The Geometric Proca–Weyl Field as a Candidate for Dark Matter
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsSee the attached file "Referee report.pdf".
Comments for author File: 
Comments.pdf
Author Response
We want to thank the referee for valuable comments and suggestions. Below we list all modifications we have done in the text in order to address the issues raised by the referee.
1) The numbering of the Sections in the paper does not correspond to the
description of the organization of the paper, given in the last paragraph of the
Introduction. Either this paragraph should be rewritten, or the rest of the Sections
should be reorganized in accordance with it.
We have rewritten the last paragraph of the Introduction.We thank the referee for calling our attention to this point.
2) The discussion in the beginning of Section 6, meant to justify the assumption
that σ₀=0 needs improvement. More precisely, the statement that these are
particles with zero momentum is confusing. The condition σ₀=0 is simply
temporal gauge for the Proca-Weyl field, which of course the authors are free to
choose. Temporal gauge is used in order to simplify the description of a system in
the Hamiltonian formalism, since the conjugate momentum of σ₀ also vanishes in
this gauge. However, this conjugate momentum is in general not the same as an
ordinary momentum, which is a distinction especially relevant for gauge theories.
The authors should clarify the discussion in the beginning of Section 6.
For the sake of clarity, we have rewritten the beginning of Section V (the old section 6). The key point here is that the Weyl field is massive, so it does not have a gauge freedom. As a matter of fact, if we assume that all components of the Weyl field depends only on t, then it follows directly from the field equations that σ₀=0 - it is easier to conclude this by considering the component ν=0 of the field equations written in the form (12). In our reasoning, the hypothesis that the field has a purely time dependence seems to be reasonable in the cosmological context.
3)The readability of the paper would be improved by writing explicitly the
expression for the quantity θ_{αβ} in eq. (25). This quantity is needed in the rest of
the paper no less than the much better known expression for the energy-
momentum tensor, which is nevertheless provided explicitly in eq. (26).
We totally agree with the referee and have included a new equation (27), in which we explicitly write the tensor θ_{αβ} in terms of the Weyl field.
4) The discussion in the paragraphs below eq. (39), justifying that π_{IJ} is negligible
seems unnecessary or confusing. The suppression factor in (39),
namely H^2/m^2, is exactly the same as the suppression factor in the ratio of the
average pressure over average energy density in Section 7.2. So why is the
argument below (39) not the same as in Section 7.2?
The old equation (39), which now corresponds to number (40),, ensures that shear stress is small compared to the energy density in the matter-dominated era if m>>H. However, to be consistent with cosmological data, we must guarantee that the degree of anisotropy introduced by shear stress cannot be greater than the level measured in the CMB radiation ( δT/T ∼10⁻⁵). To be sure of this, we must study the relationship between π_{IJ} and the temperature fluctuations δT. This interesting question is left to be analysed in another opportunity, since it can be mathematically rather laborious. Alternatively, we resort to the fact that the stress tensor is zero when the Weyl field is in an unpolarized state. We have rewritten the second paragraph below equation (40) in order to explain that the condensate (our solution) could be in an unpolarized state. The idea is that the condensate has a well-defined number of particels and that the same number of particles are in each polarization mode. As the field is massive, there are three different polarization modes. In this case, the shear tensor will be null, since a state with a well-defined number of particles has an undertemined phase.
5) There are a number of minor English language corrections to be made:
- our start point our should be starting point .
- On line 13: gave away should be gave way
- On line 20: total matter content should be total energy density
- On lines 130, 142, 155: admit should be assume
- On lines 282-283: dominated-matter should be matter dominated
- On line 294: critic should be critical
- On lines 301, 324: primitive should be primordial
- On line 306: many recent research should be a lot of recent research
We have made all the indicated corrections. We thank the referee for calling our attention to them.
We believe that all the implemented changes have improved our paper. We thank the referee for the valuable input and hope our work now meets his/her quality criteria.
Author Response File: Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsManuscript ID
universe-3412923
The geometric Proca-Weyl field as a canditate for dark matter
In this work a modification of the original Weyl unified theory is used to address the dark matter problem in terms of a purely geometrical approach. The authors propose as a model a gas of Weyl-Proca particles in a Bose-Einstein condensate state and describe its behavior on large scales.
Their findings indicate that, in a certain limit, the model is consistent with the standard cosmological model during the matter-dominated era. The authors verified that for sufficiently large particle mass, the shear stress and anisotropic pressure become negligible, even when the field is in a classical state. Furthermore, under certain conditions the Weyl-Proca condensate behaves as a dust fluid, consistent with the behavior expected of a dark matter candidate.
In the framework proposed, the Weyl field interacts with ordinary matter only through the gravitational field, thereby circumventing usual constraints applied to dark matter candidates.
The paper is well written and the explanations are clear as well as the mathematical model. The conclusions are easily followed from the hypothesis.
I also consider the paper may be interesting for a wide community and thus I recommend the paper for publication in Universe in its present form.
Author Response
We would like to thank the referee for his/her comments.
Reviewer 3 Report
Comments and Suggestions for AuthorsThe manuscript addresses the problem of dark matter in the context of the Weyl invariant gravity, where Weyl-Proca particles in a Bose-Einstein condensate perform well according to cosmological data.
The scientific content has been correctly tackled, and the little cases where further attention needs to be dealt, it is forwarded to future research/work.
The manuscript has merit to be published in the journal Universe, but I have some remarks to be addressed by the authors:
a) In line 20, it is said "matter content", however wouldn't be more correct to state "energy content" as that percentage is not from the comparison between ordinary and dark matter content, but also includes dark energy? If we were looking at matter component, we know that dark matter is about 85% of total matter content.
b) Line 53, it seems that "Weyl invariant gravity field equations" should be "Weyl invariant gravity" solely.
c) In fact, Weyl's ideas of a non-metric connection and of a conformal theory of gravity have been of great interest nowadays. In one hand, the Weyl invariant gravity is one possible way to circumvent initial problems of the model as pointed out in this paper, while other possibility also exist. It may be strengthen the paper and the advantages of revisiting and extending Weyl gravity to consider additional references:
- E. Álvarez, S. González-Martín, Weyl gravity revisited, JCAP02(2017)01 - a review on the properties of Weyl gravity and shortcomings;
- C. Gomes, O. Bertolami, Nonminimally coupled Weyl gravity, Class.Quant.Grav. 36 (2019) 23, 235016 - the Weyl connection being used in modified gravity;
- G. Kouniatalis, E.N. Saridakis, Modified gravity from Weyl connection and the f(R,A) extension, arXiv:2411.14380 [gr-qc] - the Weyl connection being used in modified gravity;
- N. Mohammedi, A note on Weyl gauge symmetry in gravity, Class.Quant.Grav. 41 (2024) 19, 195021 - some ideas on having a scale invariant theory of gravity built with a Weyl compatible connection.
d) The references. Wouldn't it be better if all the references have a single entry, instead of some sharing the same? Sometimes databases such as Scopus or Web of Science may have trouble acknowledging the citations.
Author Response
We want to thank the referee for valuable comments and suggestions. Below we list all modifications we have done in the text in order to address the issues raised by the referee.
- a) In line 20, it is said "matter content", however wouldn't be more correct to state "energy content" as that percentage is not from the comparison between ordinary and dark matter content, but also includes dark energy? If we were looking at matter component, we know that dark matter is about 85% of total matter content.
We agree that the expression "energy content" is more appropriate. The writing has been corrected to reflect this point.
- b) Line 53, it seems that "Weyl invariant gravity field equations" should be "Weyl invariant gravity" solely.
We completely agree. In the presented context, "Weyl invariant gravity" is more appropriate. We have corrected it as suggested by the referee.
- c) In fact, Weyl's ideas of a non-metric connection and of a conformal theory of gravity have been of great interest nowadays. In one hand, the Weyl invariant gravity is one possible way to circumvent initial problems of the model as pointed out in this paper, while other possibility also exist. It may be strengthen the paper and the advantages of revisiting and extending Weyl gravity to consider additional references:
- E. Álvarez, S. González-Martín, Weyl gravity revisited, JCAP02(2017)01 - a review on the properties of Weyl gravity and shortcomings;
- C. Gomes, O. Bertolami, Nonminimally coupled Weyl gravity, Class.Quant.Grav. 36 (2019) 23, 235016 - the Weyl connection being used in modified gravity;
- G. Kouniatalis, E.N. Saridakis, Modified gravity from Weyl connection and the f(R,A) extension, arXiv:2411.14380 [gr-qc] - the Weyl connection being used in modified gravity;
- N. Mohammedi, A note on Weyl gauge symmetry in gravity, Class.Quant.Grav. 41 (2024) 19, 195021 - some ideas on having a scale invariant theory of gravity built with a Weyl compatible connection.
We thank the referee for the suggested references, which have all been included in the text.
- d) The references. Wouldn't it be better if all the references have a single entry, instead of some sharing the same? Sometimes databases such as Scopus or Web of Science may have trouble acknowledging the citations.
We have carefully reviewed the reference list, ensuring that each reference has a unique entry.
We believe that all the implemented changes have improved the quality of the paper. We thank the referee for the valuable comments and suggestions. We hope the alterations satisfy the referee.
Round 2
Reviewer 3 Report
Comments and Suggestions for AuthorsThe authors have addressed all the remarks I have suggested. The manuscript has been improved in this new version. Hence, I recommend it for publication in its present form.
