4. The Destruction of Initial Baryon Number

In this section, we discuss the behavior of the baryon density in the universe at temperatures where the mass is irrelevant, and any -violating interactions should occur as fast as -conserving ones. At times (corresponding to temperatures ), it is undoubtedly not permissible to consider only background gravitational effects; quantized fluctuations in the expansion rate, and direct gravitational contributions to particle interactions (which determine the equation of state) presumably become overwhelming. Nevertheless, the perhaps overly naive estimates given below indicate that such effects should rapidly become unimportant when falls below

As mentioned in sect. 1, the likely excess of baryons over antibaryons in the present universe probably cannot be explained solely by postulating a small initial net baryon number if -violating processes occur rapidly at high temperatures, since these would tend to eradicate the initial baryon number. The purpose of this section is to estimate the rate of relaxation to a state, and to determine whether it would be complete before when the temperature fell below and -violating processes become rare. We consider the fate of a baryon excess existing at (the possible genesis of such an initial condition is, of course, quite unknown, despite Polkinghorne's program [17]). If particles carrying baryon number also carry electric charges (but the sum of the charges of all baryon species is non-zero), then to maintain the overall charge neutrality of the universe any baryon excess must be compensated by a suitable antilepton excess. Then, if for example, is absolutely conserved, destruction of an initial baryon excess would be accompanied by destruction of the corresponding (anti) lepton excess. We assume as above, purely for simplicity, that only one light neutral fermion species carrying baryon number exists, and that there is only one species of -violating boson, with

According to eq. (2.3.20), at (so that, by assumption, ) a small baryon excess in the model of subsect. 2.3 should be relaxed according to

The first term in eq. (4.1) represents the absorption of baryons by inverse decays (e.g., ); the produced eventually decay approximately symmetrically to baryons and antibaryons, and hence baryon asymmetry is diminished. The second term describes the direct destruction of baryon number by scattering processes. (As discussed in sect. 3, these act at temperatures to diminish baryon number generated in decays). Ignoring temporarily the first term in (4.1), an initial baryon concentration should relax roughly according to

assuming as above that the universe expands according to

At high c.m. energies the total cross section for two-body scattering through -channel exchange of a spin- particle behaves like

As evidenced by eq. (3.1.13), due to exchange of a vector goes to a constant value at high . (On the other hand, a scalar exchange gives a cross section asymptotically falling like ) Note that, as is typical in non-abelian gauge theories, the change in quantum numbers may be effected without any exchange of transverse momentum; the range of the violating interaction is limited only by the mass of the exchanged If then eq. (4.2) implies that any initial baryon excess would be diminished, dominantly near the Planck time, by a factor which would probably be sufficient to destroy any initial baryon excess (If then the destruction factor would only be ) A cross section results essentially from interactions with an cloud (of opacity ) surrounding a baryon, with area At high temperatures, there should be particles within the range of the interaction from a single baryon. These particles should contribute to scattering from the baryon, giving rise to processes with many-body initial states. As usual, the simple Boltzmann equation is unable to account for such effects of a ``long-range'' X interaction (21). However, as in electron-ion plasmas [18], it is presumably permissible, at least in the near-equilibrium state considered, to account for higher-body processes simply by introduction of an effective screened two-body cross section. The antibaryons surrounding a baryon typically screen its `` charge'' at distances beyond the Debye length (note that all particle species carrying any charge contribute with the same sign to the screening length)

this screening may be described by assigning the an effective mass The mean time between successive collisions of the becomes smaller than its Compton time at temperatures The effective baryon destruction cross section at high temperatures should therefore be (22) due to each species of -violating exchange. If, as in many complicated and ``realistic'' models, the number of particle species mediating -violating processes at very high temperatures is much larger than those conserving baryon number, then the total destruction effective cross section should be With this form, an initial baryon excess would be diminished by a factor which is probably so that the present could not be explained in an initially hot universe. (Recall that ). The use of an effective screened cross section will tend, if anything, to underestimate the rate of destruction. Notice that the modification to the effective mass is largely irrelevant at the temperatures considered in previous sections.

The width which governs the first term in eq. (4.1) is given at temperatures not too far above by This form assumes that the produced is on its mass-shell, and therefore that the incoming total c.m.s. energy must lie within the resonance curve; this restricts the angle of the incoming particles to be and thus introduces the ``time-dilation factor'' in However, at temperatures the mean free path of for scattering will typically be much shorter than the mean decay length. Hence the resonance should be collision broadened, and the factor resulting from the impossibility of producing with invariant masses should disappear. Then, according to eq. (4.1) inverse decay processes should diminish an initial baryon excess by a factor (If many -violating bosons exist, dominating at high temperatures, then the factor becomes ].)

Most grand unified models based on simple gauge groups [e.g., SU(5)] are asymptotically free, so that the effective gauge coupling constant falls logarithmically with increasing energy or temperature, and presumably always remains small. Nevertheless, the effective coupling constant for Higgs interactions often increases with temperature (23) , and could diverge (reach its Landau singularity (24) beyond which perturbation theory is useless at temperatures below the Planck mass. (If this occurred at then the calculations of baryon number generation given above must be modified considerably; in the standard SU(5) model this case does not occur, except in the presence of massive fermions forbidden by other considerations [20]. Nevertheless, divergence below the Planck mass could easily be achieved.) The presence of such strong interactions at very high temperatures should lead to phenomena analogous to those encountered in the Hagedorn-Frautschi model for hadronic matter (now believed to be inappropriate because of the quark composite nature of the hadron states considered). When energy is added to a strongly interacting system, it may not simply increase the kinetic energies (and hence temperature) of its constituent particles, but rather serve only to generate more massive particles with small momenta (these particles might be bound by strong Higgs interactions (25) ). Hence there may exist a maximum temperature for the universe, governed by the point at which Higgs couplings become strong. The behavior of the universe at earlier times may then be shielded: only the decay products of the massive Higgs bound states initially present will be visible when the universe has cooled below the maximum temperature; their nature should presumably be determined solely from the dynamics of the strong Higgs interactions.

Next, we consider the autolysis of a cold universe, initially consisting of a zero temperature Fermi gas of ( types of) baryons with large chemical potential The expansion rate of such a universe is given by

where and is the number of degenerate fermion species. According to eq. (2.4.19), the baryon density in a universe with should be destroyed at a rate (Pauli exclusion effects are negligible, because the phase space for the annihilation products is unrestricted by the presence of the degenerate baryon sea)

The number density is [see (C.36)]

where according to (4.6)

Just as in the high temperature case discussed above, -violating exchange at high densities should also be screened. (Although the initial baryon number of a cold universe is taken to be large, the initial SU(3) color charges, etc., were presumably zero, so that the total charge to which the couples was zero, allowing screening.) Hence the effective destruction cross section in a cold universe should be (26) so that eq. (4.7) becomes

Thus in the presence of -violating interactions, an initially cold universe with should relax to a hot universe with

Finally, we discuss the structure of the universe very close to the Planck time, and comment on the consistency of our assumption of initial equilibrium. The cross section for production or annihilation of scalar particle pairs into photon pairs at high energies is given by The corresponding cross section for production or annihilation into spin-2 graviton pairs is given at lowest order by [21] When therefore, graviton-induced interactions should be important, but at lower temperatures, they should rapidly become irrelevant. We have assumed above that the contents of the early universe behave as an ideal ultrarelativistic gas. In fact, Coulomb interactions alone should affect the properties of the gas, giving, for example, [22]

an evidently irrelevant correction. Even assuming homogeneity, gravitational interactions would provide a correction perhaps once again, the effect becomes overwhelming at but quickly becomes negligible below it. Of course, even excepting quantum corrections, the treatment of gravitational effects in the early universe is made difficult by the genuinely long range nature of gravity: since all masses and gravity is universally attractive, no screening occurs, and the Boltzmann equation becomes entirely inadequate.

In keeping with the simplest big bang cosmology, we have assumed that the universe is initially in a state of kinetic equilibrium (so that all particles follow equilibrium distributions in phase space, albeit perhaps with non-zero chemical potential). Nevertheless, of course, the gravitational field must be far from equilibrium, otherwise no expansion would occur. One might therefore be led to modify the usual initial assumptions, and postulate instead that not all particle species were in equilibrium at the Planck time [23]: they would come to kinetic equilibrium only after a few collision times. (Nevertheless, despite the fact that a finite time should be required for a signal to propagate from one part of the universe to another, it appears that the expansion of the different parts began at ``times'' closer than would have allowed a light signal to be exchanged between them. This perhaps surprising phenomenon may indicate that equilibration was more rapid in the very early universe.) In the presence of long-range interactions, one may consider only an effective collision time (Assuming not too many conservation laws, the here should probably be multiplied in practice by ) Typically, therefore, kinetic equilibrium should be achieved quite rapidly. Two effects could be thought to modify this result. First, the large effective masses of particles at high temperatures could affect their equilibrium distributions. In fact, this modification is already included in consideration of suitable screened cross sections. Second, at a time only particles within a Jeans volume could apparently be in causal contact. However, this does not necessarily provide an infrared cutoff on particle momenta (density fluctuations with larger wavelengths can exist).

We are grateful to many people for discussions, including A.D. Dolgov, E. Dwek, S. Frautschi, William A. Fowler, G.C. Fox, T.J. Goldman, S.E. Koonin, D.L. Tubbs, and R.V. Wagoner.

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