Boltzmann Equation and Its Cosmological Applications

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Dear authors,

The authors derive the (full and integrated) Boltzmann equation, treating the collision term in the light of quantum field theory. There is no assumption in finite temperature of the system and the Boltzmann equation can be applied for non-equilibrium states. The results are applied in the cosmological setting for dark matter abundance and baryogenesis.  

The results of Eqs. (73) and (75) and Fig. 1 are quite clear for the freeze-out and and final abundance in the dark matter context, as well as its Boltzmann equation.
   
For the baryogenesis scenario, the result of Eq. (107) shows that in the weakly interacting case all particle can be converted without wash-out, whereas in the strongly interacting case it is the opposite. This is consistent. 

The subject of the paper is presented clearly and in a self-consistent way, as well as the results. For this reason, I have nothing relevant to add or criticize in the discussion. 

I would like you to considerd some minor details: 

- Line 30: Sentence ends with no sense. Subtle changes can solve it.

- Line 69 and in Eq.(6) and (7): The parameter "E" is not defined.

- Section 2.2.1: This section containts the relevant information, but it can be more organized. You begin defining multiparticle states and the commutation relations for bosons without saying that these are for bosons. However, in line 89 you explicitly say define anti-commutation relations for fermions. At the and, you properly summarize what happens for bosons/fermion. The writing of this section can be subtly reconsidered.

- Lines 142, 143, 372: In the begining of a paragraph, it is more appropriate to use "Equation" instead of "Eq.". 

The paper is well written, relevant and useful for future applications. Please correct these minor details. I recommend the paper for publication in Symmetry Journal. 

Sincerely yours,

 

Author Response

Dear Reviewer,

We sincerely thank you for your positive and encouraging comments on our manuscript. We are pleased that you found our presentation to be clear, self-consistent, and relevant for future applications.

We have carefully addressed all the minor points you raised:

Comments 1: "Line 30: Sentence ends with no sense. Subtle changes can solve it."
Response 1: The sentence "The evolution equation is applied to the various analysis in cosmological or astronomical situations, e.g., the last scattering surface in the recombination epoch, the Sunyaev-Zel'dovich effect [13-16] that is a distortion of the cosmic microwave background radiation by hot electrons in galaxies, etc." has been modified as follows:
"The evolution equation is applied to various analyses in cosmological and astronomical contexts, such as the last scattering surface during the recombination epoch or the Sunyaev-Zel'dovich effect [13-16], which is the spectral distortion of the cosmic microwave background radiation caused by hot electrons in galaxy clusters."

Comments 2: "Line 69 and in Eq.(6) and (7): The parameter "E" is not defined."
Response 2: We added the definition of E above Eq. (6).

Comments 3: "Section 2.2.1: This section containts the relevant information, but it can be more organized. You begin defining multiparticle states and the commutation relations for bosons without saying that these are for bosons. However, in line 89 you explicitly say define anti-commutation relations for fermions. At the and, you properly summarize what happens for bosons/fermion. The writing of this section can be subtly reconsidered."
Response 3: We modified the sentence and the equation surrounding Eq. (10) to ensure consistency of this section.

Comments 4: "Lines 142, 143, 372: In the begining of a paragraph, it is more appropriate to use "Equation" instead of "Eq."."
Response 4: We replaced "Eq." with "Equation" at the beginning of these paragraphs to follow formal writing conventions.

Thank you again for your helpful suggestions and for recommending our paper for publication.

Sincerely,

Seishi Enomoto
On behalf of the authors

 

Reviewer 2 Report

Comments and Suggestions for Authors Report symmetry-357632 The authors present an impressively broad review of the status and achievements in applications
of the Boltzmann equation to the thermodynamics of the early universe. The article provides
a profound introduction in modern thermodynamics and it‘s successful application to cosmological
studies. Wide space is given on the integration of quantum field theoretical phenomena into the
intrinsically classical Boltzmann theory through appropriately formulated collision terms are
discussed in detail. Extensions towards a fully quanta approach via Baym-Kadanoff theory and the
Schwinger-Keldish formalism are indicated. The article ist more than a pure review: It provides a deep and well structured introduction
into modern Boltzmann theory and may even serve as as lecture notes for advanced researchers
at any level of experience as a suitable „entry point“ into cosmological thermodynamics. The
articles is recommended for publication. The authors, however, may scan the text and corrects
few typos and minor grammatical flaws.

 

Author Response

Dear Reviewer,

We appreciate your generous and supportive evaluation of our manuscript. We are delighted to hear that you found the review broad, structured, and pedagogically valuable, especially for researchers at various levels.

As per your suggestion, we carefully read through the manuscript and corrected minor typos and grammatical issues throughout the text. We hope this revised version has improved the overall readability and presentation.

We appreciate your recommendation for publication.

Sincerely,

Seishi Enomoto
On behalf of the authors

Reviewer 3 Report

Comments and Suggestions for Authors

Report 1            Dated May 12th, 2025

Authors: Seishi Enomoto, Yu-Hang Su, Man-Zhu Zheng  and Hong-Hao Zhang

Title: Boltzmann equation and its cosmological applications

Ref: symmetry-3576321-peer-review-v1

Journal:  Symmetry 

In this study, Seishi Enomoto and collaborators presented a review on the derivation of the Boltzmann equation and its cosmological applications in light of the collision term employing the quantum field theory without any assumption of the finite temperature system. While the integrated Boltzmann equation is dealt with the temperature parameter. Regarding cosmological applications of the equation, the dynamics of the dark matter abundance through the freeze-out/in process and a baryogenesis scenario are discussed. Although more time is required for the in-depth analysis of the results, to speed up the review process, I encourage authors to clarify the following points.

1. The Abstract overlaps with the source #2 as given in the similarity report attached below. Please rewrite it in your own words to enhance the originality or justify otherwise.
2. For better understanding of the novelty, objective, and advancement of the work, please clearly highlight the key factors that justify the need and advantages of this Review by emphasizing the limitations of the earlier Review works on the topic. This should be mentioned
3. As the authors stated, the nature of article is Review. Did the author first time derive the Boltzmann equation or just redrived the already established equation in literature in a different context? Please clarify and provide related references if it is redrived from the earlier sources. Otherwise clearly mention the new insights and prospects regarding the derivation of the equation with reference to the earlier studies.
4. Elaborating the above point further, is Section 2 up to Eq.18 original contribution of the authors or just presentation of the earlier literature? Please mention clearly the advancement and novelty each section before moving ahead for better understanding the same. For any original contribution beyond the critical review of the established literature for novel prospects and implications, it is essential to clearly elaborate which part of the present study is the original contribution regarding extension of the earlier work and advancement. This objective should be stated before the beginning of each Section.
5. Apparently, it seems Eq. 1 may hold true if C = L provided [f] on both sides is the same? Please elaborate more on this under the suitable conditions in light of Eqs.1-7.
6. Is Liouville operator defined on Minkowski space or Minkowski space-time? Please clarify in context of Eqs.5-6.
7. What is the mutual relationship between L and C operator apart from Eq.1, e.g. as to whether they commute or anticommute? Justify and discuss the implications in each case.
8. (Pg 7 Line 131) Please clear: ∆Nϕ is a changing number of the quantum ϕ in the process Φ,…..

9. The authors describe the role of C and L operators regarding quantum field theory and gravity; as such please discuss the implications of the derived Boltzmann equation regarding the quantization of gravity and its unification with other forces.
10. (Pg 21 Line 422): Please rephrase as follows “We hope this paper will be helpful to use the Boltzmann equation and its techniques thoughtfully.
11. Could the authors shed more light on the implications of the Boltzmann equation in simplifying (or otherwise) the calculations of the scattering/decay amplitude by common quantum field theoretical approaches such as s-matrix using Feynman diagram. This could be discussed after the Appendix C.
12. Elaborate on the Lorentz invariance of the computed properties like cross section etc.
13. Most of the references are decades-old. For better understanding on the latest development in the field and general and topic in particular, I encourage the authors to add the latest references covering 2022-2025.

 

 

 

 

Author Response

Dear Reviewer,

We appreciate your thorough reading and insightful comments on our manuscript. Below, we provide point-by-point responses and describe the corresponding revisions made.

-----------------
Comments 1: "The Abstract overlaps with the source #2 as given in the similarity report attached below. Please rewrite it in your own words to enhance the originality or justify otherwise."

Response 1: Although you referred to "source #2" as the similarity report, we could not identify it because there is no attachment in Reivewer Report we accessed. Without access to the referenced material, it is difficult to assess any specific overlap. Nevertheless, we affirm that the Abstract was written entirely in our own words and reflects the structure and content of the paper as originally prepared. The overlap, if any, may result from the use of standard terminology or conventional phrasing common in the literature on the Boltzmann equation. We hope this clarification addresses your concern.

-----------------
Comments 2: "For better understanding of the novelty, objective, and advancement of the work, please clearly highlight the key factors that justify the need and advantages of this Review by emphasizing the limitations of the earlier Review works on the topic. This should be mentioned."
Comments 3: "As the authors stated, the nature of article is Review. Did the author first time derive the Boltzmann equation or just redrived the already established equation in literature in a different context? Please clarify and provide related references if it is redrived from the earlier sources. Otherwise clearly mention the new insights and prospects regarding the derivation of the equation with reference to the earlier studies."
Comments 4: "Elaborating the above point further, is Section 2 up to Eq.18 original contribution of the authors or just presentation of the earlier literature? Please mention clearly the advancement and novelty each section before moving ahead for better understanding the same. For any original contribution beyond the critical review of the established literature for novel prospects and implications, it is essential to clearly elaborate which part of the present study is the original contribution regarding extension of the earlier work and advancement. This objective should be stated before the beginning of each Section."

Response 2, 3, and 4: We clarified that while the general form of the Boltzmann equation is known, our derivation revisits it in the QFT framework without assuming thermal equilibrium, aiming to unify its application across different regimes. We highlighted our novel aspects (e.g., formulation only assumed by the well-defined S-matrix, without any finite temperature treatment). The following sentences in the original Line 45 were replaced:

"Although there is .... For readers who want to apply the Kadanoff-Baym equation, ...."

--> Line 54 "This review highlights a novel derivation of the Boltzmann equation directly and accessibly. Especially, the derivation of the collision term is carefully discussed in section 2.2, starting from quantum field theory under the assumption that the S-matrix is well defined. We emphasize that, once the transition amplitude is known, the Boltzmann equation is derived naturally without any finite temperature system. This approach makes the physical assumptions and approximations transparent, and thus may serve as a practical entry point for cosmologists and particle physicists interested in applying the Boltzmann framework to out-of-equilibrium phenomena in the early universe. In contrast, the standard derivation of the Boltzmann equation, based on the so-called Kadanoff-Baym equation \cite{Baym:1961zz,Baym:1962sx}, can describe the system more accurately. We do not deal with it in detail to evade the complicated discussion in this paper. For readers who want to apply the Kadanoff-Baym equation, ...."

We also added a sentence at the beginning of section 2.2 concerning Comments 4:
"This section highlights our novel points of the derivation of the collision term based on quantum field theory."

-----------------
Comments 5: "Apparently, it seems Eq. 1 may hold true if C = L provided [f] on both sides is the same? Please elaborate more on this under the suitable conditions in light of Eqs.1-7."

Response 5: The equation L[f]=C[f] shold be interpreted as a dynamical evolution equation for the one-particle distribution function f(x ,p), in which the left-hand side L[f] describes the change of f along the classical trajectories in phase space (i.e., the Liouville operator), and the right-hand side C[f] represents the change due to microscopic interactions (i.e., the collision operator).sions and external expansion (as in the FLRW metric). To clarify this point, we have slightly modified the sentence surrounding Eq. (1) in the revised manuscript.

-----------------
Comments 6: "Is Liouville operator defined on Minkowski space or Minkowski space-time? Please clarify in context of Eqs.5-6."

Response 6: The Liouville operator discussed in this paper is defined on the FLRW space-time. Although Eq.(5) can be applied to an arbitrary geometry, Eq.(6) is a result of the FLRW metric. We modified the sentence below Eq.(5) to emphasize the application of the FLRW metric, as describing the evolution of the universe. 

-----------------
Comments 7: "What is the mutual relationship between L and C operator apart from Eq.1, e.g. as to whether they commute or anticommute? Justify and discuss the implications in each case."

Response 7: In the conventional formulation of the Boltzmann equation, the operators L and C represent fundamentally different physical processes. The Liouville operator L is a geometric differential operator that governs the free-streaming propagation of particles in phase space under the background space-time geometry. In contrast, the collision operator C encodes the non-geometric, statistical interaction effects among particles, often derived from scattering amplitudes in quantum field theory. Because of their distinct origins, there is no natural or meaningful notion of a commutator or anticommutator between L and C.

-----------------
Comments 8: "(Pg 7 Line 131) Please clear: $\Delta N_\phi$ is a changing number of the quantum $\phi$ in the process $\phi,…$."

Response 8: We modified the sentence to "$\Delta N_\phi$ is the change in the number of the quantum $\phi$ in the process $\phi,...$."

-----------------
Comments 9: "The authors describe the role of C and L operators regarding quantum field theory and gravity; as such please discuss the implications of the derived Boltzmann equation regarding the quantization of gravity and its unification with other forces."

Response 9: In Section 2 (particularly at the beginning of Subsections 2.1 and 2.2), we outlined the conceptual distinction between the Liouville operator L, which encodes the external background (i.e., classical spacetime structure), and the collision operator C, which represents quantum mechanical interactions at the particle level. In this sense, the formalism with quantum gravity must be L=0 and only described by the collision term.

To further address your comment, we have added a brief discussion as follows before Section 2.1.

"We note that the structural separation between the Liouville operator and the collision operator reflects a semiclassical perspective: the former encodes classical (gravitational) geometry as the long-distance force, while the latter contains the quantum process as the short-distance force. If one wants to follow the dynamics under the classical electro-magnetic field, that must be included in the Liouville operator such as the Lorentz force\footnote{In many cosmological contexts, however, the electromagnetic force effectively acts as the short-distance force because of the shielding by the other charged plasma.}. If one wants to consider quantum gravity, the collision terms associated with the graviton exchanges appear, although quantum gravity is beyond the scope of this review."

-----------------
Comments 10: "(Pg 21 Line 422): Please rephrase as follows“We hope this paper will be helpful to use the Boltzmann equation and its techniques thoughtfully.

"Response 10: This sentence has been rephrased as suggested.

-----------------
Comments 11: "Could the authors shed more light on the implications of the Boltzmann equation in simplifying (or otherwise) the calculations of the scattering/decay amplitude by common quantum field theoretical approaches such as s-matrix using Feynman diagram. This could be discussed after the Appendix C."

Response 11: We understand your point as suggesting that it would be helpful to clarify how the Boltzmann equation incorporates the results of quantum field theory, particularly how the transition amplitudes derived from the S-matrix enter into the collision term C[f]. This is indeed a central aspect of the formalism and is implicitly covered in Section 3 and Appendix C.

The Boltzmann equation does not simplify the computation of scattering/decay amplitudes, but rather provides a structured way to apply those amplitudes to the statistical evolution of distribution functions.  The formula of the collision term is similar to the formula of the reaction rate, which is integrated over the squared amplitude |\mathcal{M}|^2 in momentum space. But the actual result is "weighted" by the multiplied distribution function for each species.  In this sense, the Boltzmann framework acts as a bridge between microscopic quantum amplitudes and macroscopic dynamics in phase space.

To address your comment more explicitly, we have added a paragraph below Eq.(36), not in Appendix C, to explain clearly.

"Equation (36) is the most general result of the collision term. The amplitudes $\mathcal{M}$ for each process are computed using standard Feynman rules in perturbative quantum field theory, typically via the S-matrix formalism. Note that the Boltzmann equation itself does not modify or simplify these calculations, but serves as the framework in which these amplitudes are statistically weighted by the distribution function and integrated over phase space. In this way, it provides a bridge between microscopic quantum processes and macroscopic kinetic evolution."

-----------------
Comments 12: "Elaborate on the Lorentz invariance of the computed properties like cross section etc."

Response 12: We mentioned the Lorentz invariance in several physical quantities; cross section (above Eq.(67) and below (A.18)), Moller velocity (below Eq.(66)), partial width (below Eq.(A.12)). We also added a footnote below Eq.(A.12) as follows:

"The Lorentz invariance of the partial width is obvious because the following elements are Lorentz-invariant: $d^3k/E$, $\delta^4(k+\cdots)$, $\mathcal{M}$. This argument is also applicable to show the Lorentz invariance of the cross section."

-----------------
Comments 13:"Most of the references are decades-old. For better understanding on the latest development in the field and general and topic in particular, I encourage the authors to add the latest references covering 2022-2025."

Response 13: We have updated our bibliography to include more recent works and added the following paragraph at Line 46.

"Recent advances have broadened the use of the Boltzmann equation in cosmology. Precision calculations of relic neutrino decoupling have refined predictions [38], underscoring its continued role in linking particle interactions to cosmological observables. The Boltzmann framework also underpins studies of chiral anomalies, magnetic fields, and baryon asymmetry [39], as well as freeze-in production with time-dependent condensates [40], reflecting its expanding relevance in modern cosmology."

-----------------

We thank you again for your critical and constructive comments, which helped us significantly improve the clarity and rigor of our manuscript.

Sincerely,
Seishi Enomoto
On behalf of the authors

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

The authors have addressed the concerns of the referee raised in the earlier report, and the quality of the manuscript has improved considerably to be accepted for publication in the journal Symmetry. Regarding the first comment on the similarity, the report is attached below. Please modify the Abstract accordingly if required before the final submission.

Comments for author File: Comments.pdf

Author Response

Dear Reviewer,

Thank you for your positive comment.  We have addressed the point you raised as follows:

 

Comment: "Regarding the first comment on the similarity, the report is attached below. Please modify the Abstract accordingly if required before the final submission."

 

Response: Based on the attached document and the highlighted sections, the overlapping content appears primarily from standard phrases and terminology widely used in the literature on the Boltzmann equation and its cosmological applications.

Moreover, we noticed that the reported overlaps with "arxiv.org" (4%) and "export.arxiv.org" (4%) are very likely due to our own arXiv preprint version of this manuscript (https://arxiv.org/abs/2301.11819 and https://export.arxiv.org/abs/2301.11819). If this is the case, the match constitutes a self-overlap, not a duplication from external sources. The Abstract in the journal submission and the arXiv version reflect the same manuscript except for the modifications due to the Reviewers' Comments. Thus, such overlap is both expected and legitimate.

Given that the similarity arises from standard expressions and/or our own preprint, we do not consider any modification to the Abstract necessary, and accordingly we do not update our manuscript.

 

We hope this clarifies the issue and alleviates any remaining concerns. We thank you again for your careful reading and constructive feedback throughout the review process.

Sincerely yours,
Seishi Enomoto
On Behalf of the authors