Parameterized Evolution of the Kretschmann Scalar in Friedmann-Lemaitre-Robertson-Walker Cosmology with Torsion Contributions and Big Rip Model

Abstract: We investigate the Kretschmann scalar within the framework of Parameterized Absolute Parallelism (PAP) geometry, extending the classical understanding of spacetime curvature in General Relativity by incorporating torsion. This extension introduces a dimensionless parameter b, allowing a continuous interpolation between Riemannian geometry (b = 0) and Weitzenbock geometry (b = 1). Using the pseudo-Riemannian metric associated with Friedmann-Lemaitre-Robertson-Walker cosmology, we derive an explicit expression for the modified Kretschmann scalar, which captures contributions from standard curvature, curvature-torsion interactions, and pure torsion. Our analysis reveals that the modified scalar reduces to the classical value under specific conditions (b = 0 or b = +/- sqrt(2)), while deviations occur for other b values. This work highlights the potential role of torsion in early-universe dynamics and provides a framework for exploring deviations from standard General Relativity in cosmological models, such as the Big Bang-Big Rip model (BRM). Keywords: Gravity; Torsion; Kretschmann scalar; BRM. PACS Nos: 04.20.-q; 11.10.-z; 98.80.Es; 98.80.-k