An Alternative to Dark Matter and Dark Energy: Scale-Dependent Gravity in Superfluid Vacuum Theory

Round 1

Reviewer 1 Report

I found the paper to be quite interesting, well-written, and containing new physical results. I have no serious remarks and can recommend the paper for publication in the Universe.

Author Response

Dear Reviewer 1,

thank you for your work and positive recommendation.

Regards

Author

Reviewer 2 Report

In the present manuscript the author ventures on an interesting project in which the physical vacuum is envisaged as a cold quantum liquid. The author presents a suitable Gross-Pitaevskii-like equation which describes the dynamics of the fluid condensate wave function. The most important physical quantity in this work is the effective gravitational potential given by Eq. (19); indeed, according to the author, the existence of this effective gravitational potential demonstrates that gravity has a multiple-scale pattern and this is what could account for many puzzles arising in cosmological settings such as the accelerating expansion of the universe and the Hubble tension.

This research line has been pioneered by Volovik and others, and many pivotal results are summarized in the remarkable books quoted in Refs. [4] and [5] of the manuscript. So unquestionably this is an interesting project. However, there are some issues that should be duly considered by the author prior to any form of publication. For instance, a discussion on the Equivalence Principle is lacking. Moreover, in connection with this point, the effective metric does not necessarily obey Einstein’s equations, so it is unclear whether one can use them to "define" an induced energy-momentum tensor (incidentally, a rather unorthodox approach, to say the least), as proposed by the author. The author should elaborate more on such relevant issues (for instance, by presenting some experimental constraints) and others appearing throughout the manuscript, so that one can assess more clearly the physical consistency of the model. 

Some brief comments are also in order:

(1) There are some discussions in the text lacking necessary references that could give support to the claims vindicated by the author. In particular, I would draw the author’s attention to the assertion that a convincing theory of dark matter and dark energy should be part of a fundamental theory involving all interactions -- this subject should be better exposed, quoting more references from the current literature. In addition, the discussion on the accelerating expansion of the universe at the beginning of Section 5 should also be improved; as well known, the existence of a cosmological constant constitutes the simplest possible explanation for dark energy, and this is the idea that is employed in the LambdaCDM model. Hence this point deserves further clarification.

(2) The claim that the physical vacuum should contain "answers to such questions as what is space, time, mass (…)" (in the author’s own words) should be better explained.

(3) What does "the leading-order approximation with respect to the Planck constant" precisely mean in the present context? Is it like a semiclassical approximation?

(4) What does the author mean by "Correspondence Principle" after Eq. (4)? This terminology is widely used in discussions on the behavior of systems described by quantum mechanics and the classical limit.

(5) When the author claims that the Lorentz symmetry is "approximate", perhaps a clearer statement would be to declare that relativistic models only provide an effective description of nature (within the theory studied by the author).

(6) Usually, the derivation of an effective metric in, for instance, BEC setups, is valid under the hydrodynamic approximation. In this respect, it would be important to have a more detailed discussion on the validity of the approximations made in the calculation of the metric given by Eq. (5). The same goes for Eq. (28).

(7) The author claims that the quantity given by Eq. (14) should be envisaged as an induced gravitational potential. However, up to this point in the text this claim seems to be completely unwarranted; only after establishing Eq. (28) can this assertion be made. This should be properly elucidated.  

(8) What is the physical interpretation for the quantity M, introduced in Eq. (27)?

(9) The connection of the results presented in the manuscript with the hierarchy problem seems to be completely obscure, at least in the form exposed in the manuscript. This discussion should be properly enlarged and clarified.

(10) It is not clear whether the model proposed by the author actually solves the Hubble tension. Maybe some numbers could be given. In any case, the author should offer a more elaborated discussion on this topic.

(11) The coincidence problem should be duly discussed.

(12) English could be slightly improved. There are a few sentences throughout the text that needs a better wording. I also suggest the author to pinpoint and correct any orthographic errors that might eventually occur throughout the text.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

In the manuscript: "An alternative to dark matter and dark energy:
Scale-dependent gravity in superfluid vacuum theory" is shown an effective gravitational potential, induced by the wavefunction of the physical vacuum and the vacuum is considered as the superfluid. In this picture it is shown the explanation for the accelerating expansion of the Universe. The behaviour of rotation curves of spiral galaxies has been also explained.

The paper is well written and the arguments presented are interesting. Therefore, I suggest the publication of the manuscript on the Universe. 

Author Response

Dear Reviewer 3,

thank you for your work and positive recommendation.

Regards

Author

Round 2

Reviewer 2 Report

The author has addressed in detail all points raised previously. More elaborated (and important) discussions were given. Some relevant arguments are now clearly expressed. This is certainly an improved version of the work in comparison with the previous one. However, I would like to draw the author's attention to some minor details which I list below. 

(1) Concerning the "minor remark", I would like to clarify that I did not claim that the author employs a Gross-Pitaevskii equation but rather a "Gross-Pitaevskii-like" equation. The difference should be clear. In any case, the terminology used by the author is completely acceptable, according to his assertions.

(2) The effective metric field determined by the dynamics of the superfluid vacuum theory does not obey the Einstein's equations in the sense that it was not obtained from them; see, for instance, the general discussion in Ref.[4] of the manuscript. So once an induced metric is obtained, the author feels free to jump to the conclusion that it should surely obey Einstein's equations -- an obvious ad hoc procedure. In any case, one can still imagine an alternative interpretation of such equations by assuming their validity a posteriori and using this to define a suitable induced energy-momentum tensor -- this is essentially what the author claims. The physical justification seems to be clearer in the revised version of the manuscript. However, a little caution is necessary here; the author should keep in mind that this is an assumption of his work, but not necessarily it should hold in general. Other approach would be to reproduce the Einstein-Hilbert action for the effective metric, as briefly discussed by the author.

(3) At ordinary energies the Standard Model (with no other interactions) does not have a naturalness problem. However, the introduction of other interactions typically bring new high-energy scales and radiative corrections and as a consequence affect the scale of the Standard Model, generating the naturalness problem. Regarding the electroweak sector, the naturalness of the Higgs mass will lead to the hierarchy problem. In this sense, it is still not clear that the hierarchy problem becomes "obsolete" in superfluid vacuum theory, as argued by the author. The assertion that the superfluid vacuum theory could solve the hierarchy problem is certainly an interesting, worth-pursuing idea, but this issue should be addressed in more detail, with a clear discussion on the role of the Planck scale in the context of the superfluid vacuum theory. In any case, I understand that this is outside of the scope of the present work.

(4) The effective potential given by Eq. (14) can only be interpreted as an induced gravitational potential after the effective background metric is obtained. So a priori it is not natural to associate V_{eff} with gravitational interactions. So  a rewording is necessary in that part of the text if one wants to be very attentive to and concerned about accuracy and detail.

(5) The solution to the accelerated expansion of the universe as put forward with the assumption of a cosmological constant is somewhat unconvincing for a number of reasons, some already raised by the author. This is unquestionably a plausible assertion. However, the author should acknowledge the fact that the LambdaCDM model is based on such an assumption, and this is the simplest model that can furnish a fairly good account of many features of the cosmos, including the observed accelerating expansion of the universe. To disagree with this solution to the problem of cosmic acceleration is fine, but one should understand that the assumption of the existence of a cosmological constant is the simplest explanation for the accelerating expansion, with advantages and major difficulties, according to the vast majority of the scientific community working on this topic. So, this phenomenon is not "usually explained" (in the sense of a "standard explanation") but certainly "can be explained" by resorting to exotic forms of matter.

There were other points as well, but those aforementioned were the most important ones. Since other reviewers have already exposed their positive recommendation towards the publication of the present manuscript, I would not want to delay even more the review process. So I leave the above comments as optional revisions to be considered by the author. I think the manuscript is interesting and provides some intriguing perspectives of projects to be investigated by the author and collaborators in future studies.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf