Probing the Topology of the Early Universe Using CMB Temperature and Polarization Anisotropies

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

[Summary]
This manuscript focuses on the temperature and polarization anisotropies of the CMB as hints of early universe's topology. More specifically, it considers A Kaluza-Klein model with a compactified 5th dimension to derive infrared cutoffs for the two-point angular correlation functions, whose even/odd properties depend on the boundary conditions imposed. The effect of a warped geometry is also considered. The results are relevant to compare to future measurements of the B-mode polarization by the LiteBIRD mission. 

[General comments] (highlighting areas of weakness, the testability of the hypothesis, methodological inaccuracies, missing controls, etc.)
The manuscript is well written and its motivation is solid. The concepts and text flow well, and there is a suitable amount of relevant references. Some of the mathematical calculations are given in full detail, while others are skipped. I would recommend to include some intermediate steps in those that are skipped, to better follow the derivation. I would also encourage making the link to (future) observations more clearly, by specifying a bit more which of the derived quantities can be measured and how theoretical expectations and observational data could compare. 

[Specific comments]

Introduction: 
- Page 1, line 34: "were enormously amplified during an inflationary phase that are observable today" -- something fails grammatically in this sentence. 
- P.2, l.59: acronym "GUT" is not defined. Same goes for the abstract. 

Sec.2: 
- P.2: (1) introduces the quantity C(\theta), but (2) uses C^{TT}(\theta). If they are the same quantity, notation should be unified (here and later in the text); if not, it should be explained. 
- P.3, Fig.1: it is not referenced in the text and a more detailed explanation is needed to correctly interpret the theoretical curves. It is not clear to me if the statement in lines 82-83 referes to the results presented on this figure. Please clarify and include the meaning of k in the legend.  
- P.3: k_{min} in (5) was defined in l.88, but its physical meaning is not explicitly given. I think it would be good to include it. 

Sec.3: 
- P.6, Fig.2: I recommend to expand the explanation of the diagrams and formulas, including meaning of Z_2 / Z_2'. 

Sec.4: 
- P.8: I wish some more intermediate results were given for the tensor modes before stating the relation (17), to better understand the differences between scalar and tensor procedures. 

Sec.5: 
- P.9, l.235: It would be good to explicitly state that allowed values for q are <=1. 

Sec.7: 
- P.10, l.269-289: This part of the text reads more like part of an introduction. Maybe part of the material could be moved there, leaving here a summary to introduce the equations and results that come afterwards? 
- P.11: Are there any observational data/estimations to contrast the results in Fig.3 to? Maybe comment on this relating to Fig.1? 

Author Response

- Page 1, line 34: "were enormously amplified during an inflationary phase that are observable today" -- something fails grammatically in this sentence. 
- P.2, l.59: acronym "GUT" is not defined. Same goes for the abstract. 

Both typos have been corrected

Sec.2: 
- P.2: (1) introduces the quantity C(\theta), but (2) uses C^{TT}(\theta). If they are the same quantity, notation should be unified (here and later in the text); if not, it should be explained.

corrected!  

- P.3, Fig.1: it is not referenced in the text and a more detailed explanation is needed to correctly interpret the theoretical curves. It is not clear to me if the statement in lines 82-83 referes to the results presented on this figure. Please clarify and include the meaning of k in the legend.  

First of all I have to say that this figure, coming from a previous, paper has been upgraded in the new version and the caption rewritten.

Moreover the meaning of kmin has been quite better explained and referenced in the new version.

Fig.1  is now duly referenced (several times) in the main text 

- P.3: k_{min} in (5) was defined in l.88, but its physical meaning is not explicitly given. I think it would be good to include it. 

 

As mentioned above, the interpretation of kmin ​—either as a single IR cutoff or an IR doublet in the power spectrum—has been substantially expanded. 

Sec.3: 
- P.6, Fig.2: I recommend to expand the explanation of the diagrams and formulas, including meaning of Z_2 / Z_2'. 

I have numbered some equations in this section in order to facilitate the follow-up of the steps and formulas, and added a sligthly longer explanation

Sec.4: 
- P.8: I wish some more intermediate results were given for the tensor modes before stating the relation (17), to better understand the differences between scalar and tensor procedures.

Some equations in this section have been numbered in order to facilitate the follow-up of the steps and formulas,. New text added.

Sec.5: 
- P.9, l.235: It would be good to explicitly state that allowed values for q are <=1. 

I think there were several misprints in this subsection that I have corrected in the new version. Hope now everything is clear. 

Sec.7: 
- P.10, l.269-289: This part of the text reads more like part of an introduction. Maybe part of the material could be moved there, leaving here a summary to introduce the equations and results that come afterwards? 

I have moved part of the text from this sub-section to the Introduction. However, since a new sub-section on E-mode polarization has been added in the updated version, I considered it convenient to keep several statemenrs about the relevance of B-modes, particularly for the detection of primordial gravitational waves (PGWs)."

• P.11: Are there any observational data/estimations to contrast the results in Fig.3 to? Maybe comment on this relating to Fig.1?

As mentioned earlier, a new subsection and plot on E-mode polarization have been included. Accordingly, a general comment on the common behavior related to the possible existence of infrared cutoff(s) in the curves for temperature, E-mode polarization, and B-mode polarization angular correlations has also been added. However, to the best of my knowledge, current data on B-mode polarization at low multipoles do not allow for any meaningful comparison so far. Unfortunately, high-precision measurements from future missions, such as LiteBIRD, will have to be awaited for several years...

 

Reviewer 2 Report

Comments and Suggestions for Authors

This paper considers extra-dimensions models to explain CMB anomalies, as observed at the large angular scale of the CMB anisotropy pattern in Planck (and WMAP) data.

I find the idea intriguing, but the work misses important references and lacks fundamental analysis. Moreover, it's unclear why the author uses the two-point correlation function for B-modes but not for E-modes. The E-mode two-point correlation function has the benefit of being constrained, though not at the cosmic variance level, unlike the B-modes. Therefore, I cannot accept the manuscript in its current form.

Further details are provided below.

Omitted References. Here are important references that are currently omitted:

1. Y. Akrami et al. [Planck], "Planck 2018 results. X. Constraints on inflation," Astron. Astrophys. 641 (2020), A10, doi:10.1051/0004-6361/201833887 [arXiv:1807.06211 [astro-ph.CO]]. This paper, contrary to the author's statement, shows no statistically significant detection of kmin . See also the references therein: Contaldi et al. 2003, Cline et al. 2003, Kuhnel & Schwarz 2010, Hazra et al. 2014, Gruppuso et al. 2016.
2. Y. Akrami et al. [Planck], "Planck 2018 results. VII. Isotropy and Statistics of the CMB," Astron. Astrophys.641 (2020), A7, doi:10.1051/0004-6361/201935201 [arXiv:1906.02552 [astro-ph.CO]]. This paper performs an analysis of the even-odd asymmetry. See also the references therein.
3. P.A.R. Ade et al. [Planck], "Planck 2015 results - XVIII. Background geometry and topology of the Universe," Astron. Astrophys. 594 (2016), A18, doi:10.1051/0004-6361/201525829 [arXiv:1502.01593 [astro-ph.CO]]. This work provides constraints on topology using Planck CMB data. See also the references therein.

 

Considerations and Questions

Here are some specific considerations and questions regarding the paper:

a.  Figure 1: The error bars appear quite constant across the angular scale. They seem to not include the contribution from cosmic variance, which would increase the uncertainty at the largest angular scales. How did the author compute these errors?

b.  Figure 1: The author states that the data does not fit the expectations of the Lambda-CDM model. What estimator was used to support this statement? Was it the so-called S1/2  estimator or another one?

c.  Figure 3: How were the error bars computed for this figure? Is it made of cosmic variance in the ideal case or have some noise and incomplete sky fraction been considered?

d.  Figure 3: How do the curves depend on the considered value of r?

e.  Figure 3: To what statistical significance can the various curves, corresponding to different IR cutoffs, be distinguished? This is crucial for disentangling different topologies, as the authors suggest.

f.  Analogy to Figure 3: A similar figure could be constructed for the E-modes. The benefit of this is that the C_ell^EE  values have been measured by Planck and can therefore be confronted with the models.

Author Response

Moreover, it's unclear why the author uses the two-point correlation function for B-modes but not for E-modes. The E-mode two-point correlation function has the benefit of being constrained, though not at the cosmic variance level, unlike the B-modes. 

You are right. Initially I focused on B-mode polarization correlations from PGWs, as they would represent the smoking gun of inflation, carrying information of the very early universe. However, E-mode polarization are also sensitive to different types of early-universe topology associated with boundary conditions as addressed in this work. Following your insightful suggestion, I have incorporated into the new version an additional (sub)section with an analysis of E-mode correlations, together with an extra plot where the three cases for cutoff are considered. As shown in the revised manuscript, this points to the same effect observed in temperature anisotropies.

Omitted References. Here are important references that are currently omitted

Indeed. These and several more references have been incorporated into the new version. 

 a.  Figure 1: The error bars appear quite constant across the angular scale. They seem to not include the contribution from cosmic variance, which would increase the uncertainty at the largest angular scales. How did the author compute these errors?

A new Fig.1 has replaced the previous one, and now accounts for cosmic variance from data. The curve for the IR doublet nicely follows the observational trend within the error band, much better than for the other scenarios (no-cutoff or single cutoff). A comparison with recent work (new reference added as it was submitted just one week after my submission to Universe) has been carried out in both Figs.1 (temperature) and 3 (E-mode polarization).

b.  Figure 1: The author states that the data does not fit the expectations of the Lambda-CDM model. What estimator was used to support this statement? Was it the so-called S1/2  estimator or another one?

In both previous and current work, the main estimator has been the reduced chi-squared, \chi^2_d.o.f., derived from fits to data, either through the two-point correlation function or a parity statistic as shown in a cited reference by Sanz and myself. The S1/2 estimator should in principle provide a similar  information as the latter two. 

c.  Figure 3: How were the error bars computed for this figure? Is it made of cosmic variance in the ideal case or have some noise and incomplete sky fraction been considered?

Having added a new figure for E-mode polarization, we now refer to Figure 4 for B-mode polarization.

Indeed, the error bands were estimated assuming ideal LiteBIRD performance, so that the uncertainties arise solely from cosmic variance (i.e. the relevant C_ell ​coefficients were allowed to vary according to the corresponding errors). Other statistical or systematic sources of uncertainty, such as instrumental noise or foreground contamination, were not included in our analysis. A comment on this issue has been added to the text and in the caption of Fig.4.

d.  Figure 3: How do the curves depend on the considered value of r?

Al low-ell values (ell \leq 11) the curves are essentially proportional to r. Therefore one can easily scale up or down the curves and errors bands accordingly. 

e.  Figure 3: To what statistical significance can the various curves, corresponding to different IR cutoffs, be distinguished? This is crucial for disentangling different topologies, as the authors suggest.

Since no real LiteBIRD data, or data with sufficient precision for B-polarization correlations, are available yet, what seems clear is that, as in the cases of temperature and E-mode polarization, the main focus should be on the antipodal region, particularly at large angles where the downward tail would manifest. Only for the IR cutoff doublet would the statistical significance be high enough to allow detection. This would be especially true for a warped geometry, as the IR even cutoff is presumably much more suppressed than the odd one (as seen in Fig. 4).

f.  Analogy to Figure 3: A similar figure could be constructed for the E-modes. The benefit of this is that the C_ell^EE  values have been measured by Planck and can therefore be confronted with the models.

Yes, as I already answered before. As a final comment, current E-mode polarization plot also points to the same odd-parity preference showing up at large angles. The precision does not allow for any claim yet.

On the other hand, if analyses of the CMB angular correlations for temperature, E-mode polarization, and B-mode polarization were to collectively exhibit such an odd-parity dominance, a statistical fluke would be highly unlikely. I emphasize this point in the conclusions. And a common explanation based on the emergence of IR cutoffs is proposed in this paper.

 

Reviewer 3 Report

Comments and Suggestions for Authors

The title of the paper is: "Probing the Topology of the Early Universe using CMB Temperature and Polarization Anisotropies". It is an interesting and well written paper. The author suggests how studying polarization anisitropies in combination with temperature fluctuatuins in the CMB can lead to information about the topology of the universe in a pre-inflation era. He deduces new interesting results that can be tested againt observations with new planned equipment. 

   My main conclusion is that this paper is suitable for publication without any changes in the spacial issue of Universe which is mentioned by the editors. 

   One minor printing error can corrected in the publication process: An "a" should be removed in line 58.

Author Response

My main conclusion is that this paper is suitable for publication without any changes in the spacial issue of Universe which is mentioned by the editors. 

Thanks a lot for your support

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The idea of considering extra-dimensional models to explain CMB anomalies, as observed at the large angular scales of the CMB anisotropy pattern in Planck (and WMAP) data, is intriguing. I thank the author for modifying the manuscript, but unfortunately, I still do not consider it to be of publication quality.

Here are the reasons why I believe further modifications and clarifications are needed.

1. The manuscript still lacks several key references that are central to the field. As noted in my previous comments, the authors did not include the "references therein" from the three Planck papers I initially suggested.
2. In the new version, the author updated the constraints on u_min  (see below equation 9), finding estimates with a significance of more than 8 sigma. Typically, CMB anomalies are found at a much lower statistical significance, i.e., 2-3 sigma (see e.g., Planck papers and references therein). Why are these constraints so different from what is reported in the literature? How did the author compute them? While I understand that a chi^2 approach has been considered, given this huge discrepancy with the literature, it is essential to provide all the details of such an analysis, including the covariance considered to build the chi^2.
3. I find the analysis provided on the E-mode interesting, but it seems to be more qualitative than quantitative. For instance: to what extent are the Planck data in agreement with ΛCDM? From a statistical point of view, how do things improve with the double IR cutoffs model provided by the author?
4. The same applies to the forecast on the B-mode. The author's reply seems too qualitative and not quantitative enough. I still believe my previous question—"To what statistical significance can the various curves, corresponding to different IR cutoffs, be distinguished? This is crucial for disentangling different topologies, as the authors suggest."—can be answered. For example, the author could find the minimum r needed for LiteBIRD to be able to distinguish between the different topologies. 

Author Response

The idea of considering extra-dimensional models to explain CMB anomalies, as observed at the large angular scales of the CMB anisotropy pattern in Planck (and WMAP) data, is intriguing. I thank the author for modifying the manuscript, but unfortunately, I still do not consider it to be of publication quality.

The initial motivation of this article was to present a rather short paper: first, revisiting temperature correlations, then developing the theoretical framework behind the introduction of IR doublets into the primordial power spectra, and finally focusing on B-mode polarization. However, in light of your comments and criticisms (which I find both insightful and timely), I have extended the scope of the paper to include E-mode polarization. I also now provide quite more details about the method used to extract the new cutoff values from a parity-statistic analysis. 

Additionally, I have extended the B-mode polarization section.

In fact, there are now seven new plots compared to the previous version, totaling eight more plots than in the first submitted version, along with four additional pages.

The new text is in blue again, having removed the blue from the previous corrections/additions to make clearer the new changes.

I sincerely thank the referee for his/her interest in reviewing this work, which has greatly improved as a result.

The manuscript still lacks several key references that are central to the field. As noted in my previous comments, the authors did not include the "references therein" from the three Planck papers I initially suggested.

Yes, sorry for the oversight. I was pressed for time. I have now incorporated all these references into the Introduction, within a statement on the introduction of a cutoff in the primordial power spectrum from different motivations. Furthermore, I believe this new paragraph also helps motivate the study presented in this paper.

In the new version, the author updated the constraints on u_min  (see below equation 9), finding estimates with a significance of more than 8 sigma. Typically, CMB anomalies are found at a much lower statistical significance, i.e., 2-3 sigma (see e.g., Planck papers and references therein). Why are these constraints so different from what is reported in the literature? How did the author compute them? While I understand that a chi^2 approach has been considered, given this huge discrepancy with the literature, it is essential to provide all the details of such an analysis, including the covariance considered to build the chi^2.

I have added the discussion of the method used to extract the values of umin⁡odd/evenu_{\min}^{\text{odd/even}}uminodd/even​ in Eq.9 based on a parity-statistic, including several new figures (a new right panel in Fig. 1 and four new plots in Fig. 2). The length of the paper has grown accordingly.

I have also added, on p. 4, a caveat regarding the “evidence” of the 8–9σ obtained from the fit to the correlation functions, noting its contrast with other determinations of much lower significance.

I find the analysis provided on the E-mode interesting, but it seems to be more qualitative than quantitative. For instance: to what extent are the Planck data in agreement with ΛCDM? From a statistical point of view, how do things improve with the double IR cutoffs model provided by the author?

An additional comment has been included in the revised section on E-modepolarization. However, a more extended analysis of the E-mode polarization lies beyond the scope of the present paper, especially given the short editorial deadline (a few days). In addition, this particular study is being conducted in collaboration with colleagues, and it would be inappropriate to publish results from our ongoing joint work at this stage. Sorry for that.

The same applies to the forecast on the B-mode. The author's reply seems too qualitative and not quantitative enough. I still believe my previous question—"To what statistical significance can the various curves, corresponding to different IR cutoffs, be distinguished? This is crucial for disentangling different topologies, as the authors suggest."—can be answered. For example, the author could find the minimum r needed for LiteBIRD to be able to distinguish between the different topologies. 

I have extended the discussion on the discriminating power of the proposed method by examining the pattern of the BB correlation function, particularly in the antipodal region. Two new plots have been added to the manuscript. Thus I have provided an educated estimate for the lower value of the tensor-to-scalar ratio below which this method is likely to fail: r \simeq 10^{-3}.

As a final comment on the goal of this paper: it appears to be a common empirical though still marginal (but theoretically motivated) feature shared by temperature, E-mode, and (maybe) B-mode polarization correlations, such that odd-parity dominance emerges, leading to distinctive signatures, especially at large angular scales. This is the central message conveyed in this article and I hope with the new additions it becomes more clear. Thanks again.

 

 

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

I thank the author for his revisions. The manuscript has been improved by incorporating most of the requested changes and by providing additional details of the analysis.

The paper still contains a few points that could be clarified, such as the evaluation of the error for u_min. While this error appears to be compatible with reference [31], it's worth noting that it is not consistent with results from other publications, such as reference [22]. Despite this, the implementation of the core idea, using extra-dimensional models to explain CMB anomalies, can be assessed as a new contribution. The current presentation can be considered sufficient to recommend the paper for publication.