Quantum Fractionary Cosmology: K-Essence Theory
Round 1
Reviewer 1 Report
15 March 2023 - universe-2299746-report
Quantum fractionary cosmology: K-essence theory by J. Socorro and J. Juan Rosales
The authors discuss the use of the fractional calculus and the quantum Wheeler-DeWitt (WDW) equation for K-essence scalar field as applied to the FRW cosmology. In the Introduction they present basic definitions from fractional calculus and K-essence theory along with classification of various states of the universe based on their barotropic parameter. Then they go into the Hamiltonian formulation for a cosmological models and derive an expression for the scale factor of the universe. Finally, they apply the standard quantization methods to obtain the fractional Hamiltonian, the wave function of the Universe based on the Wheeler-DeWitt (WDW) equation, and then they discuss a broad class of solutions to their fractional Wheeler-DeWitt equation and the related probability distributions in the space of the K-essence field and scale factor related parameter Omega.
In short, the paper applies the fractional calculus to cosmology and explores the probability landscape as suggested by the fractional Wheeler-DeWitt equation. Thus, the paper applies novel mathematical concepts to cosmology. In this respect, I recommend the paper to be considered after major revision that addresses my concerns and comments below.
I) After reading the paper, I still don't understand the statement in lines 45-47: "we can introduce the idea that the kinetic energy of the scalar field should falsify the standard matter by employing the K-essence formalism." How is this falsification performed after all?
II) The authors stated in lines 56-57: "we can use this structure for the quantization program, where the ADM formalism is well known..." but it is not clear to me if they really have applied the ADM formalism and if so then an appropriate discussion of the formalism should be included, if not then the sentence may need to be revised.
III) In my opinion, the paper needs some more discussion as to the origin and meaning of equation (13b).
IV) In my opinion, the paper needs some more discussions as to derivations/connections among the equations: (13b) -> (21) -> (22) -> ? [the equation before (23) needs label] and the relation of X to dφ/dτ.
V) In my opinion, the paper needs some more discussion as to the origin and meaning of equation (24). Notice that the choice of the φ dependence is the reason for the fractional calculus later. The usual wisdom is to expand near a point up-to few powers, what motivates the specific use of this fractional power here?
VI) Equation (26) assumes specific form of H rather than defining it from (24) and the usual definition H=qv-L. It is not obvious that these two Lagrangians (24) and (27) are equivalent. If they are equivalent, then this should be demonstrated and discussed, otherwise the method may not be self-consistent!
VII) The derivation of equation (29) assumes N=1! That us, it should be tau instead of t in (29), isn't it?
VIII) On page 6 in the text before (30) include equation reference in "using the first time derivative of scalar field found previously,"
IX) In equations (40), (41), (42), (44), (45) and (46) the normalization factor ψ_0 is better to be in the denominator; this could be done just by using the negative power of 2. However, a discussion of the choice of normalization needs to be presented to justify the random choices stated in lines 130, 136 and so on.
X) Consider a contour plots for most of the figures, it may be easer to see where is the high and where is the low density probability;
XI) on page 8 lines 133-134: case 2 was not previously discussed on page 4, why? Can you include it as well in the list of case on page 4?
XII) Try to use beta and alpha as early as possible and make sure these symbols have unique use and meaning. May have to change the use of these symbols for coordinate labels to the conventional \mu and \nu in equations (13) to (15). Why using k at all in (19)?
Minor suggestions/comments about typos and errors:
1) FRW cosmology should be Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology instead.
2) It is too strong to state "all areas of science and engineering" using "many areas" is better.
3) What is the meaning of the sub index 0 in the Caputo derivative in equation (3)? Are these extra labels really needed if no other expressions are considered in this paper?
4) In equation (6) F(s) is not defined;
5) The conventional meaning of g in (1) is given in lines 77-78, why so late?
6) In the text before eq. (14) there is an undefined symbol "CE" that should be φ, isn't it?
7) between Lines 83 and 84 in the line on inflation there is "cte" what is it?
8) Lines 89 to 92 a mouthful long sentence, consider rewriting it!
9) Page 6 before (31) replace "in" with "into" in the text "promoting the classical Hamiltonian density in the differential operator";
10) Start using beta in (31) and (32) and in the text before (32).
11) Typo: before (32) use single form in "any real constants" - it is about Q, isn't it?
12) before (33) drop "order" in "in terms of the order β" to avoid confusion with the other "order" that was discussed earlier.
13) line 122 Typo: to be fixed in "become to be ";
14) on page 8 in fig caption change "from up to down" to "from top to bottom" as well as in the other fig captions!
15) use Figure # instead of plot (#);
16) Typos and unclear sentences in lines: 149 "of universe", 151 "other formalism", 167 "the momenta of the scale factor appear", 196 " found to fractional ", 197 "this class of function appears";
17) In eq (42), (45) and (46) for consistency with (39), (40), (41) it may be better to use e^ instead of Exp;
18) Rewrite the first paragraph in the final remarks!
19) In lines 169 and 193 remove "since".
20) In lines 170 to 180 use references to the figures and formulas to clarify your statements.
21) It is unclear what is meant by "huge behavior" in line 202.
Author Response
we attach a pdf file with their answers.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Background
The authors transform the K-essence theory into a conventional one and use it as an exotic matter component for the quantization program.
The paper discusses the classical and quantum solutions for the scalar field in the FRW cosmological model, including a radiation scenario.
Questions
1. The authors should provide clarification regarding the motivation for their proposed generic case Xa.
2. What is the physical meaning of all the solutions in the quantum regime? Can they be interpreted in a meaningful way?
Author Response
We attach pdf file with their answers.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
05 April 2023 - universe-2299746-report
Quantum fractionary cosmology: K-essence theory by J. Socorro and J. Juan Rosales
First I would like to thank the authors for taking the time to respond in detail to my comments, questions, and suggestions. I wish they had colored their new version the way they have written the response instead of using Draftable Comparison that is making it harder to read the new version on a laptop screen - sorry but I am on travel and don't have a large screen to view two pages at the same time. So, the manuscript seems to have been changed a lot but it is hard for me to read it. I don't think that this is how it will appear in print so I don't feel that it is sufficiently improved to warrant publication in Universe yet. There are a few minor further requests:
1) Include the answer to my Q6 as an appendix to the paper;
2) Still don't understand how the functional form of the scale factor a(t) should falsify the standard matter via the kinetic energy of the scalar field. Which scalar field? What is its kinetic energy? How is that falsifiable by the K-essence formalism?
3) The authors mention Appendix B in their response but I was not able to see it!
4) to add the explanation about index 0 in the Caputo derivative in the text of the paper...
And please don't use Draftable Comparison just submit the new version as the main file instead - Thanks
Author Response
we attach file with the answer at this revision 2.
Author Response File: Author Response.pdf
