3. Results in a Simple Model

3.1. Introduction

In this subsection, we present solutions to eqs. (2.3.10) and (2.3.20) which describe baryon number generation in the simple model of subsect. 2.3. In terms of the dimensionless variables

where the effective Planck mass was defined in (2.1.14), these equations become

For simplicity we shall henceforth take

although none of our results are sensitive to this choice. The violation parameter will be denoted by

Unitarity of the decay rates (2.3.1) requires and according to eq. (2.1.9), is formally of order . We shall write for the final baryon number density (at zero temperature); we do not include in the final factor discussed in subsect. 2.4 to account for increase in the photon number density.

If the contents of the universe were in thermal equilibrium at sufficiently early times, the solutions to eqs. (3.1.2) must satisfy the initial conditions [we assume as in sect. 2 that all particles (including ) have only one spin state and obey Maxwell-Boltzmann statistics: if has spin states (and are assigned Bose-Einstein statistics) then if it is a boson, and if a fermion]:

where is a possible initial baryon number. At lower temperatures, the equilibrium number density is given by (cf., eq. (2.2.9) and appendix C)

where is a modified Bessel function (see appendix C). We take the time-dilation factor in the effective width to be averaged over an equilibrium energy distribution, so that

For the numerical solutions of subsect. 3.3, we use

corresponding to the decay of an with spin states to two identical (spin-) fermions, coupled with strength (The factor of 2 between and in (3.1.7) arises because only half the possible final fermion spin states are accessible from a spin-0 ) We usually take The cross sections and (in which the contribution of real intermediate already included in previous terms has been subtracted off, by removing the pole part of the exchanged propagator) are equal to The high-energy behavior of the crucial for the destruction of any initial baryon number will be discussed in sect. 4. For baryon number generation, the form of at c.m.s. energies is important. In the low-energy limit, it is of the usual Fermi form

where typically (see below for specific cases). Averaging this cross section over thermal energy distributions for the incoming b gives (see appendix C)

The detailed from of as a function of depends on the couplings and spin of the exchanged For scalar exchange, one finds [using the same couplings as in (3.1.6)] (16)

In the high-energy limit, this cross section becomes

while at low energies it reduces to

Note that in (3.1.9) the contribution of -channel, as well as -channel, -exchange has been included. To obtain one must subtract from the cross section obtained by keeping only the pole part of the propagator. For a vector (with coupling ), the total cross section becomes

In the high-energy limit, this yields

and in the low-energy limit

For the numerical calculations of subsect. 3.3, these cross sections are averaged over the relevant initial energy distributions; for the purposes of analytical approximation, one may estimate the complete by replacing in by

3.2. Approximate Analytical Solutions

If and always remain small, eqs. (3.1.2) reduce simply to

corresponding to baryon-number generation by free decays, with no back reactions. In this approximation, the baryon number generated is trivially given by

where, as usual, is a possible initial baryon number. Numerical solutions in subsect. 3.3 suggest that this approximation is typically accurate for and or Note that (3.2.2) provides an upper bound on back reactions always tend to diminish the baryon density.

At high temperatures, and taking for simplicity eqs. (3.1.2) become (the necessary small expansions of etc., are given in appendix C):

where

For small the solutions to these equations are

In subsect. 3.3, we shall find that these forms are often adequate until (the largest discrepancies are usually in ).

At low temperatures (large ) the undergo exponential decay, and their number is typically negligible for Only the last term in eq. (3.1.2c) for is thus important at large Using the low-energy point form (3.1.7) for the cross sections, and taking eq. (3.1.2c) becomes

Any baryon excess generated by decay and inverse decay at high temperatures is therefore depleted at low temperatures through baryon-number violating reactions, falling roughly like

and eventually tending to a constant non-zero value. (For fixed temperature, the exponent here which arises simply from the Fermi low-energy form for the cross section.) Numerical results in subsect. 3.3 suggest that in practice, when this behavior typically sets in when rises above about 2.

If the temperature of the universe falls with time according to

(where for a radiation-dominated universe at small for a matter-dominated universe with deceleration parameter and for a closed universe [9]), then the relaxation (3.2.6) of baryon density with time due to low-energy interactions is roughly

Hence, if [as in eq. (3.2.6), for which ], cannot relax to zero even when : the age of the universe then grows faster than the time necessary to establish chemical equilibrium; the fluctuation in baryon number has been frozen by the expansion of the universe, and survives forever, albeit perhaps somewhat diminished from its high temperature value. As discussed in appendix A, this failure to destroy baryon number even after an infinite time is a consequence of the extra expansion terms in the Boltzmann equation, which invalidate Boltzmann's H theorem (17). On the other hand, in a universe with processes occur with a sufficient rate to combat expansion, and any baryon number generated at high temperatures eventually relaxes exponentially to zero. To attain would require the introduction of a cosmological term into the Einstein field equation, which can serve even to halt expansion (as in the Lemaître universe) and allow chemical equilibrium to be established. (These results are not specific to the model of subsect. 2.3 considered. In practice, however, gravitational or other clumping will drastically change the rate for B-violating interactions at large t: for example, two quarks confined within a proton have a much higher amplitude to come sufficiently close together to annihilate than would two free quarks in an ideal homogeneous gas with the same density as the proton gas (18). Note that even in the presumably physical case baryon number generated at high temperatures would be diminished to an unacceptably low level if were too large. The final baryon number usually depends, however, on the behavior of eqs. (3.1.2) in the region where simple analytical approximations fail; a numerical solution to (3.1.2) is therefore necessary.

3.3. Numerical Results

In this subsection we give numerical solutions to eqs. (3.1.2) as a function of the three dimensionless parameters and Except in considerations of the destruction of an initial non-zero at very high temperatures, for which eqs. (3.1.2) are no longer accurate (see sect. 4) the precise form for the widths and cross sections assumed is largely irrelevant; only the very model independent low-energy form (3.1.7) for the cross sections is important (these cross sections are essentially just those which should induce proton decay).

Baryon number violating interactions such as those in the simple model of subsect. 2.3 treated here should lead to proton decay, with a lifetime given by the very low-energy limit of (3.1.7) as roughly The experimental yr then implies GeV; in the SU(5) grand unified model, estimates suggest that GeV = 1 eV (19). We use eV as a standard value for our numerical results. The relevant coupling constant depends on the precise nature of the in our model. If is a gauge (vector) boson, then should presumably be the corresponding effective gauge coupling constant at an invariant mass for scatterings and for decays. A typical value obtained for this coupling constant in the SU(5) model is On the other hand, if is a scalar (presumably Higgs) boson, as is probably obligatory in generating baryon number from an SU(5) model, the relevant coupling constant is largely unknown, but it is probably rather small The value of the -violation parameter is even more uncertain. Nevertheless, all our numerical results for (and ) in fact depend linearly on to within a few percent, even when As a standard, we take the quite unmotivated value . (20) Finally, we must specify the effective Planck mass defined by eq. (2.1.14), which depends on the number of species contributing to the energy density of the universe at the temperatures considered. If no new species of particles (except ) beyond those already detected exist with masses then With this, and the choice eV, the dimensionless parameter . determines the rate of expansion in the early universe; inhomogeneities or perturbations in the metric could lead to different expansion rates for different regions of the universe. Such effects may be parametrized by different values for

Fig. 1 shows the development of the and baryon densities as a function of the inverse temperature with eV, and The dashed lines in fig. 1 are the analytical approximations for small discussed in subsect. 3.2. Note that the changes in the actual lag behind those in .

In fig. 2 we show the relative sizes of the terms contributing to with the parameters used in fig. 1. As expected, for (in this case ) all terms proportional to decrease exponentially so that the only remaining contribution is from the two-body scattering.

Fig. 3 illustrates the sensitivity of to the parameters of the model. Unless otherwise indicated, the parameters are the same as for fig. 1. Fig. 3a shows that the final is independent of an initial so long as is small. As discussed in subsect. 2.4, the destruction of a very large cannot be treated using eq. (2.3.20). Fig. 3a also exhibits the linear proportionality of on Figs. 3b-d illustrate the dependence of on and respectively.

[ Figure 1 ] Numerical solutions (solid lines) and analytical approximations (dashed lines) for the number densities in the model of subsect. 2.3. The differential equations for and are given in (3.12). The standard choices of parameters used in this and later figures are eV, and GeV).

Finally in fig. 4, we give the final value of as a function of (i.e., of ) for various values of

[ Figure 2 ] The relative magnitudes of the terms in eq. (3.1.2) contributing to the time development of the baryon density . Notice that for all terms proportional to or become exponentially unimportant, and the largest contribution to is from scattering processes.

[ Figure 3 ] The sensitivity of the baryon number development to the input parameters. Unless otherwise indicated, the parameters are the same as those used in fig. 1.

[ Figure 4 ] The final baryon to photon ratio (divided by the -violation parameter ) as a function of the ratio of the effective Planck mass to for several values of the coupling constant The upper scale shows the values of with the choice in the definition of the effective Planck mass (2.1.14).

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