Explicit equations of motion for axial perturbations in Horndeski

[Héloïse]

Hi everyone,

I am considering a specific class of scalar-tensor Horndeski theories. I have already obtained the equations of motion for the background metric and scalar field combining xTensor with xCoba using CTensors for their explicit evaluation.

I am now trying to apply a similar method to obtain the explicit equations of motion for the first-order and axial metric perturbation. However, I have many more terms in that case (around 2700 tensorial terms) and decomposing them in a list and evaluate each of them separately does not always work, as some terms take way too long, for example those containing 3 covariant derivatives of the metric perturbation (e.g. term 36 in my MWE).

Could you please help me to accelerate the explicit evaluation of those equations of motion? I would also appreciate to learn why terms like term 36 take that long within the formalism of CTensors, and I would of course be open to try another approach that is faster/more efficient.

Thank you in advance for your detailed help!

Best,

Héloïse

[Sergi Sirera]

Hi Héloïse,

These calculations can quickly become very messy so it's good to simplify the expressions as much as you can at each step to make life easier for xAct. I have worked on this extensively and I think my notebooks will be very useful to you. I have a repository using xAct to compute the axial perturbations for full HOST theories (englobing Horndeski). The results are summarised in this paper: (https://arxiv.org/pdf/2503.05651) and the repository containing the notebooks is: (https://github.com/sergisl/ringdown-calculations/tree/main/Inverting-no-hair-theorems).

While it's true that you get many tensorial terms for the perturbed action, many of those terms are either zero for axial perturbation, or they can be combined after some manipulations (e.g. integration by parts, commuting covariant derivatives). Before substituting in components, it's good to simplify as much as possible the tensorial expressions.

The approach I've taken in these notebooks (which you can also see in the paper in Eq. (3. 11)) is to write the quadratic action in terms of the axial (I use the term "odd") perturbations. If you want, you could take that action and read off the specific p_1, p_2, p_3, p_5 coefficients for your theory of interest from the notebooks.

Let me know if you have any further questions.

Best wishes,

Sergi