On the Propagation of Gravitational Waves in the Weyl Invariant Theory of Gravity
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsIn this work, the authors investigated the propagation of gravitational waves in the Weyl invariant theory of gravity. They studied the prediction of gravitational waves in the theory and obtained the new features arising in this recent modified gravity theory, in which the presence of a massive vector field appears somewhat unexpectedly. They also speculated whether the results could be examined in the context of primordial gravitational waves. In my opinion, the paper's analytic derivations and approaches seem technically correct, and the results are interesting to researchers in the field. Thus, I recommend the manuscript to be published in Universe.
Before publication, I have some comments:
(1) On p 2, I spotted ``$T_{\alpha\beta}=F_{\alpha\mu}F^{\mu}_{\beta}+\frac{1}{4}g_{\alpha\beta}F_{\mu\nu}F^{\mu\nu}$ and $T^{(m)}_{\mu\nu}$ represents the energy-momentum tensor of matter, $\kappa$ being a coupling constant" instead of ``$T_{\alpha\beta}=F_{\alpha\mu}F^{\mu}_{\beta}+\frac{1}{4}g_{\alpha\beta}F_{\mu\nu}F^{\mu\nu}$ and $T^{(m)}_{\mu\nu}$ represent the energy-momentum tensors of matter, and $\kappa$ is a coupling constant".
(2) On p 3, I spotted ``Before its detection, gravitational waves was a general relativistic phenomenum...".
(3) On p 3, I spotted ``The above considerations leads to the following consequence...".
(4) On p 4, I spotted ``...the Ricci tensor $R_{\alpha\beta}$ and the curvature scalar $R$ (9) e (10) are gauge invariant,...".
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Author Response
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Reviewer 2 Report
Comments and Suggestions for AuthorsIn the paper the propagation of gravitation waves in a version Weyl
invariant gravity theory is discussed. Unfortunately the introduction to Weyl gravity is very sketchy.
The equations of motion (1) and (2) are actually a gauge fixed
version of the full ones as discussed in ref. 4. The point is rather
important, however the authors gloss it over; indeed the
quantity Lambda is introduced by a gauge choice such that the Ricci scalar is constant and equal precisely to Lambda. Incidentally that is also
relevant for remark that in the gauge fixed version of the action
(see the formula after line 85) the vector fields gets a mass while is
massless in the gauge invariant action. Considering that only gauge
invariant quantities are physical observable quantities, the mass
of the vector field is a gauge artefact and then it can hardly be
associated to dark matter as the authors maintain in the
introduction. Moreover there is problem with the sign of
Lambda. That taking omega >0, Lambda must be negative. Contrary to what the author claim in footnote 1, the sign of omega is fixed by requiring that the kinetic term for the vector field is positive to avoid dramatic instabilities and as a result omega should be negative.
The discussion from eq. 8 to eq. 17 is well known and I think it could be easily omitted keeping the focus on the novelties on Weyl invariant theory.
Section 4 is devoted to the main topic of the paper: gravitational waves propagation. The approach is perturbative (leading order in perturbation theory) and the background chosen is the simplest: Minkowski space. Given the presence of an extra vector field additional polarisation are expected. The form of eq. (18) is a bit strange indeed it contains a term not proportional to expansion parameter epsilon that signals that the background equations of motion are not satisfied.
It is well known that at the leading order in perturbation theory different spins (spin 1 and spin 2) evolve separately. This is the case for the perturbative expansion of eqs. (4) and (5). The idea of considering the vector field of order epsilon1/2 to artificially couple the metric perturbation is not based on any physical argument. Also, as discussed before, a proper treatment of the order zero equations (background equation s) is missing.
The paper has no significant new result and some of statements are in my opinion nor correct. In conclusion the manuscript is not suitable for publication.
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Reviewer 3 Report
Comments and Suggestions for AuthorsIn the manuscript the authors investigate the propagation of gravitational waves in a Weyl-geometric setting. The phase and the group velocity of the waves are also obtained. The manuscript may be publishable in Universe if the authors would fully consider the following points:
1. It is not clear what the authors may mean under "Weyl invariant theory...". Weyl's original theory was constructed by imposing the condition of the conformal invariance on the action, and therefore the Weyl's Lagrangian was quadratic in the curvature. The action of the authors is not conformally invariant. This is an important issue which must be discussed in detail, by pointing out the difference between conformally invariant theories, and the present model.
2. The detection of the gravitational waves provided strong indication that they propagate with the speed of light. However, this is not the case in the present model. Would it be possible to obtain some restrictions on the mass m from the observations?
3. The authors solve only the linear Proca equation for the vector perturbation, but ignore solving, or discussing Eq. (20) for the tensor modes. This issue must also be fully considered.
4. The authors should also extend the Introduction and the Final remarks sections of the manuscript, by discussing in more detail the relevance of their results especially from an observational point of view.
Comments on the Quality of English LanguageQuality of English language good.
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Reviewer 4 Report
Comments and Suggestions for AuthorsIn this manuscript, the authors investigate Weyl's theory of gravity, proposing modifications to the original model. The proposal is intriguing, as it examines in detail the propagation of gravitational waves within this recently modified gravity theory, where the emergence of a massive vector field occurs somewhat unexpectedly. The authors also speculate on whether their results could be explored in the context of primordial gravitational waves. Furthermore, they make some conjectures regarding the Weyl-Proca field as a possible candidate for dark matter. The manuscript is well-written, the proposal is compelling, and the discussions are thorough. Therefore, I recommend that the manuscript be accepted in its current form for publication in Universe.
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Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsIn the revised version the authors do not address properly the critics regarding the original part of the manuscript: the propagation of gravitational waves.
1) The form of eq. (18) is suspicious: it contains a term not proportional to expansion parameter epsilon that signals that the background equations of motion are not satisfied.
2) The proposed expansion is not justified and rather arbitrary. As discussed in the previous report the authors should discuss the physical reason for such a choice. The fact the by using the "standard" expansion in which sigma is of order of epsilon does not lead to a coupling between the metric perturbation and the vector field at order epsilon is a consequence of SO(3) and plays an important role in perturbation theory in general relativity.
Unless 1 and and 2 are properly addressed I cannot agree on publication.
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