Invisible QCD as Dark Energy

Round 1

Reviewer 1 Report

The authors' answers to the review report are appreciated very much.
In the revised manuscript, the points suggested in the review report
have been reconsidered. Thus, this paper can be accepted for publication
in Universe.

Reviewer 2 Report

Once more the authors have addressed in detail all points raised previously. The manuscript is now ready to be published. Possibly being overly fastidious, I noticed the plot of w against the redshift is missing in this latest version. Furthermore, it seems that Refs. [23] and [24] are quoted in a reverse order. I would suggest the authors to contact the editor-in-charge in order to fix these issues. I would also ask the authors to read again the entire manuscript carefully in order to pinpoint additional typos in this new version.

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 paper is devoted to an interesting topic - the explanation of the dark energy phenomenon as a result of the existence of hidden sector of QCD. The idea looks original and the paper is thought-provoking.

I think that it makes sense to publish it.

Author Response

Thank you for your comments.

Reviewer 2 Report

In this manuscript the authors develop an alternative hypothetical form of dark energy. More precisely, they extend quantum chromodynamics to a SU(3) invisible sector (which they denote as IQCD). They argue that a spontaneously broken invisible chiral symmetry in the early universe is able to generate condensates of dark pions and dark gluons. A mechanism similar to the standard Nambu–Jona-Lasinio model is taken into account to describe this chiral symmetry breaking. It is shown that, in the subsequent evolution of the universe, the interaction energy between the dark gluon and dark pion condensate dominates with a negative pressure equation of state and this could explain the accelerating expansion of the universe which is seen in present times.

This is a enjoyable-reading manuscript. This subject is rather interesting, as well as the findings presented in the work. In any case, I would like to draw the authors’ attention to some comments listed below. 

(1) What is the relationship between the energy density obtained by the authors (or at least the contribution rho_{AJ}) and the critical energy density? Is it consistent with current data?

(2) Can the authors discuss with some detail the coincidence problem within the proposed setting?

(3) An explicit estimate on the order of magnitude of the effective cosmological constant obtained by the authors should be given.

(4) How would the presence of a coupling with Standard Model particles change the results?

(5) In Eq. (2) (which is a Lagrangian density, not an action), it seems that the dark quarks are not coupled with the dark gluons since, as asserted by the authors, the covariant derivative is taken with respect to the gravitational connection. If this is true, then the Lagrangian density is not gauge invariant. This should be clarified.

(6) One could expect that, after integrating out the quark fields, there should appear non-local contributions to the effective action. The introduction of curvature-squared terms is also necessary in order to renormalize the theory. Some refinement is needed concerning this matter.

(7) Could the authors discuss briefly on the possible violation of the strong energy condition? What about the null energy condition?

(8) It seems that the value of the coupling constant g is important in defining the transition from a dark-energy era to radiation era. Could the authors briefly comment on this?

(9) Does the Adler-Bell-Jackiw anomaly play any role?

(10) Is Eq. (5) correct as it stands?

(11) Is Eq. (19) correct as it stands?

(12) Could the authors provide more details on the matrix which is the result of linearization of the 1st order dynamical system? The same goes for its eigenvalues.

(13) Could the authors merge Figs. 1 and 2 into a single figure? It seems more appropriate. Other values for the coupling constant g could also be considered in the single figure.

(14) More details on how to obtain Eqs. (17) and (18) should be given. In addition, more comments on the choice for initial conditions at recombination would be most welcome.

(15) If possible, the coupling constant that parametrizes the interaction of the axial current with the gauge field should not be represented by the same letter as the determinant of the metric. It might engender some confusion.

(16) References should be updated if possible.

Author Response

Please find point-to-point response in the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

In this paper, the late time acceleration of the Universe is explored by extending the QCD color to a SU(3) invisible sector (IQCD). When the Invisible Chiral symmetry is broken in the early universe, a condensate of dark pions (dpions) and dark gluons (dgluons) forms. It is stated that the condensate naturally forms due to strong dynamics similar to the Nambu-Jona-Lasinio mechanism. The universe evolves from early times to present times the interaction energy between the dgluon and dpion condensate dominates with a negative pressure equation of state and causes late time acceleration. The discussions are interesting and the mathematical results could be helpful for the related future works. Moreover, the descriptions and mathematical analyses are given in detail. Thus, this paper can be worthy of being published. Before publication, it is recommended that the following points are reconsidered.

1. There might exist the past related works on the argument to relate QCD and dark energy in the literature. By comparing with these preceding studies, the new ingredients and significant progresses of this work should be stated more explicitly and in more detail. That is, the differences between this paper and the past ones should be described in more detail and more clearly.

2. Very naively, through the cosmological QCD Phase Transition, ordinary pions and gluons behaves like matter, but a condensate of dark pions (dpions) and dark gluons (dgluons) become a kind of dark energy component with the negative pressure. What is the physical mechanism to lead such a difference between them? Their equations of state are totally different with each other.

Round 2

Reviewer 2 Report

The authors have addressed in detail all points raised previously. Moreover, the manuscript was slightly enlarged with more elaborated discussions. This is an unquestionably improved version of the work in comparison with the previous one. I am satisfied with the science and the presentation. Therefore I recommend the present manuscript for publication. However, I would like to mention some final comments.

(1) There seems to be a latex problem with the citation of the footnotes. There are also some typos throughout the manuscript. All of these issues should be fixed.

(2) References should be arranged in numerical order throughout the manuscript for the sake of the reader-- or the list of sources cited in the text (given at the end of the manuscript) should be arranged in the order they appeared within the text. Furthermore, there seems to be references that were not quoted in the text, such as Refs. [13], [14] and [17].

(3) Is it correct the location of the imaginary unit in the Dirac sector of the Lagrangian density given in Eq. (2)?

(4) Is the sign in front of the second term on the left-hand side of Eq. (23) correct?

Author Response

Again, we thank the Referee for the attention in reading our paper, and for the feedback provided. Going through the points of the second report:

(1) There seems to be a latex problem with the citation of the footnotes. There are also some typos throughout the manuscript. All of these issues should be fixed.

We solved the issues related the footnotes appearing as citations, and fixed the typos we spotted in the manuscript. When required, further changes in the editing style will be implemented during preparation of the manuscript for publication. We also provided to remove the boldface from the previous version of the draft.

(2) References should be arranged in numerical order throughout the manuscript for the sake of the reader-- or the list of sources cited in the text (given at the end of the manuscript) should be arranged in the order they appeared within the text. Furthermore, there seems to be references that were not quoted in the text, such as Refs. [13], [14] and [17].

We arranged references consistently throughout the manuscript, and listed by the order they appear. We removed references not appearing in the text.

(3) Is it correct the location of the imaginary unit in the Dirac sector of the Lagrangian density given in Eq. (2)?

We acknowledge the Referee for spotting this editing typo. We emended the manuscript, and placed the imaginary units to multiply the covariant derivative.

(4) Is the sign in front of the second term on the left-hand side of Eq. (23) correct?

We thank the Referee for this careful check, and spotting this incorrect minus sign. We corrected the sign in the resubmitted manuscript, and checked that this was actually an editing typo, not affecting the analysis reported.