when the temperature of the universe falls to the masses of the heavy particles which mediate -violating processes. In this period, the rates for -violating reactions should be somewhat smaller than those for -conserving reactions, primarily because of the larger masses of the mediators of violation (another numerically significant effect is that in most models the number of possible -violating reactions is smaller than the number of -conserving ones by a factor between and ). Hence, for most light particles carrying baryon number, it should be adequate to assume distributions (4.3.1) in phase space, but with chemical potentials which change with time through -violating processes. If b denotes a light particle carrying (so that has ) then, for small ,

If the b, undergo the -violating reactions and , then

where denotes the cross section multiplied by the relative velocity (which cancels the flux factor in exothermic reactions at low incoming relative velocities) averaged over the c.m. energies for the collisions. The factor of 4 appears because these processes involve . Eq. (4.3.4) implies that any baryon excess introduced will be relaxed exponentially to zero by -violating interactions with a characteristic time of order of the mean free time between -violating collisions. Note that -conserving processes, such as or do not affect ; they may change the momentum distribution but not the total number density of b, .

Now consider a massive particle which decays to two-body final states containing b, and . First, for simplicity, assume invariance, so that the rates for decays and inverse decays to and from a state f are equal: . Then the number density evolves [in analogy with eq. (4.2.1)] according to

where denotes the total baryon number of the state f. If , eq. (4.3.5) reduces to eq. (4.2.1). The evolution of the number density may be obtained from (4.3.5) by charge conjugation:

where we have used invariance to write . The equilibrium distributions and are equal since they depend only on . From eqs. (4.3.5) and (4.3.6) one finds

(In later sections, we shall often use the shortened notation , .) If, for example, , so that and , but , then an excess of b over () will result in the production of more than in inverse decay processes, and therefore an increasing . If , then decays may affect . Such processes yield

Assuming for now invariance, so that , this becomes

In the cross sections it is necessary to remove the contribution from real intermediate or (e.g. ) since this is already included by iteration of the decay and inverse decay terms (see I, sect. 4). The first term in (4.3.9) illustrates that an excess of over may lead to a baryon excess when the decay. The second two terms in (4.3.9) are manifestly negative, and cause any introduced to relax towards zero through -violating interactions. It is clear from (4.3.9) and (4.3.7b) that, if invariance holds, any system with initially can never develop . As mentioned in sect. 1 (and at length in I), generation of a baryon excess from an initially symmetric state requires violation.

invariance demands that

where denotes the conjugate of the state . The unitarity condition requires

Eqs. (4.3.10) and (4.3.11) then imply

(where we have interchanged the dummy labels in the sums over all states). The equality of the first and last forms in (4.3.12) demonstrates the equality of the total decay widths for a particle and its antiparticle. If invariance were assumed, then

so that each partial width for decay into a given mode would be equal for particle and antiparticle. For baryon number generation to occur, invariance must be violated, and the equality (4.3.13) must fail. Using only eqs. (4.3.11) and (4.3.12), and not assuming invariance, eq. (4.3.8) becomes (keeping for consistency only terms of first order in both and -violating differences between rates):

The derivation of the part of the first term here is somewhat subtle [as discussed in detail in I, sect. 4, and particularly eq. (2.4.9)]; it arises from a sum of contributions from inverse decay processes and from scatterings mediated by a nearly on shell -channel or exchange. The first term in (4.3.14) allows generation of a baryon asymmetry by -, -violating decays when deviations from thermal equilibrium occur, so that . The last two terms in (4.3.14) act to relax any asymmetry produced: the final depends critically on the size of these terms. Allowance for violations requires no important changes in eqs. (4.3.7) for the development of the number densities.

If the rate for interactions is much smaller than the expansion rate of the universe around the time of decays, then large deviations in the number density from its equilibrium value may occur (as illustrated in fig. 2), and the first term in eq. (4.3.14) may dominate. In this case, comparison with eq. (4.3.7a) shows that

hence, the final baryon number density generated from an initially symmetric state is simply

This result is exact if the decay freely, with no back reactions. At high temperatures, for zero initial baryon number, the first term in (4.3.14) always dominates, so that at early times grows like [using eq. (4.2.6)]

This baryon asymmetry leads to an asymmetry in the number of and ,

for the single species considered here, this asymmetry is never important in the development of . As the temperature decreases, and increases, the third term in eq. (4.3.14) begins to counteract the first term. The ratio of these terms is roughly . For small , . This ratio falls below one, indicating that the first term in (4.3.14) no longer dominates, at . In the region , the first and third terms in eq. (4.3.14) are both large, but cancel to give a net . As the temperature decreases, decreases rapidly according to eq. (4.2.12), and the third term in (4.3.14) dominates, so that

The exponential fall off in at large for the most part neutralizes the superficially exponential relaxation of ; in practice decreases roughly as . At very low temperatures, the presence of in the third term of eq. (4.3.14) renders it negligible, and only the fourth term, which results from -violating scatterings, makes a significant contribution to . At low energies ), scattering of light particles by exchange of gives a cross section . At very low temperatures, eq. (4.3.14) thus becomes

Hence, when scatterings dominate, falls like

which tends to a constant non-zero value as . However, if the behavior (4.3.20) sets in at comparatively small , the final will be much diminished from its maximum value, attained at .

4.4. Several Massive Particle Species

Above, we have considered only a single massive particle species: in realistic theories, however, there are usually several massive vector bosons, and often huge numbers of massive scalar Higgs bosons. In this section, we first discuss a simplified case in which massive bosons decay only to light particles (and not into other massive particles), and assume that is the only quantum number with non-zero chemical potential carried by the light particles. Then the number densities of each species evolve according to eq. (4.2.1), while in eq. (4.3.14) for a sum must be performed over the possible , yielding roughly

where we have temporarily dropped numerical factors 0(1) associated with decay branching ratios.

Consider first the case of identical bosons . If each decayed freely, the final baryon number generated would be . At high temperatures, is indeed increased by a factor ; however, when , the larger generated at higher temperatures and the presence of more bosons renders the back reaction terms in (4.4.1) more important, so that the rate of destruction is higher. Writing , eq. (4.4.1a) becomes

the relaxation time for destruction of baryon number in eqs. (4.3.19) and (4.3.20) is reduced by a factor . Fig. 3 shows the factor by which the final baryon number is modified by introduction of identical boson species. When back reactions are unimportant, the effects of each boson add; when back reactions are important, the increase in the baryon destruction rate with causes the final baryon number to decrease exponentially. In eq. (4.4.2) and fig. 3 we have assumed that in all cross sections, the effects of the bosons are added incoherently: however, if the bosons are genuinely identical, their contributions add coherently, contributing a factor instead of . Eq. (4.4.1c) suggests that if a larger is generated at high temperatures by decays of bosons, then a larger for each boson will be produced by inverse decay processes. The typical contributing to eq. (4.4.2) will therefore be a factor larger than in the single boson case. Nevertheless, if the bosons have identical masses, the term will probably never be important in (4.4.2), as in the one-boson case.

[ Figure 3 ] Modification in final baryon number density produced by introduction of identical massive boson species , as described by eq. (4.4.2). Results for various typical choices of parameters are shown.

We now consider the case of several bosons with different masses and couplings. At temperatures , the total baryon number generated by their decays will behave as [cf., eq. (4.3.17)]

where we have taken . For the minimal case (see sect. 5) of two species with , , eq. (4.4.3) implies

if then the effects of the two bosons always cancel, and no baryon excess may be generated. If and are nearly degenerate, then eq. (4.4.4) implies that the final will be smaller by a factor than would result if only one of the bosons were present. If , then at high temperatures, should build up just as if the lighter boson were absent. However, when the temperature falls, eventually overtakes . When this occurs, the two ``driving terms'' in eq. (4.4.1) cancel. By this temperature, is comparatively small, so that the back reaction term is not dominant; however, is still , so that the is very large. This term is uncanceled when compensates , and causes to relax exponentially to zero. Then, as the temperature decreases further, becomes much smaller than , and baryon number is generated in the decays of the lighter boson. If is more than about 20% heavier than [so that ], then the baryon asymmetry generated by its decays is destroyed by inverse decays to , and the final is close to that obtained in the absence of . Some examples are given in fig. 4. This result suggests that the final baryon asymmetry depends only on the behavior of the lightest -violating boson species: asymmetries generated by heavier species are destroyed by inverse decays of lighter species. The result is valid, however, only in the unusual simplified case considered here where no asymmetries may occur in quantum numbers other than (cf. sects. 6 and 7).

[ Figure 4 ] Development of baryon asymmetry in the case of two supermassive bosons with masses and coupling constants and . The dotted line is the result that would obtain if the contribution of boson 1 were ignored.

Free decays of a given boson species can generate a baryon excess only if they violate invariance so that . However, even if , a -violating boson species can destroy through inverse decay processes. If the lightest -violating bosons do not violate invariance, then they will often destroy by inverse decays the baryon excess generated at higher temperatures by decays of heavier bosons. In some cases, however, this destruction is partly avoided by the terms in . When baryons are absorbed by inverse decays of lighter bosons, baryon excesses may result in , as suggested by eq. (4.4.1c), if the couples unequally to channels with opposite baryon number [see eq. (4.3.7b)]. Then, even in the absence of violation, the decays of unequal numbers of and may regenerate a baryon excess. In this way, a baryon asymmetry produced at high temperatures may be stored as a of -conserving .

We have assumed above that heavy bosons may decay only to light particles. In realistic models, however, heavy bosons may usually decay to other heavy bosons as well as to light particles. At temperatures where a given heavy boson is present at sufficient density to be significant, most of its possible decay products will still be in ``kinetic equilibrium'' by virtue of their smaller masses. The phase-space densities of these lighter bosons may thus be approximated by their equilibrium form, but with a chemical potential, , so that

where as in eq. (4.1.2), but with . Typically, reactions such as involving will only occur with sufficient rate to maintain the in ``kinetic equilibrium'' [and thereby validate eq. (4.4.5)] if : when , eq. (4.4.5) will become inaccurate, but the actual number densities in this regime will probably be so small as to be irrelevant. The inadequacy of eq. (4.4.5) at low temperatures is evident from the difference between eqs. (4.3.18) and (4.2.13). We shall use eq. (4.4.5) only in estimating the rates for inverse decays to bosons much heavier than those whose densities are approximated by eq. (4.4.5). Such rates are [cf., eq. (4.2.1)] proportional to the equilibrium number density of the heavier bosons, and are therefore negligibly small when (4.4.5) becomes inaccurate.

Using eq. (4.4.5) where appropriate, the general equation for the evolution of the number density of a species becomes

where the sums on , , but not run over both particles and antiparticles of each species. In eq. (4.4.6) denotes the number of particles of type in the state f. The previous results (4.3.5), (4.3.8), etc. may be derived as special cases of eq. (4.4.6). By charge conjugation and subtraction, one obtains from (4.4.6) the results [assuming, for consistency, ]

where now only the sums on run separately over particles and antiparticles of each species.

The evolution of the density of a quantum number may be obtained from eq. (4.4.7) using

Given a complete set of independent quantum numbers all the may be written in terms of quantum number densities by inverting the corresponding relation (4.4.8) (the inversion will be singular if the are incomplete or interdependent). In our analysis of specific grand unified models below, it will be convenient to perform such an inversion, since several equations (corresponding to different flavors, etc.) then become identical, and need not be treated separately.

In addition to describing the evolution of quantities such as baryon number which are violated only in processes involving heavy bosons, eq. (4.4.7) may also in principle be used to estimate the development of quantities such as weak isospin or flavor, which are violated by light bosons. In these cases the last two terms in eq. (4.4.7) usually dominate. The magnitude of the last term is then determined by the high temperature behavior of light boson exchange cross sections, which are difficult to define or estimate, as discussed above.

4.5. Corrections to the Boltzmann Equation

The Boltzmann transport equations discussed above describe a sequence of uncorrelated reactions between ``free'' (on-mass-shell) particles, whose cross sections are independent of the presence of other particles. In this section, we consider corrections to this picture, and discuss the limits of its validity. While important in principle, these considerations will usually be irrelevant in practice.

The basic condition for the Boltzmann equation to be valid is that the distance traveled by a particle between successive interactions should be much larger than the range of a single interaction (or than the average separation between particles) [31,32]. This condition is satisfied only for interactions involving exchange of a heavy particle, and consequently of short range. Generation of baryon asymmetries requires deviations from thermal equilibrium, and thus typically occurs at temperatures of the same order as the masses of -violating bosons. At such temperatures, the -violating reactions satisfy the condition. Other reactions involving exchanges of light bosons, do not satisfy the condition; their effects on -violating processes may nevertheless usually be approximated by effective ``screened'' cross sections [I].

We consider first processes in which one light particle species is transformed into another such species through interactions with a massive boson V. The lowest order contributing process is with exchange of a single V in the -channel. The range of this interaction is . The rate of transitions may be described by a Boltzmann equation representing a sequence of uncorrelated independent scatterings only so long as the range of these interactions is much shorter than the distance between interactions. When this condition is violated, the ``exchanged'' V may undergo several interactions between its ``emission'' and ``absorption''.

Several methods may potentially be used to account for this phenomenon.

First, one could introduce ``higher order corrections'' to the process involving additional incoming or outgoing particles. For example, to represent processes in which the ``exchanged'' V scatters once from a species one could include the process . This approach may be adequate for first-order corrections; however, it rapidly becomes impractical (and formally exhibits a plethora of divergences).

Second, the averaged effects of interactions with the V may be approximated as ``screening'' the V exchange, and may be parametrized by introducing a modified V propagator (8). In the simplest approximation, the presence of additional particles reduces the mean free path for the V, and generates an effective mass of order the inverse mean free path . This is the approximation conventionally used in attempts to apply an effective Boltzmann equation to electron-ion plasmas. It can be adequate only if roughly equal numbers of particles with positive and negative ``V charges'' are present. When this is not the case (as in self-gravitating systems), genuinely long-range effects must be included, so that the form, as well as effective mass, for the V propagator must be modified. In the early universe, one is typically concerned with situations in which both positive and negative charges are present, so that the screening approximation is expected to be adequate. (The early stages of ``cold'' universes consisting of degenerate gas of baryons constitute an exception (9) ).

A third approach is to consider a sequence of independent interactions involving off-mass-shell V. Then the rate for transitions would be given by the rate for and averaged over all four-momenta for the , and V. The ``equilibrium'' distribution of particles in four-dimensional phase space then depends not only on their energy but also on their invariant mass; unlike the case of fixed invariant mass, the distribution is not determined from the Boltzmann equation without explicit assumptions for the interaction cross sections. The resulting distribution should nevertheless qualitatively have a spread in invariant mass of order around zero, and an average energy of order . The Wigner function , where is a field operator and denotes a statistical average, may be used as a formal definition of the phase-space density when the mass-shell condition is not satisfied [33]. [The Wigner functions obey the relation .] Extension of the derivation of the standard Boltzmann equation for on-shell particles from Wigner functions to allow for off-shell particles is, however, very difficult because interaction terms in the lagrangian cannot be neglected in comparison to kinetic energy terms.

The second approach inserts corrections to amplitudes for V propagation; the third approach treats only V interaction probabilities. Interference between amplitudes for successive interactions is typically important when the distance between interactions is less than the Compton wavelengths of the interacting particles. The distance between interactions, as reflected in the average invariant mass of the particles, is probably roughly ; the Compton wavelength of the particles is , usually giving a sufficiently large ratio that quantum mechanical interference effects are small.

Most effects on say resulting from the presence of a background gas decrease if the coupling to this gas is reduced. ``Identical particle'' quantum mechanical interference effects survive, however, even in the limit of zero coupling. Consider the amplitude for a particle to propagate from a point A to B. The presence of a background gas allows processes in which an on-shell particle received at B comes from the background gas, and is not the one emitted at A. (In the usual thermodynamic approximation of weak coupling, all particles in the background gas are on their mass shells.) This suggests that the complete propagation amplitude in momentum space is given by [32,34]

where the upper sign is taken for bosons, and the lower for fermions. This form may be derived from the periodicity of the path integral in imaginary time when is an equilibrium distribution; (4.5.1) is an obvious generalization to the case of an non-equilibrium (e.g. collisionless) background. Note that in the presence of spontaneously broken symmetries, the phase-space density of the condensate field may be taken as .

All lines in a Feynman diagram carry on amplitude of the form (4.5.1). For those corresponding to ``external particles'' a discontinuity (``cut'') is taken, and only terms proportional to survive. The presence of the background gas thus gives a correction factor for each outgoing particle , regardless of its couplings. These factors account as usual (see subsect 2.4.2 of I) for stimulated emission and Pauli exclusion effects, and are important only in regions of phase space with high densities. The factors are necessary to obtain the correct Fermi-Dirac and Bose-Einstein equilibrium distributions. Corrections appear not only on explicit external lines, but also for any ``internal lines'' which may reach their mass shells in kinematic integrations. (This situation occurs in the calculation of violation described in sect. 5). Such factors are essential in maintaining the unitarity relation (I.A.22) for reaction rates, and enabling Boltzmann's theorem to be proven even though the effective interaction cross sections depend on the ambient particle density.

When interactions occur in the background gas, ``identical particle'' effects may occur not only for on-shell particles, but for any particles with counterparts in the background gas. When the interactions are rapid, ``identical particle'' effects become indistinguishable from the general scattering processes discussed above.

It is impossible to give reliable numerical estimates of the corrections to the Boltzmann equation outlined above. Several simple cases are, however, amenable to treatment.

``Identical particle'' corrections may be calculated in the approximation of a weakly interacting background gas (see subsect. 2.4 of I). For example, the Born approximation decay rate for a particle of mass is modified in the presence of an equilibrium background gas with temperature by a factor . The correction is small except perhaps for a background Bose gas close to condensation.

As a next example, we consider the first corrections to the Boltzmann equation (4.2.1) for the development of the number density due to decay and inverse decay processes. Eq. (4.2.1) includes only reactions. At , reactions such as , and (and perhaps ) may also occur (we take here for simplicity ). The processes and serve only to redistribute the in phase space, without changing their total number, and thus do not contribute to . Including the processes and , eq. (4.2.1) becomes (10)

For unstable particles (with, say, ), the second term in eq. (4.5.1) is usually irrelevant at low temperatures, since its contribution while the first term . At high temperatures, however, the second term may become large. In a formal expansion in powers of in eq. (4.5.1) should be evaluated for free and . The high-energy behavior of this ``free'' cross section depends on the spin of according roughly to (see I, sect. 3) , where is the c.m.s. energy. (In particular cases, may decrease faster with : for example, if has spin 2, and spin 1, then .) Assuming roughly equilibrium phase-space distributions, for . Using the ``free'' cross section then implies that the second term overwhelms the first at high . However, as discussed above, the ``free'' cross section is no longer adequate at large : screening effects must be included. Except for spin zero , these considerably reduce and suggest an effective cross-section rather than . Taking and , so that , eq. (4.5.2) becomes

The approximate solution to this equation at high temperatures is

For , the scattering term overwhelms the decay (inverse decay) term. Nevertheless, at such high temperatures, the total , and is thus presumably small. Baryon number generation occurs predominantly when deviations from equilibrium are maximal. A rough estimate of this temperature is given by the lowest stationary point of (4.5.3) [cf., eq. (4.2.8)]: so long as , the decay (inverse decay) term dominates in this region. Note, however, that the coupling constants , may have very different magnitudes. Any charged particles must undergo scatterings with a characteristic coupling constant . Coupling constants for decays of gauge vector bosons must be of the same order: however, for Higgs bosons or heavy fermions, it is conceivable that the effective . It appears (see sect. 5) that for Higgs bosons, this possibility is probably not realized. Nevertheless, as discussed in ref. [35] and mentioned above, it may well occur for weakly interacting heavy fermions.

4.6. The Boltzmann Equation in Non-standard Cosmologies

In most of this paper, we approximate the early universe as homogeneous and isotropic. However, observation of structures at least up to clusters of galaxies in the present universe demonstrates that this Friedmann approximation is not exact. Most theories for the origin of galaxies suggest that any density fluctuations should nevertheless be small at the times when baryon asymmetry would be generated. In this section we discuss consequences of possible corrections to the Friedmann approximation at high temperatures.

We first discuss the form of the Boltzmann transport equation for an arbitrary cosmology. The Boltzmann equation may be written schematically in the form

The right-hand collision term depends only on the phase space density at a particular space-time point, and is therefore independent of the large-scale properties of space-time. On the other hand, the left-hand term depends on space-time derivatives of the phase space density, and is therefore potentially sensitive to the properties of space-time. The general relativistic covariant form of the Liouville operator is [41]

where is the proper time, and the are the Cristoffel symbols (affine connections) which enter the covariant derivative. In a comoving frame, . and the Liouville operator is

In the simplest approximation of a homogeneous isotropic universe, and setting the curvature parameter , the metric has the Robertson-Walker form, in which the , where , , and is the Robertson-Walker scale parameter. In addition, the phase space distribution depends only on the magnitude of (or equivalently ), and the time . In this case, the Boltzmann equation thus becomes

In a comoving frame, , so that

The total number density is obtained as an integral over momenta . Inserting this into eq. (4.6.3), integrating by parts, and dropping the and surface terms, one obtains

This result has the simple physical interpretation that the expansion of the universe affects the number density only by increasing the volume containing a fixed number of particles.

The simplest non-standard modification to the Friedmann-Robertson-Walker cosmology consists in allowing anisotropy but retaining spatial homogeneity. The Bianchi classification [2] gives the possible metrics in such cases. For example, in Bianchi I cosmologies . (When all the scale parameters . this metric reduces to the Robertson-Walker metric.) The derivation of the Boltzmann equation in this case is analogous to that given above, except that the volume expansion term is replaced by where . It appears in fact that in all homogeneous cosmologies, expansion of the volume element is the sole effect of the expansion of the universe on the form of the Boltzmann transport equation. The rate of expansion of the volume is given in general by , where is the fluid velocity in a comoving frame, and the semicolon denotes covariant differentiation. In a comoving frame, the complete velocity 4-tensor at fixed time may be decomposed into a traceless symmetric part (the shear tensor), an antisymmetric part (the vorticity tensor) and a trace term (the volume expansion rate). The Einstein equations may be used to relate these quantities to the fluid energy density and the local Ricci curvature scalar at fixed times (on spacelike hypersurfaces orthogonal to the fluid flow with ) [38]

where and is a possible cosmological constant. With the standard assumptions . this equation reduces to the usual Robertson-Walker result , and , where is the Robertson-Walker scale parameter, and is the curvature scale inserted to scale the curvature constant . The fact that this curvature term is negligible in the present universe shows that it may be entirely ignored in the early universe. Although the cosmological constant is now small, it is possible that restoration of spontaneously-broken symmetries in the very early universe could have allowed it to be important then. This possibility has been discussed at length elsewhere [43,44].

Changes in the expansion rate which enters the Boltzmann equation may be parametrized by modifying the temperature-time relation, and may often be accounted for simply by changing the effective Planck mass. The consequences of such changes were discussed briefly in I and treated in detail in [38]. While they affect the rates of baryon number production, the changes cannot lead to deviations from thermal equilibrium in situations where such deviations would otherwise not occur.

In the discussion above, we have made the assumption of homogeneity which implies that all phase space densities depend only on and and not on . In the generic case, one must allow inhomogeneity. Unless any inhomogeneity initially present or generated in the early universe is rapidly damped out, it will evolve into a universe far more inhomogeneous than is observed. Nevertheless, for a brief period, perhaps around the time of a symmetry-breaking phase transition, inhomogeneity may have existed. In such a case, the Boltzmann equation in general involves the spatial and momentum derivatives of the phase-space densities. These additional terms lead to deviations from thermal equilibrium even for massless particle densities. However, in the ideal gas approximation of infinitely rapid collisions, the modification to all massless boson and all massless fermion densities will be identical. For baryon number generation to occur, it is necessary not only that there should be deviations from equilibrium phase space densities, but also that these deviations should be different for different species of particles. It is possible that the deviations may be different for massless bosons and fermions, so that baryon number production may occur as a result of interactions involving both bosons and fermions.

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