Analysing Distinctive Kinematic Behaviours in Cosmic Models across Modified Gravitational Theories

Abstract Observational evidence for late‑time cosmic acceleration has motivated numerous modified gravity frameworks that can mimic ΛCDM at the level of the Hubble parameter yet diverge in higher‑order kinematic descriptors. This study undertakes a comparative kinematic analysis of representative cosmic expansion histories emerging from f(R), f(T) teleparallel, scalar–tensor (including Horndeski), Gauss–Bonnet, and massive/braneworld scenarios. Working in a model‑agnostic cosmographic formalism, we reconstruct redshift evolutions of the deceleration q(z), jerk j(z), snap s(z), statefinder {r,s}, and Om(z) diagnostics directly from data combinations (SNe Ia distance moduli, BAO scale measurements, cosmic chronometer H(z) points, CMB shift parameters, and growth‑rate fσ8 constraints propagated via effective background relations). Bayesian hierarchical inference with flexible nonparametric expansion bases (Padé, Chebyshev, and Gaussian‑process priors) yields posterior ensembles of kinematic trajectories for each theory class after marginalizing over nuisance calibration and spatial curvature. We then apply unsupervised manifold learning and clustering metrics to map theory spaces to regions in a reduced kinematic feature plane, highlighting where observationally allowed trajectories separate. Distinctive signatures emerge: viable f(R) reconstructions tend to track ΛCDM in q(z) but show broadened j(z) posteriors at z≲1; common f(T) parameterizations favour an earlier acceleration transition redshift; selected scalar–tensor/Horndeski sectors permit evolving effective Planck mass that imprints curvature in Om(z); while braneworld and dRGT‑like massive gravity realizations populate extreme corners of the statefinder plane with extended phantom‑crossing arcs. Forecast analyses incorporating Stage‑IV survey precisions (e.g., Rubin/LSST, Euclid, Roman) indicate that joint constraints on {q0, j0, s0} plus the Om slope can rule out broad swaths of parameter space currently degenerate with ΛCDM. We conclude that a data‑driven kinematic atlas provides an efficient, theory‑transparent discriminator that complements field‑equation‑level analyses and helps prioritize modified gravity models for detailed perturbation studies. The framework and diagnostic dataset are released to enable rapid consistency checks and machine‑learning surrogates for forthcoming multi‑probe cosmological analyses.