Modelling Dark Energy as a Gravitational Effect of Matter Beyond the Observable Universe

We develop a quantitative gravitational model in which the additional cosmological expansion conventionally attributed to dark energy results from gravitational interaction with matter beyond the observable Universe. The gravitational acceleration produced by external sources is calculated using Newtonian gravitation and related to an additional cosmological expansion term. In a present-epoch analysis, configurations containing one or more external gravitational sources reproduce the target cosmological constant Λ and the corresponding Hubble parameter H0. The solutions are repeatable but non-unique: for a single source, the present gravitational effect constrains M/R², allowing different masses and distances to produce the same cosmological contribution. Extending the model to redshift-dependent evolution produces expansion histories that describe cosmic chronometer, DESI BAO, and Pantheon+ observations. The fitted matter contribution, Hm,0 = 37.77 km s⁻¹ Mpc⁻¹, is close to the value expected from standard cosmological parameters without imposing this value in the optimization. Increasing the number of external sources from N = 1 to N = 3 reduces the total chi-square from 1424.02 to 1416.35. Since additional sources introduce additional model parameters, this improvement alone does not determine the number of external sources. The results provide a quantitative gravitational explanation for the phenomenon attributed to dark energy, in which the additional cosmic expansion is the observable effect of gravitational interaction with matter beyond the observable Universe. If the external sources are associated with other universes, this gravitational effect constitutes an observational signature of a multiverse.