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.
Comment: 6 pages, 2 figures. Adjusted criterion now allows for a geometric identification of the MOND a0 constant, is consistent with rotation curve and lensing observations at galactic scales, and with the recent accelerated expansion of the universe at cosmological scales. Accepted for publication in MNRAS