Mass of Cosmological Perturbations in the Hybrid and Dressed Metric Formalisms of Loop Quantum Cosmology for the Starobinsky and Exponential Potentials

Round 1

Reviewer 1 Report

In this paper, the authors discuss some important differences between the scalar and tensor masses at the bounce from the hybrid and the dressed metric approach for the Starobinsky and the exponential potentials. I think comparing the effective masses from two major approaches  to perturbations in LQC serves as a good starting point to differentiate these two approaches from the observational signals in the future. The manuscript will benefit from the following revisions.

 

1.  This is a minor suggestion. As one can see from Eq. (4). the dimension of the scalar mass (similar to the tensor mass) is actually mass squared. As a result, in some  previous papers, for example arXiv: 1809.03172, it is called the effective mass function  (see 2.20 and 2.21 in their paper). I am not sure why the authors refer to this term as "mass" when it does not have its dimensions.

 

2. The importance of the positivity of the effective mass at the bounce is carefully addressed in the second paragraph of the conclusions. For benefit to the readers, I would recommend to move this whole paragraph to the introduction so that readers will understand this point from the very beginning.  I think a good place to add this content is in  the last paragraph on page two which starts with "Since relevant quantum geometry effects occur only in a narrow period.....". But if the authors think it is better to leave the relevant discussion in the conclusions, I don't have any objections. 

 

3. In Eq. (7), I don't see any particular reason to add a tilde on the last term $\tilde V_{\phi}$. Although it is the same as $V_{\phi}$, there is no tilde on $V_{\phi \phi}$, so I suggest to remove this tilde from  $\tilde V_{\phi}$. 

 

4. The authors explain the origins of the difference between tensor masses  (given in Eqs. (5) and (6)) in the second last paragraph on page 2 of Sec. 1.  The authors should expand more discussion on what precisely caused the difference between the potential terms given in Eq. (7) and (8) which should originate from the same classical expression. In a recent paper,  arXiv:2206.12434, those authors explain where these differences come from in hybrid and dressed approaches, essentially related to the choice of phase space variable used to write the Hamiltonian of perturbations and the polymerization of the momentum of the scale factor. It will be useful for the readers to know how your analysis fits in light  of these arguments. Whether or not authors agree with above argument, it will be useful to discuss this in brief detail.

 

5. To be honest, I am not fully convinced with the significance of the positivity of the effective mass at the bounce. My reasons are as follows. 1. This difference can affect the adiabatic vacuum, but so far as I know, for the choice of the adiabatic states as the initial states, neither the hybrid nor the dressed metric approach can result in  a prediction consistent with the observations for small $l$,  the results from two approaches are very similar as discussed in arXiv: 1601.01716, they lead to an amplified power spectrum in the intermediate regime  instead of the suppressed one required by observations. So does that mean adiabatic states are not the right choice? 2. It is also recommended in some paper that the initial states should be chosen in the contracting phase when most of the interesting modes are inside the horizon initially. In this case, what is the significance of a positive mass at the bounce. One may think that the positive mass ensures the validity of the adiabatic condition, however, in arXiv:1809.03172, those authors show that adiabatic conditions are violated at the bounce in both the hybrid and the dressed metric approach (their Fig. 3). 3. The initial states can also be chosen in the Planck regime satisfying some particular principles as discussed in arXiv: 1608.04228. Does the sign of the effective  mass in the Planck regime also make a difference in the choice of those initial states? For the above three points, can the authors add a few of their own opinions on each of them in the conclusions? Authors should comment on above points and express their opinion on arguments in above works.

 

Author Response

Reply to Reviewer 1

We thank the reviewer for their kind comments and constructive suggestions. In order to take them into account, we have introduced the following changes (every indication below refers to the updated version).

1) A footnote has been inserted in the third paragraph of the Introduction, and a slight change in the line after Eq. (4), in order to introduce and explain the terminology used to refer to the mass term in the dynamical equations of the perturbations.

2) After the first sentence of the fifth paragraph of the Introduction, a new sentence has been included to, at least, point out succintly some of the reasons why the positivity of the mass term at the bounce is important, mentioning that further details can be found in the Conclusions.

3) The typo in Eq. (7) has been corrected.

4) In order to cope with the suggestions of both of the reviewers 1 and 2, further explanations and clarifications about the difference between the hybrid and the dressed metric formalisms have been included at the end of the fourth paragraph of the Introduction, incorporating some of the results that were stated in Ref. [50], where this issue was discussed.

5) The second paragraph of the Conclusions (before the last line) contains now further arguments supporting the naturalness of the choice of initial conditions for the vacuum state at the bounce, as well as the importance of the positivity of the mass term there if we want to derive a set of positive frequency solutions and, e.g., investigate their Hadamard behavior, possibly in relation with a certain limit of higher-order (i.e., beyond low-order) adiabatic states. A footnote has been inserted and four new references have been added (two of them cited by the reviewer).

List of all changes (all indications refer to the new version)

- A footnote has been inserted in the third paragraph of the Introduction.

- Two symbols have been defined in the fist sentence of the fourth paragraph of the Introduction, to denote the classical and the effective Hamiltonians.

- Six sentences have been added at the end of the fourth paragraph of the Introduction.

- Fifth paragraph of the Introduction: A sentence has been introduced after the first one. The new third sentence has been slightly changed. Minor changes have been made in the next to last sentence.

- A minor modification has been made in the first sentence of the seventh paragraph of the Introduction.

- A minor modification has been made in the first sentence of Sec. 2.

- A sentence has been included at the end of the paragraph after Eq. (2).

- In the line after Eq. (4), the wording “(square)” has been introduced.

- A typo has been corrected in Eq. (7).

- Several sentences and a footnote have been introduced just before the last sentence of the second paragraph of the Conclusions.

- Third paragraph of the Conclusions: A sentence has been added after the first one. The last sentence has been modified.

- All over the article: the symbol $s_a$ has been replaced with $\sigma_a$.

- Eight new references have been added: [65-68] and [75-78].

Reviewer 2 Report

In this paper the authors extend previous work on the differences between the mass term for cosmological perturbations in the dressed and hybrid approaches to Loop Quantum Cosmology (LQC). I believe that the work is of high quality, well written and covers an important topic. The subject of perturbations in LQC is vital to the attempt to connect theory with observations and to extend the interesting background behaviour of simple models into more physically relevant models. The possibility of the mass term to become negative at the bounce is, as the authors point out, crucial in this analysis. With the above in mind, I believe that the paper should be accepted and published.

Below I list some suggestions for the authors, which should not be considered necessary for publication, but which may help improve the manuscript in minor ways.

1) The justification for the Starobinsky potential has, to my knowledge, been motivated by its origin as the effect of introducing an R^2 term in the action of GR. However, this does not appear to be the origin here, as the background quantization is of the standard Einstein-Hilbert action (following the polymer quantization modelled after holonomy-flux methods). The authors could acknowledge that the choice in LQC is thus somewhat ad-hoc, and include some more note or motivation for the choice of the potentials that they use.

2) Similarly to the above, if the purpose is to generalize the investigation of the mass difference beyond simple potentials, it may be helpful to have some more discussion of whether the results which have now been generalized to include these potentials should be expected to be generic features. Do the authors believe that any potentials (within some well-defined class) will cause negative mass terms at the bounce in the dressed metric case, or are there qualitative features of the potentials that determine this? Or do the authors have some plan to do followup (possibly numerical) work to explore the bounds of the result?

3) "s" has been used throughout as the mass term. It was a little confusing to see "s_a" used as a sign term in equation 8. There may be a change of notation that makes this read a little easily.

4) A little further discussion about the differences in how one arrives at the dressed metric and hybrid quantization results may be helpful to readers who are not familiar with the prior paper (ref 50). I had to go back and re-read this to recall the truncate-constrain-quantize differences between the two. A sentence or two in the introduction/discussion may clear this up quickly for a reader who does not go back to the prior paper.

As noted above these are suggestions for the authors to consider including should they wish to do so, but not requirements - the paper is fit for publication as it stands. The paper was a pleasure to read and review.

Author Response

Reply to Reviewer 2

We thank the reviewer for their nice comments and very useful suggestions. In order to incorporate them, we have introduced the following changes (every indication below refers to the updated version).

1) A sentence has been included at the end of the paragraph after Eq. (2) to comment on the relation between the Starobinsky potential and the introduction of corrections to the Einstein-Hilbert theory that are quadratic in the scalar curvature. Four new references have been added.

2) A sentence has been added after the first one of the third paragraph of the Conclusions, and the last sentence of that paragraph has been modified, in order to clarify that the results for the mass of the tensor perturbations are generalizable to generic potentials, whereas the results for the scalar mass show that the positivity or negativity is maintained in intervals of values of the potential that nonetheless depend on its specifications. We are not planning a continuation of our work by considering other particular classes of potentials.

3) We have replaced the symbol $s_a$ with $\sigma_a$ in the whole paper.

4) Further explanations and clarifications about the difference between the hybrid and the dressed metric formalisms have been introduced at the end of the fourth paragraph of the Introduction, following the suggestion of the reviewer.

List of all changes (all indications refer to the new version)

- A footnote has been inserted in the third paragraph of the Introduction.

- Two symbols have been defined in the fist sentence of the fourth paragraph of the Introduction, to denote the classical and the effective Hamiltonians.

- Six sentences have been added at the end of the fourth paragraph of the Introduction.

- Fifth paragraph of the Introduction: A sentence has been introduced after the first one. The new third sentence has been slightly changed. Minor changes have been made in the next to last sentence.

- A minor modification has been made in the first sentence of the seventh paragraph of the Introduction.

- A minor modification has been made in the first sentence of Sec. 2.

- A sentence has been included at the end of the paragraph after Eq. (2).

- In the line after Eq. (4), the wording “(square)” has been introduced.

- A typo has been corrected in Eq. (7).

- Several sentences and a footnote have been introduced just before the last sentence of the second paragraph of the Conclusions.

- Third paragraph of the Conclusions: A sentence has been added after the first one. The last sentence has been modified.

- All over the article: the symbol $s_a$ has been replaced with $\sigma_a$.

- Eight new references have been added: [65-68] and [75-78].

Round 2

Reviewer 1 Report

Although in the revised version the authors have addressed some of the comments in my last report, I do not think that they have completely answered the most important parts which are the last two comments of my previous report. My reasons are as follows.

 

1.  In the last comment of my previous report, I raised three questions: are the adiabatic states not practically viable to produce a power spectrum consistent with the observations? Why not choose the initial states of the perturbations in the contracting phase so that the problem with the negative mass at the bounce can be avoided in some sense? And the authors' opinions on the other initial states imposed in a neighborhood of the bounce which can lead to desired suppression at certain scales. I think the first and the last of these three questions have been properly addressed  in the conclusions of  the revised version. However, the second question about imposing the initial states in the contracting phase is completely ignored.  In the revised version, it is only emphasized that it is very natural to impose the initial states at the bounce in order to make the vacuum state "optimally adapted to the background evolution and capture the quantum geometry effects". But I do not think this argument is strong enough. In LQC, the big bang singularity is generically replaced by the big bounce. However, the bounce is not the beginning of the time, there is still history coming from the contracting phase.  What if there exists tremendous particle creation when the universe is in the contracting phase. Can the same vacuum state at the bounce still be attained? Besides, even with the positivity of the mass, the adiabatic conditions are still violated as pointed out explicitly in arXiv: 1809.03172 which signifies particle creation in the bouncing regime. Furthermore, if anisotropies are taken into consideration, how to choose  the bounce point to impose initial states  when the scale factors in three different direction reaches their minimum values at different instants of time.  Finally,  imposing the initial states at different times can also help check the robustness of the results.  Therefore, it is highly non-trivial to state the naturalness of the choice of initial conditions for the vacuum state at the bounce. I think the authors should at least mention the choice of the initial states in the contracting phase as an option but not exclude this possibility by without even mentioning it. 

 

2. Quite unfortunately, my second last comment in the previous report is completely left unaddressed. Maybe the authors misunderstood this comment as they have added more discussions in the introduction of their revised version on the differences of the scalar masses in two approaches which originate from the evaluation of $a^{\prime\prime}$.  Although these discussions are quite in detail and reader-friendly, they do not serve the purpose to answer my question.  What I want to ask is whether there is any fundamental reason to explain these differences. Or equivalently, why in the dressed metric approach we use $H^\mathrm{eff}_0$ to evaluate the Poisson bracket while in the hybrid approach we first use $H_0$ and then evaluate the resulting expression on the effective solutions of LQC. In the paper that I mentioned in my previous report, i.e. arXiv:2206.12434,  those authors attribute these differences as originating from using different variables in two approaches which implies that two approaches are essentially coming from quantization ambiguities. I think it will be useful if the authors can think about and then comment on this point. 

 

Based on the above reasons, I think this revised version needs a further revision to cover the points mentioned above. It is a little disappointing to see that people avoid citing their colleagues' papers even when these papers are relevant to the topic.  I think at least two additional papers mentioned above in the current report must be cited somewhere in the revised version of the manuscript. 

 

Author Response

Reply to the reviewer

To take into account the points made by the reviewer in their second report we have proceeded as follows.

1. We have included two new footnotes in the second paragraph of the Conclusions. One of them reflects the possibility of choosing initial conditions for the vacuum at times other than the bounce, and comments on the extension of our analysis to such times. The other footnote includes one of the references mentioned by the reviewer and clarifies that positivity of the mass does not guarantee that the conditions for a WKB approximation are met.
2. A footnote has been included in the fourth paragraph of the Introduction, clarifying the reason for the appearance of different Hamiltonians in the Poisson-bracket terms of the hybrid and dressed metric formalisms. This footnote contains the second reference mentioned by the reviewer.

List of changes

We have introduced a footnote in the fourth paragraph of the Introduction and two footnotes in the second paragraph of the Conclusions. These footnotes take into account the points made by the reviewer. They contain three new citations (Refs. [52], [75], and [77]).