Rainbow <italic>f</italic>( <italic>R</italic>) Inflation: de Sitter-Quintessence Phases

In this study, we investigate the inflationary dynamics of Rainbow f( R) gravity characterized by f( R) ≃  αRⁿ, with the rainbow deformation f˜(H)∼Hλ and g˜=1. The effects of the rainbow parameter λ and the curvature exponent n on inflationary observables are analyzed under the Atacama Cosmology Telescope (ACT) constraint 0.9709 <  ns < 0.9777. Numerical results reveal a clear correlation among n, λ, and the tensor-to-scalar ratio r, such that decreasing n corresponds to increasing λ, accompanied by a growth in r within observational limits. Negative values of λ (typically for n ≈ 3) yield a quasi–de Sitter inflationary phase with a nearly constant Hubble parameter and suppressed tensor modes, while positive λ (for n ≲ 2.2) drive a quintessence–like inflationary regime with enhanced expansion and a time-dependent curvature. Using power–law solutions a( t)∝ t^p, we relate the expansion exponent to the number of e-folds N ≃ 50–60. Small deviations between the matter and effective equations of state indicate a mild breaking of we=w, suggesting the emergence of matter–like behavior within the quintessence–like inflationary field. At low curvature, we further show that a logarithmic Nojiri–Odintsov correction f( R) ≃  α′ln ( R/ μ ²) naturally dominates the dynamics, giving rise to a late–time quintessence phase ( −1−1) that asymptotically approaches a de Sitter attractor ( we→−1 as ϵ 1 → 0 or λ=−1). Thus, the combined Rⁿ and logarithmic contributions provide a unified geometric framework in which early–time inflation and late–time cosmic acceleration naturally arise from different curvature regimes of Rainbow f( R) gravity, consistent with ACT observational data.