Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors introduce quantum memory effect and  slow-roll entropy  as vacuum imprint to understand dark energy. They also give numerical simulations based on the Planck 2018, BAO and Pantheon+. This study is interesting and meaningful.  It could be added some discussion on what physical distinct differences between QMM, entropy and material sectors to dark energy in conclusion. It could be considered for publication.   

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This paper argues that a finite Hilbert-space capacity $d_max$​ of Planck cells regularizes the zero-point imprint energy and leaves a residual vacuum-imprint energy with stress-energy of cosmological-constant form, that (for their estimate of $d_max$) matches the observed vacuum energy without fine tuning. The same coarse-grained entropy field has two regimes: gradient-dominated and potential-dominated. The authors derive modified Friedmann equations, linear perturbation equations, provide a single reproducible notebook that reproduces figures and shows how next-generation surveys can test the slow-roll imprint.
Nevertheless, the work exhibits significant weaknesses, particularly in its theoretical foundation.Addressing these shortcomings is essential before the work can be considered for publication.

1) The bibliography is incomplete or inappropriate. For instance, the references related to the heat kernel literature are not appropriate.

2) The discretisation is introduced in a largely phenomenological manner. The authors should more clearly discuss its relation to other, more fundamental UV complete approaches to discrete quantum gravity (causal sets, loop quantum gravity, causal dynamical triangulations); otherwise, the discretisation and the results derived from it appear rather arbitrary. All the results depend on $d_max$. 

2a) Especially, many of these quantum gravity approaches also make predictions for dark energy (see everpresent \Lambda).

3) The use of the heat kernel is done in an inappropriate way. Are the authors using Euclidean heat kernel? For cosmological applications Lorentzian heat kernel should be used.

3a) The heat kernel results presented in appendix A and their dependence on $d_max$ is assuming that $S$ is a tensor with $d_max$ components. This choice is not motivated at all.

4) The derivation neglects $O(R^2l_{Pl}^2)$ corrections which could renormalizthe vacuum enrgy  in high-curvature regimes. This is again important for UV completion and for robustness of the numerical coincidence.

5) The model has to be embedded into the full cosmological evolution. The two regimes are imposed ad-hoc.

6) Derivative couplings of $S$ to metric perturbations could change the value of $f_{NL}$. Why is this not considered?

7) If I understand it correctly, in this setup, the Planck-cell capacity has already saturated, so the “writing” of imprints continues only sluggishly. The dynamics are then overdamped by Hubble expansion, hence the analogy to slow roll. It’s a way to parametrize residual dynamics of a field that is nearly frozen but not exactly constant. How is this mechanism justified? How far is that from other dark energy models (quintessence or others)?

For these reasons I reccomend a through revision of the paper before being reconsidered for publication.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

No comments

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have addressed some of my concerns. Still, there are important points which I feel have to be addressed before publication.

1) I suggested to revise the references on the heat kernel but still references 19 and 20 are AI generated, this is not acceptable.

2) The connection to Lorentzian is unclear and the comment about Wick rotation not to be found in the text. In particular, some of the coefficients will be complex, as shown in 2404.18220 [hep-th] and 2507.14296 [hep-th].

4) Reference 22 is not Group Field Theory, but Asymptotic Safety. Either the authors should choose an appropriate GFT reference or mention the renormalization group flow of asymptotic safety.

There is also an additional point. Holographic principle: how is this used? Make a clear distinction with the known approaches in AdS/CFT.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have addressed my concerns and the paper is ready for publication in its present form.