Scale Symmetry in the Universe

Round 1

Reviewer 1 Report

I recommend publication of this paper in Symmetry. This paper reviews and further develop the use of symmetry in gravitational systems, the paper obtains very interesting results and shows scale symmetry  is a very powerful tool to analyze such a system,

There is only one typo that should be corrected, I checked the correct scale covariance  equation is not
φ_λ(x) = φ(λx) + 2log λ, but 

φ_λ(x) = φ(λx) + 2Tlog λ,

 

instead.

I suggest this paper be included in the special issue 

 "Global and Local Scale Symmetry in Gravitation and Cosmology"

https://www.mdpi.com/journal/symmetry/special_issues/Global_Local

since the paper is exactly on this subject. After the small typo I have pointed out is corrected, this paper can be published.

Author Response

Thanks a lot for pointing out that inconsistency, coming from mixing the conventions of two different references. In checking the scale covariance equation, I have found that the sign was also wrong. Does the reviewer agree?

I have introduced in the paper other changes and comments to satisfy the requirements of the other reviewers.

As regards the suggestion of a different special issue, thanks again. I did not know that issue, which is certainly interesting. However, I already submitted the paper for the issue "Symmetries in the Universe" and I think that the editor is interested in the subject. I suppose that it is not possible to publish a paper in two different special issues, so it is difficult for me to know how to proceed. I would like to leave the decision to the editors, possibly including the Editor-in-Chief, who has a qualified and overall vision of the functioning of the journal Symmetry.

Actually, the suggestion has motivated me to have a cursory look at the many open special issues of Symmetry and I have found a couple of them in which my paper could fit as well: "Scale Relativity and Fractal Space-Time Theory" and "Astronomy and Symmetry". I was not aware of the many possibilities and now I think that it is best to let the editors decide.

Reviewer 2 Report

The paper deals with the scale symmetry ruled by gravity and giving rise to, on astronomical or cosmological scales, to frsctsl or multifractal geometries. The paper is interesting and  written with great care. There is only a double "the" in the last paragraph of pag. 4. I would also ask the author to define the variable T in equation (5). Actually, should it be kT instead of T, due to dimensional reasons? I would also suggest to number all equations.

Author Response

Thanks for the review. I have corrected the double "the" and I have numbered all equations. Sorry that the variable T was not defined. I had in mind a common convention that measures the temperature in energy units, so kT -> T (as Landau and Lifshitz, for example). I have introduced two explanatory sentences in the paragraph.

I have introduced in the paper other changes and comments to satisfy the requirements of the other reviewers.

Reviewer 3 Report

Dear Author,

 

In my opinion, the paper does not have logical errors and the presentation is fine. Therefore, I recommend to publish it.

 

Regards, Referee

Author Response

Thanks for carefully reading my paper. I have introduced in the paper a few changes and comments to try to improve the presentation (some of them to satisfy the requirements of the other referees).

Reviewer 4 Report

This work is rather interesting and important. The author has studied the large-scale structure of the Universe with the use  of the stochastic-processes theory methods, in particular the nonlinear Poisson-Boltzmann-Emden  equation. In line with the formulated problem, the author considers the fractal solutions of this equation and their relation to the multifractal mass distributions in the  Universe. The language of this work is clear, the methods presented are correct, and the results agree with those from analysis of cosmological simulations and from observations of the galaxy distribution. I recommend its publication in Symmetry. However there is a minor drawback. The used simple model for mass distributions is associated  with the methods of Newtonian gravity. By my opinion, the proposed approach looks very promising. It seems expedient to supplement the work with a  small paragraph  to introduce discussion of the potentialities for generalization of the proposed model considering  the results of General Relativity.

Author Response

Thanks for carefully reading my paper and for your good opinion. As regards your question on the role of General Relativity, I think that it is not a simple question. There were a few remarks in the paper about quantum gravity, which is necessarily relativistic. Presumably, the reviewer is thinking about the role of just classical General Relativity. I have included a paragraph about it, as the reviewer suggests. It is the four paragraph of Sect. 3 (it seems to me a suitable place to put it). There are also a new relativistic example in the first paragraph of Sect. 3 and a footnote in the same section. I hope that it is sufficient, although much more can be said about this question, of course.