What are some interesting f(R) models?

Question

\(f(R)\) 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?

Answer 1

Drawing from my knowledge acquired mostly by reading the Living Review on this topic, it is obvious that many equivalences can be built via a conformal transformation. However, it should be noted that equivalences built by conformal transformation are only formal; mathematically, the field solutions of the theory will be the same, but the physical content of the theory becomes different. The non-equivalence can be easily seen e.g. through the change in behaviour of time-like geodesics under a conformal transform.

Furthermore, the theories will mostly not  be equivalent even in terms of mathematical form and solutions in the presence of matter because the coupling becomes non-minimal "$\mathcal{L}_m(g_{\mu \nu},...) \to \mathcal{L}_m(\Omega^{-2} g_{\mu \nu},...)$". The transformation to the Einstein frame is only a mathematical trick and a useful tool to compare the behaviour of the degrees of freedoms of different theories.

The only true equivalence seems to be acquired by the transform $f'(R) = \Phi$ (assuming $f'' \neq 0$) which yields the O'Hanlon action (not even general Brans-Dicke, only $\omega=0$!)

$$S = \int d^4 x \sqrt{-g}\left( \frac{1}{2} \Phi R - V(\Phi) \right) + S_m $$

I.e. $f(R)$ theory seems to not be even equivalent to general dilatonic or Brans-Dicke theory.

For further references on what these theories might mean I recommend the paper by O'Hanlon and the previous works by him and Fujii referred therein. (I do not know any reference from the string-theoretical side.)

Comments

Thank you, interesting.