A Boundary-Invariant Derivation of the Friedmann Matter Equation

In this theoretical work, we derive the standard matter-dominated Friedmann equation (H^2 = 8πGρ/3) by applying a previously established, universal radiative-gravitational boundary invariant (X) to the cosmological horizon.

The invariant X was shown in earlier work to take the same constant value (4πσ/c^4) across very different physical systems, including stellar photospheres, the CMB last-scattering surface, and black hole horizons. This universality suggests that X captures a deeper geometric and thermodynamic constraint within general relativity.

By modeling the observable universe as a boundary system with radius R = c/H, and using only the standard general relativistic relations for luminosity (L), enclosed mass (M), and effective surface gravity (g), we impose the universal value of X and obtain the matter Friedmann equation directly.

Significance:

This derivation does not assume the  FRW metric, does not assume large-scale homogeneity, and does not rely on isotropy. Instead, the expansion law emerges as a boundary condition derived from a fundamental, dimensionless radiative–gravitational relation. This interpretation is consistent with principles of horizon thermodynamics and may indicate a deeper structural origin for cosmological evolution.