A generalization of unimodular gravity with vacuum-matter energy   exchange
  An effective theory of gravity in the infrared is proposed, which involves the determinant of the metric relative to the determinant of a prior metric taken to be that of Minkowski spacetime. This effective theory can be interpreted as a generalization of unimodular gravity. In a cosmological context with ultrarelativistic or cold matter, the resulting field equations have only one solution, empty Minkowski spacetime (selected by the prior metric of the theory). The introduction of energy exchange between vacuum and matter gives rise to nonstatic cosmic solutions. It is found that Minkowski spacetime (from the prior metric) appears as an attractor of the dynamic equations. A further result is that energy-momentum conservation of any localized material system is violated in a nonconstant gravitational background. The impact for experiment appears, however, negligible if the vacuum-energy mass scale is of order meV. 
F.R. Klinkhamer
hep-th gr-qc