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.
