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This is a question regarding a calculation in perturbative GR. We have :

$g_{\mu\nu} = \eta_{\mu\nu}+h_{\mu\nu}$

where $h_{\mu\nu}$ is a small perturbation around the flat spacetime metric. In linearized theory, we ignore terms which grow as $h^2$ and higher.

Can anyone please provide me with the Ricci scalar to cubic powers in h?

No, none can provide you with the Ricci scalar to cubic powers in h. You will see that I am right, unfortunately.

Computation by hand will prove to be a bit tedious, but xACT software can do this in a matter of seconds. Here, is the expansion of the Ricci scalar to cubic orders around a generic background :

I suppose that $R=R(\epsilon=0)$, as well as $\eta_{\mu\nu}$, still are functions of "exact coordinates" $x_{\mu}$.

@fermionic_tushar Sorry can you tell me what xACT software is, provide a link or something? I'm curious but I didn't find much online.

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