Elucidating the Dark Energy and Dark Matter Phenomena Within the Scale-Invariant Vacuum (SIV) Paradigm

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors developed a very interesting paper. Moreover, one of the authors is also familiar with the underlying topic that appears quite interesting.  

I have some comments that may be useful for improving the overall quality of the manuscript. 

1) I would suggest the authors to explain the physics of the cosmological constant in view of the fact that from a quantum field theory perspective there's absolutely the need of introducing vacuum fluctuations that actually disagree with the observed cosmological value. I would ask the authors to better explain how to reconcile the two worlds, cosmology with SIV and QFT. 

2) The dark energy paradigm seems to favour a cosmological constant, albeit DESI suggests for a slightly evolving scalar field, very much criticised to be honest (see e.g. 2404.12068, 2404.07070), and cosmological tensions are currently plaguing the model itself. Is there a possible explanations of such problems that can be found within SIV?

3) Perturbations within SIV could shed light on the possibility to disentangle the two approaches. Is there any chance to explain them and to relate the stuff done here with them?

After the changes (I invite the authors to add them in boldface in the manuscript) I'd like to give a final look at the paper.

Comments on the Quality of English Language

English and style may deserve a refresh.

Author Response

We are grateful to the reviewers for taking the time to review our manuscript and highly appreciate the reviewer’s comments, questions, and suggestions. Our responses are in color text below as well as in the text of the new manuscript.

Comment 1: I would suggest the authors to explain the physics of the cosmological constant in view of the fact that from a quantum field theory perspective there's absolutely the need of introducing vacuum fluctuations that actually disagree with the observed cosmological value. I would ask the authors to better explain how to reconcile the two worlds, cosmology with SIV and QFT.

Response 1: The following text have been added to the paper on page 6:

“The Quantum Field Theory (QFT) predicts an enormous value of the vacuum energy when viewed as zero-point energy of the matter fields (i.e. c^7/hbar/G^2 ~ 10^{114} erg/cm^2), while anthropic considerations a la Weinberg, and even simple dimensional estimate using the relevant physical constants (i.e. H_0^2c^2/G ~10^{-8} erg/cm^2) seems to arrive at the correct order of magnitude for the vacuum energy related to the cosmological constant. Thus, one can conclude that quantum effects involving Planck’s constant \hbar have nothing to do with the observed cosmological constant. Therefore quantum vacuum fluctuations are just that -- fluctuations whose mean value is zero. Within SIV this is reflected in removing the Cosmological Constant from the Freedman equations as seen in (13) in favor of an early-dark-energy term defined by (14) that involves the conformal factor \lambda. This can be interpreted as a choice of parameterization that brings the GR equations into the true co-moving frame with no Cosmological Constant as extra energy density. This is similar to what happens when choosing a special motion such that the kinetic energy of a system is zero, but in this case, it is the time parametrization controlled by \lambda that controls the amount of extra energy.”

 

Comment 2: The dark energy paradigm seems to favour a cosmological constant, albeit DESI suggests for a slightly evolving scalar field, very much criticised to be honest (see e.g. 2404.12068, 2404.07070), and cosmological tensions are currently plaguing the model itself. Is there a possible explanations of such problems that can be found within SIV?

Response 2: We are grateful to the reviewer for bringing to our attention the above two papers that are exploring model independent kinematic cosmographic constraints using three different data sets and their combinations to determine key cosmological parameters such as deceleration, jerk, and snap along with the corresponding model specific parameters that are supposed to shade light on the time variation of the dark energy and its equation of state. Performing a similar assessment within SIV is clearly one of the many future studies that needs to be done on the way of determining the relevant SIV model parameters, in this case only \Omega_m since just like LCDM the model has very few parameters. We have added the following text on page 9.

“However, one first will have to determine the model parameter $\Omega_m$ and the validity of the SIV paradigm. This could be done by determination of the relevant SIV $\Omega_m$ from the cosmological parameters such as deceleration, jerk, and snap, which were recently constrained using model-independent kinematic cosmographic study utilizing three different data sets and their combinations \cite{2024A&A...690A..40L, 2024arXiv240412068C}.”

As to the questions on the “possible explanations of such problems”: The SIV theory, which considers the possibility of an additional symmetry in the gravitation theory, solves a number of problems currently found in Astronomy. Regarding first the accelerated expansion of the Universe, it occurs quite naturally by the presence of an additional term in the cosmological scale invariant equations (see eqs. (1) & (2) in general and (6) & (7) for the SIV theory). Also, the modified Newton equation, which contains one small additional term (see eq. (19)) allows to account for the mass excess in galaxy clusters, for the flatter rotation curves in galaxies, for the deviations of classical mechanics recently observed in the motions of very wide binaries. Moreover, SIV theory is able to account for the departure usually found between the mass from lensing and the estimated stellar content of galaxies [Symmetry 2024, 16(11), 1420]. Concerning, the tension between H_0 from the CMB fluctuations and from distant supernovae SNIa, we strongly believe that the Hubble tension could be resolved via the new early-dark-energy term as identified in the discussion paragraph following eq. (13) and (14) and explicitly seen in RHS of eq. (6) & (7) within the SIV theory. However, substantiating this claim depends on future work that should involve the growth of density fluctuations [Physics of the Dark Universe 25 (2019) 10031] but with focus on the CMB phenomenon.

Finally, we point to the fact that the Einstein Cosmological Constant within the SIV framework is a true constant as indicated in the text between eq (15) and (16).

Comment 3: Perturbations within SIV could shed light on the possibility to disentangle the two approaches. Is there any chance to explain them and to relate the stuff done here with them?

Response 3: We are pleased with this question since it was a success of the SIV theory. Traditionally, in the standard model, the density fluctuations from the Big Bang do not have a growth rate sufficient to allow galaxy formation in the radiation era. This was a significant problem presumably solved by the costly hypothesis of a dark matter (DM), sensitive to gravitation but not to radiation pressure. This way, the baryons would set in the potential wells kindly prepared by DM. This hypothesis is not necessary in SIV theory. As shown by the two present authors [Physics of the Dark Universe 25 (2019) 10031] the modified equation of the weak fields naturally favors the fast growth of the density fluctuations at a rate that even allows the formation of massive galaxies in the very early stages of the Universe, as recently found by the JWST.

Reviewer 2 Report

Comments and Suggestions for Authors

The paper seems interesting, although some subjects are left unexplained, for exaple, the authors observed that scale invariance of the vacuum for flat cosmology (k = 0), which is broken 78

only by the presence of source fields charactarized by the energy-density of matter and its 79 pressure.

Well , that means that the scale invariance IS NOT A SYMMETRY, Therefore the choice of lambda IS NOT A GAUGE as the authors claim, Another choibe would lead to ADIFFERENT PHYSICAL THEORY. All this is very unclear, if the authors could expand on this, becuse due tho the breaking of dcale invariance the choice of lambda is not a gauge, I ask the authors to clarify this point in their paper before publication

Author Response

We are grateful to the reviewers for taking the time to review our manuscript and highly appreciate the reviewer’s comments, questions, and suggestions. Our responses are in color text below as well as in the text of the new manuscript.

Comment 1: “… the authors observed that scale invariance of the vacuum for flat cosmology (k = 0), which is broken only by the presence of source fields characterized by the energy-density of matter and its pressure.”

Response  part 1: We believe that the referee is concerned by the simultaneous presence of a symmetry, of matter and of flatness, a remark which we well understand.

General Relativity and Maxwell equations are scale invariant in absence of matter and charges. Matter is breaking scale invariance, a result already shown by Galileo Galilei and by Feynman. An interesting result of the SIV cosmological models is that as soon as matter exists in the Universe, scale invariance is drastically reduced to totally disappear in model Universe with a mean density equal or higher than the critical density r_c. Since our Universe is currently considered to have W_m= 0.20-0.30, scale invariance is not completely killed and some minor effects are still present. They appear both in the cosmological models and in the weak fields particularly on very long time scales.

Regarding flatness, the equations of the cosmology containing one more term than the Friedmann’s equations (see (6) and (7)) allow flat solutions for a variety of W_m values, not necessarily W_m =1 as in Friedmann’s, the balance of the equations being assured by the additional term. We emphasize that these equations have also been derived from an action principle [see Symmetry 2023, 15(11), 1966].

Response part 2: As to the question on the choice of lambda and that “another choice would lead to A DIFFERENT PHYSICAL THEORY”. Once scale invariance is restored by an arbitrary conformal transformation with a conformal factor \lambda(t), that is transferring the equations from EGR into WIG frame, then the extra terms containing \lambda appear, as seen in (1) and (2) upon a→a \lambda and dt → \lambda dt applied to the standard Freedman equations while utilizing that G\rho → G\rho/\lambda^2 due to its time units. Thus, any particular choice of \lambda(t) would define a particular WIG frame but not a different physical theory. We are interested in one particular frame given by (3) where when applied to (1) and (2) one can see that scale invariance is valid upon a rescaling with a constant factor. One can expand the choice for \lambda by using only the \lambda terms in (1) and (2) to define equations for \lambda and in doing so arriving at (4) and (5), or in general by considering the equations (13) and (14). This can be interpreted as a choice of parameterization that brings the GR equations into the true co-moving frame with no Cosmological Constant as extra energy density. This is similar to what happens when choosing a special motion such that the kinetic energy of a system is zero, but in this case, it is the time parametrization controlled by \lambda see the text added on page 6.

Reviewer 3 Report

Comments and Suggestions for Authors

The authors tried to explain the cosmological constant problem and the dark matter problem by arguing the scale invariant vacuum paradigm. Although I didn't check the correctness of the derivation, especially Eqs. (4) and (5), It seems to me that they are correct. The explanation of the small cosmological constant is interesting if the argument is correct. It will be useful to get the modified Newtonian dynamics from the Weyl scaled GR in the Newtonian limit, I am not sure if this is possible. The authors cited many works they wrote before which are relevant to this paper. However, they didn't give the journal name in the references. 

Author Response

We are grateful to the reviewers for taking the time to review our manuscript and highly appreciate the reviewer’s comments, questions, and suggestions. Our responses are in color text below as well as in the text of the new manuscript.

Reviewer 3 Comment: The authors tried to explain the cosmological constant problem and the dark matter problem by arguing the scale invariant vacuum paradigm. Although I didn't check the correctness of the derivation, especially Eqs. (4) and (5), It seems to me that they are correct. The explanation of the small cosmological constant is interesting if the argument is correct. It will be useful to get the modified Newtonian dynamics from the Weyl scaled GR in the Newtonian limit, I am not sure if this is possible. The authors cited many works they wrote before which are relevant to this paper. However, they didn't give the journal name in the references.

Response: We are grateful to the reviewer for reading and commenting on our paper. We appreciate the remark about the journal name in the references and have removed the references that were only on the preprint server in favor of the later papers that have been published in peer-reviewed journals. As for the MOND comment we have to point out that the weak field limit of the GR, which leads to Newtonian gravity, is still too strong of gravity to allow the manifestation of the MOND behavior. However for extremely weak Newtonian gravity, which is below the MOND cutoff acceleration a_0, the deep MOND limit has been derived in section 4.3.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The paper studies a theory that has local conformal invariance in thegravitational sector, however, as the author acknowledge, this symmetry is broken by the matter, they fix however a gaugefixing through a gauge fixing that  is not related to the natureof thebreaking in the theory.

This is not how gauge symmetry breaking for fixing the appropriate gauge fixings. Take for example a massles U(1) gauge theory, this model has the U(1)  gauge symmetry, Add a gauge symmetry breaking mass term, and the consistency of equations of motion force the  condition that the 4Divergence of the gauge potential is zero, showing that breaking term  induces also a gauge fixing tat is associated to this symmetry breaking, this contradicts the independent choice of gauge fixing and breaking term, that means the procedure followed is incorrect, I told theauthors to look at this , instead they provided a non sense response.

OK , so until the authors figure out a way out to resolve this issue, the paper cannot be accepted. 

 

Author Response

We are grateful to the reviewer for expressing his opinion that “they fix however a gauge fixing through a gauge fixing that is not related to the nature of the breaking in the theory” which has indicated to us that we have not been successful in explaining how the “gauge” choice is made and what is its significance. So, we have added more text to the paper that should be clarifying the situation. The main point we would like to stress here is that the “gauge” symmetry of the SIV theory, is not the usual local gauge symmetry we are all familiar with from particle physics. Our response is in the blue color text below as well as in the text of the new manuscript.

The following text has been added to the paper on pages 6 & 7:

“This is similar to what happens when identifying the co-moving frame such that the kinetic energy of a system is zero and therefore there is no relative special motion. However, we do leave in 4D spacetime, which brings up the question about relative time parameterizations; that is, what if the coordinate time of the observer is different from the proper time of the system under study? It seems the relative time parametrization controlled by λ also controls the amount of extra energy that there could be.

Another way to understand the situation is to recognize that the positive cosmological constant ΛE on the LHS of (11) indicates extra energy density as part of the RHS (11). The presence of ΛE explicitly breaks the global rescaling symmetry with the ρ, p, and k/a² terms in the Freedman equations. The breaking is still there even for the macroscopic vacuum, characterized by ρ = p = k = 0 if ΛE is non-zero. This can be viewed as a manifestation of unproper time parametrization, since for proper time parametrization one expects zero energy density instead. To correct the time parametrization one can apply global conformal transformation λ(t) instead of the commonly discussed local conformal gauge λ(x). The use of λ(x) would imply the presence of a physical field whose excitations should manifest as particles, which is not permissible [22]. Thus, the idea of using λ(t) is well justified in order to preserve isotropy and homogeneity of space. It is aligned with the above discussed idea about the role of time parametrization. Therefore, the existence of λ(t) as defined by (17) removes ΛE from the Freedman equations and results in (6) and (7), which are clearly scale invariant when ρ = p = k = 0. This demonstrates the relationship between the scale-breaking therm ΛE, and its relation to the symmetry-restoring WIG frame defined by λ(t) given by (17).

Therefore, the “gauge” symmetry of the SIV theory is not like the usual local gauge symmetry, which we are familiar with from particle physics; as such, one can circumvent the earlier mentioned problems by showing that ΛE is an actual constant within SIV. ”

 

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

Paper can be published after the revisions answering my questions

Comments on the Quality of English Language

Enghis is fine