Painlevé Analysis of the Cosmological Field Equations in Weyl Integrable Spacetime
Round 1
Reviewer 1 Report
In this paper, the author considers the possibility of constructing exact solutions to the equations of the gravitational field in Weyl Integrable Spacetime for a spatially flat FRWL universe filled with ideal gas and the cosmological constant.
The main result of the work is the statement about the existence of exact solutions of the cosmological dynamics equations in the form of the Puiseux expansions.
The analysis of exact solutions of the cosmological dynamics equations is an important direction in modern cosmology, and the presented results are of interest in this area of research.
However, I will make a few comments:
1.Reference [30] does not contain any imprint of the article.
2. Can the presented solutions of the gravitational field equations describe phenomenologically correct cosmological models?
3. In my opinion, it is necessary to supplement the article with arguments in favor of the fact that the proposed solutions are relevant for building the realistic cosmological models.
After taking into account these comments, I can recommend the article for publication.
Author Response
I want to thank Reviewer for the positive comments on my work.
My reply on the specific comments follows.
Point 1: Ref. [30] has been submitted in a Journal and it is under consideration.
Point 2 & 3: I have included two new paragraphs in the text, where I discuss the physical properties of this cosmological model and of the existence of the solutions. The first paragraph is given in Section 2, where I review previous analysis on the subject. The second paragraph is given in the conclusions where we discuss how the analytic solutions can be used for the study of the cosmological parameters.
Reviewer 2 Report
In the manuscript “Painlevé analysis for the cosmological field equations in Weyl Integrable Spacetime”, the author constructs a new integrable model that generalizes and connects two known integrable models. The previously known integrable models corresponds to either \Lambda=0 and an arbitrary \gamma (see [29,30]) or \gamma=2 (that corresponds to an additional scalar field without any potential) and an arbitrary \Lambda (see V.R. Ivanov, S.Yu. Vernov, Eur. Phys. J. C, 81 (2021) 985). The result of this papers is important, because the analytic solutions and integrable models play an essential role in the cosmology (see for example, the review by V. Faraoni, S. Jose, St. Dussault, Gen. Rel. Grav., 53 (2021) 109). At the same time, it is not clear, how the obtained solutions in terms of Puiseux series can assist in the studying of the cosmological evolution. Does the obtained Puiseux series converge? Can the author clarify the behavior of the Hubble parameter that corresponds of the obtained solutions? Is it possible to get the Hubble parameter in the form of the Puiseux series? I think that answers on these questions should be included in the paper.
Remarks:
1) det (A) = 0 does not mean the corresponding linear system has infinite number of solutions. Maybe it has no solution. This point should be clarified.
2) The constants x_0 and \Omega_{L0} are connected by the constraint equation after Eq. (25), but the author does not use this relation in (26)-(28). Also, the authors includes both x_0 and \Omega_{L0} in (32). It would be better to present coefficients x_1 and \Omega_{L1} via x_0 only.
3) The solution is presented as a Puiseux series in \tau, whereas in \sqrt{\tau} the Laurent series solutions will be obtained. Maybe the choice of \sqrt{\tau} as a parametric time can simplify the system of equations and allows to find an additional integral of motion by the Noether symmetry analysis. It would be very interesting result to demonstrate how the singularity analysis can be used in combination with the Noether symmetry analysis.
4) After (25) should be q=-1, instead of p = -1. Right?
5) It would be better to write about the Painleve test in more detail or/and to give a reference to the corresponding textbook, for example, R. Conte, M. Musette, The Painlevé Handbook, Springer, 2008
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DOI |
10.1007/978-1-4020-8491-1 |
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Author Response
I want to thank the reviewer of the valuable comments on my work.
My reply to the specific points of the report follows.
Reviewer: At the same time, it is not clear, how the obtained solutions in terms of Puiseux series can assist in the studying of the cosmological evolution. Does the obtained Puiseux series converge? Can the author clarify the behavior of the Hubble parameter that corresponds of the obtained solutions? Is it possible to get the Hubble parameter in the form of the Puiseux series?
Reply: A new paragraph has been introduced in the conclusion where I discuss a simple application for the study of the physical properties of these solutions. Recall that the analytic solutions are presented in terms of the parameter τ=ln a and the physical components are written in terms of ln a. Last but not least, from the dynamical analysis of the cosmological model we know that the future attractor is the de Sitter universe, where easily we can infer that the Puiseux series converge.
On the specific points:
1) The details on detA=0 have been added.
2) The expression are too complex when we expressed them by using the constraint equation. For simplicity of the text I selected to presented the coefficients in terms of xâ‚€ and Ω_{Λ0}.
3) That is a very interesting point mentioned by the referee and it has been widely investigated before in the literature. However, there has been a specific answer yet to the subject. What we know is that in order the first step of the singularity analysis to work, the dominant terms should admit a scaling solution. For more details see the discussion in arXiv:1605.04164 .
4) The typo has been corrected.
5) New references have been added in the text.
Reviewer 3 Report
In this paper, the author makes the Painleve analysis of isotropic cosmological models filled by an ideal fluid and a cosmological constant in the modified Weil Integrable Spacetime gravity. The paper is rather mathematical than a physical one, and needs more connection with usual physics and cosmology. In particular, its equations have a simple analytic closed-form solution (20) in the absence of cosmological constant. So, why the Painleve analysis is needed in this case at all? On the other hand, there no closed-form analytic solution in terms of known functions in the presence of Lambda. The author is even unable to write the general form of coefficients of the Laurent series beyond the first two ones. So, while a mathematician can say that an analytic (in his sense) solution exists in the form of the Laurent series, a physicist would say that there is no explicit solution in the closed form, even in the form of a series with fully and explicitly defined coefficients. Thus, the abstract of the paper strongly misleads a typical reader of such physical journal as Universe in this respect.
Moreover, going closer to physical cosmology, I strongly disagree with the author’s statement at the end of the first paragraph on page 2 that the movable singularity in the Painleve analysis should not be confused with the cosmological singularity. Just the opposite takes place: the cosmological singularity, which may occur at any moment of time t_0 due to homogeneity of time, is just the movable singularity from the point of the Painleve analysis. So, to make contact with physical cosmology, the author has to present the temporal behaviour (for N=1) of the scale factor, the field \phi and \rho in this singularity which corresponds to the first term in his Puiseux series (23,30,31). From this behaviour, it will become clearer to a reader why the presence of the cosmological constant affects the structure of singularity in this case, contrary to what takes place in EGR. The behaviour of the model involved at late times when a\to\infty is of interest, too, both in the presence and absence of Lambda.
Another minor comment is that \rho_{m0} disappeared in formulas of Sec. 3.1, so it is not possible to check what happens in the absence of matter. Finally, to make contact with massless scalar field cosmology in EGR, the author has to show explicitly that his conservation law in the first line of page 4 just corresponds to the textbook result \dot \phi \propto a^{-3} for N=1.
As a whole, I suppose that the paper need major revision to make it content more physical.
Author Response
I want to thank the reviewer for the detailed comments on my work.
My reply on specific comments of the report follows.
A) In particular, its equations have a simple analytic closed-form solution (20) in the absence of cosmological constant. So, why the Painlevé analysis is needed in this case at all?
Reply: The only reason that I apply the singularity analysis in absence of the cosmological constant term is to have a consistency on the presentation of this work.
B) On the other hand, there no closed-form analytic solution in terms of known functions in the presence of Lambda. The author is even unable to write the general form of coefficients of the Laurent series beyond the first two ones. So, while a mathematician can say that an analytic (in his sense) solution exists in the form of the Laurent series, a physicist would say that there is no explicit solution in the closed form, even in the form of a series with fully and explicitly defined coefficients. Thus, the abstract of the paper strongly misleads a typical reader of such physical journal as Universe in this respect.
Reply: The analytic solution is presented in terms of Laurent expansion which in general has infinity terms. There is not need to present the infinity coefficients since they can be easily derived. However, closed-form solutions are only a specific part of analytic solutions. In terms of physics and specifically of cosmology the importance of integrability has been discussed in various works, some of them are cited in the text.
C) Moreover, going closer to physical cosmology, I strongly disagree [...]
Reply: In my study, I work on dimensionless variables and parameter τ is ln a. There should not be any confusion on this.
D) Another minor comment is that \rho_{m0} disappeared in formulas of Sec. 3.1 [...]
In the case where ρ_{m0}=0, the model is reduced to a simple scalar field model with a constant potential. Which has been widely studied before.
