Derivations and Structural Consequences of the R-Universe Framework

Despite its remarkable phenomenological success, the ΛCDM model does not provide a microscopic explanation for the observed value of the cosmological constant. In particular, the origin of the present dark-energy fraction ΩΛ ≃ 0.69 remains unexplained within standard cosmology. Here we propose a framework in which the late-time cosmological evolution can be interpreted as an effective renormalization–group (RG) flow of the vacuum sector driven by the Hubble scale. Within this description the dimensionless dark-energy fraction obeys a closed evolution equation that takes the form of a logistic RG flow connecting the fixed points ΩΛ = 0 and ΩΛ = 1, corresponding to matter-dominated and vacuum-dominated cosmological regimes. Consistency between the RG scaling of the cosmological coupling and the Friedmann background dynamics implies a non-trivial stationary solution for the dark-energy fraction: ΩΛ = θ / (θ + 1) which relates the present cosmological composition directly to the relevant critical exponent θ of the underlying gravitational fixed point. Using the observed value ΩΛ ≃ 0.69 yields θ ≃ 2.2, numerically close to critical exponents obtained in functional renormalization–group studies of asymptotically safe gravity.