Quantum Modified Gravity at Low Energy in the Ricci Flow of Quantum Spacetime postprint

Abstract: Quantum treatment of physical reference frame leads to the Ricci flow of quantum spacetime, which is a quite rigid framework to quantum and renormalization effect of gravity. The theory has a low characteristic energy scale described by a unique constant: the critical density of the universe. At low energy long distance (cosmic or galactic) scale, the theory modifies Einstein's gravity which naturally gives rise to a cosmological constant as a counter term of the Ricci flow at leading order and an effective scale dependent Einstein-Hilbert action. In the weak and static gravity limit, the framework naturally contains a MOdified Newtonian Dynamics (MOND)-like theory first proposed by Milgrom. When local curvature is large, Newtonian gravity is recovered. When local curvature is lower or comparable with the asymptotic background curvature corresponding to the characteristic energy scale, the baryonic Tully-Fisher relation can be obtained. For intermediate general curvature, the interpolating Lagrangian function gives a similar curve to the observed radial acceleration relation of galaxies. The critical acceleration constant $a_{0}$ introduced in MOND is related to the low characteristic energy scale of the theory. The cosmological constant gives a universal leading order contribution to $a_{0}$ and the flow effect gives the next order scale dependent contribution beyond MOND, which equivalently induces the "cold dark matter" to the theory. $a_{0}$ is consistent with galaxian data when the "dark matter" is about 5 times the baryonic matter. The Ricci flow of quantum spacetime is proposed as a possible underlying theory of MOND and related acceleration discrepancy phenomenon at long distance scale.