Higher-order gravity

Gravitational theories are possible which are built from curvature_invariants more complicated than the Ricci scalar of the Einstein-Hilbert lagrangian. In four dimensions there are two possible lagrangians to the next order corresponding to the Ricci scalar squared $R^{2}$ and the Weyl tensor squared $C_{abcd}\,C^{abcd}$. Varying these gives the Pauli and Bach tensors repectively, these involve fourth dervatives of the metric whereas the Einstein tensor has only second derivatives of the metric. In more than four dimensions the situation becomes more complicated as lagrangians can be built from the Gauss-Bonnet and Lovelock invariants. Higher-order gravities have been applied to comology and galactic rotation. The reason that higher-order gravities are important is that they occur as corrections to general relativity in virtually all quantum theories of gravity and string theory.

Reference

Mark D. Roberts,Galactic Rotation Curves and Quadratic Lagrangains. Mon.Not.R.astro.Soc.249(1991)339-342.