Statistical mechanics and the description of the early universe. (I). Foundations for a slightly non-extensive cosmology

We analyze how the thermal history of the universe is influenced by the sta tistical description, assuming a deviation from the usual Bose-Einstein, Fe rmi-Dirac and Boltzmann-Gibbs distribution functions. These deviations repr esent the possible appearance of non-extensive effects related with the exi stence of long-range forces, memory effects, or evolution in fractal or mul ti-fractal space, In the early universe, it is usually assumed that the dis tribution functions are the standard ones. Then, considering the evolution in a larger theoretical framework will allow to test this assumption and to place limits to the range of its validity. The corrections obtained will c hange with temperature, and consequently, the bounds on the possible amount of non-extensivity will also change with time. We generalize results which can be used in other contexts as well, as the Boltzmann equation and the S aha law, and provide an estimate on how known cosmological bounds on the ma sses of neutrinos are modified by a change in the statistics. We particular ly analyze here the recombination epoch, making explicit use of the chemica l potentials involved in order to attain the necessary corrections. All the se results constitute the basic tools needed for placing bounds on the amou nt of non-extensivity that could be present at different eras and will be l ater used to study primordial nucleosynthesis. (C) 2001 Elsevier Science B. V. All rights reserved.