Linear Stability of Einstein and de Sitter Universes in the Quadratic Theory of Modified Gravity

1) The problem under study and the method of solution are clearly stated.

1) There are frequent typos and grammatical errors
2) Some of the equations are related by symmetry so instead of repeating them three times they could have been better organised to only appear once.
3) The results are just summarised in the conclusions without discussing where they leave us in terms of the relevance of the given theory for cosmology and no outlook for future studies is provided.
4) Although the reference list quite extensive, some standard references, for instance to the early work by Stelle on similar theories, could perhaps also have been included.

The paper is appropriate for publication in SciPost Physics proceedings as it reports on a contribution to the relevant conference.

The work examines the appropriateness for cosmology of an alternative theory of gravity which is purely quadratic in the curvature tensors. In particular, it studies the linear stability of the Einstein static universe and de Sitter space in this theory. As in standard GR, but unlike other modifications of gravity which include both the standard and higher derivative terms, the Einstein static universe appears to be unstable. Assuming the validity of certain conditions stated in the paper, de Sitter space appears to be stable under linear perturbations.

This article is part of an ongoing effort to understand the validity and relevance of various modified gravity theories to current questions in cosmology, and as such can help to guide future studies.

The work can be published after the author addresses the changes/comments below.

1) In several places, "anistropic" should be replaced by "anisotropic"
2) In section 3: non-isotroposity->anisotropy (or non-isotropy?)
3) Above eq.16: "yield to"->"arrive at"
4) above eq.28: "redeuced"->"reduced"
5)The derivation and meaning of (28) is not clear. In particular, should the $\xi$'s be $\delta\xi$'s? Also the relevance of this equation is not clear as it doesn't appear to be necessary for the discussion that follows.
6) In (43) there is probably a typo as the first term on the third line contains an $h$ which should be either $h_1$,$h_2$ or $h_3$.

7) End of section 6: The main result of the contribution hinges on the parameters $A_{1,2},C_{1,2},E_{1,2}$ being positive at the same time. Since these are complicated functions of the parameters $h_i$, it is not immediately clear that such a solution exists. A proof of this claim, or an example of a domain where it is true, should be included here.

8) A related confusing point is that the lagrangian (5) and the following equations of motion (6)-(8) are symmetric under exchange of the 1,2,3 labels, and the fluctuation parameters are also introduced in a symmetric way at the beginning of section 6. So one would expect the final answer for the fluctuations (29),(34),(39) to reflect that symmetry, in other words the $A_{1,2},C_{1,2},E_{1,2}$ parameters to be mapped to each other under suitable exchanges of the $h_i$ parameters. This does not seem to be the case. Is it possible that some terms are missing in (29),(34),(39)? Otherwise, the author should comment on where this symmetry breaking arose during the process of solving the fluctuation equations.

9) In the conclusions, the relevance of the stability condition mentioned in point 7) above should be discussed further. Is it a natural condition and what are its implications for the validity of this alternative gravity theory? Similarly, do we learn anything from the instability for the Einstein static universe, since this is not a valid cosmological solution anyway? A short outlook could also be included.