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Until recently the study of the gravitational field of dark matter was primarily concerned with its local effects on the motion of stars in galaxies and galaxy clusters. On the other hand, the WMAP experiment has shown that the gravitational field produced by dark matter amplifies the higher acoustic modes of the CMBR power spectrum, more intensely than the gravitational field of baryons. Such a wide range of experimental evidences from cosmology to local gravity suggests the necessity of a comprehensive analysis of the dark matter gravitational field per se, regardless of any other attributes that dark matter may eventually possess. In this paper we introduce and apply Nashs theory of perturbative geometry to the study of the dark matter gravitational field alone, in a higher-dimensional framework. It is shown that the dark matter gravitational perturbations in the early universe can be explained by the extrinsic curvature of the standard cosmology. Together with the estimated presence of massive neutrinos, such geometric perturbation is compatible not only with the observed power spectrum in the WMAP experiment but also with the most recent data on the accelerated expansion of the universe. It is possible that the same structure formation exists locally, such as in the cases of young galaxies or in cluster collisions. In most other cases it seems to have ceased when the extrinsic curvature becomes negligible, leading to Einsteins equations in four dimensions. The slow motion of stars in galaxies and the motion of plasma substructures in nearly colliding clusters are calculated with the geodesic equation for a slowly moving object in a gravitational field of arbitrary strength.
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We report on precision resonance spectroscopy measurements of quantum states of ultracold neutrons confined above the surface of a horizontal mirror by the gravity potential of the Earth. Resonant transitions between several of the lowest quantum states are observed for the first time. These measurements demonstrate, that Newtons inverse square law of Gravity is understood at micron distances on an energy scale of~$10^{-14}$~eV. At this level of precision we are able to provide constraints on any possible gravity-like interaction. In particular, a dark energy chameleon field is excluded for values of the coupling constant~$beta > 5.8times10^8$ at~95% confidence level~(C.L.), and an attractive (repulsive) dark matter axion-like spin-mass coupling is excluded for the coupling strength $g_sg_p > 3.7times10^{-16}$~($5.3times10^{-16}$)~at a Yukawa length of~$lambda = 20$~{textmu}m~(95% (C.L.).
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