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Phonon redshift and Hubble friction in an expanding BEC
by Stephen Eckel, Ted Jacobson
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Submission summary
Authors (as registered SciPost users):  Stephen Eckel · Theodore Jacobson 
Submission information  

Preprint Link:  https://arxiv.org/abs/2009.04512v3 (pdf) 
Date accepted:  20210217 
Date submitted:  20201112 20:38 
Submitted by:  Eckel, Stephen 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We revisit the theoretical analysis of an expanding ringshaped BoseEinstein condensate. Starting from the action and integrating over dimensions orthogonal to the phonon's direction of travel, we derive an effective onedimensional wave equation for azimuthallytravelling phonons. This wave equation shows that expansion redshifts the phonon frequency at a rate determined by the effective azimuthal sound speed, and damps the amplitude of the phonons at a rate given by $\dot{\cal V}/{\cal V}$, where $\cal{V}$ is the volume of the background condensate. This behavior is analogous to the redshifting and "Hubble friction" for quantum fields in the expanding universe and, given the scalings with radius determined by the shape of the ring potential, is consistent with recent experimental and theoretical results. The actionbased dimensional reduction methods used here should be applicable in a variety of settings, and are well suited for systematic perturbation expansions.
Published as SciPost Phys. 10, 064 (2021)
Author comments upon resubmission
Dear Editor,
Thank you for considering our manuscript for publication on SciPost.
We thank the reviewers and include replies to their comments here. Specifically, with respect to the second reviewer:
 In section 4.1, for the expanding ring BEC, they mention that the static approximation is surprisingly accurate even though in [19] the condition \dot R << c is violated (and they show in the appendix that the corrections to the static approximation are indeed small). Do the authors understand why?
The condition \dot R << c would certainly imply adiabaticity, but it is not strictly required. As discussed in the appendix, the accuracy actually comes from a combination of factors. First, in the adiabatic limit where no additional Bogoliubov modes are excited by the expansion, the effect of the additional noninertial forces on the relevant azimuthal modes enters at lowest order with the thin ring parameter. Thus, the thin ring used in Ref. [19] helps to suppress any effect. For some of the expansions in [19], there is clear evidence of nonadiabatic evolution in the form of oscillations of the center of mass of the condensate after the expansion stops. The effect of this nonadiabatic motion is suppressed by of the odd symmetry of the first radial Bogoliubov mode, which has no impact on the relevant azimuthal phonon modes at first order in the excitation amplitude.
In our resubmission, we have updated the statement regarding the condition \dot R << c to make it clear that while it is sufficient for adiabaticity, it is not strictly required. We also summarized the arguments of the appendix in a new, expanded discussion at the start of Sec. 4.2.
 in the definition of the speed of sound c_theta^2 in (4.19), isn't the factor (1+rho/R) in the numerator?
Referring to Eq. (2.5), there are two contributions to this factor: r from the measure of the integrand \sqrt{h} and 1/r^2 from the inverse metric h^{ij}. Combined, these two produce a factor of 1/r = 1/(1+\rho/R).
In our resubmission, we have updated the definitions after (2.1) in order to make explicit the distinction between the covariant and contravariant (inverse) components of the metric tensor.
With these changes, we respectfully resubmit our improved manuscript for your consideration.
Best regards,
Stephen Eckel and Ted Jacobson
List of changes
As a result of the reviewer comments, we have made the following changes to the manuscript:
1. We have updated the definitions after (2.1) in order to make explicit the distinction between the covariant and contravariant (inverse) components of the metric tensor.
2. We have updated the statement regarding the condition \dot R << c to make it clear that while it is sufficient for adiabaticity, it is not strictly required. We also summarized the arguments of the appendix in a new, expanded discussion at the start of Sec. 4.2.
In addition to these two changes, we also updated the text after (B.11) to remove a superfluous double minus sign and added a footnote with the calculated $a_\mu$ parameters of Ref. [19]. These two changes help to make the argument of the appendix clearer for the reader.
Submission & Refereeing History
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Reports on this Submission
Report 3 by Matthew Davis on 2021213 (Invited Report)
 Cite as: Matthew Davis, Report on arXiv:2009.04512v3, delivered 20210213, doi: 10.21468/SciPost.Report.2550
Strengths
1. Presents an actionbased method that helps understand the results of an important experiment on analogue cosmology.
2. Paper is comprehensive, and clearly set out the details of the calculations.
3. Informative abstract/introduction/conclusion.
Weaknesses
1. Perhaps the manuscript is a little longwinded in the main text, but this is a matter of personal taste.
Report
This manuscript corrects an analysis of the experiment performed in Ref. [19], and hence derives a result for the "Hubble friction" that is in agreement with the experimental results and an alternative theoretical approach of Ref. [20]. It is generally straightforward to follow, and well written. The methodology and dimensional reduction techniques could prove useful in application to alternative scenarios, and will be of interest to several in the field. I don't think there are any further significant changes following those from the first round of refereeing. Therefore I feel that this manuscript meets the acceptance criteria for SciPost Physics.
Requested changes
1. After Eq. (3.17) there appears $kr_\mu<1$  I suspect this is the condition for the validity of Eq. ( 3.17)?
2. The conclusion refers to the "short" and "long" appendix  I think it would be best to say "Appendix A" and "Appendix B"
Anonymous Report 1 on 20201115 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2009.04512v3, delivered 20201115, doi: 10.21468/SciPost.Report.2194
Strengths
Clear paper
Weaknesses
None
Report
The authors discuss in a detailed way the phonon wave equation in an expanding ring BEC by dimensional reduction , showing characteristic effects one finds in cosmology , namely redshifting and Hubble friction experienced by quantum fields in an expanding universe. So far the study of analogue gravity models were mostly devoted to find the analogue of Hawking radiation in black holes. The application to cosmological issues is a developing field extremely interesting. The paper is clearly written in a rather pedagogical way that allows even not expert in the field to appreciate it.
Author: Stephen Eckel on 20210222 [id 1259]
(in reply to Report 3 by Matthew Davis on 20210213)We thank Prof. Davis for his helpful comments. In the published version, we will be sure to eliminate the spurious reference to $k r_\mu < 1$ after (3.17) and will properly reference the appendices.