The Einstein equations assume the spacetime dynamics to be carried by a rank 2 tensor only, the metric, together with a metric compatible connection. If we release part or all of these assumptions, the resulting models can be as diverse as one can think of.

For instance, Palatini’s approach assumes an independent connection. Starting with the Einstein-Hilbert action, it follows that this independent connection is metric compatible. Starting with another action, such as in the f(R) or EiBI models, this is no longer true. Furthermore, a Palatini treatment of an action should in priniciple make no a priori assumption on torsion, and in principle the connection should be allowed to contain an antisymmetric part. Doing so, the resulting metric-affine model does contain extra degrees of freedom. On another hand, for instance, the metric approach to f(R) gravity leads to a model equivalent to a scalar-tensor model.

A scalar-tensor model of gravity is a model containing the metric (tensor) coupled in a non-minimal way to a scalar field. The matter field is coupled minimally to the metric (Jordan frame). The theory can be recast into a model where gravity decouples from the scalar (Einstein frame), but then the matter sector becomes non minimally coupled to the scalar field. These two formulations are equivalent and related to each other by transformation of the metric and scalar field.

From another perspective, scalar-tensor models are motivated from higher dimensional theories. In the Kaluza-Klein reduction process, a scalar field appears and becomes non minimally coupled to the metric. In string theory, this scalar field is usually called the dilaton (or an axion, depending on the starting model and reduction process).

String theory also justifies the so-called dilaton-Gauss Bonnet (dBG) or dynamical Chern Simons (dCS) model. These models contain the Einstein-Hilbert term, with a correction quadratic in curvature. The Gauss-Bonnet and Chern-Simons term are in fact topological terms (total divergence), so they wouldn’t contribute alone to the spacetime dynamics. Once they are coupled to a scalar field, they contribute in a highly non-trivial way to the field equations.

The most general way to write down an action coupling a tensor and a scalar and leading to at most second order equations (note that dCS doesn’t belong this class) has been studied by Horndeski some times ago. The Horndeski model has regained interest recently, but has been forgotten for some decades before.