The Formulation of Scaling Expansion in an Euler-Poisson Dark-Fluid Model
Round 1
Reviewer 1 Report (New Reviewer)
In the reviewed artical, the authors studied a dark fluid model with spherical symmetry. They solved the induced coupled non-linear partial differential equation system by using a self-similar time-dependent ansatz. They also showed that these kinds of solutions can provide new solutions that are consistent with the Newtonian cosmological frame-work. The manuscript is clearly written and the analysis is correct.
Author Response
We sincerely appreciate the time and effort you dedicated to reviewing our manuscript. We really appreciate your positive comments and how they match with what we were trying to do.
Reviewer 2 Report (New Reviewer)
This paper considers the self-similar solutions to the Euler-Poisson equations with rotation terms for Dark-fluid. By comparing the rotating model with the non-rotating model, the self-similar solutions demonstrate the enhancing effect of rotation on the expansion rate.
The result is interesting, and has potential significant impacts on the scientific community.
Author Response
We're really thankful for the time you spent going through our work. Your kind words and support mean a lot to us.
Reviewer 3 Report (New Reviewer)
The paper under review studies self-similar solutions for the gravitational Euler-Poisson system, which is an important and well studied model to describe the motion of self-gravitating inviscid gaseous stars in the universe. The authors studied self-similar solutions proposed by Sedov and Taylor to the spherically symmetric Euler-Poisson system. By the self-similar transformation, the PDEs can be reduced to a nonlinear ODEs. Various kinds of numerical solutions to the ODEs are given. The authors also showed that these solutions are consistent with the Newtonian cosmological frame-work and can be applied to describe normal-to-dark energy on the cosmological scale.
The paper studies an interesting topic both in cosmology and in Mathematics. The result is new and the derivation is rigorous as far as i see. So, i recommend the publication of the manuscript.
Author Response
We would like to thank you for reviewing our paper so thoughtfully. Your positive feedback and the way you saw the value in our work make us really grateful. Your suggestions made our paper clearer and better, and we're glad you saw the same potential in our research. Your time and expertise are greatly appreciated.
Reviewer 4 Report (New Reviewer)
In the present manuscript, the authors study non-relativistic fluid under the assumptions of spherical symmetry and the existence of self-similar structure for the crucial quantities like the fluid velocity, the energy density and the gravitational potential. In addition they assume that the fluid in some cases is rotating i.e. the observer is not comoving but rather tilted. The equation of state is assumed to be linear (not polytropic as the authors say since they set $n=1$ in eq. (3)) with state parameter $w=p/rho $ equal to $-1$ (the so-called phantom divide or $\Lambda -$term equation of state). Then making additional assumptions on the scale parameters (e.g. the standard conjecture in self-similar spherically symmetric models with the use of the dimensionless variable $\zeta =r/t-$assumption $\beta =1$ in their computations) they try to find (numerically) solutions of the resulting system of differential equations.
The homothety problem in General Relativity as well as in the Newtonian limit is well known and studied over the years and many theorems have been established by Sedov, Barenblatt, Zeldovich, Cahill, Taub and others especially in spherically symmetric configurations (either relativistic or Newtonian). Although, from a mathematical point of view, the manuscript perhaps contains some instructions for new researchers of how the structure of the equations is reduced to ordinary differential equations) it is not clear, from a physical point of view, what is the new constituent added to our knowledge regarding the different phases of the evolution of our Universe. Note also that it is well known, self-similar models are playing a key role in Astrophysics and Cosmology since they represent the asymptotic states (not evolving) of general (evolving) models (see e.g. Carr B J and Coley A A, \emph{Self-similarity in general relativity}, 1999 Class. Quantum Grav. \textbf{16} R31-R71 for a review) therefore I cannot see how one can use self similar models to describe the evolution of our Universe even with non-relativistic matter.
In view of the above my opinion is that the authors must first address, with some detail, the previous points before the manuscript can be potentially accepted for publication in the special issue of \textquotedblleft Universe\textquotedblright.
The authors should check the grammar and the syntax of their manuscript for minor errors and typos.
Author Response
We sincerely appreciate the time and effort you dedicated to reviewing our manuscript. Your insightful comments and suggestions have significantly contributed to enhancing the quality and clarity of our work. We acknowledge the valuable perspectives and remarks you shared and the constructive feedback that you provided. We have carefully considered each of your points and have implemented a series of modifications to address your comments. We highlighted these major changes in red in the manuscript.
Referees' comment:
In the present manuscript, the authors study non-relativistic fluid under the assumptions of spherical symmetry and the existence of self-similar structure for the crucial quantities like the fluid velocity, the energy density and the gravitational potential. In addition they assume that the fluid in some cases is rotating i.e. the observer is not comoving but rather tilted. The equation of state is assumed to be linear (not polytropic as the authors say since they set $n=1$ in eq. (3)) with state parameter $w=p/rho $ equal to $-1$ (the so-called phantom divide or $\Lambda -$term equation of state).
Authors' reply:
As the referee accurately noticed, we have used only the $n=1$ special case of the polytropic equation of state. Therefore, we added "(linear)" specification to the text everywhere it was needed.
Referees' comment:
Then making additional assumptions on the scale parameters (e.g. the standard conjecture in self-similar spherically symmetric models with the use of the dimensionless variable $\zeta =r/t-$assumption $\beta =1$ in their computations) they try to find (numerically) solutions of the resulting system of differential equations.
Authors reply:
On the one hand, the referee correctly stated that we assumed that the relevant physical fields are functions only of the $\zeta =r/t^{\beta}$ as the similarity variable. On the other hand, the value of the $\beta$ is obtained by solving the algebraic equation for the similarity exponents. Since $\beta$ was a free parameter in this model the form of the equation narrowed down to unity. This idea is comes from the previous similar model of the authors from Ref. [16]. As it was pointed out by the Referee, one of the key aim is to provide (numerical) quasi-analytical solution for the problem.
Referees' comment:
The homothety problem in General Relativity as well as in the Newtonian limit is well known and studied over the years and many theorems have been established by Sedov, Barenblatt, Zeldovich, Cahill, Taub and others especially in spherically symmetric configurations (either relativistic or Newtonian).
Authors' reply:
Thank you for the notice; We included a short section in the Introduction to mention the important work of Cahill, Taub, and others on the homothety problem in general relativity.
Referees' comment:
Although, from a mathematical point of view, the manuscript perhaps contains some instructions for new researchers of how the structure of the equations is reduced to ordinary differential equations) it is not clear, from a physical point of view, what is the new constituent added to our knowledge regarding the different phases of the evolution of our Universe.
Authors' reply:
Cosmological hydrodynamical simulations play an important role in today's theoretical astrophysics [1]. The downside of these simulations is that they require a lot of resources. Our model provides a quasi-analytic solution for the Euler-Poisson equation. Our method is a different type of approach to Newtonian Dark Fluid cosmology which provides a basis of comparison for these kinds of simulations.
[1] Complementary Cosmological Simulations. A&A 672, A59 (2023)
Referees' comment:
Note also that it is well known, self-similar models are playing a key role in Astrophysics and Cosmology since they represent the asymptotic states (not evolving) of general (evolving) models (see e.g. Carr B J and Coley A A, \emph{Self-similarity in general relativity}, 1999 Class. Quantum Grav. \textbf{16} R31-R71 for a review) therefore I cannot see how one can use self similar models to describe the evolution of our Universe even with non-relativistic matter.
Author's reply:
The referee correctly noted that homothetic solutions in general relativity represent asymptotic states. In classical hydrodynamical models, self-similar solutions represent so-called "intermediate-asymptotics", they describe the solution as no longer dependent on boundary or/and initial conditions [2]. In this paper, we only studied local solutions of Euler-Poisson equations where the boundary and initial effects no longer appear. In this region, the self-similar solution provides relevant information about the evolution.
[2] G. I., Barenblatt, YA B. Zel'dovich, Self-similar solutions as intermediate asymptotics
In view of the above my opinion is that the authors must first address, with some detail, the previous points before the manuscript can be potentially accepted for publication in the special issue of \textquotedblleft Universe\textquotedblright.
We sincerely hope that the above adjustments we've undertaken meet your expectations and address both the Referee and the Editor concerns.
Round 2
Reviewer 4 Report (New Reviewer)
No further comments.
This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.
Round 1
Reviewer 1 Report
The authors investigate the dynamics of a nonviscous, nonrelativistic, rotating and selfgravitating fluid. A particular focus of the paper is on a question if a specific type of solutions, given by Sedov-Taylor Ansatz, could be useful in describing cosmic dark fluids, i.e. dark matter and dark energy.
Although the authors do not fully argument why specific solutions of the Sedov-Taylor Ansatz type should be a dominant description of the fluid dynamics, it would certainly be very interesting to learn if a specific particular, yet geometrically well motivated solution of a general fluid dynamics could contribute to the understanding of puzzling phenomena such as dark matter and dark energy. The approach and considerations presented in the paper, however, have serious inconsistencies.
1. The authors treat the rotating fluid as spherically symmetric. Describing an essentially axially symmetric system as spherically symmetric is justified by the authors by the assumption that the effects of rotation are very small or negligible. Then the authors fix the value of the polar angle and continue their analysis. The results, however, exhibit not only the dependence on the parameter \omega describing rotation intensity, but even big sensitivity on this parameter. One would expect that if rotation is negligible then all its effects should be negligible.
2. The results depicted in Figure 2 say that the potential \phi is not dependent on time. On the other hand, the density \rho is time dependent. The potential and the density are connected by the Equation (6). For the solutions presented in Figure 2, the l.h.s of (6) is independent on time and the r.h.s. of (6) is dependent on time. This is a serious inconsistency that casts doubt on the results presented in Figure 2.
3. The considerations in section 5.1. use a radially dependent solution which is at odds with the Cosmological principle. The authors do not show at all how the solutions they study could be relevant in describing the dynamics of a FRWL universe.
4. In section 5.2. the authors attempt to use the properties of their solutions to explain the rotation curves of spiral galaxies. However, they use the profile of the radial speed obtained in their solutions as an explanation of the orbital speed presented in the rotation curves of spiral galaxies. The authors provide no arguments why the radial speed profile should be a relevant description of the orbital speed profile in galaxies (apart from not so ideal functional similarity).
I believe that this manuscript is not suitable for publication in the journal Universe.
Author Response
Referee's comment:
The authors investigate the dynamics of a nonviscous, nonrelativistic, rotating, and selfgravitating fluid. A particular focus of the paper is on the question if a specific type of solution, given by Sedov-Taylor Ansatz, could be useful in describing cosmic dark fluids, i.e. dark matter and dark energy.
Although the authors do not fully argue why specific solutions of the Sedov-Taylor Ansatz type should be a dominant description of fluid dynamics, it would certainly be very interesting to learn if a specific particular, yet geometrically well-motivated solution of general fluid dynamics could contribute to the understanding of puzzling phenomena such as dark matter and dark energy. The approach and considerations presented in the paper, however, have serious inconsistencies.
Authors' reply:
First, we would like to thank the referee's hard work and the important comments.
Referee's comment:
The authors treat the rotating fluid as spherically symmetric. Describing an essentially axially symmetric system as spherically symmetric is justified by the authors by the assumption that the effects of rotation are very small or negligible. Then the authors fix the value of the polar angle and continue their analysis. The results, however, exhibit not only the dependence on the parameter \omega describing rotation intensity but even big sensitivity on this parameter. One would expect that if rotation is negligible then all its effects should be negligible.
Authors' reply:
Thank you for the question. We assumed that if the rotation of the system is small compared to the radial dynamical properties of the system, the spherical symmetry violation will be also small. Nevertheless, the dynamic properties of the dark fluid can change significantly.
Referee's comment:
The results depicted in Figure 2 say that the potential \phi is not dependent on time. On the other hand, the density \rho is time-dependent. The potential and the density are connected by Equation (6). For the solutions presented in Figure 2, the l.h.s of (6) is independent of time and the r.h.s. of (6) is dependent on time. This is a serious inconsistency that casts doubt on the results presented in Figure 2.
Authors' reply:
We thank the referee for this very relevant remark. We checked the whole calculation and found a banal mistake. One of the exponent's gamma is equal to two. So it had to be changed. The obtained coupled ODE system was not changed just the final form of the three dynamical variables. Therefore the temporal and spatial dependences of the third equation became clear and consistent.
Referee's comment:
The considerations in section 5.1. use a radially dependent solution which is at odds with the Cosmological principle. The authors do not show at all how the solutions they study could be relevant in describing the dynamics of a FRWL universe.
Authors' reply:
The referee is right. We start not from cosmological principles but from classical spherical symmetric Euler equations which ensure the preservation of the impulse (which is a very relevant constraint) and apply the most simple EOS which is capable to model dark fluid. Our final results have an expanding property that shows qualitative agreement with the Universe. From the results, one can see that density distribution becomes asymptotically flat at large distances which can be interpreted as physical.
Referee's comment:
In section 5.2. the authors attempt to use the properties of their solutions to explain the rotation curves of spiral galaxies. However, they use the profile of the radial speed obtained in their solutions as an explanation of the orbital speed presented in the rotation curves of spiral galaxies. The authors provide no arguments why the radial speed profile should be a relevant description of the orbital speed profile in galaxies (apart from not so ideal functional similarity).
Authors' reply:
We thank the referee's comment. Yes, we made a mistake and mixed up the two kinds of velocities. We removed the corresponding part of the manuscript together with the figure and all the wrong conclusions and statements which were written.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The paper titled 'The Formulation of Scaling Expansion in an Euler-Possion Dark-fluid Model' authored by Dr. Endre Szigeti et.al. studies a kind of dark fluid model with the polytropic equation of state by solving the coupled non-linear partial differential equations which have self-similar time-dependent solutions. It would be interesting, but I cannot recommend the possible publication in the current form due to the possible fowling flaws:
1. In cosmology the effective fluid with the equation of state w=-1 corresponds to the cosmological constant, also dubbed as dark energy. In this simple case, it is just the de Sitter exponent expansion universe. Of course, in this paper the authors consider another possibility where the energy density and pressure are time t and scale r dependent functions. So the question is what's meaning for the time t, the scale r, the energy density and pressure in cosmology? And Eq. (4) for the w=-1 case, c^2_s is negative, the instability of the system will appear.
2. Why should we consider the self-similar solutions in these dark fluid models. This solution can solve what problems in cosmology or astrophysics? If one considers the dark fluid models evolution in cosmology, the background expansion effect should be involved.
3. In Eq. (8), the rotation term is singular at initial time t=0, is it physical?
4. The authors have already discussed too much on the initial conditions of the self-similar ansatz, but it would be still not enough. The real physical system should be considered.
5. The last but more important question, what's the relation between the model presented in this paper and the spherical collapse model which is discussed extensively in cosmology?
Author Response
Referees' comment
The paper titled 'The Formulation of Scaling Expansion in an Euler-Possion Dark-fluid Model' authored by Dr. Endre Szigeti et.al. studies a kind of dark fluid model with the polytropic equation of state by solving the coupled non-linear partial differential equations which have self-similar time-dependent solutions. It would be interesting, but I cannot recommend the possible publication in its current form due to the possible fowling flaws:
Authors' Reply:
We would like to emphasize, that the main focus of our manuscript is not to provide a better physics explanation of the big questions of today's cosmology and astrophysics but to present an analytic mathematical solution, which is applicable in specific physical relevance. We think in this sense we failed with the presentation, although this was explicitly written in the Abstract, in the Introduction, and in the Summary sections. For highlighting this, we re-phrased these parts, with a more straightforward message. We hope this way the presentation of the content in the manuscript is suitable to accept for publication.
Referees' comment
1. In cosmology the effective fluid with the equation of state w=-1 corresponds to the cosmological constant, also dubbed as dark energy. In this simple case, it is just the de Sitter exponent expansion universe. Of course, in this paper, the authors consider another possibility where the energy density and pressure are time t and scale r dependent functions. So the question is what's meaning for the time t, the scale r, the energy density and pressure in cosmology? And Eq. (4) for the w=-1 case, c^2_s is negative, the instability of the system will appear.
Authors' Reply:
We present an analytic hydrodynamic model. For the equation of state, we apply the p = -rho relation which is an accepted formula for dark matter in the community. (The p = rho is the EOS for ordinary matter.) This is not our investment, we just use it, it can be found in numerous books. This EOS even gives a constant for the speed of sound. Our publication clearly shows that the minus sign in the EOS gives reasonable results for the dynamical variables for the density, velocity, and gravitational field.
Referees' comment
2. Why should we consider the self-similar solutions in these dark fluid models? This solution can solve what problems in cosmology or astrophysics? If one considers the dark fluid models' evolution in cosmology, the background expansion effect should be involved.
Authors' reply:
As we pointed out in our answers above, our deduction is the opposite: we present self-similar hydrodynamical solutions which can be tested in certain physical problems. The authors here presented two specific cases, which are compatible with the given physical cases. Our goal was not to solve the complete cosmological problem but to present a consistent hydrodynamical model without contradiction that is capable to describe the expansion of the universe.
Referees' comment
3. In Eq. (8), the rotation term is singular at initial time t=0, is it physical?
Authors' reply:
The referee has the right, that equation contains singularity at t=0 and indeed r=0 as well. This is a natural consequence of the chosen coordinate system, as was shown among others [J. Math. Kyoto Univ. (JMKYAZ) 44-1 (2004), 129–171]. However, the existence of local solutions for the isentropic case (similar to our case) was proven outside the origin. We wanted to define and investigate a clear model which has self-similar symmetry, so the original PDE system can be reduced to an ODE system. Self-similarity is a kind of symmetry that necessarily occurs in systems that have spreading runaway and dispersive or dissipative properties. The universe or a galaxy is such an entity. On the other hand, to fulfill the constraints among the exponents Eq. 9a - 9c we have to introduce an extra 1/t^2 factor in the rotation term. The singularity of such a function in the origin is not a significant problem, because with a slight shift from t to t-t_0 moves the solution to the regular domain.
Referees' comment
4. The authors have already discussed too much on the initial conditions of the self-similar ansatz, but it would be still not enough. The real physical system should be considered.
Authors' reply:
Nobody really knows the real physical system of the universe with ordinary and dark matter. As we clearly mentioned above we just present a simple hydrodynamical model, which contains information about the dark fluid, via the given EOS, has no internal contradiction and as it can be clearly seen from our analysis gives a reasonable expanding property. We do not state, that the universe is so, we just state that such a model describes some physically relevant properties.
Referees' comment
5. The last but more important question is, what's the relation between the model presented in this paper and the spherical collapse model which is discussed extensively in cosmology?
Authors' reply:
Our model tries to answer the expanding properties of the universe and not the collapsing one. This model with this EOS has an expanding property, to investigate spherical collapse probably a completely different model should be defined, which is out of the scope of the present study.
Thank you for the point. The recent manuscript is a key point in this way, which we would like to follow in a forthcoming publication.
Round 2
Reviewer 2 Report
Dear editor,
The authors tried to clarify the questions raised by the referee, but apparently it was failed. Thus, I cannot recommend the possible publication in the current form of this manuscript.
