A Boundary Closure Condition for Flat FLRW Cosmology

The Einstein field equations determine local gravitational dynamics, while the Friedmann equations determine the homogeneous cosmological evolution once the density fractions are specified. In the standard flat matter plus vacuum background, the present vacuum fraction remains a fitted parameter. This work formulates a global boundary closure condition at the present Hubble radius. The Einstein Friedmann background gives

Ω_Λ,0 = Λ₀ R_H,0² / 3

with

R_H,0 = c / H₀

so that the dimensionless combination

Λ₀ R_H,0²

remains unfixed by the local equations of motion.

A spherical radiative boundary has the dimensionless normalization

𝓧 = [ L / (R² T⁴) ] · [ ℏ³ c² / k_B⁴ ]

and therefore

𝓧 = π³ / 15.

The proposed closure condition applies this boundary normalization at the present Hubble radius:

Λ₀ R_H,0² = 𝓧 = π³ / 15.

It follows that

Ω_Λ,0 = π³ / 45

and

Ω_m,0 = 1 − π³ / 45.

The boundary relation is not a modification of the Einstein field equations. It is a global closure condition selecting the present epoch Hubble scale solution of the Einstein Friedmann background.