Cosmological dynamics of a generalized Klein-Gordon equation

In this paper, we present a purely geometric generalization of the Klein–Gordon (KG) equation within the framework of Parameterized Absolute Parallelism (PAP) geometry. Starting from the standard relativistic wave equation for a spin-0 field on Riemannian backgrounds, we replace the Levi-Civita connection by the PAP connection, which introduces an adjustable parameter b controlling the contribution of torsion through the contortion tensor. This yields a modified d’Alembert operator containing an additional first-order derivative term that couples the scalar field gradient to a geometric vector current constructed from the trace of the contortion tensor. The resulting equation reduces to the conventional KG equation when [Formula: see text], while for [Formula: see text] the propagation of the scalar field becomes sensitive to torsion in a manifestly geometric way, without invoking non-minimal matter couplings or modifying the gravitational field equations. As an application, we study a spatially flat Friedmann–Robertson–Walker (FRW) background induced by a diagonal tetrad and show that the effective friction term in the homogeneous mode equation is rescaled by a factor [Formula: see text], leading to either enhanced or suppressed damping depending on the sign of b. We discuss the physical implications of this result, including its impact on scalar field dynamics in the early universe and the potential of PAP geometry to encode torsion effects in scalar cosmology and quantum field theory in curved spacetime.