Consider an action of the form
$$ S = -\frac{2}{\kappa^2}\int d^4x\sqrt{-g}~\left(\phi R + \phi\mathcal{L}_{matter}\right). $$
Expanding this to second order in $h_{\mu\nu}$ and including a harmonic gauge fixing term $\mathcal{L}_{G F}= \partial_{\mu} \bar{h}^{\mu \nu} \partial^{\lambda} \bar{h}_{\lambda \nu}$, we find $$ S = -\int d^4x\left(\frac{2\phi}{\kappa^{2}}\left(\partial_{\mu} \partial_{\nu} h^{\mu \nu}-\square h\right)-\frac{\phi}{2}\left[\partial_{\lambda} h_{\mu \nu} \partial^{\lambda} \bar{h}^{\mu \nu}-2 \partial^{\lambda} \bar{h}_{\mu \lambda} \partial_{\sigma} \bar{h}^{\mu \sigma}\right] + \phi\mathcal{L}_{matter}\right), $$ where $\bar{h}_{\mu\nu} = h_{\mu\nu} - \frac14\eta_{\mu\nu}h$.
Now let's say I want to find the propagator: do I vary this action with respect to $h_{\mu\nu}h_{\rho\sigma}$? That would seemingly give me something akin to the graviton propagator but now multiplied by a scalar field, e.g. some kind of 3pt interaction. Do I vary wrt $\phi h_{\mu\nu}$? What is such an object?
Can I simply define $\tilde{h}_{\mu\nu} = \sqrt{\phi}h_{\mu\nu}$ and vary with respect to that? This would seemingly give me the usual de Donder gauge propagator, but what exactly is $\tilde{h}_{\mu\nu}$? Is it a graviton?
Alternatively, perhaps the $\phi$ contribution just drops out during the inversion of the operator, is that legitimate?
