Structural Origin of Cosmological Density Ratios: $\Omega_m^2 = 3\Omega_\Lambda \Omega_b$

Abstract

This work investigates a simple algebraic relation among cosmological density parameters $$\Omega_m^2 \approx 3\,\Omega_\Lambda\,\Omega_b$$ suggested by representative late-time cosmological parameter estimates reported by Planck and DESI analyses. The framework uses the fixed structural ratio $$R=\frac{Y_2^+}{Y_1^0}=\frac{13}{6},$$ adopted from the companion foundational construction and not fitted to cosmological data. The cosmological density hierarchy is written as $$\Omega_\Lambda:\Omega_m:\Omega_b=(Y_2^+)^2:Y_1^0Y_2^+:Y_2^0=169:78:12.$$ Equivalently, $$\Omega_\Lambda:\Omega_m:\Omega_b=1:\frac{1}{R}:\frac{1}{3R^2}.$$ After normalization, the framework predicts $$\Omega_\Lambda=\frac{169}{247},\qquad\Omega_m=\frac{78}{247},\qquad\Omega_b=\frac{12}{247}.$$ Using the standard decomposition $\Omega_m=\Omega_b+\Omega_{dm}$, the corresponding residual matter component is $$\Omega_{dm}=\frac{66}{247}.$$ These values satisfy the exact quadratic closure relation $$\Omega_m^2=3\,\Omega_\Lambda\,\Omega_b,$$ or equivalently $$Q=\sqrt{\frac{\Omega_m^2}{3\,\Omega_\Lambda\,\Omega_b}}=1.$$ Using representative parameter combinations from Planck 2015, Planck 2018, and DESI 2024 cosmological analyses, the inferred values cluster near this predicted closure relation. The proposed relation is a structural constraint among standard cosmological density parameters. It does not by itself specify the microscopic nature of dark matter or dark energy, nor does it replace standard FLRW expansion dynamics. The results suggest that the cosmological density sector may exhibit a simple structural organization at the level of parameter ratios.