Bayesian Analysis of Hybrid Neutron Star EOS Constraints Within an Instantaneous Nonlocal Chiral Quark Matter Model

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In the present paper the authors consider a two model hybrid equation of state and using a Bayesian inference calculation determine the likelihood of the different EoS imposing observational constraints. They present a quite complete introduction. They conclude that the most favorable EOS have a quite low onset of the quark phase corresponding to stars with a mass below one solar mass  and maximum masses that can go above 2.15M_Sun. They analyze separately the effect on the likelihood of different observations on top of the ones considered as basic which correspond to the NICER observations and to J0348, in particular, results from GW170817, for the BW J0952, and the HESS observation J1731. For the quark model they take a constant speed of sound EOS that is fitted to non-local chiral quark with a vector term and a two flavor color-superducting term. The study has interesting results but before I can recommend it for publication I would like the authors to address the following points:
- the conclusions drawn with the present study are closely connected to the choice of a quite hard hadronic EOS, in particular hybrid stars with masses below 0.35M_Sun.  This should be discussed.
- Is Fig. 4 showing the prior of the Bayesian analysis? How where the range of parameters defined?
- The authors state in the abstract ``a generic class of hybrid neutron star equation of state with color superconducting quark matter on the basis of a recently developed nonlocal chiral quark model'' is used. In fact an EOS with a constant speed of sound is used,  and beta-equilibrium is only considered at the hadronic level. Could the authors comment how realistic is this approximation?
- One expects that at high densities  strangeness sets in either in the hadronic or quark phase. How could this affect the present conclusions?
- In order to better understand the different scenarios considered it would be nice if a plot with the probability distributions of the value of the corresponding log likelihoods.
- Are the present results compatible with pQCD constraints? A probability distribution that takes into account the results of Komoltsev & Kurkela (PRL 128,202701). are available at zenodo.org/records/7781233

Other comments
- why did the authors choose only one set of NICER for pulsar J0030 and did not consider the average of both analyses?
- is the value of the tidal deformability in line 239 correct? shouldn't it be 190^+390_120?
- Fig 4 and 12 show most of observations used but it also includes J0030 by Riley (not used) and excludes J0437 by Choudhury (used).
- According to ref 57 the GW170817, R_1.4= 11.9±1.4 km, the top limit being above the one indicated in line 353.
- Some figures have too small labels, in particular, Fig. 2 and Fig. 8. In Fig. 8 it is not very easy to distinguish the green and the blue.
- Could you specify which is the Seidov criterion?
- Fig. 6 indicates 68% CI region, but all the rest of the paper refers to 60%. Could the authors verify?
- In Table 2, is E really inside the 60% CI? Looking at Fig. 7 eta_D=1.18 seems to be out of this interval.

 

 

Author Response

We thank the referee for comments and criticism, which we repeat below in black color and provide our answers in italics and blue color. We attach the pdf of the difference file, where new text is highlighted in blue color and removed text is striked through in red color. 

In the present paper the authors consider a two model hybrid equation of state and using a Bayesian inference calculation determine the likelihood of the different EoS imposing observational constraints. They present a quite complete introduction. They conclude that the most favorable EOS have a quite low onset of the quark phase corresponding to stars with a mass below one solar mass  and maximum masses that can go above 2.15M_Sun. They analyze separately the effect on the likelihood of different observations on top of the ones considered as basic which correspond to the NICER observations and to J0348, in particular, results from GW170817, for the BW J0952, and the HESS observation J1731. For the quark model they take a constant speed of sound EOS that is fitted to non-local chiral quark with a vector term and a two flavor color-superducting term. The study has interesting results but before I can recommend it for publication I would like the authors to address the following points:

(1) the conclusions drawn with the present study are closely connected to the choice of a quite hard hadronic EOS, in particular hybrid stars with masses below 0.35 M_sun. This should be discussed.

Originally, in the previous paper [G.A. Contrera et al., Phys. Rev. C 105 (2022) 045808], we found that there were no crossings between the softer hadronic EOS DD2F and the QM EOS. We obtained crossings with DD2 at high densities. Then, we added a density-dependent bag pressure B(mu) to the quark matter EOS and were able to manipulate the crossing region. Using the same QM EOS with the softer hadronic EOS DD2F, we obtain meaningful Maxwell constructions only for a strongly restricted small region in the QM parameter space.

(2) Is Fig. 4 showing the prior of the Bayesian analysis? How where the range of parameters defined?

Yes, Fig. 4 shows all hybrid star configurations which were obtained by solving the TOV equations with hybrid EOS that were obtained from Maxwell construction with the chosen hadronic baseline EOS DD2 and the set of two-parameter quark matter EOS. The region in the 2D parameter plane where a successful Maxwell construction is possible is shown by points in, e.g., Fig. 2 or 7. The parameter space of eta_D and eta_V was chosen in such a way as to have a phase transition to color superconducting QM.

(3) The authors state in the abstract ``a generic class of hybrid neutron star equation of state with color superconducting quark matter on the basis of a recently developed nonlocal chiral quark model'' is used. In fact an EOS with a constant speed of sound is used, and beta-equilibrium is only considered at the hadronic level. Could the authors comment how realistic is this approximation?

The EOS for quark matter is based on a nonlocal chiral quark model with a density dependent bag pressure superimposed to it, in β-equilibrium with electrons and muons according to the publication by G.A. Contrera et al., Phys. Rev. C 105 (2022) 045808. It turned out that the constant-speed-of-sound (CSS) EOS of Eq. (6) provides an excellent fit to this EOS. The dependence of the CSS parameters on the dimensionless vector and diquark couplings (η_V, η_D) is given in Eqs. (7)-(9) and Tab. 1. Because of this equivalence with the microscopic nonlocal CQM the CSS fit EOS can be considered as a realistic quark matter EOS in the range of its application in our work.

(4) One expects that at high densities strangeness sets in either in the hadronic or quark phase. How could this affect the present conclusions?

One of the main conclusions of our study is that an early onset of deconfinement, in the mass range 0.5-0.7 M_sun is favored, with maximum masses of the most favorable hybrid star sequences reaching 2.15-2.22 M_sun. We expect that both conclusions would not be affected by the inclusion of strangeness in the hadronic and quark matter phases, since the onset mass of hyperons in the hadronic phase is well above the onset of deconfinement and therefore would not affect this part of the conclusions. What concerns the maximum mass of the hybrid neutron stars with strange quark matter core, we would expect that a readjustment of the vector meson and diquark coupling strength would be necessary, but they would result in basically the same favorable range of maximum masses as in the 2-flavor quark matter case.

Such a conclusion can be motivated by looking at Fig. 4 of arXiv:2409.05859v2. It can be observed that a maximum mass of 2.3 M_sun is obtained within a hybrid configuration where a SU(3)_f version of the nonlocal model was considered, with a quite strong vector coupling constant. It can be seen that the inclusion of strangeness in the model does not substantially modify the results.

An analogous Bayesian study that would be based on the three-flavor generalization of the present hybrid EOS approach is planned for a forthcoming manuscript.

We added a corresponding discussion in the revised manuscript at the end of Sections 4 and 5.

(5) In order to better understand the different scenarios considered it would be nice if a plot with the probability distributions of the value of the corresponding log likelihoods.

We appreciate this suggestion. However, we find that overlaying the contours of two levels of confidentiality (60% and 90%) to the diagram with the color coded regions of different phase transition scenarios as done in Figs. 7-10 is more instructive than to plot just the probability distributions.

(6) Are the present results compatible with pQCD constraints? A probability distribution that takes into account the results of Komoltsev & Kurkela (PRL 128,202701). are available at zenodo.org/records/7781233

We have checked the compatibility of the EOS in our study with the pQCD limit following the work of Komoltsev & Kurkela (PRL 128,202701). For that we have applied the program available at zenodo.org/records/7781233 starting from the highest energy densities relevant for neutron stars, namely the values of energy density, pressure and baryon density at the center of the maximum mass configuration of a given (hybrid) EOS parametrization. We found that all EOS shown in Fig. 1 and the corresponding M-R relations in Fig. 4 (up to the maximum mass) are compatible with the pQCD constraint in the sense that the EOS at the center of the maximum mass configurations can be joined with the pQCD limit by an EOS that fulfills causality and thermodynamic stability.

We added corresponding text and references on page 8 when discussing Fig. 4.

Other comments
(7) why did the authors choose only one set of NICER for pulsar J0030 and did not consider the average of both analyses?

We do not agree that taking an average of both analysis results would be appropriate. We decided to prioritize the analysis of the team published by Miller et al. since it appeared to us more conservative and thus the better choice for our Bayesian analysis.

(8) is the value of the tidal deformability in line 239 correct? shouldn't it be 190^+390_-120?

Yes, indeed. We corrected the typo in the revised manuscript.

(9) Fig 4 and 12 show most of observations used but it also includes J0030 by Riley (not used) and excludes J0437 by Choudhury (used).

We updated the constraint template for Figs. 4 and 12 according to this observation. We removed also the Riley constraint for J0740 because it was not used.

(10) According to ref 57 the GW170817, R_1.4= 11.9±1.4 km, the top limit being above the one indicated in line 353.

The radius limits R_1.4=10.94 – 12.61 km from Ref. [81] is a result reported as multi-messenger radius determination that takes into account other observational constraints besides tidal deformability for GW170817.

(11) Some figures have too small labels, in particular, Fig. 2 and Fig. 8. In Fig. 8 it is not very easy to distinguish the green and the blue.

We improved the figures where labels were too tiny.

(12) Could you specify which is the Seidov criterion?

The Seidov criterion states that a hybrid star configuration shall be gravitatuionally unstable when the jump in energy density Δε at the phase transition fulfils the criterion 2Δε > (ε_onset+ 3p_onset), where ε_onset and p_onset are the energy density and the pressure at the onset of the phase transition.

We added a footnote with the reference explaining this.

(13) Fig. 6 indicates 68% CI region, but all the rest of the paper refers to 60%. Could the authors verify?

We agree with the referee that it is better to use the same confidence level throughout the paper. Therefore, we changed Fig. 6 so that it also shows the 60% CI region instead of the 68% one.

(14) In Table 2, is E really inside the 60% CI? Looking at Fig. 7 eta_D=1.18 seems to be out of this interval.

We thank the referee for finding this typo in the caption of Table 2. Instead of 60% it should be written 90% CI. We corrected this.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

see the report below: 

 This manuscript studied the Bayesian analysis of hybrid neutron star (NS) EoS，and they presented  the typical mass range of these NSs 1.2−2.0M⊙, with the radii   between 11.9 and 12.4 km,  which is an interesting result.  Before it is recommended for publication, the following  issues would be clarified.  

1.  On the EOS of quark  matter NS, the original idea should be discussion and some related  references should be mentioned. 
2.  for the millisecond pulsars (MSPs), which are recycled in binary systems,  their mass average is about 1.6 solar masses by accretion of 0.2 solar mass, could you imply that the EOS of some MSPs are related to the quark matter?  which should be discussed based on the statistics of pulsar mass, e.g. , the mass of double NS is about 1.3 solar masses and  MSPs' mass can be as high as 2.0 solar masses. 

3. Authors might discuss  the transitions between different phases of neutron or quark matter, which may be  corresponding to    the transition fron NS to quark star ? 

4.  The radii of quark matter NS are about 12 km, which are independent of NS mass, could you explain it in detail in discussion section. 

5.  Before the merge of double NSs, their EOSs should be different from that of the new formed NS  if there  is no black hole formed?

 

Comments on the Quality of English Language

The English quality of this paper is understandable. 

Author Response

We thank the referee for questions and suggestions which we repeat below in black color and to which we provide our replies in italics and in blue color. The changes made in the revised manuscript are highlighted in the attached difference file.  

 

This manuscript studied the Bayesian analysis of hybrid neutron star (NS) EOS，and they presented the typical mass range of these NSs 1.2−2.0M⊙, with the radii between 11.9 and 12.4 km, which is an interesting result. Before it is recommended for publication, the following issues would be clarified.

(1) On the EOS of quark matter NS, the original idea should be discussion and some related  references should be mentioned.

We added a set of references related to the discussion of EOS for quark matter in NS.

(2) For the millisecond pulsars (MSPs), which are recycled in binary systems, their mass average is about 1.6 solar masses by accretion of 0.2 solar mass, could you imply that the EOS of some MSPs are related to the quark matter? which should be discussed based on the statistics of pulsar mass, e.g. , the mass of double NS is about 1.3 solar masses and MSPs' mass can be as high as 2.0 solar masses.

We are going further and state that the most likely case is the very early onset of deconfinement, at masses well below a solar mass. This would entail that all observed pulsars for which we know the masses (including the MSPs) are hybrid neutron stars.

This feature, however, may not be robust if the quark matter EOS would have a strongly medium dependent stiffness as in the relativistic density functional model of Kaltenborn et al. [19], where the twin stars occur in the MSP mass range. In this case, the deconfinement transition may be triggered by mass accretion onto a MSP in a binary which may entail observable features such as eccentric orbits, as discussed in Chanlaridis et al. [4].

(3) Authors might discuss the transitions between different phases of neutron or quark matter, which may be corresponding to the transition fron NS to quark star ?

We never intended to include pure quark matter stars in the discussion. Our model is hybrid. The stars considered are composed of a hadronic crust and a core of nuclear matter with a possible transition to color superconducting quark matter.

From Fig. 12 it can be observed the onset of the QM in the hybrid star core at the point where the thick solid green curve (pure hadronic EOS) connects to the hybrid configurations (coloured thin solid lines).

(4) The radii of quark matter NS are about 12 km, which are independent of NS mass, could you explain it in detail in discussion section.

We thank for this suggestion. In the revised manuscript, we added a discussion of this feature of the favorable M-R relations corresponding to hybrid NS in relation to the recent work by Constanca Providencia et al. and added this reference.

(5) Before the merge of double NSs, their EOSs should be different from that of the new formed NS  if there is no black hole formed?

During a binary NS merger, the matter will heat up to temperatures beyond the Fermi temperature of nuclear matter T=50 … 100 MeV and populate a large region of the QCD phase diagram, very likely including the regions of a mixed phase and the deconfined quark matter phase. The finite temperature EOS is different from the T=0 EOS considered in the present analysis. However, in the inspiral phase of the NS merger, from which tidal deformability can be extracted as in the case of GW170817, the neutron stars did not yet have contact and can well be described by the T=0 EOS.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This is a solid entry in the set of papers which use astronomical observations to constrain the equation of state of cold catalyzed matter beyond nuclear saturation density.  The nuclear setup seems reasonable given the goals of the paper (i.e., the equation of state is not fully general but it does focus on the appearance of deconfined quarks).  I will therefore focus on the use of astronomical constraints, which is not quite up to date.

1. The paper https://arxiv.org/pdf/2412.02850 reanalyzes data from PSR J0348+0432 and finds a mass of 1.806(37) solar masses rather than the previously reported 2.01 (not 2.10 as indicated in the current paper) +- 0.04 Msun.  Mmax>2.0 Msun is probably still a reasonable criterion, but the reference and number should be corrected.

2. It is not clear in the current paper whether Mmax>2.0 Msun is a hard yes-or-no limit (Mmax<2.0 eliminates the EOS, but any Mmax>2.0 is equally good) or whether the maximum mass is implemented as a weight (as in Alvarez-Castillo, Ayriyan, et al. 2016 and Miller, Chirenti, and Lamb 2020).

3. Line 232: the constraints are mis-stated for PSR J0030+0451.  They should be M=1.44(+0.15,-0.14) Msun and R=13.02(+1.24,-1.06) km.  Was the current statement of constraints just a typo, or were the incorrect numbers actually used in the analysis?

4. The updated measurements for PSR J0740+6620, from Dittmann et al. (2024), are M=2.08+-0.07 Msun (still dominated by the radio constraints) but R=12.92(+2.09,-1.13) km.  

5. I like the approach taken by the authors to divide fairly reliable constraints (e.g., radio-derived mass measurements and NICER-derived radius measurements) from ones that are less certain (PSR J0942 and HESS J1731).

6. In terms of phrasing, I recommend that in Figure 12, when the GW 170817 tidal deformability constraints are included, the word "disfavored" be used rather than "excluded".  The tidal deformability measurements are measurements like any other, with uncertainties, so one cannot talk about a hard exclusion.

Comments on the Quality of English Language

The English is pretty good, but it would be helpful if a native English speaker were to go through the manuscript one more time.

Author Response

We appreciate the detailed and constructive feedback of the referee. The comments have helped us refine the manuscript. Below, we repeat the referee's comments in black color provide in italics and blue color our responses to each points and outline the corresponding revisions made to the manuscript. We attach a difference file of our manuscript, where all changes are highlighted.

This is a solid entry in the set of papers which use astronomical observations to constrain the equation of state of cold catalyzed matter beyond nuclear saturation density. The nuclear setup seems reasonable given the goals of the paper (i.e., the equation of state is not fully general but it does focus on the appearance of deconfined quarks). I will therefore focus on the use of astronomical constraints, which is not quite up to date.

(1) The paper https://arxiv.org/pdf/2412.02850 reanalyzes data from PSR J0348+0432 and finds a mass of 1.806(37) solar masses rather than the previously reported 2.01 (not 2.10 as indicated in the current paper) +- 0.04 Msun. Mmax>2.0 Msun is probably still a reasonable criterion, but the reference and number should be corrected.

Thank you for bringing this to our attention. This manuscript was published in December of last year, by which time most of our calculations had already been completed. We are awaiting confirmation of the new estimate through its publication in a journal. For now, we will retain the original values while also mentioning the new estimate and its reassessment. The misprint has been corrected.

We also appreciate the referee for highlighting paper 2412.02850, which we have now included in the reference list of the revised version. At this stage, we prefer to adhere to the optical data analysis of Antoniadis et al. (2013), which yielded a mass of 2.01±0.04 M_sun. We thank the referee for noticing the typo in the mass value and have corrected it. The mass measurement remains an open issue within the community, and there are indications of a third body in the PSR J0348+0432 system. A corresponding publication is currently in preparation.

(2) It is not clear in the current paper whether Mmax>2.0 Msun is a hard yes-or-no limit (Mmax<2.0 eliminates the EOS, but any Mmax>2.0 is equally good) or whether the maximum mass is implemented as a weight (as in Alvarez-Castillo, Ayriyan, et al. 2016 and Miller, Chirenti, and Lamb 2020).

We were among the first to apply Bayesian analysis to neutron star equations of state using mass-radius constraints, and we have never used a strict yes-or-no condition for the maximum mass. From the beginning, we have implemented a more statistically appropriate likelihood approach, similar to what was described in Alvarez-Castillo, Ayriyan, et al. (2016) and Miller, Chirenti, and Lamb (2020). This method ensures that uncertainties in the observational data are properly accounted for and is based on the normal cumulative distribution function, where the mathematical expectation is taken as the mean value of the measurement and the deviation is determined by the uncertainties (see Eq. (13)).

(3) Line 232: the constraints are mis-stated for PSR J0030+0451. They should be M=1.44(+0.15,-0.14) Msun and R=13.02(+1.24,-1.06) km. Was the current statement of constraints just a typo, or were the incorrect numbers actually used in the analysis?

You are absolutely right; this was a misprint in the manuscript. Thank you for noticing it. The correct values of uncertainties in the 68% credibility region are indeed as you pointed out. However, we did not use these values directly in our analysis. Instead, we used the corresponding data provided by Miller et al. (2019) and made available in Zenodo. The data was used to construct the probability density function (PDF) for mass and radius using the kernel density estimation method. This PDF was then used to build the likelihood function in our Bayesian framework. Specifically, we applied the likelihood as the marginalization of the mass-radius curve in the M-R plane, as described in Eq. (14).

(4) The updated measurements for PSR J0740+6620, from Dittmann et al. (2024), are M=2.08+-0.07 Msun (still dominated by the radio constraints) but R=12.92(+2.09,-1.13) km.

The numbers have been corrected, and the reference has been added.

(5) I like the approach taken by the authors to divide fairly reliable constraints (e.g., radio-derived mass measurements and NICER-derived radius measurements) from ones that are less certain (PSR J0942 and HESS J1731).

Thank you for supporting this approach. Yes, we used fairly reliable constraints to determine the coupling parameter values through Bayesian analysis. At the same time, we also found it interesting to examine the impact of constraints from PSR J0942 and HESS J1731 to explore possible outcomes and their implications.

(6) In terms of phrasing, I recommend that in Figure 12, when the GW 170817 tidal deformability constraints are included, the word "disfavored" be used rather than "excluded". The tidal deformability measurements are measurements like any other, with uncertainties, so one cannot talk about a hard exclusion.

We agree with this observation and have corrected the wording accordingly (please, see fig. 4 and 12).

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

In the paper entitled “Bayesian analysis of hybrid neutron star EoS constraints within an instantaneous nonlocal chiral quark matter model”, the authors have studied the equation of state constraints using observational data for masses, radii and tidal deformability of pulsars and a generic class of hybrid neutron star equation of state with color superconducting quark matter on the basis of a recently developed nonlocal chiral quark model by a physics-informed Bayesian analysis. This is an impressive research work combined astrophysics and machine learning. The results obtained in this study are useful for the community of high-energy astrophysics. I recommend its publication after some revisions.

Comments:

(1) In the abstract, do you need to explain DD2?

(2) L115-116: “the relativistic density functional EOS with density-dependent nucleus-meson couplings, denoted as DD2”. L135: “This GDRF EOS which we call ‘DD2’ throughout this work”. Are the two descriptions the same? You may give a uniform description or use different abbreviation if necessary.

(3) L131: “the generalized RDF” and “GDRF” are not uniform in abbreviation.

(4) In tables 1 and 2, some parameters are presented. What are the uncertainties of these parameters? Please discuss this issue and list the uncertainties. Also please check the text to solve the same issue.

(5) L143: You have used m_u = m_d = 2.3 MeV. This value for u quark is suitable. But is it too small for d quark? Please discuss this issue and provide the related references.

(6) In Figs. 2 and 6-10, are there any units or dimensions for the parameters eta_V and eta_D? Please clarify this issue.

(7) In Fig. 11, the squared speed of sound is much greater than 1/3 which is the limitation of gas matter in three-dimensional space. How to explain the difference?

 

Author Response

We thank the referee for carefully studying our manuscript and pointing out weaknesses which helped us to improve the quality of our work. We repeat below the comments by the referee in black color and provide our answers to the referee's comments in italics and blue color. We attach a difference file where the changes made in our revised mansucript are highlighted. 

In the paper entitled “Bayesian analysis of hybrid neutron star EOS constraints within an instantaneous nonlocal chiral quark matter model”, the authors have studied the equation of state constraints using observational data for masses, radii and tidal deformability of pulsars and a generic class of hybrid neutron star equation of state with color superconducting quark matter on the basis of a recently developed nonlocal chiral quark model by a physics-informed Bayesian analysis. This is an impressive research work combined astrophysics and machine learning. The results obtained in this study are useful for the community of high-energy astrophysics. I recommend its publication after some revisions.

 

Comments:

(1) In the abstract, do you need to explain DD2?

We thank the referee for your suggestion. Since our abstract slightly exceeds the maximum word limit, we have taken this opportunity to reduce the explanation of DD2 while ensuring clarity. The meaning of the abbreviation „DD2“ is explained in Section 2 just before subsection 2.1 and once more in the first sentence of this subsection.

(2) L115-116: “the relativistic density functional EOS with density-dependent nucleus-meson couplings, denoted as DD2”. L135: “This GDRF EOS which we call ‘DD2’ throughout this work”. Are the two descriptions the same? You may give a uniform description or use different abbreviation if necessary.

We thank the referee for pointing this out. These two descriptions refer to the same equation of state. To maintain consistency and avoid redundancy, we have removed the explanation in line L135.

(3) L131: “the generalized RDF” and “GDRF” are not uniform in abbreviation.

We thank for pointing out this typo which we have corrected. The abbreviation should be GRDF.

(4) In tables 1 and 2, some parameters are presented. What are the uncertainties of these parameters? Please discuss this issue and list the uncertainties. Also please check the text to solve the same issue.

We do not compute uncertainties for the parameters listed in Tables 1 and 2.

In Table 1, the presented values are derived from a fitting procedure using a set of data points that best reproduce the equation of state (EOS) within the relevant coupling parameter space for our model. The fitting procedure was performed using microscopically calculated quark EOSs with the following objectives:

1) Assessing whether the nonlocal Nambu-Jona-Lasinio (NJL) model for two-flavor color superconducting quark matter can be effectively represented by a constant speed of sound parametrization.

2) Densifying the grid within the given ranges of coupling parameter values to enable a more detailed study of the coupling parameter region.

However, we have added the mean squared error (MSE) and the sum of relative squared errors (RSE) to Table 1 to indicate the quality of the fit.

In Table 2, the values are the results of a numerical simulation, namely, the numerical solution of the Tolman-Oppenheimer-Volkoff (TOV) equations.

(5) L143: You have used m_u = m_d = 2.3 MeV. This value for u quark is suitable. But is it too small for d quark? Please discuss this issue and provide the related references.

Thank you for your comment. The values of quark masses depend on the physical context in which they are used. While the commonly quoted current quark masses are m_u = 2.2 MeV and m_d = 4.7 MeV, these values are particularly relevant for processes involving electromagnetic corrections. However, in the context of strong interactions, particularly within effective models such as NJL, the masses of the up and down quarks are often assumed to be the same, because the mass difference is small compared to the energy scale of QCD and the dynamically generated masses in chiral models. By setting the masses equal, the calculations are simplified without significantly affecting the physical predictions of the model.

(6) In Figs. 2 and 6-10, are there any units or dimensions for the parameters eta_V and eta_D? Please clarify this issue.

The coupling parameters eta_V and eta_D are dimensionless because they are defined as ratios of coupling constants having the same units η_V=G_V/G_S and η_D=G_D/G_S,

as they are defined below Eq. (1).

(7) In Fig. 11, the squared speed of sound is much greater than 1/3 which is the limitation of gas matter in three-dimensional space. How to explain the difference?

The difference arises from the constant speed of the sound model used in our study. However, the conformal limit of 1/3 should be reached at sufficiently high energy densities, there is no strict constraint on the specific energy density at which this must occur.
Usually the conformal limit is not necessarily reached within the interior of neutron stars, since the maximum mass of a stable neutron star typically occurs before the conformal limit. This suggests that realistic equations of state for dense matter do not necessarily obey the conformal constraint in the density range relevant for compact stars.

We have checked the compatibility of the EOS in our study with the pQCD limit following the work of Komoltsev & Kurkela (PRL 128,202701). For that we have applied the program available at zenodo.org/records/7781233 starting from the highest energy densities relevant for neutron stars, namely the values of energy density, pressure and baryon density at the center of the maximum mass configuration of a given (hybrid) EOS parametrization. We found that all EOS shown in Fig. 1 and the corresponding M-R relations in Fig. 4 (up to the maximum mass) are compatible with the pQCD constraint in the sense that the EOS at the center of the maximum mass configurations can be joined with the pQCD limit by an EOS that fulfills causality and thermodynamic stability.

 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have considered my comments and, whenever appropriate, have made changes in the manuscript. I recommend the present paper for publication in Universe.