Plausible Sources of Dark Energy and Associated Vacuum Properties: Implications for Newton’s G, MOND’s a₀, Vacuum Invariants, and the Tully–Fisher and Faber–Jackson Galactic Relations

We believe that the origin of the universal dark energy may crucially depend on the behavior of the gravitational constant \(G\). If \(G\) is constant throughout the universe, then the dark energy density u₀ is supported by the vacuum via its constant and evolving properties. On the other hand, in varying-\(G\) gravity (whose low-acceleration limit is MOND), u₀ is a manifestation of radial \(G\)-gradients in the source of gravity. We estimate the present-day dark energy density of the universe in these two independent cases without using conventional Planck-2018 modeling in deriving the expressions for u₀. The constant-\(G\) derivation uses dimensional analysis, vacuum constants, and a newly discovered evolving bridge between vacuum mechanical and electromagnetic quantities. The varying-\(G\) derivation relies on the MOND critical acceleration \(a_0^{}\), the source of gravity that falls off as \(\sim r^{-2}\) at large distances \(r\), and the assumption that the present-day energy density \(u_0^{}\) can be estimated by spatially averaging this radial profile over the MOND volume \(\frac{4\pi}{3}r_0^{~3}\), where \(r_0^{}=c^2/a_0^{}\) and \(c\) is the speed of light in vacuum. The agreement between these two determinations and the Planck-2018 results from \(\Lambda\)CDM modeling is at the percent level, so that the results cannot distinguish clearly between constant-\(G\) and varying-\(G\) gravity. The analysis further indicates that the Newtonian constant G₀, the MOND constant \(a_0^{}\), and the Planck units of force, power, voltage, and current should be accepted as empirical vacuum invariants of the same stature as the well-known resistive properties of the vacuum; whereas charge, capacitance, inductance, and various fields are scale-dependent properties evolving in the expanding universe since the Stoney era. Ultimately, the vacuum behaves as a remarkably stiff elastic medium under stress, irrespective of the presence or absence of matter. This property has long been encoded, albeit inconspicuously, in the much discussed Tully–Fisher and Faber–Jackson relations, as well as in the Casimir effect and the field equations of General Relativity.