Dynamics of cosmological phase crossover during Bose-Einstein condensation of dark matter in Tsallis cosmology

During the cosmic evolution process, as the temperature of a cosmological boson gas falls below a certain threshold, a Bose-Einstein condensation process can occur at various points throughout the cosmic history of the Universe. In this model, dark matter, conceptualized as a non-relativistic, Newtonian gravitational condensate is governed by the Gross-Pitaevskii-Poisson system. In our present study, we investigate the Bose-Einstein condensation process of bosonic DM by treating it as an approximate first-order phase transition within a modified cosmological framework, known as Tsallis cosmology. We examine the evolution of relevant physical quantities characterizing the evolution dynamics of the Universe, including energy density, temperature, redshift, scale factor, Hubble parameter, and dimensionless deceleration parameter before, during, and following the Bose-Einstein condensation phase transition takes place. Additionally, we especially investigate the specific era of the evolution of the Universe characterized by a mixture of normal and condensate phases of dark matter. We analyze the behavior of temporal evolution of an important time-dependent parameter, called the condensate dark matter fraction throughout the condensation process and find the time duration of condensation of dark matter in the Tsallis cosmological model. We see that the presence of Bose-Einstein condensate dark matter in the framework of Tsallis-modified cosmology significantly alters the cosmological evolution of the Universe as compared to the standard model of cosmology. We also find for a typical value of Tsallis non-extensive parameter beta=0.35\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =0.35$$\end{document}, the model could explain an accelerated Universe without invoking any additional energy component and solve the age problem of our Universe.