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Peer-Review Record

Frozen Coherence in an Emergent Universe with Anisotropy

Universe 2025, 11(6), 185; https://doi.org/10.3390/universe11060185

by Helder A. S. Costa^*

and Paulo R. S. Carvalho

Reviewer 1: Anonymous

Reviewer 2: Anonymous

Reviewer 3: Anonymous

Universe 2025, 11(6), 185; https://doi.org/10.3390/universe11060185

Submission received: 23 April 2025 / Revised: 30 May 2025 / Accepted: 4 June 2025 / Published: 8 June 2025

(This article belongs to the Section Cosmology)

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

REVIEW REPORT:

The manuscript Quantum coherence and entanglement in an emergent universe with anisotropy presents a rather clear study in a subject of interest mainly from the point of view of quantum theory and how its properties as entanglement and coherence are affected in curved space-time.

As already indicated in the questions to answer by the referee, the paper is of current interest to the reader.

I find the paper publishable and appropriate for the Journal Universe, both in subject, content and presentation.

I recommend the following revision/improvement before publication:

In the Introduction, line 38, in "previous investigations" Refs are missing and some Refs must be quoted.
How much the results of the paper depend or not of the model of "emergent universe" the authors are using ?
I such a model the early universe before inflation is considered a classical space-time, the authors should mention a justification for that (even if such a model is not their own model, but just used to explore the quantum field theory effects in such curved space time).
In the Conclusions/results The differences between the known results for the isotropic case and the anisotropic of this paper should be more explained
The freezing effect/regime conditions of coherence and entanglement for high H0 and m (H0 and m higher than certain values): It seems these conditions require a small universe and massive quantum scalar field, back reaction or self-consistent cosmological model (with back reaction included) should be considered, or at least to mention the approximation of just using purely classical space-time.

END OF THE REPORT

Author Response

We would like to express our gratitude to the referee for their constructive comments, which have contributed to enhancing the clarity and quality of our manuscript. We have carefully addressed all the points raised. Please find below our detailed responses and the corresponding modifications made to the manuscript.

In the Introduction, line 38, in "previous investigations" Refs are missing and some Refs must be quoted.

Response: We agree with the referee. We have revised the introduction in the new version.
How much the results of the paper depend or not of the model of "emergent universe" the authors are using?

Response: We have added the following comment at the end of Section 6 to clarify this point:

“… It is important to emphasize that the results presented here are model-dependent. Nevertheless, the qualitative features observed, such as the directional dependence of quantum coherence and the emergence of freezing regimes, are expected to be generic properties of non-singular cosmological models exhibiting asymptotically behavior with small anisotropic perturbations.”
I such a model the early universe before inflation is considered a classical space-time, the authors should mention a justification for that (even if such a model is not their own model, but just used to explore the quantum field theory effects in such curved space time).

Response: We agree with the reviewer. The paragraph at the beginning of section 3 has been revised to clarify this point:

“In this study, the emerging model adopted is based on the proposal of Ellis et al. [47,48], in which the universe begins from an Einstein static state and then evolves into an inflationary epoch. An important feature in this model is that there is no time-like singularity. This scenario provides a well-defined spacetime for the semi-classical treatment of quantum field theory. In particular, this approach allows us to define asymptotic vacua and perform the parturbative particle creation analysis using well-established Bogoliubov transformations.”
In the Conclusions/results The differences between the known results for the isotropic case and the anisotropic of this paper should be more explained

Response: We appreciate this important observation. The paragraph at the end of section 6 has been revised to clarify this point:

“In contrast to isotropic models, such as those investigated in Refs. [35,39–41], our findings suggest that the presence of anisotropy introduces a directional dependence in the evolution of quantum features. In particular, coherence exhibits sensitivity to the azimuthal angle ϕ, which serves as a direct probe of the spacetime anisotropy.”
The freezing effect/regime conditions of coherence and entanglement for high H0 and m (H0 and m higher than certain values): It seems these conditions require a small universe and massive quantum scalar field, back reaction or self-consistent cosmological model (with back reaction included) should be considered, or at least to mention the approximation of just using purely classical space-time.

Response: We have added the following paragraph at the end of section 6 to clarify this point:

“In addition, the freezing regime identified in this work are obtained within the approximation of a purely classical background spacetime. In such regimes, the quantum scalar field does not significantly alter the geometry (i.e., backreaction is neglected). Our findings offer a preliminary characterization of the behavior of quantum coherence in an anisotropic expanding spacetime. ...”

We hope that these clarifications and revisions adequately address all concerns raised by the referee. We are grateful for the thoughtful comments and remain at your disposal for any further improvements required.

Sincerely,
The authors

Departamento de Física
Universidade Federal do Piauí

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The authors study the dynamics of entanglement and coherence of a scalar field in an anisotropic Bianchi type I spacetime. This follows in the footsteps of other seminal works in QFT in curved spacetime which study the entanglement dynamics outside black holes, expanding universes etc. I don’t have any strong objections to the publication of this work; my main comment is that the authors finish the numerical analysis section without much comment on the interpretation of the results, e.g. a discussion on the physical reason for the observed behaviour of the entanglement as a function of, for example, the expansion parameter. Providing some physical intuition (e.g. by analysing the behaviour of mode functions or the like) for why the various quantities behave as they do, would improve the impact / contribution of the paper.

Other minor comments:

Line 95: the sentence (where the variables in the equation are defined seems to be incomplete).

Line 128: conforme-> conformal factor? conformally? Illustred -> illustrated

Line 130: express as -> expressed as

Line 179: the authors discussion that Alice and Bob’s detector interact with the field in localised regions via time-dependent switching functions. But where are the switching functions modelled? There does not seem to be any explicit reference to these in the text / any equations.

Line 202: desinty -> density

Line 208: Logarithmic negativity is not only sufficient, but necessary for detecting entanglement

Line 236: enconding -> encoding

Line 238: conforme -> Conformal?

Line 261: Can the freezing effect be shown analytically? This would provide a bit more rigour and intuition to this effect. Also, it is a little difficult to discern from the graph, perhaps the y-axis can be extended slightly above 1.0 to show this.

Line 283: mportantly -> Importantly

Author Response

Main Comment: “My main comment is that the authors finish the numerical analysis section without much comment on the interpretation of the results, e.g. a discussion on the physical reason for the observed behaviour of the entanglement as a function of, for example, the expansion parameter. Providing some physical intuition (e.g. by analysing the behaviour of mode functions or the like) for why the various quantities behave as they do, would improve the impact / contribution of the paper.”

Response: We thank the referee for this important observation. In response, we have expanded the discussion bellow of Fig. 4 in the Section 6 by introducing the comment:

“The results depicted in Fig. (4) demonstrate that coherence freezing occurs when the values of the parameters { a0 , H0 , m, k} lie within the specific regimes outlined in table (1). Within these regimes, the l1 -norm coherence becomes effectively insensitive to variations in the expansion parameters, suggesting that the quantum features of the bipartite state shared by comoving observers remain preserved throughout the cosmic evolution of the emergent universe. The origin of this robustness of coherence can be attributed to the suppression of particle creation (| β k | → 0). This corroborates the fact that the weak curvature of space-time or high-frequency modes minimize the impact of curvature induced mode mixing, thus protecting quantum resources.”

Line 95: “the sentence (where the variables in the equation are defined seems to be incomplete).”

Response: The sentence has been revised for completeness and clarity. It now reads:

“Here, dΣ denotes the volume element of the hypersurface Σ, and n^μ is a future-directed timelike unit vector orthogonal to Σ.”

Line 128: conforme-> conformal factor? conformally? Illustred -> illustrated