Inverse-Power-Law Scalar-Field Cosmology in (n+2) Dimensions: Slow-Roll Evolution and de Sitter-Like Asymptotics

This study investigates a homogeneous canonical scalar field with an inverse-power-law potential in an (n+2)-dimensional spatially flat Friedmann-Robertson-Walker/Friedmann-Lemaître-Robertson Walker (FRW/FLRW) spacetime. The approximately 5% baryonic-matter, 25% dark-matter, and 70% dark-energy fractions are used only as present-day observational motivation and are not imposed as fixed weights in the scalar-field dynamics. Using the metric signature (+,-,...,-), the scalar-field definition and the higher-dimensional Friedmann, acceleration, Klein-Gordon, and conservation equations are formulated consistently with the gravitational coupling Gₙ₊₂. For K > 0, α > 0,ϕ> 0, and integer n ≥ 2, the scalar-field-dominated late-time regime is analyzed under the slow-roll approximation. The solution obeys an asymptotic power-law scaling with exponent 2/(α+4). Substitution back into the field equations shows that the kinetic-to-potential-energy ratio, the second slow-roll ratio, and the Hubble slow-roll parameter all tend to zero. Consequently, the scalar-field equation-of-state parameter approaches -1 and the dimension-dependent acceleration condition is satisfied asymptotically. The resulting expansion is de Sitter-like in its equation-of-state and slow-roll behavior, although H remains time dependent and the solution is not an exact de Sitter fixed point. The analysis is restricted to homogeneous background evolution; observational parameter fitting, perturbations, stability, and structure formation are left for future work.