Parameterized Absolute Parallelism: Stability, Singularity Avoidance, and Accelerated Expansion of the Universe

This article explores the dynamics and stability of cosmological models within the framework of Parameterized Absolute Parallelism (PAP) geometry, with particular emphasis on the effects of spin-torsion interactions. Utilizing the Parameterized Raychaudhuri Equation, we derive stability conditions for fixed points and examine their influence on the evolution of the universe’s expansion scalar. Our findings indicate that the parameter b associated with spin-torsion coupling plays a pivotal role: when it fosters stability, it prevents singularities and facilitates smooth exponential expansion reminiscent of inflation. Conversely, when $$b<1$$ , instability and divergence occur. This framework also addresses singularity avoidance through nonsingular bounces and establishes a connection between spin-torsion effects and accelerated expansion, effectively mimicking dark energy. Numerical simulations support these conclusions, illustrating evolution of the scale factor, the expansion scalar, and stability conditions. The results highlight the potential of the PAP geometry in tackling significant cosmological challenges, including singularity resolution and the mechanisms driving accelerated expansion. We apply these findings to a comprehensive model encompassing the Big Bang, Big Rip, and Big Crunch scenarios.