Inflation in Myrzakulov $F(R,T)$ Gravity: A Comparative Study in Metric, Symmetric Teleparallel, and Weitzenböck Formalisms

Abstract: We present a unified treatment of cosmic inflation within the framework of Myrzakulov Gravity, exploring its realization in three different formalisms: metric (curvature-based), teleparallel (torsion-based), and symmetric teleparallel (non-metricity-based). For each case, we derive the corresponding field equations in a flat FLRW background, study inflationary solutions driven by a scalar field, and compute observable quantities such as the scalar spectral index ( n_s ) and the tensor-to-scalar ratio ( r ). In addition to these geometric sectors, we extend our analysis to the more general and dynamically richer Myrzakulov ( F(R,T) ) gravity, which incorporates both curvature ( R ) and torsion ( T ) in a unified action. We derive the inflationary dynamics in this hybrid model and investigate how it interpolates between pure ( f(R) ) and ( f(T) ) behaviors. The resulting framework allows for enhanced flexibility in matching Planck and BICEP/Keck observational constraints. We present analytic estimates and schematic predictions in the ( n_s )--( r ) plane, demonstrating that appropriately chosen parameters in ( F(R,T) ) models can produce viable and distinguishable inflationary signatures. This comparative and extended study highlights the potential of Myrzakulov Gravity and its generalizations to provide a consistent and geometrically motivated description of the early universe, with predictive power across different formulations of spacetime geometry.