2026/07/24 by Amin Rezaei Akbarieh, Mohammad Amin Bolouri, Yaghoub Heydarzade
#gr-qc
We investigate cosmic evolution in generalized Nash's theory of gravity involving the quadratic Ricci invariant χ=RμνRμν. The analysis is divided into two complementary branches. First, we study the power-law family f(R,χ)=Rα+βχ as a reduced autonomous system in a flat FLRW background. Because the adopted variables become singular at the Einstein--Hilbert limit α=1, the phase-space analysis is restricted to α≠1, with α=2 used as a representative quadratic benchmark. This benchmark contains radiation-like boundary configurations, restricted scaling saddles, and de Sitter-like accelerating endpoints (a stable node away from α=2 and non-hyperbolic at the benchmark itself), but not a complete regular radiation-to-matter-to-de Sitter sequence. Second, we constrain the regular observational branch f\rm obs(R,χ)=R-2Λ+βχ, which reduces exactly to flat ΛCDM when β→0. The Hubble rate is obtained from the reduced ΛCDM-connected background branch, integrated over 0≤ z≤10 and matched at higher redshift to a standard radiation+matter+Λ background. Using SNe~Ia, BAO, and Planck~2018 compressed CMB distance priors, we find an expansion history very close to ΛCDM, with the quadratic correction tightly constrained around the nested standard-model limit. The resulting bound on β should be interpreted as a background-level constraint within this reduced prescription, not as a perturbation-level viability test of the full higher-derivative theory.