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

TL;DR

This paper extends the classical Kretschmann scalar in cosmology by incorporating torsion within Parameterized Absolute Parallelism geometry, revealing how torsion influences curvature and cosmological models like the Big Rip.

Contribution

It introduces a parameterized framework for the Kretschmann scalar in torsion-inclusive cosmology, bridging Riemannian and Weitzenbock geometries, and explores implications for early-universe dynamics.

Findings

Modified Kretschmann scalar depends on torsion parameter b

Classical scalar recovered at specific b values

Torsion effects could influence Big Rip cosmology

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