The action for linearized gravity in a curved background

Question

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?

Answer 1

When you say 'linearized gravity', it means linearized at the level of the equations of motion (EOM). To get linearized EOM, you need an action that is expanded to the quadratic level (which is what you've written).

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.

You can look at eq (3.7) in this paper by Duff and Christensen for the Einstein-Hilbert action with a cosmological constant at the quadratic level.