  This paper proposes a toy model where, in the Einstein equations, the
right-hand side is modified by the addition of a term proportional to the
symmetrized partial contraction of the Ricci tensor with the energy-momentum
tensor, while the left-hand side remains equal to the Einstein tensor. Bearing
in mind the existence of a natural length scale given by the Planck length,
dimensional analysis shows that such a term yields a correction linear in h to
the classical term, that is instead just proportional to the energy-momentum
tensor. One then obtains an effective energy-momentum tensor that consists of
three contributions: pure energy part, mechanical stress and thermal part. The
pure energy part has the appropriate property for dealing with the dark sector
of modern relativistic cosmology. Such a theory coincides with general
relativity in vacuum, and the resulting field equations are here solved for a
Dunn and Tupper metric, for departures from an interior Schwarzschild solution
as well as for a Friedmann-Lemaitre-Robertson-Walker universe.
