  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 gives rise to a transition trend away from Newtonian gravity and
similar to the MOdified Newtonian Dynamics (MOND) around the characteristic
scale. When local curvature is large, Newtonian gravity is recovered. When
local curvature is low enough to be comparable with the asymptotic background
curvature corresponding to the characteristic energy scale, the transition
trend produces the baryonic Tully-Fisher relation. For intermediate general
curvature around the background curvature, the interpolating Lagrangian
function yields a similar transition trend to the observed radial acceleration
relation of galaxies. When the baryonic matter density is much lower than the
critical density at the outskirt of a galaxy, there may be a universal
"acceleration floor" corresponding to the acceleration expansion of the
universe, which differs from MOND at its deep-MOND limit. 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 it and the flow effect gives the next
order scale dependent contribution, 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.
