Cosmology with Scalar-Euler form Coupling

Understand

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.