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We derive an effective gravitational potential, induced by the quantumwavefunction of a physical vacuum of a self-gravitating configuration, whilethe vacuum itself is viewed as the superfluid described by the logarithmicquantum wave equation. We determine that gravity has a multiple-scale pattern,to such an extent that one can distinguish sub-Newtonian, Newtonian, galactic,extragalactic and cosmological terms. The last of these dominates at thelargest length scale of the model, where superfluid vacuum induces anasymptotically Friedmann-Lemaitre-Robertson-Walker-type spacetime, whichprovides an explanation for the accelerating expansion of the Universe. Themodel describes different types of expansion mechanisms, which could explainthe discrepancy between measurements of the Hubble constant using differentmethods. On a galactic scale, our model explains the non-Keplerian behaviour ofgalactic rotation curves, and also why their profiles can vary depending on thegalaxy. It also makes a number of predictions about the behaviour of gravity atlarger galactic and extragalactic scales. We demonstrate how the behaviour ofrotation curves varies with distance from a gravitating center, growing from aninner galactic scale towards a metagalactic scale: a squared orbital velocity'sprofile crosses over from Keplerian to flat, and then to non-flat. Theasymptotic non-flat regime is thus expected to be seen in the outer regions oflarge spiral galaxies.

An alternative to dark matter and dark energy: Scale-dependent gravity in superfluid vacuum theory