Prediction of Super-Exponentially Accelerated Universe in a Friedmann–Lemaitre–Robertson–Walker Metric

A coupled model of variable-modified Chaplygin gas (as a candidate of dark energy) and dark matter is assumed in the background of the Friedmann–Lemaitre–Robertson–Walker universe. Firstly, a two-dimensional autonomous dynamical system is considered for a flat universe after determining some suitable dimensionless parameters. The evolution of those parameters along with the deceleration parameter is observed. Next, the evolution of matter and energy density parameters is studied for variational coupling parameters. Phase portrait analysis is performed to explain the present as well as future expansion of the universe. Secondly, both three- and four-dimensional dynamic systems are constructed when the evolution of spatially homogeneous and isotropic model of the universe is considered, which includes the cosmological constant and three-curvature symmetry surfaces. For the three-dimensional system, updates of dimensionless parameters along with deceleration parameters are analyzed for variational coupling parameters with respect to progression of the universe. Again, the evolution of the deceleration parameter is studied for different values of \(n\). Stability analysis is carried out for the concerned dynamical system with the help of eigenvalues. The four-dimensional dynamical system is remarkable as it helps to study the evolution of \(\Lambda\) density along with other parameters. Lastly, for the concerned dynamical system, the evolution of matter and energy density parameters is analyzed for different values of \(c\) to study whether or not the coupling parameters affect the ultimate evolution of the universe.