Extensive and Intensive Aspects of Astrophysical Systems and Fine-Tuning

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Dear Editors,

I must admit that I was initially quite skeptical after reading this paper. However, after numerically solving the TOV equations for a degenerate Fermi gas, I was surprised to find that varying the value of G in the simulation did not affect the value of the Newtonian potential at the surface of the stellar object, Φ, referred to as the "compactness" by the author.

Subsequent discussions with a colleague revealed that, for the TOV equations, this invariance can be explained through the following rescaling symmetry of the TOV equations:

G→a G, R→a^{−1/2} R, M→a^{−3/2} M, Φ→Φ, ρ→ρ ,p→p ,c→c,

where R and M are the radius and mass of the stellar object respectively, ρ and p its density and pressure, c the speed of light in vacuum.

I find the discussion in the paper intriguing and potentially worthy of publication, since the dependence of Φ on microphysics – and not on G – appears to be underappreciated (at least by myself and the colleagues I consulted). That said, I have several concerns that currently prevent me from recommending the article for publication:

1. Novelty vs. Prior Work
   It is unclear what is genuinely new in this paper relative to references [13-16]. I would appreciate clarification, particularly regarding the observation in lines 237-240, which seems potentially valuable, but whose originality relative to the cited literature is uncertain.

2. Physical Significance and Practical Relevance
   While the claim that Φ is independent of G in virialized systems might be correct, its practical importance appears limited. One still needs to know the value of the radius R to infer the Newtonian potential U via U=Φ R/r, with r the distance from the center of mass of the stellar or astrophysical object. As Table 1 shows, R always depends on G, through its relationship to λ, the electromagnetic-to-gravitational coupling ratio. Hence, observables still explicitly depend on G, contrary to what is suggested in the paper. For instance, gravitational lensing and redshift depend on U, not Φ, and therefore remain G-dependent. The manuscript should tone down its claims accordingly. In particular, the statements at lines 112-124 seems incorrect to me, as seems most of Section 3 and the discussion in lines 243-247 of the conclusion.

3. Unjustified Assertion (Lines 51-53)
   The assertion made here lacks justification. It needs to be either explained, supported with argumentation, or referenced appropriately.

4. Equation (2) Explanation
   The first two equalities in Eq. (2) are definitional. However, the final equality requires justification or citation.

5. Dependence on G (Line 91)
   The statement regarding M and R scaling with powers of G is confusing. G is a constant, whereas M and R are not. Please clarify the intended meaning.

6. Holographic Bound (Line 228)
   The mention of a "holographic bound" is unclear and appears unrelated to the core discussion. Please elaborate on its relevance and provide proper context and references.

Best regards,

A referee.

Author Response

Please check the attachment.

     

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Dear editor and author,

The manuscript by Meir Shimon explores what sets the scale of the gravitational potential Φ/c² in virialised astrophysical systems. I find the manuscript to be very interesting and well written. It argues convincingly that this is set by various other dimensionless quantities like the fine structure constant and the ratio of electron to proton mass. I have just a few minor comments regarding the article.

 

In the second-last sentence of the abstract, perhaps the authors could write ".....equilibrium between self-gravity and microphysical processes."

 

The author only briefly talks about how cosmological observations can be used to measure G. The author should discuss the role played by the CMB anisotropies and cite this study:

https://doi.org/10.1051/0004-6361/202451602

 

The author may also like to cite this study in relation to the good agreement between the baryon density inferred from the dispersion measures of localised fast radio bursts and standard LCDM and BBN expectations for the present baryon density based on early universe considerations:

https://www.nature.com/articles/s41550-025-02566-y

 

The author talks about galaxies, but here it is less clear whether their arguments apply, because galaxies may well follow a non-Newtonian gravity law. The author should discuss the possibility that they lack any invisible mass and instead follow the Milgromian law of gravity (MOND), citing this extensive review into the paradigm:

https://arxiv.org/abs/2110.06936

 

Most of the arguments raised by the author are not related to galaxies, so just a brief mention will do. It might be interesting to consider how the a₀ scale in MOND might arise in the ΛCDM context, and whether this can explain the order of magnitude coincidence with cH₀. However, these are optional comments that the author does not need to consider.  What is important is to at least mention that cold dark matter might not exist and galaxies might actually be purely baryonic, requiring them to follow a gravity law similar to MOND.

 

In summary, a few minor additions to the manuscript would be helpful.

 

Yours sincerely,

Referee

Author Response

Please check the attachment.

     

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Dear Editors,

The authors have successfully addressed the objections I previously raised. However, in the meantime, the paper has changed significantly and is now almost a new paper.

I will not oppose the publication of the manuscript, as the author has the right to communicate their current understanding of a complex issue. Nevertheless, I believe there may be flaws in some parts of the current version, particularly in the entirely new Section 3, which I find somewhat sketchy. Specifically, the content prior to Section 3.3 does not seem to motivate the construction of a Weyl-invariant action in Sec. 3.3—on the contrary. As I mentioned in my previous report, I verified that the TOV equations are indeed invariant under a rescaling, but this holds within the framework of General Relativity, not in alternative theories in general. Therefore, since this scaling symmetry (which leaves Φ unchanged under a change of G) already exists in GR, there appears to be no clear motivation to modify the action of GR. Consequently, Section 3 seems unnecessary and tends to obscure the main message of the paper.

Furthermore, there are subtleties with Weyl-invariant actions that must be treated carefully (see, e.g., [Hertzberg, Physics Letters B 745 (2015) 118–124] or [Jackiw, Pi, Physical Review D 91, 067501 (2015)]), and I am not sure the author has thought about these points with sufficient care.

Overall, I would suggest that the author consider deleting or rewriting Section 3, but I will not oppose publication in its current form, although I believe the manuscript, overall, could be significantly improved.

Best regards,
A referee.

Reviewer 2 Report

Comments and Suggestions for Authors

The revised manuscript by Shimon addresses my earlier comments and should now be accepted.

 

Yours sincerely,

Refereeee