Was GW170817 a Canonical Neutron Star Merger? Bayesian Analysis with a Third Family of Compact Stars

Round 1

Reviewer 1 Report

The authors propose an interpretation of the GW170817 event in terms of the merging of an hadronic star and an hybrid star or a hybrid star with another hybrid star. The EoSs adopted in the paper are the relativistic mean field DD2 and a particular version of the NJL model which includes superconductivity and a mechanism which mimics the effect of confinement. 

I think the results of the paper deserve publication. In particular they are consistent with both the GW170817 data analysis as well as with recent NICER data on Mass-radius constraints.The existence of quark matter in neutron stars is therefore compatible with present astrophysical data.

I ask to the authors to adress just a few minor points:

1) The authors use a Maxwell construction to join the hadronic and the quark phases. This is of course acceptable but I think they should at least comment on the existence of another possibility like the Gibbs construction.

2) I would also appreciate to have a comment on the possible presence of instabilities that may arise using the Maxwell construction in the Mass-radius relation. Is there the appearence any of such instability spanning the parameter space of the quark EoS? ( Journal of Physics G: Nuclear and Particle Physics, Volume 37, Number 2)

3) In the conclusions when referring to the hyperon puzzle I think appropriate to quote also results from microscopic calculations:

Phys.Rev. Lett. 114, 092301

Eur. Phys. J. A 55:207  

Author Response

We thank the referee for comments and suggestions which we repeat below in black color. Our replies are in red color. We attach a file "diff.pdf" that highlights the changes in the manuscript.

1) The authors use a Maxwell construction to join the hadronic and the quark phases. This is of course acceptable but I think they should at least comment on the existence of another possibility like the Gibbs construction.

As we have pointed out already in the Abstract of our manuscript, we have performed a mixed phase construction which agrees well with a pasta phase construction that would be even preferable over the Gibbs construction (so-called “Glendenning construction”) which would assume a vanishing surface tension.

As we have shown in Maslov et al. PRC100 (2019), this Glendenning construction case would correspond to our mixed phase parameter Δ_P~5-6%. We vary in this work Δ_P=0 … 8 %, thus fully covering and including pasta and Gibbs constructions. In subsection 2.2 we have described the mixed phase construction in detail, including references.

2) I would also appreciate to have a comment on the possible presence of instabilities that may arise using the Maxwell construction in the Mass-radius relation. Is there the appearence any of such instability spanning the parameter space of the quark EoS? ( Journal of Physics G: Nuclear and Particle Physics, Volume 37, Number 2)

We are sorry to have the impression that the referee has overlooked some major aspects in the contents of our work. We have not only addressed the aspects of the instability due to the Maxwell construction which lead to the formation of the third family branch in the M-R diagram, but we also quantified in terms of the mixed phase parameter Δ_P what it takes to remove this instability and make the third family branch join again to the second family branch. In that case the formation of hybrid stars is no longer associated with the occurrence of the gravitational instability leading to the separate third family branch.

3) In the conclusions when referring to the hyperon puzzle I think appropriate to quote also results from microscopic calculations:

Phys. Rev. Lett. 114, 092301

Eur. Phys. J. A 55:207

We thank for pointing out further references which we included in the revised version.

Author Response File: Author Response.pdf

Reviewer 2 Report

The authors of “Was GW170817 a canonical neutron star merger? Bayesian analysis with a third family of compact stars” discuss the possibility that GW170817 did not involve two conventional neutron stars but at least one hybrid star. I find that the paper is interesting and clearly written. Below my comments and questions.

Comments/questions

1) For the considered models it seems that the analysis suggests the onset of deconfinement for low stellar masses, around 0.5 M_\odot. Which is the corresponding value of the central density? Please comment on this aspect and on the corresponding stellar radius.

2) It seems that even including the tidal deformability, it is hard to discriminate standard neutron stars from hybrid stars. The authors show the effect of a particular fictitious Nicer measurement on their analysis, but it is not clear whether this would really help to distinguish hybrid equation of states from a nuclear ones.
It would be interesting if the authors could indicate a set of Nicer measurements that  may discriminate nuclear equation of states from hybrid EoSs.

Author Response

We thank the referee for comments and questions that we repeat below in black color. Our replies are in red color. We attach a file "diff.pdf" where the changes in the manuscript are highlighted (red: removed text, blue: added text).

Comments/questions

1) For the considered models it seems that the analysis suggests the onset of deconfinement for low stellar masses, around 0.5 M_\odot. Which is the corresponding value of the central density? Please comment on this aspect and on the corresponding stellar radius.

In Fig. 2 we add selected points of phase transition onset for which we add the values of baryon density in the mass-radius plot. Then, at the same time the corresponding radius values can be read off. The lowest onset density in our model and shown in Fig. 2 is 1.44 n₀.

In Figs. 8, 10 and 11 we also show the EoS constraint region by Hebeler et al. as hatched grey region, which starts to spread at 1.1 n₀. All our onset densities for deconfienment therefore lie well above the nuclear saturation density n₀. We added a corresponding comment in the text.

2) It seems that even including the tidal deformability, it is hard to discriminate standard neutron stars from hybrid stars. The authors show the effect of a particular fictitious Nicer measurement on their analysis, but it is not clear whether this would really help to distinguish hybrid equation of states from a nuclear ones.

It would be interesting if the authors could indicate a set of Nicer measurements that may discriminate nuclear equation of states from hybrid EoSs.

Our aim was to point out that in a situation when the LVC measurement of compactness (radius) on GW170817 would be in contradiction with the NICER measurement of radius in the same mass range, still both can be right! Then, a strong phase transition like it is modeled in our class of hybrid EoS, could offer a solution to such a contradiction because of the presence of mass twin stars, i.e. stars with significantly different radii in the same mass range. We illustrate this with new suggestions for fictitious measurements of NICER.

Such a solution would not be possible with purely hadronic EoS or hybrid EoS with a gentle phase transition that does not produce the third family branch. Most of the other, seemingly more complete classes of EoS are based on multipolytrope EoS and they in most cases miss out on the twin stars because either they to only a three-polytrope approximation to the high-density EoS [e.g., Hebeler et al.] and/or their variation of the matching densities for the piecewise polytropic parts are not appropriate to capture large density jumps as required for the occurrence of the twins and third family solutions.

Author Response File: Author Response.pdf

Reviewer 3 Report

In the manuscript under consideration, the authors perform a Bayesian analysis to make a statement on
the likelihood that their model is able to describe several neutron star observables. In particular the
authors discuss the (according to the Bayesian analysis) most likely set of parameters and the resulting
equation of state for dense nuclear and quark matter.
The paper addresses an important question and tries to highlight a method to quantitatively compare models
for the equation of state of neutron star matter with experimental observations.

It is reasonably well written, with a few shortcomings that I will address below.

I think the manuscript can be accepted for publication if the following points are addressed by the authors:
My main concern is about what actually the conclusion of this study is. It is clear that this should
be the first statistical analysis that includes the present set of observational constraints which is important.
On the other hand the set of possible equations of states considered is limited by the construction.
Therefore I would recommend considering the following points:

1. The analysis favors an early phase-transition. To me this looks like the analysis actually disfavors the DD2 EoS
and thus favors an equation of state which goes from the DD2 to another EoS quickly. This may indicate that the DD2
is not very reliable above nuclear saturation density. If one would take a different equation of state for the high
density hadronic matter, one which eventually leads to smaller radii, the most likely transition may move also to a larger density.
Thus I think this statement may be just a result of the specific construction used in this work.
2. Why did the authors choose to limit their EoS on the DD2 + Quark phase construction? One could have done a more 'free'
selection of the EoS as done in other works. Again, the results seem to be biased by the choice of EoS.
3. One could show the Posterior probabilities also for the APR EoS. Does it give an even higher probability than all the
possible parametrizations of the combined EoS?
4. A study which tries to answer the same question, in essence, is Phys.Rev.D 101 (2020) 5, 054016. It should at least
be mentioned and possibly the results on c_s^2 can be compared.
5. It appears that, for the actual experimental observations, there is no (or only very weak) dependence of the posterior
probability on the Delta-P parameter. On the other hand this dependence is strong for the 'arbitrary point' chosen by hand.
The authors seem to think this is due to the smaller error. But this could also be due to the specific choice of the mean
of the fictitious measurement. In other words there may be combinations of M-R measurements which put a stronger constraint
on the EOS than other combinations, at least in the context of the model that is used in this paper.
6. The above comment is also related to the observation, that for the present paper all the possible maximum masses are
either in the M-R region of 14 km and 2 solar masses (for DD2) or 10-11 km and 2 solar masses for a transition to a quark phase.
Intermediate combinations are very rare. Thus the observation with larger radii will put a very strong constraint on the parameters
simply because this particular model does not allow for large radii. On the other hand there may be other models which do.
7. Since only the space of EoS which is defined by the model is investigated, one should probably consider the posterior
probabilities as relative probabilities, since there could be an infinite number of EoS which can be more probably but are
not allowed within the presented model.
8. The authors introduce two step functions for the vector coupling strength. Why does one need two as both tend to reduce
the vector coupling strength towards zero. Also only one of these switching function is varied as a free parameter for the
Bayesian analysis. Is there a reason why the other one should be fixed?
9. The authors introduce a 'constant speed of sound' extrapolation. From what I can read this has to do with the fact that
the speed of sound can become superluminal. It is never explained what causes this behavior. Is it the switching functions
of the vector coupling? Does that lead to a thermodynamically inconsistent EoS? This result clearly violates the Stefan-Boltzmann
limit of a free gas of quarks.
10. It is assumed that considered constraints are independent of each other. Is this a certainty? Are the different
observational results fully independent or may they rely on certain common assumptions. This is more a naive question a
reader may have, so a short sentence on this can be helpful.

Once the above comments are addressed the paper can be accepted for publication.

Author Response

We thank the referee for insightful comments and questions that we repeat below in black color. Our replies are in red color. We attach a file "diff.pdf" where the changes in the manuscript are highlighted (red: removed text, blue: added text).

1. The analysis favors an early phase-transition. To me this looks like the analysis actually disfavors the DD2 EoS and thus favors an equation of state which goes from the DD2 to another EoS quickly. This may indicate that the DD2 is not very reliable above nuclear saturation density. If one would take a different equation of state for the high density hadronic matter, one which eventually leads to smaller radii, the most likely transition may move also to a larger density. Thus I think this statement may be just a result of the specific construction used in this work.

Our choice of DD2p40 was done so that this hadronic EoS was not necessarily in accordance with GW170817 but keeps the possibility of a large radius star at approximately the same mass range as GW170817. Therefore, our class of hybrid EoS is constructed such as to have third family branches and mass twin stars.

We demonstrate that in this scenario also the stiff hadronic EoS DD2p40 could not be excluded, even in the presence of the radius constraint from GW170817 like

R_1.4<13.5 km [Annala et al. (2018)].

2. Why did the authors choose to limit their EoS on the DD2 + Quark phase construction? One could have done a more 'free' selection of the EoS as done in other works. Again, the results seem to be biased by the choice of EoS.

We admit that a more complete analysis should consider also a systematic variation of the hadronic EoS like it was done, e.g., by our Bayesian analysis in EPJA (2016) when varying the nucleonic excluded volume parameter in the DD2 class of EoS. Imagine we would vary the stiffness of the hadronic EoS in addition to varying the onset density of the phase transition. Then, for a softer hadronic EoS the hybrid EoS would not necessarily produce a third family and thus mass twin stars.

However, the striking result of the present work, namely to solve the possible tension between a lowest value of a NICER radius measurement which exceeds the maximum radius that follows from the LVC analysis of tidal deformability, requires a rather stiff hadronic EoS like DD2p40, with a radius R_1.4 > 14 km.

3. One could show the Posterior probabilities also for the APR EoS. Does it give an even higher probability than all the possible parametrizations of the combined EoS?

We cannot just apply the Bayesian analysis to the APR EoS as such because the resulting probability would not be comparable to those of the hybrid EoS set because it is outside that class. However, we may regard the hybrid EoS with μ_<=1000 MeV as a proxy for APR because these two EoS are sufficiently close to each other in the range where we have constraints. Consequently we could read off the posterior probability from the rightmost LEGO plot in Fig. 7 for μ_<=1000 MeV.

Note, that the hybrid EoS with smaller μ_< value result in higher posterior probabilities simply because they result in a better fulfillment of the constraint on the tidal deformability from GW170817, see Fig. 5.

4. A study which tries to answer the same question, in essence, is Phys. Rev. D 101 (2020) 5, 054016. It should at least be mentioned and possibly the results on c_s^2 can be compared.

New constraints from the multimessenger era can strengthen the limits on EoS and eliminate an argument that, e.g. c_s² < 1/3.

Our speed of sound also requires to be above the “conformal limit”, like in the reference mentioned. We add this reference and the recently published Annala et al., Nature Physics (2020) with a corresponding comment.

5. It appears that, for the actual experimental observations, there is no (or only very weak) dependence of the posterior probability on the Delta-P parameter. On the other hand this dependence is strong for the 'arbitrary point' chosen by hand.

The authors seem to think this is due to the smaller error. But this could also be due to the specific choice of the mean of the fictitious measurement. In other words there may be combinations of M-R measurements which put a stronger constraint

on the EOS than other combinations, at least in the context of the model that is used in this paper.

The mixed phase solutions are disfavored in the case of the fictitious measurements with the radius parameter 14 km which we discuss, because the narrow cigar shape and its orientation strongly prefer the solution in the Maxwell construction case.

In the case of the early onset of the phase transition there are no observational constraints in the region of the mixed phase construction. This entails a flat distribution of the posterior probabilities in the direction of the mixed phase parameter Δ_P, see Fig. 7.

6. The above comment is also related to the observation, that for the present paper all the possible maximum masses are either in the M-R region of 14 km and 2 solar masses (for DD2) or 10-11 km and 2 solar masses for a transition to a quark phase.

Intermediate combinations are very rare. Thus the observation with larger radii will put a very strong constraint on the parameters simply because this particular model does not allow for large radii. On the other hand there may be other models which do.

As we mentioned in the answer to point 1, it was not our aim to formulate a most general class of EoS that would cover all theoretically possible shapes of EoS (and thus M-R curves). We are focussed on the discussion of the possibility that the measurement of seemingly contradicting radii within a narrow region of masses addressed by GW170817 and NICER is possible when the class of EoS concerns

strong phase transitions with a third family of hybrid stars and mass twin stars in that very mass range.

7. Since only the space of EoS which is defined by the model is investigated, one should probably consider the posterior probabilities as relative probabilities, since there could be an infinite number of EoS which can be more probably but are not allowed within the presented model.

According to the Bayesian formalism all probabilities are relative, i.e. subject to the choice of the class of models. We do not exclude that in Nature other classes of EoS would be possible that would be favorable. But those classes cannot easily be compared with our class of EoS unless they are special cases of a more general class of EoS that includes, e.g., our class of hybrid models as a subclass.

8. The authors introduce two step functions for the vector coupling strength. Why does one need two as both tend to reduce the vector coupling strength towards zero. Also only one of these switching function is varied as a free parameter for the

Bayesian analysis. Is there a reason why the other one should be fixed?

The main question for the present work is the position of the onset of the phase transition.Therefore, we have chosen to vary the parameter μ_< which is related to the onset of deconfinement.

We did not make use of other parameters like for the position and width of the stiffening of the quark matter EoS because it is sufficient for us that the hybrid EoS fulfills the 2M_⊙ mass constraint which is the case with the once chosen set 2 parameters from Ref. [Alvarez-Castillo et al., PRD (2019)].

9. The authors introduce a 'constant speed of sound' extrapolation. From what I can read this has to do with the fact that the speed of sound can become superluminal. It is never explained what causes this behavior. Is it the switching functions

of the vector coupling? Does that lead to a thermodynamically inconsistent EoS? This result clearly violates the Stefan-Boltzmann limit of a free gas of quarks.

The reason for the superluminality of the quark matter EoS at high densities is not the behavior of the switch functions. It was identified to stem from the nlNJL EoS which at high densities exhibits a backbending of the pressure versus energy density curve. There is a maximum energy density that can be reached with this nonlocal NJL model and at that point c_s²=dP/dε → ∞. That it is this artifact of the nlNJL model which entails the necessity to correct the high-density behavior of the quark matter EoS has been spelled out clearly at the beginning of subsection 2.2, where the CSS extrapolation has been introduced. The reason for the artifact itself can be traced back to the use of covariant (energy-dependent) formfactors in such a model. Roughly speaking, the argument of the formfactor contains the square of the shifted Matsubara frequency (ω_n+i μ)² = ω_n² + iω_n μ - μ². Therefore, at large chemical potentials μ the argument of the formfactor drops with increasing μ and thus the energy density gets reduced, which results in the backbending of P(ε).

The Stefan-Boltzmann limit is for asymptotic densities, not relevant for dense neutron star matter.

10. It is assumed that considered constraints are independent of each other. Is this a certainty? Are the different observational results fully independent or may they rely on certain common assumptions. This is more a naive question a reader may have, so a short sentence on this can be helpful.

The three measurements are conditionally independent. Common assumptions (like validity of General Relativity) do not make them dependent. They concern three different objects (PSR J0740+6220, GW170817, PSR J0030+0451) and are performed with different methods of measurement (Shapiro-delay measurement in close binary, GW from binary neutron star merger, timing residuals from millisecond pulsar).

Once the above comments are addressed the paper can be accepted for publication.

Author Response File: Author Response.pdf