MOND-Like Acceleration in Integrable Weyl Geometric Gravity

Abstract

In this paper a Weyl geometric scalar tensor theory of gravity with scalar field $\Phi$ and scale invariant cubic ("aquadratic") kinetic Lagrangian is introduced. Einstein gauge (comparable to Einstein frame in Jordan-Brans-Dicke theory) is most natural for studying trajectories. In it, the Weylian scale connection induces an additional acceleration which in the weak field, static, low velocity limit acquires the deep MOND form of Milgrom/Bekenstein's gravity. The energy momentum of $\Phi$ leads to another add on to Newton acceleration. Both additional accelerations together imply a MOND-ian phenomenology of the model. It has unusual transition functions. They imply higher phantom energy density than in the case of the more common MOND models with transition functions $\mu_1(x), \, \mu_2(x)$. A considerable part of it is due to the scalar field's energy density which, in our model, gives a scale and generally covariant expression for the self-energy of the gravitational field.