Matter annihilation in the late universe

We discuss the annihilation of electrons and positrons left over after baryon decay in a $k=0\mathrm{}$ Friedmann universe. It will take at least ${10}^{116}$ yr for most annihilations to occur, producing not less than ${10}^{14}$ times the present entropy of the universe in photons which will then exceed the blackbody background density by a factor of ${10}^{70}$ or more. Enough electrons and positrons will always remain to overwhelm the red-shifting blackbody background energy and be at least comparable to the energy in the radiation produced by the ever-continuing annihilation. An alternate scenario with large-scale but never universal gravitational collapse may keep the energy of the universe dominated by ever-larger black holes. If instead black holes cease to form or grow after some late time, their evaporation might temporarily give radiation domination. However, if electrons or other massive particles are absolutely stable, the universe cannot become and remain completely radiation dominated forever. In either scenario a $k=0\mathrm{}$ universe never reaches thermal equilibrium, and the entropy may increase without limit.