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 statistical description, assuming a deviation from the usual Bose–Einstein, Fermi–Dirac and Boltzmann–Gibbs distribution functions. These deviations represent the possible appearance of non-extensive effects related with the existence of long-range forces, memory effects, or evolution in fractal or multi-fractal space. In the early universe, it is usually assumed that the distribution 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 change 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 Saha law, and provide an estimate on how known cosmological bounds on the masses of neutrinos are modified by a change in the statistics. We particularly analyze here the recombination epoch, making explicit use of the chemical potentials involved in order to attain the necessary corrections. All these results constitute the basic tools needed for placing bounds on the amount of non-extensivity that could be present at different eras and will be later used to study primordial nucleosynthesis.