Chaotic cosmologies and the topology of the Universe

Topologically multiply connected universes with finite volume naturally produce the special initial conditions required by the Rees chaotic cosmology model. Such universes are especially appealing since with a soft equation of state at early times they produce an entropy per baryon of |N (c/Gm²ₚ) ^1/4 10^9. 5| in rough agreement with the observations. Since at present we would see blackbody photons coming from many images of the same fundamental volume the isotropy of the blackbody photons is explained in an extraordinarily simple way. One can set lower limits on the proper radius of the fundamental volume in such a model by our failure to find multiple images of galaxy clusters in the Shane–Wirtanen sample, i. e. |R_ H 400 (H₀/50 km s^-1 Mpc^-1) ^-1|⁠. For | 1| models topological constraints dictate even larger values of RH, In such models the proper radius of the universe cannot be much smaller than its radius of curvature. Because of these observational and topological constraints, multiply connected universes with chaotic initial conditions are not capable of thermalizing the cosmic black-body radiation with normal thermalization processes. If large amounts of dust are produced at early epochs thermalization is possible for chaotic multiply connected |₀ 1| models.