Constraint on the Cosmic Curvature in a Model with the Schwarzschild–de Sitter Metric from Supernovae and Gamma-Ray Burst Observational Data
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThis paper considers a universe described by the Schwarzschild-de Sitter metric, and uses various observational data to constrain the parameters in the cosmic model. The presentation in the paper is clear. I only have one major question regarding the paper:
Schwarschild-de Sitter metric describes a black hole or the exterior of some spherically symmetrically distributed mass distribution in de Sitter spacetime. What is the matter content meant to be in the cosmic model that the author considers? In particular, the author considers the effect of the model in a very large scale, so it is difficult to imagine that some locally distributed matter (like a supermassive black hole) could have very significant effects.
A minor comment: at the end of the paragraph right below eqn. (6), the author says "This prediction was experimentally confirmed ten years later." There is a missing period at the end of the sentence in the paper. Moreover, I think there needs to be a reference here.
Author Response
Comment-1: I only have one major question regarding the paper:
Schwarschild-de Sitter metric describes a black hole or the exterior of some spherically symmetrically distributed mass distribution in de Sitter spacetime. What is the matter content meant to be in the cosmic model that the author considers? In particular, the author considers the effect of the model in a very large scale, so it is difficult to imagine that some locally distributed matter (like a supermassive black hole) could have very significant effects.
Response-1: In lines 100-105, I have tried to explain the fact that in elliptical space, the central symmetry of the Schwarzschild-de Sitter metric is transformed into the spherical symmetry without
any local centre of mass distribution.
In elliptical space with its antipodal points identified, the center of a local mass distribution (as well as any other point around the observer) is geometrically transformed into a huge sphere around the observer. The radius of this sphere is so large that the local differences in the mass centres locations are negligible in terms of the relative displacements of their corresponding large sphere surfaces.
In elliptical space, with its identified antipodal points, the center of the local mass distribution
(as well as all other points of this mass distribution, including the points of the Schwarzschild horizons
of each mass) are geometrically transformed into a huge sphere around the observer. The radius of this sphere is so large that any local differences in the positions of the mass distributions
correspond to negligible relative displacements of the distant sphere surfaces, including
the surfaces of the Schwarzschild horizons.
Thus, a collective spherically symmetrical horizon of radius r_g is formed around
each point in space, and all points in space are equivalent in this respect. So the universe looks
the same to any arbitrarily chosen observer.
In such a scheme, all masses within some vicinity of the observer contribute to the global
collective gravitational redshift through their antipodal points. From the point of view of an observer,
the size of this neighbourhood can be enormous. It includes matter in the form of local and distant galaxies, as well as possibly distant clusters of galaxies. However, compared to the gravitational radius r_g, the size of this region around the observer is negligible.
Comment-2: A minor comment: at the end of the paragraph right below eqn. (6), the author says "This prediction was experimentally confirmed ten years later." There is a missing period at the end of the sentence in the paper. Moreover, I think there needs to be a reference here.
Response-2: In this paragraph I was referring to the discovery of the cosmological redshift by Lemaitre in 1927 and by Hubble in 1929. There is really a lack of relevant links here. I have made the appropriate corrections to the manuscript (lines 46-47 of the revised version).
Reviewer 2 Report
Comments and Suggestions for AuthorsThis ms proposes the analysis of an cosmological model as an alternative to the commonly (and for a wealth of good reasons) accepted FLRW model in terms of a Schwarzschild-de Sitter model with antipodal identification. The latter does not allow for time dependent cosmological expansion with a Big-Bang singularity, does not address the various, theoretically and observationally strongly supported cosmological phase transitions (e.g., BBN, e+e- annihilation, recombination and reionisation), does not address the process of structure formation, and models the redshift to a source solely by its static distance. Even though the analysis performed by the author could be seen as an interesting curiosity by some audience I deem it to be too irrelevant for the field of cosmology to be published in a scientific journal. (The match with data performed by the author indeed yields a ridiculously small curvature of space.) Therefore, this manuscript should not be published in Universe.
Author Response
Comment-1: This ms proposes the analysis of an cosmological model as an alternative to the commonly (and for a wealth of good reasons) accepted FLRW model in terms of a Schwarzschild-de Sitter model with antipodal identification. The latter does not allow for time dependent cosmological expansion with a Big-Bang singularity, does not address the various, theoretically and observationally strongly supported cosmological phase transitions (e.g., BBN, e+e- annihilation, recombination and reionisation), does not address the process of structure formation, and models the redshift to a source solely by its static distance. Even though the analysis performed by the author could be seen as an interesting curiosity by some audience I deem it to be too irrelevant for the field of cosmology to be published in a scientific journal. (The match with data performed by the author indeed yields a ridiculously small curvature of space.) Therefore, this manuscript should not be published in Universe.
Response-1: My response to this Reviewer is lengthy and includes some references. So I am attaching a PDF file with this response.
Author Response File: 
Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsSe the attached document
Comments for author File: Comments.pdf
Author Response
Comment-1: this comment includes some formulae and was attached as a file.
Response-2: My response also contains a set of formulae, so I am attaching it in the form of a PDF file.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsDear Author,
even though I do not agree with your statement that the JWST data necessitates a departure from a (spatially flat) FLRW in the way you suggest it (my belief is that the rhs of the Friedmann equation must be modified at high z as compared to what LambdaCDM posits, also I maintain that a static model does not do justice to what we know about expansion history - your statement about mathematical equivalence between Schwarzschild-de Sitter and McVittie does not cover the CDM part of LambdaCDM at low z and at high z not the radiation sector of Lambda CDM) and even though there are technical errors in the manuscript (e.g., the statement on q in line 183 is wrong - no r_s dependence since (15) indeed is a cubic in r_s) I believe this manuscript should have a stage. (Data analysis is fine, but subject to an "approximated" SdS model.) Therefore, I changed my mind and recommend publication in Universe now.
Comments on the Quality of English LanguageThere are some minor spelling problems: e.g. Schwartzschild
Author Response
Comments 1: My belief is that the rhs of the Friedmann equation must be modified at high z as compared to what LambdaCDM posits, also I maintain that a static model does not do justice to what we know about expansion history - your statement about mathematical equivalence between Schwarzschild-de Sitter and McVittie does not cover the CDM part of LambdaCDM at low z and at high z not the radiation sector of Lambda CDM
Response 1: There are already many modifications of Friedmann's equations, but so far none of them have led to the possibility of explaining the JWST observational facts. By and large, such modifications inevitably break the entire coherent concept of the standard cosmological model. In this case, if the static model is insufficient to explain what we know about the expansion history of the universe, then the expansion model becomes even more insufficient. After all, it is this supposed knowledge about the history of the extension that is derived from the model with an expanding metric. That is, such knowledge is purely model-dependent, which means that one cannot be sure of this knowledge if the whole concept collapses. The history of cosmology shows that static models can be quite successful if the blinders are removed from sight.
As for the CDM sector of the LCDM model, experimental confirmation of the existence of CDM is still lacking. On the other hand, the concept of gravitational redshift discussed in this manuscript assumes that there is a spherically symmetric external gravitational influence for each point in space. In particular, such an impact exists for every galaxy. And, if we take this external influence into account in the rotation curves of galaxies, it turns out that flat rotation curves will extend to very large distances. By the way, a report about just such flat rotation curves was published recently [Mistele T. et al., Indefinitely Flat Circular Velocities and the Baryonic Tully–Fisher Relation from Weak Lensing, ApJ Lett. 2024, 969:L3]. And the authors of this publication find that their are difficult to understand in terms of LCDM.
The Schwarzschild-de Sitter metric model can only explain the flat rotation curves of galaxies at very large distances from these galaxies. In the closer vicinity of galaxies, there should be a halo of their inconspicuous or invisible matter. But it does not have to be dark matter unknown to modern physics from a new type of elementary particles that cannot be detected in any way. This may well be quite physically possible so-called "grey" coarse-grained dust, the possibility of the existence of which has already been reported in several publications [e.g., Yershov V. N. et al. Distant foreground and the Planck-derived Hubble constant; MNRAS, 2020, 492, 5052].
In intergalactic space, such dust can consist of solid hydrogen with impurities that prevent sublimation. This idea was proposed by F. Hoyle [Nature, 1968, 218, 661] and was discussed by many other authors [e.g., Pfenniger D., 2004, Proc. IAU Symp. 220, 241; Walker M. A., 2013, MNRAS 434]. There are even publications substantiating the possibility of the existence of macroscopic bodies based on solid hydrogen the size of asteroids in intergalactic space. The existence of such bodies made of common material, like hydrogen, is much more reasonable than the dark matter particles unknown to science but taken as the basis of the LCDM model.
Comment-2: There are technical errors in the manuscript (e.g., the statement on q in line 183 is wrong - no r_s dependence since (15) indeed is a cubic in r_s)
Response-2: I have corrected misprints and fixed some clumsy clauses.
Reviewer 3 Report
Comments and Suggestions for AuthorsSee the attached document
Comments for author File: Comments.pdf
Author Response
Comments 1: Unfortunately I made a printing error. Of course Sr should not appear in the expression for q. The correct expression is q=r_g R^2 /2. I suggest that the author discusses these solutions and use the relevant ones in section 3. This will strengthen the paper.
Response 1: Initially, I did not realise that it is the coefficient q that needed to be corrected. After fixing this coefficient, I have, indeed, calculated the proposed exact solution, finding the distance-redshift relationship. Accordingly, I have re-calculated all the parameters of the model under discussion. The result is placed to the second part of Table 1.
As proposed, I have discussed this formalism in Sections 2 and 3, and modified the references accordingly. As it was expected, the exact solution turned out to be better than my initial approximate solution.
Both solutions are presented in Figure 1 with the red (approximate) and black (exact) dashed curves. They coincide at high redshifts. But at low redshift the exact solution performs much better that the approximate solution and the LCDM solution.
In the revised version of my manuscript, I am acknowledging the Reviewer's contribution who found an exact solution to the redshift-distance problem.
