A possible correction of the Saha curve for non-equilibrium states

TL;DR

This work introduces a Tsallis-statistics–based generalization of the Saha equation to describe primordial hydrogen recombination out of equilibrium. By treating a time-dependent Tsallis parameter $q(z)$ as an effective temperature, the authors connect non-Gaussian statistics to the evolving thermodynamic state of the plasma, particularly noting recombination via excited states. A four-step strategy—derive $x_e(z)$ from Peebles and HYREC-2, extract a $q(z)$, fit it with a polynomial, and apply the fitted $q(z)$ to a generalized Saha equation—yields a time-varying $q$ that can reproduce the non-equilibrium ionization history with high fidelity, achieving a minimum $q_{ ext{min}} \approx 1.17$ during recombination. This framework provides a physically meaningful, phenomenological bridge between non-extensive statistics and cosmological ionization histories, with potential implications for observable signatures in the CMB and beyond.

Abstract

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 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. The implications of this approach might go beyond its immediate applications, as the Saha equation is widely used in various scientific fields like astrophysics, cosmology, plasma physics, and condensed matter physics.

Figures (3)

• Figure 1: The curves depict the evolution of the ionization degree considering the Peebles (solid line) and Saha (dashed line) approaches and the result provided by the HYREC-2 code (dotted line).
• Figure 2: Numerical solution for the $q$-parameter as a function of redshift $z$ from step 2 of our strategy. The black solid line was obtained using the Peebles model. The blue and red dashed lines were yielded via the HYREC-2 code and seventh-degree polynomial fit, respectively.
• Figure 3: Evolution of the degree of ionization in the hydrogen recombination period. Three curves are shown: Peebles' approach (black solid line), Saha's ionization corrected (red dotted line), and Saha's 7th-degree polynomial fit (blue dashed line).