*Cosmo Comments*. The following comments can also be viewed as annotations on the paper via Hypothesis.

This article reveals an interesting mechanism, which seems to be at work in Horndeski theories featuring kinetic braiding, such as the cubic Galileon model. Namely, the scalar field standing for dark energy is unstable in the presence of gravitational waves (GWs) with a sufficiently high amplitude. The article concludes that Horndeski models whose EFT parameter $\alpha_B$ is larger than 1% are excluded.

As a non-specialist of Horndeski theories, I tried my best to understand the physics of this destabilization mechanism, but I ended up quite puzzled. Thus, I would like to ask the authors a few questions; hopefully, this dialogue will be useful for other members of the community.

Let me first try to summarize what I think is the origin of the instability – again, to the best of my understanding, which may be incorrect. The important term seems to be the last one of Eq. (3.2), which is quadratic in $\Gamma_{\mu\nu}$. If this term is large enough, the perturbations of the scalar field, denoted $\delta\pi$ feature a ghost. This appears quite clearly in Eqs. (3.19) and (3.30), where the second-order differential operator of the equation of motion for $\delta\pi$ becomes elliptical in that case.

Since $\Gamma_{\mu\nu}\propto\dot{h}_{ij}$, the time derivative of the GW amplitude, $\Gamma^{\mu\nu}\Gamma_{\mu\nu}$ may be understood as the energy density of the GW. Whatever its interpretation, it is a space-time curvature term, because it is the square of Christoffel symbols. However, what surprises me is that the cubic Galileon $\pi$ does not directly couple to curvature. Thus, how can curvature appear in its effective Lagrangian, Eq. (3.2)?

I guess that the reason is the following. While $\pi$ is not directly coupled to curvature, the gravitational potentials $\Phi$ and $\Psi$ are. However, in the sub-Hubble regime, $\pi\propto\Phi=\Psi$ according to Eq. (2.5) of the article. Replacing $\Phi$ and $\Psi$ by $\pi$ would then lead to $\pi$ being effectively coupled to curvature.

My questions to the authors are the following:

- Is the above sound or erroneous?
- What is the domain of validity of Eq. (2.5)?
- If (2.5) is very general, then $\pi$ should inherit all the couplings of $\Phi$ and $\Psi$, including with matter. In other words, I suppose that if one had included the matter action in this work, one would get a term $\propto \pi T$, where $T$ is the matter energy-momentum tensor, in Eq. (2.8). In other words, there would now be a significantly larger source of instability for the model: the powerful curvature provoked by the presence of matter. This would significantly improve the constraint on $\alpha_B$, wouldn’t it?

*[These comments were shared with us by a member of the cosmology community. They do not necessarily reflect the opinion of the Cosmo Comments team.]*