Bounce Cosmology in a Locally Scale Invariant Physics with a U(1) Symmetry

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

 

The paper, ``Bounce Cosmology in a Locally Scale Invariant Physics with a U(1) Symmetry,'' proposes a new model of primordial cosmology (solving the usual horizon and flatness problems, explaining the origin of structures, etc.), where the action is Weyl invariant and where the phase of a complex ghost field allows for a non-singular `bouncing background' (viewed in the appropriate `frame'). Stability of the perturbations through the `bounce' is analysed.   The paper is interesting and novel. I am sympathetic of the idea and the attempt. However, there are several things that do not appear to make sense. The manuscript could be reassessed if the author can clarify their work.   \begin{enumerate} \item Starting with Eq.~(1), $\phi$ appears as a ghost field (wrong-sign kinetic term). [This is not critical, but the observation that $\phi$ is a ghost appears about one page after Eq.~(1); since this is rather important, I would have mentioned it earlier, perhaps saying a more thorough discussion would be found later.] Ghosts can give non-singular cosmologies like bounces, but generally, ghosts are bad. This paper claims that the ghost is non-dynamical and that it is classical, hence it is not a problem. I struggle with both assertions, which are related to further issues: \begin{enumerate} \item It is said that $\phi$ and $\phi^*$ are non-dynamical due to the fact that the theory is Weyl invariant. I do not think that this is accurate. If the background cosmology has some dynamics given by the scale factor evolution $a(\eta)$, then the scalar field will have dynamical evolution according to $\phi(\eta)\propto 1/a(\eta)$. Here, the author \emph{chooses} to pick the solution $a(\eta)=1$, hence $\phi$ must also be constant by symmetry. This is a choice, but any other function $f(\eta)$ allows for dynamics according to $a(\eta)=f(\eta)$, $\phi(\eta)\propto 1/f(\eta)$, so this remains a dynamical theory with a propagating ghost field, which is pathological. \item Even if one picks the solution $a(\eta)=1$ and correspondingly $\phi=$ constant, how can it then be that $\chi$ and $\Psi$ in $\phi=\chi e^{i\Psi}$ have dynamics? Clearly this is critical since the ghost nature of $\Psi$ is what gives the `bounce' in $\chi$. Those must be dynamical, hence so must be $\phi$ and $a$. \item Even if the ghost nature of the scale factor is only classical in nature (and hence unproblematic), it would still be a challenge quantum mechanically (and one would still want to be able to have a quantum version of the theory, e.g., quantum cosmology, where the ghost nature of the scale factor is an issue). Likewise, even if $\phi$ is a `classical field', one would want a well-defined quantised version of the theory --- our world is quantum, though it looks classical in some limits. Therefore, if $\phi$ is a classical ghost, it's fine, but that is not satisfactory --- the full theory would need to be quantum and explain how the ghost can be handled. One has a similar story in quadratic gravity, which has a ghost degree of freedom. \end{enumerate} \item Minor points: the starting action has a Higgs field in addition to the Standard Model Lagrangian. Is this the Standard Model without the Higgs? Or is the theory meant to have two Higgs? The starting action also has a self-interaction term, $\lambda'\phi^4$. Are the Higgs, this self-interaction term, the respective potentials for $H$ and $\phi$, as well as the SM, essentially all shoved in Eq.~(5)? \item The theory has a global U(1) symmetry? What does the author do with the lore that quantum gravity may not admit global symmetries? \item It is said of Eq.~(11) that the energy-momentum tensor is not conserved. Considering the matter action is later taken to be akin to a perfect fluid, the energy-momentum tensor should be conserved in and of itself. Shouldn't that entail that the matter Lagrangian cannot have any $\phi$ (or equivalently $\chi$) dependence? In fact, the whole discussion around and below Eq.~(15) is unclear. Why would the EoS take only integer multiples of $1/3$? How is the Lagrangian derived really? Eq.~(15) is some kind of potential for $\chi$ such that the equations of motion are consistent with matter having a perfect fluid EoS $w$? \item Something akin to the matter-bounce scenario is discussed on page 9. It is mentioned that a dust-like component can be obtained with a scalar field with potential $V\sim m_\Phi^2\Phi^2$. Note that this would only be true for $m_\Phi\gg$ Hubble scale, but is the idea that the background is still $a=1$ (so Hubble $=0$) there? Why can't you get your `dust' through $\chi$ as per Eq.~(15) with $w=0$? Regardless, it is said that close to the bounce shear dissipation would dump entropy from $\Phi$ into the CMB. How is this consistent with the fact that the theory's Weyl symmetry does not allow shear (and hence avoids the issue of the unbounded growth of shear anisotropies as the bounce is approached)? \item The matter-bounce scenario is mentioned in the discussion section again, as one way of generating the desired perturbations. It is said that the mechanics does not necessarily involve quantum fluctuations of the metric field. Note that the matter-bounce realisation is precisely like inflation, with the quantum fluctuations of the metric acting exactly as in inflation. Also, because of that, the matter bounce generates tensor modes, just as in inflation. The discussion there should be more clear. Either it is like the matter bounce (which similar in its process and its predictions to inflation), either it is something completely different (e.g., the seeds are thermal fluctuations, but I don't see how this could be the case in the current work). \item In the last paragraph of page 9, the matter-bounce scenario is mentioned and one reads in parenthesis ``or in ekpyrotic models''. Ekpyrosis is not introduced anywhere, there are no references, and the mechanism is rather different from the matter bounce, so this is highly confusing. Why this comment? \end{enumerate}

Author Response

Dear Referee,

The thorough and useful report is very much appreciated. All the comments made in the report are fully addressed in the attached pdf file and the revised manuscript.

Sincerely,

The Author

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The generalization of the Standard Model into a Weyl invariant theory is particularly appealing since its gravity sector, that is a Brans-Dicke theory with additional couplings to matter, could potentially solve fundamental problems in cosmology. The main idea is to describe the expansion of the universe using the dynamical evolution of the gravitational coupling (that depends on a scalar) instead of the scale factor included in the metric ansatz. In the specific case of this manuscript, the scalar field is complex so that the theory is endowed with an additional U(1) symmetry that, as expected, produces a constraint on the scalar field phase, leading to the bouncing driven by the rate change of the modulus of the scalar field. The authors addressed many aspect of the proposed theory in order to reproduce all the achievements of the 𝛬CDM model and solve the typical problems of the hot big bang scenario leaving apart the description of a possible mechanism for primordial density perturbation. The change of cosmological paradigm in this work is not new, but the presence of the bouncing makes the proposed research interesting among the alternatives to inflationary cosmology and relevant for future developments. Few text editing required. I recommend for publication.

Author Response

Dear Referee,

I would like to thank you for taking the time to review my work and for the positive report.  Some minor text editing has been carried out in the revised manuscript (in addition to addressing other referees' comments).

Sincerely,

The Author

Reviewer 3 Report

Comments and Suggestions for Authors

In this work, the authors present a model of gravity where the Einstein-Hilbert Lagrangian is generalized to a Weyl invariant Lagrangian, including the Higgs field and a complex scalar field.
They derive the field equations and discuss the viability of a spatially homogeneous and isotropic cosmological solution of them as a bouncing model candidate, with some interesting results.
This work can certainly be worthy of publication, although I think it would be beneficial if some points were further explained.

1) Are the equations of geodesics for this model the same as the ones of general relativity?

2) In line 245, the authors claim that the U(1) symmetry of the matter Lagrangian results in a spatially isotropic energy-momentum tensor. Why is that? Are unitary transformations of the fields of the theory related to spatial direction?

3) Equation (19) is the analog of the classical Friedmann equation. In this model, it governs the dynamics of the modulus of the scalar field. It is not clear to me how the dynamics of the modulus can be used to describe a cosmological bounce, especially since the metric tensor in the specific coordinate system and frame seems static. Can the authors elaborate on this?

Author Response

Dear Referee,

The useful report is much appreciated. All the comments made in the report are fully addressed below in the attached pdf file and in the revised manuscript.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The author satisfactorily addressed my questions and concerns. The manuscript is a relevant addition to the literature, so I recommend publication.

Reviewer 3 Report

Comments and Suggestions for Authors

The author has adressed my comments so I consent to the publication of the manuscript on the Universe journal.