Non-Gaussian Effects of the Saha's Ionization in the Early Universe

TL;DR

This paper investigates how non-Gaussian statistical effects, modeled by Tsallis statistics, influence the Saha ionization equation and cosmological recombination, revealing new effects on hydrogen binding energy and related phenomena.

Contribution

It introduces a generalized Saha equation incorporating non-Gaussian effects via Tsallis statistics, highlighting two new chemical equilibrium conditions and the concept of binding $q$-energy.

Findings

Identified two generalized chemical equilibrium conditions for relativistic and non-relativistic regimes.

Discovered that hydrogen binding $q$-energy exhibits symmetrical behavior around standard energy.

Analyzed the impact of non-Gaussian effects on deuterium bottleneck and particle-antiparticle excess.

Abstract

Tsallis' thermostatistical has received increasing attention due to its success in describing phenomena that manifest unusual thermodynamic properties. In this context, the generalized Saha equation must follow a condition of generalized thermal equilibrium of matter and radiation. The present work aims to explore the non-Gaussian effects on Saha's ionization via Tsallis statistics. To accomplish this, we generalized the number density taking into account a non-Gaussian Fermi-Dirac distribution, and then set out the Saha equation for the cosmological recombination. As a result, we highlight two new non-Gaussian effects: $i$) two generalized chemical equilibrium conditions, one for the relativistic regime and the other for the non-relativistic one; and $ii$) the hydrogen binding $q$-energy. We demonstrated that to yields smooth shifts in the binding energy, the $a$-parameter must be very small. We also showed that binding $q$-energy exhibits symmetrical behavior around the value of the standard binding energy. Besides, we used the $q$-energy in order to access other hydrogen energy levels, and we ascertained the values of the $a$-parameter that access those levels and their relationship to temperature. Finally, we employed these results to examine the non-Gaussian effects of the deuterium bottleneck, recombination and the particle anti-particle excess.