Deriving the modified Poisson Equation using xAct

[Nat Woodcock]

Hi I am a new user of mathematica and I am trying to create a code which derives the modified poisson equation from the action (images attached). I have tried several ways in xAct to try this but it does not seem to work. I am curious if there is someone who has already produced this or could help me.

The pipeline goes as;

• Vary the action to find how $\varphi$ contributes to the energy-momentum tensor.

• Perturb the metric and the scalar field around a flrw background.

• Apply the Quasi-Static limit, stripping away $1/c$ terms and time-evolutions that are slower than the speed of light.

• Combine the equations to isolate the spatial Laplacian of the potential $\Phi$.

Thank you for your time