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Experimental lower limits on the lifetime of the proton indicate that the bosons which should mediate its decay in grand unified gauge models have masses [1]. In this paper we derive cosmological constraints on the properties of of heavy fermions () which appear in many such models [2,3] The presence of baryon-number violating bosons implies that any net baryon number introduced as an initial condition in the standard hot big bang early universe should have been destroyed (e.g., [4]). A baryon asymmetry may subsequently be generated by - and -violating decays of heavy bosons (or fermions). In order for a sufficient asymmetry to survive to the present, it is necessary that the number of photons generated at later times not be excessive. Any deviations from thermal equilibrium, which would generate entropy, and hence increase must be small. However, very weakly interacting particles will not remain in thermal equilibrium when the universe cools. If they survived for long enough between decoupling from equilibrium and decaying, then the entropy released by their decays would dilute to below its observed value (even if before decay). (The expansion of the universe is by assumption adiabatic; its entropy nevertheless increases unless deviations from equilibrium are relaxed away sufficiently quickly for the expansion to be ``quasistatic.'') To avoid this phenomenon heavy fermions in grand unified theories must have masses and lifetimes which lie outside the shaded region delineated in fig. 1.

[ Figure 1 ] Masses and lifetimes of weakly interacting particles lying within the shaded region are inconsistent with the standard cosmological model. The decays of such would generate a large entropy and result in excessive dilution of . The solid contour lines give the before decay necessary to avoid dilution below the observed . The wavy lines give lifetimes for decaying through coupling to a boson of mass . The dashed lines give the effective lifetime for which decay at a fixed transition temperature .

According to the standard hot big bang model, any should be in thermal equilibrium in the sufficiently early universe, with a number density about that of photons (1) . If the interact with the thermal bath of photons through two-body scattering mediated by the exchange of a particle of mass and coupling , they are maintained in equilibrium until , when their interaction rate falls below the expansion rate of the universe (where the effective Planck mass is the Planck mass is the number of particle species with which is typically for in grand unified models). If the remain in thermal equilibrium, then their number density would eventually become . If , the decouple from equilibrium while they are still relativistic (2) with a number density . After decoupling the behave as a collisionless gas and their number density remains , falling as the universe expands. The expansion redshifts all momenta , so that the energy density of photons . However, the energy density is dominated by the rest mass, so that ; for the may dominate the energy density of the universe. Eventually the decay, thereby converting their potentially large energy density into light particles and reheating the universe. If the survive out of thermal equilibrium for a long time before decaying, then a large amount of entropy is generated which when shared with the rest of the universe may dilute to below its observed value. The temperature of photons in a universe whose energy is dominated by falls with time according to (3)

so that at the time when the decay,

where is the effective (4) lifetime of the (5). The decays convert this rest energy into a cascade of light particles which quickly become thermalized [5]. The number density of light particles produced is given by , so that the original light particle density is increased by a factor

Any present prior to the decays (or even produced in the decay) would be diluted by this factor through the entropy produced in the decays. The solid contour lines in fig. 1 give the prior to the decays which would be required in order for the final to be as observed at present. Detailed investigations [6] suggest that grand unified models could give a maximum before decay of about .

A heavy fermion which decays only through a virtual boson should have a width given, in analogy to decay, by

where represents the sum over ``virtual decay modes'' weighted by requisite mixing angles (including a factor for the number of X states), and typically . This is shown by the wavy lines in fig. 1 for various . If eq. (4) obtains, the light particle density is increased through decays by a factor [cf. eq. (3)]

For example, if this dilution is to be less than a factor of (allowing an initial ) then

so that for , .

Grand unified models based on extensions of SU(5) (e.g., SO(10), E(6)) almost inevitably involve heavy fermions associated with SU(5) singlet (neutral) components in the fermion representations. In some cases, the may be introduced as a heavy right-handed partner for the light ; in other models, the may be independent. The former possibility is realized in a class of SO(10) models which provide a natural explanation of small light neutrino masses [7] (where is the mass of the relevant charge quark). In these models the dominant decay modes of the are , , , (where is the usual Higgs doublet (6) ) yielding a decay width

which completely swamps eq. (4), and prevents useful bounds on . Limits on the light neutrino masses nevertheless provide some bound on . In other models the need not be associated with although it may still decay to a light doublet (e.g., [9]), thereby avoiding our bounds.

In models with dynamical symmetry breakdown and without explicit Higgs fields, another decay mechanism may be important. The may mix with light (virtual) fermions which then ``decay'' by emission of a light W boson. Typically, such mixing occurs only when the universe has cooled below the critical temperature at which the symmetry which forbids the mixing (through mass terms) is spontaneously broken. The lifetime of the in the hot early universe is shown by the dashed line in fig. 1 if the decay proceeds through mixing to light fermions at the indicated transition temperature .

In other possible schemes, the cannot decay by mixing, but only by emission of (usually -violating) bosons. The lifetimes of such should follow the wavy lines in fig. 1. If the relevant bosons are more massive than the SU(5) gauge boson , our bound indicates that in such schemes .

We are grateful to Pierre Ramond for helpful conversations.

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