Dynamical and Observational Analysis of Generalized Nash's Theory of Gravity

We investigate cosmic evolution in generalized Nash's theory of gravity involving the quadratic Ricci invariant $\chi=R_{\mu\nu}R^{\mu\nu}$. The analysis is divided into two complementary branches. First, we study the power-law family $f(R,\chi)=R^{\alpha}+\beta\chi$ as a reduced autonomous system in a flat FLRW background. Because the adopted variables become singular at the Einstein--Hilbert limit $\alpha=1$, the phase-space analysis is restricted to $\alpha\neq1$, with $\alpha=2$ 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 $\alpha=2$ 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 $f_{\rm obs}(R,\chi)=R-2\Lambda+\beta\chi$, which reduces exactly to flat $\Lambda$CDM when $\beta\to0$. The Hubble rate is obtained from the reduced $\Lambda$CDM-connected background branch, integrated over $0\le z\le10$ and matched at higher redshift to a standard radiation+matter+$\Lambda$ background. Using SNe~Ia, BAO, and Planck~2018 compressed CMB distance priors, we find an expansion history very close to $\Lambda$CDM, with the quadratic correction tightly constrained around the nested standard-model limit. The resulting bound on $\beta$ 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.