Anisotropic Fractional Cosmology: K-Essence Theory
Round 1
Reviewer 1 Report
Comments and Suggestions for Authors
I read this paper with interest because using fractional calculus is rapidly becoming a tool among researchers in quantum cosmology and the early universe.
Using Caputo's derivative is undoubtedly one step in advancing Fractional Quantum Cosmology. Other authors have mainly focused on other derivatives. The authors have however seen how important it is to address quantum cosmology by discussing either derivative, some other papers and other references (there is a book now with a chapter devoted to this) as well.
On the whole, The results in this paper are curious and of interest. The paper is sound and robust, and I saw no errors or inconsistencies. I consider it a fair contribution to the research community to have in the paper and the Special issue a discussion and contrast of methods plus results (Caputo, K-essence) with, say, other approaches (Riemann-Liouville ie Riesz, GR minisuperspace). Maybe as the investigation in fractional (quantum) cosmology progresses, one may study models with either derivatives and contrast, i.e., Bianchi-I has not yet been studied with the Riesz-Riemman approach. Or some Bianchi with curvature and LRS i.e. rotational symmetry (besides Bianchi-I). But the scope is still young regarding fractional cosmology. The authors have also discussed items upon which one may appraise and with worthiness. This paper will enhance the Special Issue and, mostly, the emergent recent domain of fractional Calculus in gravity and cosmology.
There are a few open avenues that I share with the authors:
Study the cosmic no-hair conjecture i.e. how does fractional calculus change a DeSitter attractor (if any);
Can fractional features assist the generality of inflation?
What would fractional calculus 'pick up': the no boundary or vilenkin?
All are (in my view) worthy queries which may or not bring another paper. It will depend on the results.
So, I do endorse and support the publication of this paper in this Special issue of Fractional and Fractional.
Comments on the Quality of English Language
none
Author Response
comments in attached file
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for Authors
Title: Exploring the Universe: A Look at K-essence Theory
Authors: J. Socorro, J. Juan Rosales, and Leonel Toledo-Sesma
In this research, the authors dive into the Wheeler-DeWitt quantum equation. They find that in certain situations, a new kind of equation related to the scalar field appears on its own. This new equation can change based on specific parameters.
Moving deeper into the world of quantum physics, the authors encounter a challenge, which results in another new equation for the scalar field. They solve this using a method called the fractional power series expansion. They discuss solutions that apply to both classic physics and quantum physics.
Comments and Recommendations
The conclusion of the paper talks about two main alternatives. Diving deeper into these ideas in the main part of the paper could help readers understand better.
In the presented research, where mathematical formulations and technical methodologies are utilized, it's essential to provide a contextual interpretation of these findings. The results should be explained in terms of their significance within the broader framework of current cosmological understanding.
Typo after line 112 '=' is missing from gττ = gττ =− 1
Author Response
comments in attached file
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for Authors
The authors perform a detailed study of anisotropic scalar field models with fractional derivatives. This subject has drawn the attention of cosmologists the last years since it can provide a potential way to solve the cosmological tensions. On the other hand, the universe is a great laboratory to understand the concept of fractional derivatives in nature.
Furthermore, the authors introduce the problem of quantum cosmology and they determine exact solutions for the wave function of the universe.
For me it is not clear what is happening to the anisotropy in large scales, and how the anisotropy contributes in the early universe. A discussion similar to that presented in 2301.07160 should be mentioned. Moreover, the authors should discuss if fractional cosmology can be used to avoid the initial singularties, see also the discussion in 1511.08382. Last but not least, they should discuss the application of the fractional derivatives in the semi-classical limit, see for instance: 2103.00802 ; 2012.108814 ; 1710.0666 .
After the inclusion of such discussion I recommend the present article for publication.
Author Response
comments in attached file
Author Response File: Author Response.pdf
