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

Abstract

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 and Milgrom in Astrophys J 286:7–14, 1984) and a second order derivative term introduced by Novello et al. in the context of a Weyl geometric approach to cosmology (Novello et al. in Int J Mod Phys D 1:641–677, 1993; de Oliveira et al. in Class Quantum Gravity 14(10):2833–2843, 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 in the deep MOND regime. It implies a corresponding “non-metrical” contribution to the acceleration of free fall trajectories. In contrast to pure RAQUAL, the second order derivative term of the Lagrangian leads to a non-negligible contribution to the energy momentum tensor and an add-on to the light deflection potential in beautiful agreement with the dynamics of low velocity trajectories. Although the model takes up important ingredients from the usual RAQUAL approach, it differs essentially from the latter.—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.