Commutatively Deformed General Relativity: Foundations, Cosmology, and Experimental Tests

An integral kernel representation for the commutative $\star$-product on curved classical spacetime is introduced. Its convergence conditions and relationship to a Drin'feld differential twist are established. A $\star$-Einstein field equation can be obtained, provided the matter-based twist's vector generators are fixed to self-consistent values during the variation in order to maintain $\star$-associativity. Variations not of this type are non-viable as classical field theories. $\star$-Gauge theory on such a spacetime can be developed using $\star$-Ehresmann connections. While the theory preserves Lorentz invariance and background independence, the standard ADM $3+1$ decomposition of 4-diffs in general relativity breaks down, leading to different $\star$-constraints. No photon or graviton ghosts are found on $\star$-Minkowski spacetime. $\star$-Friedmann equations are derived and solved for $\star$-FLRW cosmologies. Big Bang Nucleosynthesis restricts expressions for the twist generators. Allowed generators can be constructed which account for dark matter as arising from a twist producing non-standard model matter field. The theory also provides a robust qualitative explanation for the matter-antimatter asymmetry of the observable Universe. Particle exchange quantum statistics encounters \emph{thresholded} modifications due to violations of the cluster decomposition principle on the nonlocality length scale $\sim10^{3-5} \,L_P$. Precision Hughes-Drever measurements of spacetime anisotropy appear as the most promising experimental route to test deformed general relativity.