# The action for linearized gravity in a curved background

I'm familiar with the Lagrangian for linearized gravity about a flat background,

$$\mathcal{L} = \frac{1}{2}[(\partial_\mu h^{\mu\nu} \partial_\nu h - \partial_\mu h^{\rho \sigma} \partial_\rho h^\mu_\sigma + \frac{1}{2} (\partial_\mu h^{\rho\sigma})^2 - \frac{1}{2} (\partial_\mu h)^2] + \mathcal{O}(h^3)$$

See, for example, Carroll's textbook, $$\S 7.1$$.

I am interested in the Lagrangian for linearized gravity around a curved background. My background metric is diagonal and the only source is a cosmological constant. In principle I know this can be done by simply expanding the Einstein-Hilbert action to order $$h^2$$ about the background, but this is tedious and I would hope there is a reference where this has already been worked out.

This review by Flanagan and Hughes gives the Einstein and Ricci tensors to linear order in $$h$$ ($$\S 5$$) but not the action. Additionally, this comment on a question about linearized gravity mentions that in a curved background the action has a term of the form $$R_{\mu\nu\rho\sigma} h^{\mu\nu} h^{\rho\sigma}$$, where $$R_{\mu\nu\rho\sigma}$$ is the curvature of the background metric, but does not elaborate further.

Is there a reference that clearly states the action for linearized gravity about a curved background?

I suppose by 'gravity', you mean GR given by the Einstein-Hilbert action. You can find the expression for Ricci scalar $$R$$ up to $$\mathcal{O}(h^2)$$ in this answer of mine.
Another thing you will need is the expansion of $$\sqrt{-g}$$ up to $$\mathcal{O}(h^2)$$: $$\sqrt{-g} = \sqrt{-\bar{g}} \left(1+ \frac{h}{2} + \frac{h^2}{8} - \frac{h_{\mu \nu}h^{\mu \nu}}{4} \right)$$ where $$\bar{g}$$ is the background (non-flat) metric.