Saha ionization for non-equilibrium states: an application to primordial hydrogen recombination

It is widely known that the Saha equation is not suitable for describing plasmas out of thermodynamic equilibrium. The primordial hydrogen recombination plasma is an example of this. In this work, we propose a theoretical modification to the standard Saha curve by introducing a non-Gaussian generalization of the Saha equation, motivated by Tsallis statistics. In particular, we explore the possibility that a time-dependent q-parameter may serve as an effective proxy for the evolving thermodynamic conditions during recombination, especially considering that hydrogen recombination occurs from excited states. Within this framework, the q-parameter could be interpreted as encoding departures from equilibrium and could play the role of effective time-dependent temperature. This indicates that the time evolution of the q-parameter could provide a phenomenological mechanism for incorporating non-equilibrium effects into the recombination history. Our findings suggest that the Tsallis parameterization provides an alternative path to fit the distribution of free electrons by using an effective temperature. Our approach enables us to reproduce hydrogen recombination results predicted by the Peebles kinetic approach, HYREC-2 code, or any other recombination model.