Combined Studies Approach to Rule Out Cosmological Models Which Are Based on Nonlinear Electrodynamics

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Authors considered contributions to r.h.s. of Einstein equations due to the Lagrangians of non-linear electrodynamics with only a magnetic field presented, L(B).  They studied the restrictions on the form of L(B) arising from the requirement that variation rate of B-pressure with B-energy is a positive number less than speed of light (stability and causality). For obtained permitted forms, they then studied critical points/cosmological phases of the scale-factor dynamics in the magnetic-universe cosmological models, with the conclusion that none of them can satisfactory explain the current acceleration expansion. 

 

Authors should improve numerous missprints in the text, in particular,  pointed out in the list below.

 

Comments to the authors:

 

Line 40 - reference should be improved.

Lines 49-56 - the same sentence is repeated twice.

Eq. (6) - should be improved.

Lines 133-138 - the same sentence is repeated twice.

Lines 161-162 - English should be improved.

Eq. (20) - should be improved.

Lines 485-487 - English should be improved.

Line 688 - references should be improved.

 

Comments on the Quality of English Language

Line 40 - reference should be improved.

Lines 49-56 - the same sentence is repeated twice.

Eq. (6) - should be improved.

Lines 133-138 - the same sentence is repeated twice.

Lines 161-162 - English should be improved.

Eq. (20) - should be improved.

Lines 485-487 - English should be improved.

Line 688 - references should be improved.

 

Author Response

Comments to the authors:

Line 40 - reference should be improved.

Reply: The reference has been replaced with a more recent one.

Lines 49-56 - the same sentence is repeated twice.

Reply: The repeated statement has now been removed. As a matter of fact, large parts of the text of the manuscript have been rewritten.

Eq. (6) - should be improved.

Reply: The equation has been modified to make it more transparent.

Lines 133-138 - the same sentence is repeated twice.

Reply: The wording of this paragraph has been improved.

Lines 161-162 - English should be improved.

Reply: The wording of this paragraph has already been improved.

Eq. (20) - should be improved.

Reply: Eq.20 has been reviewed and corrected

Lines 485-487 - English should be improved.

Reply: The wording of this paragraph has been improved. As we have already said, large parts of the text in the original manuscript have been rewritten.

Line 688 - references should be improved.

Reply: All references have been checked (also updated) and written correctly. Several lacking references have been added.

We appreciated the reviewer's comments and suggestions, which allowed us to ensure the manuscript's readability and accuracy.

Reviewer 2 Report

Comments and Suggestions for Authors

The stability and causality of cosmological models that incorporate nonlinear electrodynamics are the main topics of investigation in this work. The power law and the Rational Lagrangian are two particular NLED models that the authors examine. They seek to comprehend the viability of these kinds of models in explaining the evolution of the cosmos by combining Bayesian inference with dynamical systems theory. The theoretical foundation of NLED in conjunction with general relativity is presented, and then the stability and causality of several NLED Lagrangians are examined. The ranges where these models are causal and stable are subsequently determined by the authors through a thorough dynamical investigation. The findings demonstrate how the Maxwell radiation-dominated state of the Power-Law Lagrangian model evolves into a matter-dominated state as it moves through different cosmic phases.

The dynamical equations for the FLRW model linked with dark matter and NLED are formulated by the authors in Section 2. They have shown that important physical insights can be obtained from this simplified framework. They have concentrated on magnetic universes with only the invariant $F=2 B^2$ determining the electromagnetic Lagrangian densities driving them. Their derivation of $\rho_m$ and $f$ in terms of scale factor $a$ follows as a result. This material is accurate in terms of science and presented well.

The authors have provided a thorough explanation of the cosmological parameters under their model in Section 2.2. They have provided detailed descriptions of the squared speed of sound, density parameter, EoS parameter, and deceleration parameter in this regard. 

Here, my specific suggestion is to incorporate statefinder parameters.

Stability and Causality Analysis of Several NLED Lagragians have been shown in the next section. They have provided an expression for SSS in Eq. (28). By stating that "the analysis to determine the parameter values for which this Lagrangian density is stable and causal is quite complex," the authors have eschewed a thorough examination. This isn't going to happen. I've observed that the writers are able to provide a comprehensive examination of the expression for stability analysis following a hard work. The same is true for Equation (30).  

So, my specific suggestions are:

a) Incorporation of detailed statefinder analysis;

b) Incorporation of in-depth and thorough analysis of SSS for stability.

c) My final suggestion is to establish novelty of the study with respect to existing literature. 

I suggest a major revision. 

Author Response

So, my specific suggestions are:

- Incorporation of detailed statefinder analysis.

Reply: We have incorporated the statefinder analysis and the parameters $r$ and $s$ have been evaluated at the critical points of the dynamical system, in addition to other relevant cosmological parameters. All of the modifications made in response to this and other recommendations of the referees have been highlighted in red color in the improved version of the manuscript.

- Incorporation of in-depth and thorough analysis of SSS for stability.

Reply: In response to this referee's suggestion, we added an in-depth qualitative and quantitative analysis of the squared sound speed (SSS). We have explicitly written the SSS equation for each of the analyzed models in terms of appropriate variables, including plots of the SSS vs the parameters of the models. In this regard new figures FIG. 1, FIG. 2 and FIG. 3 have been incorporated to illustrate the results of our study.

 

- My final suggestion is to establish novelty of the study with respect to existing literature. 

Reply: The novelty of the present investigation relies in the combined study we perform through using different complementary tools: 1) the tools of the dynamical systems theory, which are useful to expose the asymptotic cosmological dynamics of the models, 2) the study of the instabilities which are due to small perturbations of the cosmic background filled with non-Maxwellian radiation, as well as of the tachyonic instabilities which arise due to violation of causality and 3) parameter fitting investigation through implementing a Bayesian analysis of the relevant cosmological parameters.

The importance of the present study lies on the power of the combined study we perform here. There are separate parameter fitting studies as well as dynamical systems and stability investigations of several cosmological models which are based in non-linear electromagnetic fields as the sources of gravitation. According to these studies some models can successfully explain the inflationary stage of the early times cosmic dynamics, while others can explain the present stage of accelerated expansion of the universe without the need for dark energy. As we show in our manuscript for two diffferent NLED-based models, after the application of the combined study we are proposing here, we consistently show that none of these are acceptable models of our cosmos despite published work leading to contrary results. Hence, our approach serves as a consistent way to rule out cosmological models which, otherwise are investigated as serious candidates for either the inflaton or the dark energy.

These arguments have been incorporated in the introduction of our revised manuscript and in the “discussion and conclusion” section while the obtained results are being discussed. The title as well as the abstract of the manuscript have been changed to contribute towards a clearer statement of the novelty of our work.

- I suggest a major revision. 

Reply: In conclusion, we have made significant improvements to the manuscript based on the reviewers' insightful comments, which have helped us refine and enhance the overall quality of our work. These revisions have led to a more comprehensive and robust study, aligning the analysis with current scientific standards and ensuring that the manuscript presents a more complete and detailed exploration of the nonlinear electrodynamics models in cosmology. We believe that the changes made will greatly contribute to our findings' clarity, rigor, and impact, positioning the manuscript as a valuable addition to the existing literature.

Reviewer 3 Report

Comments and Suggestions for Authors

see attachment

Comments for author File: Comments.pdf

Comments on the Quality of English Language

see attachment

Author Response

line 60: The Lagrangian L is here wrongly defined, compared with line 58.

Reply: We want to make clear that in our manuscript we investigate well-known Lagrangians that have been previously investigated in the bibliography in different contexts, so we adhere to the form of Lagrangians as these were proposed in previous works. Our aim in this work is, precisely, to rule out these Lagrangians as a potential description of our universe.

Having said that, we would like to clarify that the different expressions for $L$ are intentional and are used to explore distinct electrodynamics models depending on the parameter $\sigma$ value. Each Lagrangian formulation is tailored to achieve specific properties in the nonlinear electrodynamics’ framework. Thus, they are not erroneously written but are rather designed to illustrate the variations in electrodynamics behavior under different conditions.

line 121: The authors should mention why a cosmological constant is omitted in L.

Reply: The primary goal of our study is to investigate whether nonlinear electrodynamics (NLED) can contribute to modeling the current accelerated expansion of the universe without relying on the cosmological constant. By excluding $\Lambda$, we aim to explore the potential of NLED as an alternative mechanism for driving this expansion. This approach allows us to assess the capability of NLED to account for the observed cosmological phenomena independently, thus providing a novel perspective on the dynamics of the universe without the inclusion of a dark energy component.

line 124: I don´t understand where the term G comes from.

Reply: The term $G$ you mentioned refers to the second electromagnetic invariant, often defined as $G = \mathbf{E} \cdot \mathbf{B}$, which represents the scalar product of the electric and magnetic fields. In our study, we have chosen not to include this term because our focus is on cosmological models where the electromagnetic Lagrangian depends solely on the first invariant, $F = \mathbf{B}^2$, which aligns with the specific nonlinear electrodynamics (NLED) models we are analyzing. The omission of $G$ simplifies the analysis and is consistent with the scenarios we aim to explore, where the magnetic field dominates. Thus, the term $G$ is not relevant in the context of our study.

lines 127–1140: The treatment of F is rather confusing. Why not just write the conventional 1/FF?

Reply: Our manuscript's treatment of the invariant $F$ is intended to maintain a general framework for nonlinear electrodynamics (NLED) models. While the conventional form $\frac{1}{F}$ is often used, our approach allows for a broader exploration of NLED Lagrangians, which may involve different dependencies on $F$. This flexibility is crucial for investigating various possible forms of electrodynamics beyond the linear regime. Additionally, the form we use aligns with the specific models and physical scenarios we are analyzing, where the behavior of $F$ under certain conditions is particularly interesting. We hope this clarifies our approach.

Eqs (5), (6): The notation of the commas is unclear. For example, F^{\mu\nu}, L …

Reply: The notation using commas and semicolons in some expressions has been intentionally chosen to avoid confusion with the subscripts that regularly appear throughout the text. Specifically, derivatives and covariant derivatives are denoted by commas and semicolons, respectively. This distinction ensures clarity and consistency in the mathematical notation used in the manuscript, helping to differentiate these operations from the indices.

The authors switch-off the electric field. Why not consider directly the Melvin-type universe?

Reply: We acknowledge the reviewer's suggestion regarding considering a Melvin-type universe. However, the primary focus of our study is on the cosmological implications of nonlinear electrodynamics (NLED) within a simplified framework where the electric field is switched off. This approach allows us to isolate and analyze the magnetic field's effects more clearly without the additional complexities introduced by an electric field. While a Melvin-type universe is indeed an interesting scenario, it involves a highly specific configuration with electric and magnetic fields, which is beyond the scope of our current investigation. Our objective is to explore the broader cosmological consequences of NLED under more general conditions, and we believe that this approach provides valuable insights into the model's dynamics and stability. Future work could extend this analysis to include scenarios like the Melvin-type universe.

lines 160-161: The simplification of the FLRW is, in my opinion, inappropriate in the case of nonlinearity.

Reply: Thank you for your insightful comment regarding simplifying the FLRW model in our study. We acknowledge that nonlinearity introduces complexities that may challenge the applicability of the standard FLRW assumptions like symmetries. However, we intended to explore the effects of nonlinear electrodynamics within a well-established cosmological framework, so we employed the FLRW model. By doing so, we aim to understand the first-order effects of these nonlinearities before considering more complex geometries (including those mentioned before about the Melvin-type universe). We believe this approach provides a useful baseline, allowing for a clearer comparison with the standard model of cosmology. Nevertheless, we are open to exploring alternative frameworks in future work to address the nonlinearities more comprehensively.

lines 276-283: The authors introduce the notions “causality” , “unitarity” and “ghosts”. These are used in quantum-effects in the Lagrangian. This goes far beyond the considered model.

Reply: Thank you for your observation. While it is true that terms like "causality," "unitarity," and "ghosts" are frequently discussed in the context of quantum field theory, these concepts also have relevance within the framework of classical general relativity, especially when analyzing the stability and physical viability of nonlinear electrodynamics models. In our work, we employ these terms to ensure that the classical theories we study do not exhibit unphysical behaviors, such as superluminal signal propagation or instabilities against small perturbations of the cosmic background, which would violate fundamental principles of general relativity. Therefore, their usage in our study is appropriate and necessary for a comprehensive analysis of the model under consideration.

Reviewer 4 Report

Comments and Suggestions for Authors

The authors investigate the dynamics of cosmological models with nonlinear electrodynamics. The stability and causality are studied. This is an interesting work but some points should be clarified.

1. In lines 57,58 the authors consider Born-Infeld Lagrangian but name it as Born-Infeld-type Lagrangian which was proposed in [22]. In lines 60,62 the author consider Born-Infeld-type Lagrangian but site it in [21],22].In [21] the authors considered another Lagrangian! 

2. In line 63 the authors mention Lagrangian with two parameters which generalizes Euler-Hesenber Lagrangian and it was considered also in Phys.Rev.D 75 (2007) 117301 but not sited by the authors. In line 86 the authors consider Born-Infeld Lagrangian but not Born-Infeld type Lagrangian.

3. In Ref. [56] the journal, volume and year are absent.

4. In eqs. (29) and (31) $b=1$ (?).

 

Comments on the Quality of English Language

 Minor editing of English language required.

Author Response

In lines 57,58 the authors consider Born-Infeld Lagrangian but name it as Born-Infeld-type Lagrangian which was proposed in [22]. In lines 60,62 the author consider Born-Infeld-type Lagrangian but site it in [21],22].In [21] the authors considered another Lagrangian! 
Reply: We thank the referee for this comment. The referee is right. The Lagrangian considered in [21] differs from the one we referenced in our manuscript. We apologize for the oversight. The Lagrangian in [21] is different. In the revised version of the manuscript, we made the necessary corrections to accurately reflect the distinct Lagrangian forms discussed in the respective references.

In line 63 the authors mention Lagrangian with two parameters which generalizes Euler-Hesenber Lagrangian and it was considered also in Phys.Rev.D 75 (2007) 117301 but not cited by the authors. In line 86 the authors consider Born-Infeld Lagrangian but not Born-Infeld type Lagrangian.
Reply:  In the revised version of our manuscript we have included the reference to Phys.Rev.D 75 (2007) 117301, which discusses the generalized Euler-Heisenberg Lagrangian in the context of nonlinear electrodynamics. However, we would like to clarify that in our manuscript, we have mentioned this Lagrangian in the context of FRW cosmological models. In contrast, the reference primarily deals with nonlinear electrodynamics phenomena. Additionally, we have removed the mention of Born-Infeld-type Lagrangians from line 86, as it was indeed a duplication. We appreciate the attention of the referee to these details and believe the corresponding changes will improve the clarity and accuracy of our manuscript.

In Ref. [56] the journal, volume and year are absent.
Reply: We have improved Ref. [56] by including the journal, volume, and year.

In eqs. (29) and (31) $b=1$ (?).
Reply: We have not specifically mentioned that parameter $b=1$. Interestingly, the squared sound speed does not depend on this parameter, simplifying our analysis. However, as the manuscript later discusses, parameter b does play a significant role in the overall dynamics of the cosmological model.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

With reference to the previous comment, I observed that a detailed description of the novelty is incorporated. Another suggestion was to check the stability of the model through the squared speed of sound. I observed that it is also incorporated.  In view of another suggestion, the authors have incorporated statefinder analysis as well. In my opinion, the paper is now in a better shape and can be considered acceptable in this revised form.