Rainbow f(R) 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)∝tp, 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/μ2) naturally dominates the dynamics, giving rise to a late–time quintessence phase (−1<we<−1/3 for λ>−1) that asymptotically approaches a de Sitter attractor (we→−1 as ϵ1 → 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.