Epicyclic Oscillations and Circular Orbits in Hairy Black Holes: Testing by High-Frequency Quasi-Periodic Oscillations Observed in Microquasars
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe manuscript studies the high-frequency quasi-periodic oscillations (QPOs) in the accretion disk around a class of static spherically symmetric black holes with scalar hair. The QPOs are described within the geodesic resonance models using a simple resonance between the radial and the vertical epicyclic frequencies. The theoretical predictions for the QPO frequencies are compared with the observational data for three microquasars with available mass and spin measurements thus imposing constraints on the solution parameters associated with the black hole hair.
The paper explores some possible phenomenological manifestations of the black hole hair and their compatibility with the experimental data. In this respect it contributes to the broader efforts of testing the Kerr hypothesis and constraining the modified theories of gravity. I consider the paper suitable for publication in Universe with the following recommendations:
1) I would recommend that the author represents the metric in a dimensionless form introducing a dimensionless radial coordinate and a dimensionless parameter l. In the theoretical analysis the problem is circumvented by working with a unit mass. However, when comparing with experimental data this is no longer possible.
2) The metric contains two parameters given by alpha and l. I guess that the value of alpha is fixed to alpha =1 throughout the paper. If this is true, it should be stated more clearly in the text and mentioned in the captions of all the figures. Since this is a particular case, some comments should be included discussing how the variation of alpha would influence the conclusions in the paper.
3) Table 1 contains a fit of the ISCO position to the three microquasars studied in the paper. Since there is no information in the text how this fit is performed, some discussion should be added. In particular, does the author use the Kerr metric in order to access the ISCO for the microquasars?
4) In order to improve readability the values of the experimentally measured QPO frequencies for the microquasars should be mentioned in the text including the appropriate references.
5) How are the inflection points of the effective potential calculated in section 3.2? The condition that the first and second derivative of the effective potential vanish is a necessary but not a sufficient one. In order to ensure that these are really inflection points the higher order derivatives should be also studied proving that an odd order derivative becomes non-zero for the first time. If such analysis is not performed the conclusions about the existence of two inflections points in Fig. 6 could be wrong.
6) Shouldn't 'l=-7' be 'l=-10' in line 118 since there is no such value in Fig. 5?
Comments on the Quality of English LanguageThe text is understandable but it contains many errors and imprecise terminology. A thorough language editing is necessary.
Author Response
Thank you to the referee for their careful reading of my paper and their valuable feedback and assistance in improving it.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThe manuscript contains a study of phenomenological properties of a certain hairy black hole solution. The results might be relevant for a future analysis of astrophysical observations of black holes. There is only a couple of remarks. 1) The author considers motion of massive particles around a hairy black hole. Meanwhile at least a comment on what happens with massless photons for the given black hole metric is certainly required. Moreover, it would be important to know how does the presence of hairs affect gravitational lensing by the black hole. 2) Readers will be curious to learn why in Eq.(8) the energy is squared while the potential is to the first power (in the chosen system of units).
I suggest publication of the article upon a reply for the above remarks.
Comments on the Quality of English LanguageThe manuscript is written in good English. Only minor style corrections are needed. For example. the last sentence comes after a list of several options and says "All of it has to be investigated in the future." Presumably "All of it" should be replaced by "All this" (or similar).
Author Response
Thank you to the referee for their careful reading of my paper and their valuable feedback and assistance in improving it.
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsThe paper applies what is alleged to be a new solution for spherically symmetric black holes to the origin of quasi-periodic oscillations in quasars. The first problem with the paper is Birkoff's theorem which designates the Schwarzschild solution as the only spherically symmetric vacuum solution. I assume then that the Ovalle solution is not a vacuum solution. If so, this should be stated, otherwise there should be a coordinate transformation from Ovalle to Schwarzschild.
The paper is otherwise technically competent, but the results are inconclusive. However the worst aspect is the presentation.
1. For example, Units are not discussed clearly. By setting c=G=1, we have only cm or sec as units. So setting M=1 is a physical choice and should be discussed.
2. \ell\rightarrow -\infty is frequently discussed, but said to be unphysical. Yet it is used in equation (5). Similarly, it is frequently unclear whether \alpha is 1 or not. See for example equation (4).
3. Equations 8-9 are essential to the argument and the connection should be explicit. \ell\rightarrow-\infty again in equation (11).
4. the figure captions are terrible. quantities such as l_p are introduced before definition. It is rarely clear that alpha=1, and L=3\sqrt(3) is given before explanation. figure 3 seems to have L rather than \ell in the different panels. figures 6 and 7 need caption clarification! Explain the squiggle on the right of the left panel. The orange line and right panel are not described until the discussion in the text, which is also not clear. The curves in figure 7 do not meet smoothly.
5. Part of the problem is that the definitions of ISCO, standard ISCO, IPEP, LSCO are not clearly defined, or the definitions are not readily accessible.
In the key orbital section, one refers to circular orbit with L=0. It may be that a limit is intended. Is eqation (7) the right reference for \omega_\phi?The 1/2 in the Hamiltonian (presumably mass) looks odd and should be commented.
6. Figure 9 has a poor notation on the ordinate. Why is the 3:2 with Schwarzschild relevant?
7. Table 1 needs a better caption. I don't see where the bounds -3 to -0.7 come from.
8. Finally, as the author admits, the advantage of all this is not clear. It adds one other approach to QPO's, but magnetic fields and accretion discs add much complexity to the system.
Author Response
Thank you to the referee for their careful reading of my paper and their valuable feedback and assistance in improving it.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsThe author has taken into account all my comments and significantly improved the text of the paper. I recommend publication in the current form only with minor language editing.
Comments on the Quality of English LanguageThe text is understandable and need only a minor language editing.
Reviewer 3 Report
Comments and Suggestions for AuthorsPaper presentation is improved. One should emphasise that the Ovalle solution must have significant surrounding mass to avoid the Birkhoff theorem in a vacuum. Otherwise the effects are small.
Comments on the Quality of English LanguageIt is acceptable with minor faults.
