Derivation of the PBH Well Contribution to the Friedmann Equation

Presents a modification to the standard Friedmann equation where early-universe expansion is driven by energy density from matter–antimatter annihilation and primordial black hole (PBH) gravitational wells, eliminating the need for a hypothetical inflaton field. Derives the additional term from Buchert’s averaging formalism, linking it to integrals over the PBH density power spectrum and shear. Includes sigma₈-normalized Eisenstein–Hu modeling of , parameter sweeps over , and comparative plots of with and without the PBH-well term. Predicts observable non-Gaussian CMB anisotropies and a possible dark matter connection via PBH remnants.