Loosely speaking, in flat spacetime, one defines the effective Lagrangian by writing down all possible operators compatible with the symmetries and suppressed by some energy scale, and one usually truncates this series depending on the accuracy one seeks. Also, in flat spacetime, one can arrange the various operators in a hierarchy depending on their mass dimension.
I am having trouble understanding these ideas in curved spacetime. My first question is, how would one define energy scales on an arbitrary background? Moreover, is it still possible to arrange the various operators in a hierarchy? Wouldn't having different backgrounds reshuffle the operators in terms of their "importance"?
I am also confused about EFTs in the context of classical modified gravity. Usually, it is stated that these theories are effective theories in the strong field regime, e.g., near a black hole. In what sense is classical modified gravity an EFT? Consider, for instance, Horndeski's theory there are multiple terms appearing in the Lagrangian. Is it possible to arrange them in a hierarchy given that the background is generic and, in general, it is not possible to define energy?
Finally, I suppose one would need to distinguish between effective field theories that deal with the scattering of quanta and those that deal with large classical objects. The issue is that I couldn't find a discussion of these ideas in the literature.
I hope these questions are not too vague. It is just that I cannot wrap my head around EFTs in curved spacetime. Any clarifications are very much appreciated. Also, if anyone knows of any references tackling these questions, please mention them.
