Description
We propose in this letter a relativistic coordinate independent
interpretation for Milgrom's acceleration $a_{0}=1.2 \times 10^{-8}\hbox{cm/s}^{2}$ through a geometric constraint obtained from the product of the Kretschmann invariant scalar times the surface area of 2--spheres defined through suitable characteristic length scales for local and cosmic regimes, described by Schwarzschild and Friedman--Lemaître--Robertson--Walker (FLRW) geometries, respectively. By demanding consistency between these regimes we
obtain an appealing expression for the empirical (so far unexplained) relation between the accelerations $a_0$ and $c H_0$. Imposing this covariant geometric criterion upon a FLRW model, yields a dynamical equation for the Hubble scalar whose solution matches, to a very high accuracy, the cosmic expansion rate of the $\Lambda$CDM concordance model fit for cosmic times close to the present epoch. While these results are very preliminary and strictly valid only at a toy model level, we believe that they could provide relevant information in the search of alternative gravity theories or even within General Relativity itself.
