Approaching the Dark Sector through a bounding curvature criterion

Understanding the observations of dynamical tracers and the trajectories of lensed photons at galactic scales within the context of General Relativity (GR), requires the introduction of a hypothetical dark matter dominant component. The onset of these gravitational anomalies, where the Schwarzschild solution no longer describes observations, closely corresponds to regions where accelerations drop below the characteristic $a_{0}$ acceleration of MOND, which occur at a well established mass-dependent radial distance, $R_{c}\propto (GM/a_{0})^{1/2}$. At cosmological scales, inferred dynamics are also inconsistent with GR and the observed distribution of mass. The current accelerated expansion rate requires the introduction of a hypothetical dark energy dominant component. We here show that for a Schwarzschild metric at galactic scales, the scalar curvature, K, multiplied by $(r^{4}/M)$ at the critical MOND transition radius, $r=R_{c}$, has an invariant value of $\kappa_{B}=K(r^{4}/M)=28Ga_{0}/c^{4}$. Further, assuming this condition holds for $r>R_{c}$, is consistent with the full spacetime which under GR corresponds to a dominant isothermal dark matter halo, to within observational precision at galactic level. For a FLRW metric, this same constant bounding curvature condition yields for a spatially flat spacetime a cosmic expansion history which agrees with the $\Lambda$CDM empirical fit for recent epochs, and which similarly tends asymptotically to a de Sitter solution. Thus, a simple covariant purely geometric condition identifies the low acceleration regime of observed gravitational anomalies, and can be used to guide the development of { extended} gravity theories at both galactic and cosmological scales.