Stability Analysis of Dilaton‐Inspired Scalar Field within the Geometrical Trinity of Gravity

The dynamics of the dilaton‐inspired scalar field are investigated, formally rewritten by means of a Brans‐Dicke Lagrangian, within the framework of geometrical trinity of gravity . In this respect, a stability analysis is performed by adopting a non‐flat Friedmann–Robertson–Walker (FRW) metric and considering the well‐established exponential potential in three distinct gravitational frameworks: general relativity, teleparallel gravity, and symmetric‐teleparallel gravity. By comparing the scalar field behaviors across these theories are compared, the role of curvature, torsion, and non‐metricity in shaping cosmic evolution is highlighted. The analysis reveals that, both in general relativity and teleparallel gravity, the dilaton‐inspired field can drive the accelerated expansion of the universe, effectively behaving as cosmological constant at late times. In contrast, within the symmetric teleparallel gravity scenario, the performance of a complete linear stability analysis is prevented by the use of the non‐coincident gauge. Nevertheless, the latter paradigm introduces complexity into the autonomous system, resulting in a structurally different analysis. For general relativity and teleparallel scenarios, the regions of attractor solutions and unphysical domains in which we do not expect the viability of our dilaton‐inspired Lagrangian are remarked. However, within the framework of symmetric‐teleparallel gravity, the stability analysis reveals no attractor points for the chosen set of free parameters. In support of these findings, physical conclusions, kinematical studies, and consequences on Friedmann dynamics are thus explored.