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

[Sergi Sirera]

Hello Nat,

Welcome to the great land of Mathematica! I've actually just released an xAct package which will allow you to do this quite easily: https://github.com/sergisl/xAlpha/tree/main.

In the example notebook, I use the package to calculate, for luminal Horndeski theories (of which your cubic Galileon is an example), the metric and scalar equations to second order in perturbations on a FLRW background. You can easily apply this notebook to your case by simplifying the action at the start. The notebook will vary the action and give you in an organised manner the perturbation equations. If you want the modified Poisson equation, you will only need the linear perturbation so you can ignore the quadratic ones. Once you have the full unapproximated equations, you can very easily apply the quas-static limit and combine the equations to get the Poisson equation.

Hope you find the package useful, but let me know if you have any questions in the implementation. You might also find useful the paper where we explain these calculations in a bit more detail: https://arxiv.org/pdf/2605.04154.

Best wishes and good luck with your project!

Sergi