Neutron Star–Dark Matter Admixed Objects in the Mass Gap Region
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThis manuscript presents the still not much considered scenario of a DM admixed NS where DM is the majority. This scenario is worthy of study, but I find the discussion in this manuscript not sufficiently clear and in depth. In general the results must be presented in a clearer fashion. In particular
1) minor point, but the author should clarify the assumptions on the interactions that allow to separate the hadronic and DM pressures, and write two separate TOV equations (this is generally not the case, so it would be useful to point this out for the reader)
2) The figures are unclear. What do the two curves correspond to? the part on stability presumably does not correspond to the first panel? What is the red square on the plots? What does the the red cross signify? what are the number next to it? They do not seem to correspond to what is on the axis?
3) Apart from an in depth re-analysis of the figures, the configurations should also be explained in detail, as in some of the plots there appears to be a standard NS branch at lower masses, but then a standard less compact branch, but it is not explained exactly what is being plotted
4) It is also unclear what the consequences of such configurations would be. The author mentions gravitational waves. Presumably one would see a merger in the mass gap, but the two objects would touch well before the point a BH would merge. Hence there should be a signature at lower frequency when the DM halos begin to interact (depending on the strength of the interaction)...or if they do not, then a counterpart for the hadronic merger which would be unexpected for the high masses being measured.
Author Response
I want to thank the reviewer for their kind and on-point comments concerning the research. I tried my best to fix the issues and include the information needed.
Comment 1) minor point, but the author should clarify the assumptions on the interactions that allow to separate the hadronic and DM pressures, and write two separate TOV equations (this is generally not the case, so it would be useful to point this out for the reader)
Respone 1) The derivation of the Two Fluid model can be found in the relative references. I could include some parts of it, but I think that it would overload the paper and drift the aim of the specific paper which is the existence of such objects in the mass gap region. But I will be happy to include it if the reviewer insists.
Comment 2) The figures are unclear. What do the two curves correspond to? the part on stability presumably does not correspond to the first panel? What is the red square on the plots? What does the the red cross signify? what are the number next to it? They do not seem to correspond to what is on the axis?
Response 2) The reviewer is correct and I happily entered more text and in depth analysis of the figures and the research in total.
Comment 3) Apart from an in depth re-analysis of the figures, the configurations should also be explained in detail, as in some of the plots there appears to be a standard NS branch at lower masses, but then a standard less compact branch, but it is not explained exactly what is being plotted
Response 3) I corrected everything by entering more detailed text concerning the figures and their analysis.
Comment 4) It is also unclear what the consequences of such configurations would be. The author mentions gravitational waves. Presumably one would see a merger in the mass gap, but the two objects would touch well before the point a BH would merge. Hence there should be a signature at lower frequency when the DM halos begin to interact (depending on the strength of the interaction)...or if they do not, then a counterpart for the hadronic merger which would be unexpected for the high masses being measured.
Response 4) I included some text in the end of the concluding remarks talking about the possible detection. If further information is required I will be happy to include it.
Reviewer 2 Report
Comments and Suggestions for AuthorsThe author postulated a mixture of DM and normal matter for possibly large masses of neutron stars with the inclusion of the repulsive DM interaction. This may add some insights into understanding the mass gap. However, the manuscript can not be accepted in the present form for the unclear physics and poor formulation. Specifically, they are given as follows.
1) Though lots of the references about the DM-admixed NS were quoted, it is not appropriate to state that " But all of the aforementioned work is limited to working with a small percentage of DM in NS" in line 26, since the work with a large percentage of DM in NS can also be found in the literature quite often. For instance, see the recent work "Fermion Proca Stars: Vector Dark Matter Admixed Neutron Stars" by Jockel et al.
2) DM may have interactions between themselves. While a strong long-range repulsive interaction is postulated to contain the more mass in the gravitational well, the author should discuss other possible observable effects for these DM. In addition, it deserves to discuss whether there is any constraint on the interaction strength of DM.
2) According to Eq.(7), the DM fraction was given by the exetrma of the DM number as a function of the central pressure.
However, in addition to the central pressure, the conditions to determine the N_DM should also include the DM interaction if the DM particle mass is fixed. If the stability is established on the condition that the extremum of DM number is obtained, it is just a point, not the curve that N_DM decreases with the increase of N_b. The confusion needs to be clarified.
3) The main text (section 4) was not formulated acceptably. There were no corresponding analyses and discussions of the results shown in figures. No quotations to these figures can be found in the text. The writing should abide by appropriate norm. The author can not expect the readers to understand more than unclear information in figures. For instance, the red cross on the curve in Fig. 1 which stated the values R_star=11.9 km and M_star=1.4_sun can not have a right extraction from the axis scale.
4) The derivation of eqs.(1-4) does not seem to be obvious, especially when there is strong interactions. A derivation would be necessary unless the quotation directs to some right literatures.
Comments on the Quality of English Language
not so bad.
Author Response
I want to thank the reviewer for their kind and on-point comments concerning the research. I tried my best to fix the issues and include the information needed.
Comment 1) Though lots of the references about the DM-admixed NS were quoted, it is not appropriate to state that " But all of the aforementioned work is limited to working with a small percentage of DM in NS" in line 26, since the work with a large percentage of DM in NS can also be found in the literature quite often. For instance, see the recent work "Fermion Proca Stars: Vector Dark Matter Admixed Neutron Stars" by Jockel et al.
Response 1) I removed this line of text, it was bold indeed and it was my mistake. Thanks to the reviewer for this crucial correction.
Comment 2) DM may have interactions between themselves. While a strong long-range repulsive interaction is postulated to contain the more mass in the gravitational well, the author should discuss other possible observable effects for these DM. In addition, it deserves to discuss whether there is any constraint on the interaction strength of DM.
Response 2) The strong repulsive interaction is required for the large accumulation of DM and the formation of these objects. A relevant discussion is cited (ref. 51)
Comment 3) According to Eq.(7), the DM fraction was given by the exetrma of the DM number as a function of the central pressure.
However, in addition to the central pressure, the conditions to determine the N_DM should also include the DM interaction if the DM particle mass is fixed. If the stability is established on the condition that the extremum of DM number is obtained, it is just a point, not the curve that N_DM decreases with the increase of N_b. The confusion needs to be clarified.
Response 3) The confusion is clarified by the addition of new text. This was indeed a mistake by my side and I included all the work done and discussed the figures in more detail.
Comment 4) The main text (section 4) was not formulated acceptably. There were no corresponding analyses and discussions of the results shown in figures. No quotations to these figures can be found in the text. The writing should abide by appropriate norm. The author can not expect the readers to understand more than unclear information in figures. For instance, the red cross on the curve in Fig. 1 which stated the values R_star=11.9 km and M_star=1.4_sun can not have a right extraction from the axis scale.
Response 4) I corrected it and added all the work done and an extensive discussion of the figures.
Comment 5) The derivation of eqs.(1-4) does not seem to be obvious, especially when there is strong interactions. A derivation would be necessary unless the quotation directs to some right literatures.
Response 5) The derivation of these equations of the Two Fluid model is not the aim of this work. But the references given explain the derivation of these equations in more detail. I think that adding more information on this subject could overload this study and drift it from the main goal, which is the supermassive compact object of NS-DM admixture in the mass gap region. If the reviewer insists, I would be more than happy to include some relevant information.
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsThe author has clarified some of the points I had raised, and better discussed the conclusions, and I thus believe that as a preliminary study this work can be published. I still thing, however, that the captions of the figures need to be more extensive (e.g. explaning what the cross is, it is only explained in the text, and arrows are mentioned, but I cannot see them)
Comments on the Quality of English LanguageThe English writing is generally of good level, and the paper reads well.
Author Response
Thank you for your second round of reviews. To ease your concern, I added a little bit of extra in the caption of the figure, but I think that since I give a thorough explanation of the figures in the results and discussion, the size of the caption of the figures is enough. Nonetheless, I added some text to clarify the results.
Reviewer 2 Report
Comments and Suggestions for AuthorsThe manuscript was partially improved according to the last review report. However, it still does not satisfy the average criterion for acceptance, as some confusions remain and the previous review report was not considered very seriously. Considering that the manuscript is an extension of their previous work (ref.1), more specific demonstration of the present study seems to be necessary. Additional improvements should be made according to following points.
1) As said in the Introduction, this work is an extension of ref.1 and focuses on the mass gap between black holes and neutron stars. A review for various efforts studying the gap should be necessarily added somewhere in the Introduction. This is rather informative to readers and can increase the significance of this work, since it is so obvious to increase mass gap by adding the arbitrary repulsion that lacks of constraints.
2) The stability issue seemed to be better formulated. However, the author should discuss the case that the stability is somehow dependent on the stances of normal matter or dark matter. On the DM side, decreasing the DM number with the rise of central pressure also means the instability. On the other hand, the stability curves in figures also contains the instability region. It is certainly not simply the stability curve but a stability discrimination.
3) Confusions remain in the text. Confusions and contradictions are involved in lines 115-119 and in fig.1. a) Is it 3.54 M_sun or 3.56 M_sun? b) While the APR gives a maximum NS mass of 2.16 M_sun, how can DM fraction f=0.03 be for a compact star mass of 3.56 M_sun? c) In this case, how can the NS with R_star=11.9 km and M_star=1.4_sun support f=0.03? d) The arrow mentioned in the caption of fig.1 is not shown in fig.1.
4) In results, the DM mass is relative small and DM-admixed NS has a large size which indicates the boson mediates the DM is light. Such a light boson (say, named U-boson) may also couple to nuclear matter. This possibility was extensively discussed in the past. The derivation of eqs.(1-4) does not seem to be obvious, especially when there is strong interactions. Therefore, a pertinent discussion on this issue is very necessary if the derivation is absent. In fact, nice derivations of two-fluid TOV equations can be found in the literature [ref.34, Naidu et al. PRD104,044014] and appropriate quote is also needed.
Comments on the Quality of English Language
English is readable, but can be improved.
Author Response
Thank you for the second round of reviews. The reviewer made some excellent points and I have tried my best to answer them.
Comment1: As said in the Introduction, this work is an extension of ref.1 and focuses on the mass gap between black holes and neutron stars. A review for various efforts studying the gap should be necessarily added somewhere in the Introduction. This is rather informative to readers and can increase the significance of this work, since it is so obvious to increase mass gap by adding the arbitrary repulsion that lacks of constraints.
Response1: I previousy cited some works on the mass gap for its significance on the final chapter, but I also added a paragraph in the first chapter with the references and added two more references.
Comment2: The stability issue seemed to be better formulated. However, the author should discuss the case that the stability is somehow dependent on the stances of normal matter or dark matter. On the DM side, decreasing the DM number with the rise of central pressure also means the instability. On the other hand, the stability curves in figures also contains the instability region. It is certainly not simply the stability curve but a stability discrimination.
Response2: I added some text explaining how one can receive the stability curves, I also stated that these are the stability discrimination for a singe configuration and explained that in a future project I aim to derive the stability curve
Comment3: Confusions remain in the text. Confusions and contradictions are involved in lines 115-119 and in fig.1. a) Is it 3.54 M_sun or 3.56 M_sun? b) While the APR gives a maximum NS mass of 2.16 M_sun, how can DM fraction f=0.03 be for a compact star mass of 3.56 M_sun? c) In this case, how can the NS with R_star=11.9 km and M_star=1.4_sun support f=0.03? d) The arrow mentioned in the caption of fig.1 is not shown in fig.1.
Response3: a) I corrected it, my apologies, b),c) The compact object is highly influenced by the dark matter particle parameters. The fraction is a secondary parameter that only serves in the creation of the M-R diagram. With these parameters chosen and the M-R diagram provided, one can see that due to the dm particle mass being 200 MeV, the compact object becomes very large in dimensions and in size since the less heavy the particles are, their repulsion plays a bigger role, and this is why we chose a relatively small interaction because the interaction also drives the object to become larger. We first create the pure DM model, spot a mass that suits our research purposes and then slowly add neutron matter to see what happens with the stability. Once we reach a favorable point where we can discriminate the stability from the instability, we stop and we choose a configuration that contains in the center a neutron star of 1.4 solar masses because these are the most common. Then we derive the fraction of the DM depending on the configuration that gives us the 1.4 solar mass neutron star in the center. Now there are some extreme cases that the NS never reaches 1.4 solar masses (fig 2 and 3) and this is due to the massive dm particle and to the relatively strong interaction. c) I changed it to crosses, that was my typo.
Comment4: In results, the DM mass is relative small and DM-admixed NS has a large size which indicates the boson mediates the DM is light. Such a light boson (say, named U-boson) may also couple to nuclear matter. This possibility was extensively discussed in the past. The derivation of eqs.(1-4) does not seem to be obvious, especially when there is strong interactions. Therefore, a pertinent discussion on this issue is very necessary if the derivation is absent. In fact, nice derivations of two-fluid TOV equations can be found in the literature [ref.34, Naidu et al. PRD104,044014] and appropriate quote is also needed.
Response4: I still think that the point of this work is not the derivation of the Two Fluid equations. I cannot add the whole derivation of them, but I have some citations to it. I also included the citation you mentioned if this helps.
Round 3
Reviewer 2 Report
Comments and Suggestions for AuthorsThe manuscript was clearly improved according to the last review report. After following minor but necessary points are revised, I can recommend acceptance of the manuscript that will also be on the decisions of other reviewers and the editor.
1) Line 38, Citation should be added to the heaviest neutron stars (2.5 M_sun). Such a heavy NS should be associated with the fast rotation. It was reported by Romani et al in 2022 that it is about 2.19 M_sun.
2) As the author persisted the direct usage of eqs.(1-5) without derivation, upset may exist reasonably in the results. I strongly suggest to add the causality lines in all the figures of mass-radius trajectories.
3) Ref. 34 presented a nice derivation of the two-fluidic TOV equations if there is no interaction. I suggest to add its citation on line 64 above eqs.(1-5) that should be informative and inclusive.
4) The author should mention in the text with a word that while the APR gives a maximum NS mass of 2.16 M_sun, the small DM fraction f=0.03 supporting a large compact star mass of 3.56 M_sun is due to the strong repulsion.
5) Line 77, the citation was missing. Line 29, the citations are not at right order.
There can be other problems or typos, be cautious and have a thorough check, as my focus has just been on Fig. 1.
Comments on the Quality of English Languagereadable.
Author Response
I want to once again thank the reviewer for their help and fruitful comments. I have tried my best to include all their notices.
Comments1:Line 38, Citation should be added to the heaviest neutron stars (2.5 M_sun). Such a heavy NS should be associated with the fast rotation. It was reported by Romani et al in 2022 that it is about 2.19 M_sun.
Response1: Citation added. Thank you for the suggestion
Comments2: As the author persisted the direct usage of eqs.(1-5) without derivation, upset may exist reasonably in the results. I strongly suggest to add the causality lines in all the figures of mass-radius trajectories.
Response2: If the reviewer means the Rs = 2GM line that is a border, I added it, but in order for it to be more seen, the scale should change to log-log. But changing it to log-log, we lose the true scale of the objects because only powers of 10 are shown as a scale. I really think that for the reader to have a full image of what is going on, this scale should remain. But if the reviewer persists, I will be more than happy to change the scales.
Comment3: Ref. 34 presented a nice derivation of the two-fluidic TOV equations if there is no interaction. I suggest to add its citation on line 64 above eqs.(1-5) that should be informative and inclusive.
Response3: I added the citation, thank you for noticing.
Comment4:The author should mention in the text with a word that while the APR gives a maximum NS mass of 2.16 M_sun, the small DM fraction f=0.03 supporting a large compact star mass of 3.56 M_sun is due to the strong repulsion.
Response4: Text was added to the discussion of the results. I tried to inform the reader about what is going on by using repulsive interactions such as the reviewer suggested.
Comment5: Line 77, the citation was missing. Line 29, the citations are not at right order.
Response5:Firstly, I want to thank the reviewer for their notice, it slipped from my check. I changed the order and tried to make it as clear as possible. Either way, if the reader clicks on the citation they will be referenced there so I think that there will be no confusion.
