Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Report on Ref.:  Ms. No. Symmetry 3936679

Title: Source Energy Field Theory: A Novel Framework for Cosmic Structure Formation

 

This study discusses the application of Source Energy Field Theory (SEFT) framework to discuss the cosmic structure formation. The authors provided the framework of nonlinear wave equations and provided a structure based on these equations. They also performed some calculations and obtained specific post optimized parameters via two specific Software. The authors calculated different statistical parameters to validate their results in comparing with Lambda CDM model.

Though it is a small mathematical exercise, yet there are some recommendations which can make this exercise better:

The plagiarism and AI content (64% according to turnitin) of the manuscript should be reduced.

Important details about the foundation of the SEFT model are currently missing from the Introduction section. Whether the model was created from scratch or modified from pre-existing literature must be made clear by the authors. If modified, appropriate citations for its background, mathematical structure, and basic ideas must be provided. A stronger defense is needed if the model is new and its validity is supported by a single example and comparison with the Lambda CDM model alone.

The authors did not mentioned respective parameters in different equations, i.e., mu, lambda, and gamma in Eq. 1.

The physics and geometry of energy momentum tensor in Eq.11. should be explained properly. Which type of energy momentum tensor author want to discuss, dust, fluid, radiation or something else?

Are they discussing waves in vacuum via nonlinear wave equation?

It should be explained how the coefficient of determination ($R^2$) is calculated. The authors should explain why they only focus on $R^2$ as a validation metric and argue that it is sufficient in comparison to a more thorough statistical approach. Either a nonlinear least-squares analysis should be conducted and included, or the reasons why it was not done should be addressed.

Figure 1 should have proper and detailed explainations.

The authors should specify some other traits to compare and validate the results of their SEFT model.

The significant error related to the developed SEFT model must be explicitly stated and discussed in the manuscript.

If the authors can accommodate all these points mentioned above, the work will probably be accepted.

Author Response

We have attached our detailed responses to the reviewers' comments and a revised manuscript with all changes highlighted. Please see the attached files.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

Dear authors,

I read with great interest your work titled "Source Energy Field Theory: A Novel Framework for Cosmic Structure Formation." In this paper, you introduce a theoretical proposal called Source Energy Field Theory (SEFT), which aims to reinterpret phenomena related to cosmic structure and cosmological redshift. Additionally, the authors derive a nonlinear scalar field equation from a proposed Lagrangian and connect its properties to resonance phenomena. They also interpret dark matter and dark energy as non-resonant manifestations of this universal field. Nevertheless, I have some comments and questions, implying revisions to your work:

(i) Firstly, I recommend that the abstract not consider acronyms (SEFT). I suggest adding all the phrases and then explaining them throughout the document starting from the introduction.

(ii) Where are the references in the first paragraph (lines 29-36)?; for example, it is important to consider the reference to special relativity! For example, from Inspire:

\bibitem{Einstein:1905ve} A.~Einstein, %``On the electrodynamics of moving bodies,'' Annalen Phys. \textbf{17} (1905), 891-921 doi:10.1002/andp.200590006 

The same issue for the third paragraph (lines 41-47).

(iii) At the end of the introduction, it would be well to add a paragraph showing us the structure of your manuscript.

(iv) The eq. (1) proposed here: Is it Lorentz invariant? Does it arise from a well-defined, covariant Lagrangian density ? The terms have unclear dimensional consistency and no connection to standard field-theoretical formulations. Explain more about these points, please.

(v) Unfortunately, the theory lacks any coupling to spacetime geometry or Einstein’s equations. From the above, which is the interpretation within general relativity or cosmological dynamics? Please explain about this.

(vi) Are the definitions of dark matter and dark energy supported by physical or observational arguments in terms of resonance or non-resonance with electromagnetic or gravitational waves?

(vii) Although the SEFT model produces slightly better statistical indicators, unfortunately, the numerical improvement is marginal and likely due to overfitting, due to the model using more free parameters than ΛCDM. How do you justify this point?

(viii) On pages 5-7. What is the physical meaning of the parameters a1-d3 ? 

I would be willing to review a revised manuscript if the author resolves these issues.

Author Response

We have attached our detailed responses to the reviewers' comments and a revised manuscript with all changes highlighted. Please see the attached files.

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

Dear Editor,

I have carefully reviewed the manuscript entitled "Source Energy Field Theory: A Novel Framework for Cosmic Structure Formation" by Sakihara and Nabeshima. The paper presents an alternative cosmological framework attempting to explain redshift through energy field modulations rather than cosmic expansion. While the authors demonstrate statistical improvements over $\Lambda$CDM using Pantheon+ and SH0ES data, several technical concerns require clarification before publication can be recommended.

The fundamental equation of SEFT is given as:
\begin{equation}
\Box\Psi + \mu^2\Psi + \lambda|\Psi|^2\Psi - \gamma\nabla \cdot (|\Psi|^2\nabla\Psi) + \gamma|\nabla\Psi|^2\Psi = 0
\end{equation}
However, the transition from this general nonlinear wave equation to the redshift-distance relation $z(d) = \mu d^2 + \lambda d^4 + \gamma d^8$ (Eq. 13) lacks mathematical rigor. How do the field parameters $\mu$, $\lambda$, and $\gamma$ in Eq. 1 directly map to the distance-dependent coefficients in Eq. 13? The dimensional analysis appears problematic, as $\mu^2$ has units of inverse length squared while the coefficient in Eq. 13 requires different dimensions.

The dispersion relation derived in Eq. 10:
\begin{equation}
\omega^2 = c^2(\mu^2 + \lambda|A|^2 + \gamma|\nabla A|^2)
\end{equation}
assumes spatial averaging that eliminates the divergence term. What boundary conditions justify this assumption for cosmological scales? Additionally, the claim that mass emerges from solitonic solutions needs explicit demonstration. Can the authors provide specific soliton solutions to Eq. 1 and calculate their effective masses?

The final distance formula (Eq. 15) incorporates terms $\cos(\delta)$ and $\cos(\alpha)$ for directional dependence, yet cosmological observations typically show isotropy at large scales. What physical mechanism in SEFT generates this anisotropy, and how does it reconcile with CMB isotropy constraints? The authors acknowledge this formula is phenomenological rather than an exact solution, which weakens the theoretical foundation significantly.

Regarding the statistical analysis, while $\Delta$AIC $= -41.753$ and $\Delta$BIC $= -19.997$ favor SEFT, the model employs six free parameters compared to two in $\Lambda$CDM. The marginal improvement in RMSE (145.521 vs 147.665 Mpc) represents approximately 1.5\%, which may not justify the additional complexity. Have the authors tested their model against independent datasets such as BAO measurements or CMB power spectra?

The Lagrangian density (Eq. 3) includes the term $\frac{\gamma}{2}|\Psi|^2|\nabla\Psi|^2$. Is this formulation gauge-invariant? What symmetries does this Lagrangian possess, and are there corresponding conserved currents via Noether's theorem?

The authors should consider enriching their theoretical framework by consulting Phys.Rev.Lett. 81 (1998) 2008-2011; e-Print: astro-ph/9810409 [astro-ph]; Eur.Phys.J.C 85 (2025) 5, 554; Physica D 77 (1994) 354; Mod.Phys.Lett.A 37 (2022) 14, 2250085; ASP Conf.Ser. 148 (1998) 21.

Despite these concerns, the paper presents an interesting alternative perspective. With substantial revisions addressing the mathematical rigor, physical justification for the distance formula, and broader observational tests, this work could make a valuable contribution. I recommend major revisions before reconsideration.

Comments for author File: Comments.pdf

Author Response

We have attached our detailed responses to the reviewers' comments and a revised manuscript with all changes highlighted. Please see the attached files.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I recommend that this work is worthy of being accepted for publication in the journal Symmetry. 

Author Response

We sincerely thank the reviewer for the positive evaluation and recommendation for acceptance of our manuscript.
We are very grateful for the reviewer’s constructive comments, which have helped us further improve the clarity and consistency of the paper.

Following the reviewer’s suggestions, we carefully reviewed the manuscript to refine the descriptions of the methods and conclusions, and to ensure that all figures, tables, and equations are clearly presented and consistently formatted.

We deeply appreciate the reviewer’s recognition that this work is worthy of publication in Symmetry, and we are encouraged by this supportive assessment.

Reviewer 2 Report

Comments and Suggestions for Authors

Dear authors, thank you for taking the time to address the comments and suggestions. The second version of your manuscript demonstrates an improvement over the first. However, I believe that SEFT still has weaknesses regarding its compatibility with general relativity and field theory. Specifically, the physical interpretation of the field $Psi$ and the mathematical connection to spacetime curvature remain insufficiently rigorous in this new version of your work. Concretely:

(i) The paper could be accepted (after minor corrections) if you demonstrate at least a schematic tensorial derivation showing how the SEFT field couples to geometry and how the stress-energy tensor arises from the Lagrangian density.

(ii) Together with the above, there are some inconsistencies of redaction present in this new version, and the equations must be highly improved.

 

Author Response

We sincerely thank the reviewer for this valuable comment.
To clarify the theoretical linkage between the SEFT field and spacetime geometry, we have added a schematic formulation in Section 2.3 (Origin of Space).
Specifically, we now describe how the stress–energy tensor TμνT_{\mu\nu}Tμν​ arises from the SEFT Lagrangian density through variation with respect to the metric tensor, and how, in the macroscopic limit, this formulation corresponds to the Einstein field equation Gμν=8πG⟨Tμν⟩ΨG_{\mu\nu} = 8\pi G \langle T_{\mu\nu} \rangle_{\Psi}Gμν​=8πG⟨Tμν​⟩Ψ​.
This addition explicitly demonstrates how spacetime curvature emerges from the intrinsic modulation of the source energy field.

Furthermore, following the reviewer’s suggestion, we have carefully revised all equations to improve clarity and consistency, addressing minor inconsistencies in notation and formatting.
We also unified the equation numbering and reference style throughout the manuscript to ensure full coherence between the text, figures, and appendices.

Reviewer 3 Report

Comments and Suggestions for Authors

    Dear Editor,
I have reviewed the revised manuscript. he revisions have adequately addressed the concerns raised in the previous review round.
I recommend acceptance of this manuscript for publication in Symmetry.

Author Response

We sincerely thank the reviewer for the positive evaluation and recommendation for acceptance of our manuscript.
We are deeply grateful for the reviewer’s time and effort in assessing our work throughout the review process.
We are pleased that the revisions have satisfactorily addressed the previous concerns and that the current version meets the standards for publication in Symmetry.

Once again, we sincerely appreciate the reviewer’s supportive feedback and encouraging comments.