Scaling Invariance of Perturbations in k-Inflation Models

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors The manuscript studies using the Hamilton formalism two classes of $k$-inflation models giving rise to an enhancement of the curvature perturbations on small scales, being thus relevant for primordial black hole (PBH) formation. In particular, the authors find a scaling invariance of the background equations used in order to tune the model parameters and the initial values in a way that the shape and the normalisation of the curvature perturbation spectrum are not affected. To my opinion, it is a well written paper with some parts needing improvement in terms of presentation to the reader. The main novelty of the paper, according to the authors, is the presence of scaling properties of k-inflation models relevant for PBH formation. However, I am not sure if the scaling properties derived for $k$-inflation models is something unique for such models or they are also present in other inflationary setups. Consequently, I am asking for the authors to address the following questions/comments.   1. I would strongly suggest that the authors expand the Introduction by giving more of motivation for $k$-inflation models and in general for non-canonical inflation. It would be nice as well to include a discussion on some fundamental theoretical setups from which one can get $k$-inflation theories, e.g. string theory. 2. Are the scaling properties found for $k$-inflationary classes unique? Are they also present for other inflationary models? I would expect that in general the parameters of any particular inflationary scenario can always be redefined so as to give the correct normalisation of the curvature power spectrum. If these scaling properties found for $k$-inflationary classes are unique to them, then this should be evidently demonstrated and emphasized within the paper, in particular within the abstract and the conclusions. 3. It is not clear to me how the authors arrive to Eq. (94) which is the key equation of the paper. I would strongly suggest to explain in detail how they arrive to Eq. (94). 4. The authors claim that the rescaling of the Hubble parameter $H\rightarrow c^{-1/2}_0H$ does not affect Eq. (85). I would say the contrary since Eq. (85) contain the Hubble flow slow-roll parameters which are functions of the Hubble parameters. One then would expect that Eq. (85) would be in principle affected by the rescaling of the Hubble parameter. I would just suggest that the authors write explicitly how Eq. (85) transforms under the rescaling of the Hubble parameter $H\rightarrow c^{-1/2}_0H$. 5. Is the curvature power spectrum fine-tuned under the variation of the input parameters? This fine-tuning issue is generically present in single-field inflationary models leading to PBH formation~\cite{Cole:2023wyx}. Thus, it would be nice to have a small discussion regarding this issue within the inflationary set up adopted in this work. 6. I would propose that the authors comment as well on the backreaction issue of small-scale one-loop corrections to the large-scale curvature power spectrum, which could potentially alter the curvature perturbation amplitude measured by Planck being extensively studied in~\cite{Inomata:2022yte, Kristiano:2022maq,Choudhury:2023rks,Ballesteros:2024zdp,Franciolini:2023lgy,Firouzjahi:2023ahg}. Did the authors check that their inflationary setup is protected against this backreaction problem?
Typos Line 27: Add ``point" after ``near inflection". Line 28: Replace ``scales above" with ``scales below" since we need to have a peak on small scales so as to be relevant for PBH formation. In Eq. (15) use the definition symbol to define $H$ instead of equality. Why $\alpha<1$ in the PLLS model? Add ``factor" after ``scale". Line 287: Correct to ``a priori". Line 325: Correct to ``will be represented". Line 327: Add ``point" after ``inflection". Line 331: Add ``point" after ``inflection".

Comments for author File: Comments.pdf

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Reviewer 2 Report

Comments and Suggestions for Authors

The paper presents a technically sound analysis of scaling properties in k-inflation models. The discussion on parameter redefinition to achieve the correct normalization conditions is insightful and valuable for cosmologists working on inflationary perturbations and primordial black hole formation in such scenarios. However, several aspects should be improved before recommending the paper for publication: 

1. The paper introduces two classes of k-inflation models (Class A and Class B), but the rationale behind their selection is not fully developed. Why were these specific models chosen over other k-inflation scenarios in the literature? To what extent are the observed scaling relations an inherent feature of k-inflation models? Aren't they merely the result of a deliberate engineering of the framework?

2. While the paper carefully addresses power spectrum normalization, it does not explicitly compare the predicted scalar spectral index ns and the tensor-to-scalar ratio r with current observational constraints. This omission makes it difficult to assess the observational viability of the proposed scenarios. 

3. Additional details on the numerical implementation of the Mukhanov-Sasaki equation would be beneficial. Specifically, how sensitive are the results to initial conditions and the choice of numerical integration methods? Are the slow-roll conditions violated at the field values leading to a significant amplification of the primordial power spectrum, entering an ultra-slow-roll regime? Also, Figure 1 presents the computed power spectra but lacks a clear discussion on the significance of the quantitative differences between the exact numerical solution and the approximation.

I will reconsider the possibility of recommending the paper for publication once these issues have been adequately addressed.

Author Response

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Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Dear authors,

Please find attached the report.

Yours sincerely,

The referee

Comments for author File: Comments.pdf

Author Response

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Reviewer 2 Report

Comments and Suggestions for Authors

The authors have addressed all the suggestions from the previous report, enhancing the overall quality of the paper and clearly defining its scope and limitations. Therefore, I recommend the paper for publication in its current form.

Author Response

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Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have addressed al the points raised. I am thus happy to recommend the paper for publication.