A scalar field inducing a non-metrical contribution to gravitational acceleration and a compatible add-on to light deflection

A scalar field model for explaining the anomalous acceleration and light deflection at galactic and cluster scales, without further dark matter, is presented. It is formulated in a scale covariant scalar tensor theory of gravity in the framework of integrable Weyl geometry and presupposes two different phases for the scalar field, like the superfluid approach of Berezhiani/Khoury. In low acceleration regimes of static gravitational fields (in the Einstein frame) with accordingly low values of the scalar field gradient, the scalar field Lagrangian combines a cubic kinetic term similar to the ``a-quadratic'' Lagrangian used in the first covariant generalization of MOND (RAQUAL) (Bekenstein/Milgrom:1984) and a second order derivative term introduced by Novello et al. in the context of a Weyl geometric approach to cosmology (Novello/Oliveira_et al:1993, Oliveira/Salim/Sautu:1997). In varying with regard to $\phi$ the latter is variationally equivalent to a first order expression. The scalar field equation thus remains of order two. In the Einstein frame it assumes the form of a covariant generalization of the Milgrom equation known from the classical MOND approach. It implies a corresponding ``non-metrical'' contribution to the acceleration of free fall trajectories. The second order derivative term of the Lagrangian leads to a non-negligible contribution to the energy momentum tensor and an {\em add-on to the light deflection potential} in beautiful agreement with the dynamics of low velocity trajectories. -- In higher sectional curvature regions, respectively for higher accelerations in static fields, the scalar field Lagrangian consists of a Jordan-Brans-Dicke term with sufficiently high value of the JBD-constant to satisfy empirical constraints. Here the dynamics agrees effectively with the one of Einstein gravity.