  A coupling between the spacetime geometry and a scalar field involving the
Euler four-form can have important consequences in General Relativity. The
coupling is a four-dimensional version of the Jackiw-Teitelboim action, in
which a scalar couples to the Euler two-form in two dimensions. In this case
the first order formalism, in which the vierbein (or the metric) and the spin
connection (or the afine connection) are varied independently, is not
equivalent to the second order one, in which the geometry is completely
determined by the metric. This is because the torsion postulate (T=0) is not
valid and one cannot algebraically solve the spin connection from its own field
equation. The direct consequence of this obstruction is that the torsion
becomes a new source for the metric curvature, and even if the scalar field is
very slowly varying over cosmic scales as to have no observable astronomical
efects at the galactic scale, it has important dynamical efects that can give
rise to a cosmological evolution radically diferent from the standard FRWL
model.
