I am interested in linearizing actions that are constructed out of geometrical objects. By this I mean perturbing the metric (or vielbein) and keeping in the action terms which are quadratic in the perturbation.
For the purpose of this question, let's consider the well known Einstein-Hilbert action, $$S_{\text{EH}}=\kappa\int\text{d}^4x\sqrt{-g}R~.$$ and perturb the metric around an arbitrary background, $$ \tilde{g}_{mn}=g_{mn}+h_{mn}~,$$ where $g_{mn}$ is the background metric and $h_{mn}$ is the perturbation, $|h_{mn}|\ll 1$.
As I said, we need to keep terms quadratic in the perturbation. It seems to me that this would require us to expand the scalar curvature $R$, and hence the Riemann tensor $R_{kpmn}$ to quadratic order. Expanding to linear order is not that bad, but expanding to quadratic order (particularly around an arbitrary background and not flat) is quite an arduous task. So I would like to know if there is an easier way.
We know that the equation of motion that ought to result from the linearized action is $$R_{ab}^{\text{lin.}}=0~.$$ So by expanding $R_{ab}$ to linear order (much easier than expanding to second order), we can then deduce that the variation of the action (w.r.t the perturbation) is of the form $$\delta S_{\text{EH}}^{\text{lin.}}=\kappa\int\text{d}^4x\sqrt{-g}\delta h^{ab}R_{ab}^{\text{lin.}}~.$$ This already gives us some information on what the action should look like when expanded to second order. But I am not sure where to take it from here, or if there is an even easier way to proceed.
Do you know of a shortcut to obtaining the quadratic expansion of the action? Is your method applicable to a broader range of action functionals (not just EH)? For illustrative purposes, it would be fine if an answer expands around a flat background instead.
Edit: See comments for a little more detail on what I am looking for.
