${\mathbf{\Omega }}_{\mathbf{1}}{\mathbf{\Omega }}_{\mathbf{2}}$-$\mathbf{\Lambda }$CDM: A promising phenomenological extension of the standard model of cosmology

We investigate a phenomenological extension of the standard $\mathrm{\Lambda }$CDM framework, the ${\mathrm{\Omega }}_1{\mathrm{\Omega }}_2$-$\mathrm{\Lambda }$CDM model, in which the total energy density of the universe is expanded in powers of $1+z$. This parameterization recovers $\mathrm{\Lambda }$CDM and introduces two additional terms, ${\mathrm{\Omega }}_1(1+z)$ and ${\mathrm{\Omega }}_2(1+z{)}^2$, in the dark energy sector alongside the cosmological constant, leading to a physically interpretable and observationally testable effective dark energy dynamics. Using Planck cosmic microwave background (CMB) data, we find that the model allows additional freedom in the late-time expansion history while preserving standard early-universe physics. When DESI baryon acoustic oscillation (BAO) data are included, the inferred Hubble constant is $H_0=69.74\pm 0.77\mathrm{k}\mathrm{m}{\mathrm{s}}^{-1}\mathrm{M}\mathrm{p}{\mathrm{c}}^{-1}$ (68% CL), consistent with Tip of the Red Giant Branch (TRGB) measurements from the Carnegie–Chicago Hubble Program (CCHP). The reconstructed dark energy equation of state exhibits a smooth transition across the phantom divide followed by asymptotic de Sitter behavior, modifying late-time dynamics while maintaining standard early-time cosmology. Overall, controlled late-time deviations from $\mathrm{\Lambda }$CDM improve cosmological concordance.