\(f(R)\) gravity is a class of gravitational theories similiar to General Relativity in that the Lagrangian is a function of the Ricci scalar. Standard General Relativity with a cosmological constant is a special case of \(f(R)\) gravity where \(f(R)=R+\Lambda\). Expanding a general \(f(R)\) yields an expansion like \(f(R)=\sum\limits_{n=0}^\infty a_nR^n\) with the first term \(a_0 \) being the cosmological constant, and \(a_1R\) is the standard term that is the basic Einstein-Hilbert Lagrangian.

The Brans-Dicke theory with \(\omega=-3/2\) was found to be an \(f(R)\) theory with a connection-independent \(\mathcal{L}_M\) term. **What are some other interesting gravitational models that can be (unexpectedly) written as an \(f(R)\) theory? **

Of course, there are a number of other theories which can be written in an \(f(R)\) form, if we do not restrict ourselves to scalar generalisations of General Relativity. For example, Gauss-Bonnet gravity and the effective gravitational action of standard string theories all contain terms like \(R^{\mu\nu}R_{\mu\nu}\) and \(R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}\)).

**Are there any theories of quantum gravity that predict a scalar generalisation of General Relativity?**