Strongly Interacting Dark Matter admixed Neutron Stars

Dear editor and referees,

we have overhauled completely the presentation of our results, to bring them inline with the requests of the referees. In particular, direct comparison of the results to observational constraints, without need to rely on any concrete amount of dark matter in the neutron star, has been made. Moreover, we carefully studied other literature on the same concept - testing different dark matter models in the context of compact objects - and ensured that our presentation and overview as well as presentation of the results is comparable to other published work, to ensure all necessary quality standards.

With kind regards,

the authors

1 Reply to referee A

1. Figure 1 plots the equation of states used in the paper. How many data points are used to make each curve? Without additional information, from just looking at the figure, each equation of state seems like it could be just a handful of points connected by straight lines. This should be addressed in the paper, perhaps by mentioning roughly how many points are used in each curve. The data points should be plotted in Fig. 1 as dots on the curves.

Answer: These data points are already included in the plot in figure 1 (this remark may erroneously refer to the first version on the arXiv, where this was not yet the case). We also extended the caption to make it more transparent.

1. Tidal deformability analysis is too limited (only the softest EOS and the DM mass corresponding to the core configuration have been presented).

Answer: We have extended the discussion. In addition, in appendix A.2, figure 10 provides this information for multiple systems now.

1. Figures are not easily readable, and the corresponding texts are not clarifying (the connection to M–R phenomenology is unclear.)

Answer: We have improved the figures throughout, also with respect to the comparisons. Using now explicit examples of observed compact objects should highlight how M-R behavior is tested with our data. Using two extreme and two average neutron stars should give an improved picture compared to our earlier cut-based analysis. The latter can still be easily performed with the plots, but also from the data provided from a data release, which will become available together with the eventual publication of this work, if desired so.

1. While the broad range of observables is valuable, it does not establish the consistency of the selected parameters with current astrophysical measurements. As a result, the dark matter parameters are not effectively constrained by the presented calculations. Consequently, the main conclusion stated in the abstract, that dark matter masses of a few hundred MeV to a few GeV are consistent with the latest neutron star observations, appears overstated, since the latest observational constraints were not actually incorporated into the analysis.

Answer: We have updated the comparison fundamentally, and made now an explicit connection to representative measurements, which tests multiple parameters simultaneously. In particular, this shows explicitly how current uncertainties very well fit with our data in the relevant mass range. While this still presents only a subset of all available observational data, the envelope characterized by the presented measurements is fairly representative, especially when it comes to the less constrained radius values.

1. Lack of standard fixed central-pressure ratio or fixed dark matter fraction sequences for a full M-R line, which are widely used in neutron star literature.

Answer: The new plot 3 should provide this information: Fixed colors here correspond to the fixed dark matter fraction, and can now be fairly easily tracked as a curve in the mass-radius diagram.

1. Unclear treatment of dark matter self-interaction (implicit in lattice EOS but not clarified).

Answer: In this respect the self-interaction is fixed entirely by the underlying Lagrangian, which is discussed at length in the literature on our dark matter candidate theory. As for an effective low-energy description, this would require a major undertaking, akin to our previous work [60] and [94]. This would require new lattice simulations to be performed, which would be a major computational endeavor, beyond our current abilities. We have added a comment on this in section 2.2.

1. Lack of any report about the speed of sound.

Answer: This information is now provided as a panel in figure 1. In particular, it shows that the behavior between the ordinary matter speed of sound has similar features as those of the dark matter one, though quantitatively quite different.

1. In one instance, the authors cite around 30 references immediately after the sentence “The amount of dark matter may vary depending on the age, history, and location of the neutron star, but most estimates conclude that it is very unlikely to exceed 1% of the total neutron-star mass.” This is unusual and should be reconsidered.

Answer: The 1% is a very hotly contested quantity in the literature. While our results are independent of the actual number, as we scan the full range, its phenomenological impact is nonetheless also tied to this number. We aimed here at representing the full breadth of this very controversial discussion. We have made this clearer in the text, but see also no other option to fully represent this issue in a way which conforms with the expectation of a fair representative of the state of the art. However, and eventually, this is a point beyond our ability to address within this work, and the answer to it may very well depend strongly on the actual dark matter physics. Thus, while we argue from the literature why this ma be an interesting number, our results and presentation do not rely on it in any way, and it is used only in an illustrative purpose.

1. First of all, I would like to emphasize that the authors should compare their results directly with the most recent observational constraints on neutron stars if they intend to propose their parameter space for dark matter (even if it is only the dark matter particle mass) as a consistent range. In particular, they need to present both mass–radius and tidal deformability relations in a way that clearly illustrates the behavior of their models relative to the data. They may clarify this with some plots with exact observational constraints included, or by providing tables or texts which explain explicitly which range of parameters and which EOSs are consistent with the observational data (For example, Λ1.4<580). This is the main missing part of the paper.

Answer: We have followed this suggestion by the now explicit plots for the mass-radius relation Figure 3 and Figure 5 for the tidal deformability. The later is, of course, relatively limited due to the lack of data yet. Also, other plots have been including this information now.

1. The authors should elaborate on the role of dark matter self-interactions in their frame-work.

Answer: Astronomical data at least allow it. Their role is explicitly that we consider here, for the first time, a consistent self-interaction due to a gauge interaction, and test its consequences. Due to coupling universality, there is very little freedom, up to the mass of the the dark matter particle, in such a setup. It is not a necessary feature for dark matter. However, what we show is that such a feature is consistent, without needing to resort to guesses about the structure of such an interaction. We emphasized this in the text.

1. The authors should at least provide some comments on the speed of sound in the employed EoSs.

Answer: We have added a panel in Figure 1 showing the speed of sound for all equations of state employed.

1. The authors state that they interpolate piecewise polytropes between NChPT at low densities and pQCD at high densities. However, the maximum energy density reached in their models is well below the regime where pQCD is valid. This statement should therefore be revised to accurately reflect the interpolation procedure actually used.

Answer: We have added a sentence in Section 2.1 to address the idea that ultimately one single equation of state should be able to describe both regimes and therefore an interpolation limited by constraints like cs < 1 should be able to capture at least the qualitative features. However, we refer here to the original literature [71], where this approach has been justified.

1. On page 7, the authors claim that the crust would hardly affect the radius. This is not correct, as the crust has a non-negligible impact on neutron star radii. The statement about not including a crust should be modified accordingly.

Answer: We have modified the corresponding statement.

1. For clarity and consistency, I recommend revising the use of terminology and abbreviations. Some common terms (e.g., "equation of state," "neutron stars") are repeated many times without abbreviation, while certain abbreviations (e.g., WIMPs) appear without definition, or are defined only after their first occurrence (e.g., NChPT). Defining each abbreviation upon its first use and employing it consistently thereafter would improve readability and flow.

Answer: We have improved the manuscript correspondingly.

1. Correct the thermodynamic identity to nμ = ε + p (instead of nμ = εp) before Eq. (6.1).

Answer: We have corrected for this typo. Thanks.

1. Correct the expression for the chemical potential.

Answer: We have corrected this typo in Eq. 6.3. Thanks.

1. Review the manuscript for typographical errors such as “we use choose . . . ” or “is a a system . . . ” and correct them throughout.

Answer: We have proofread the whole document independently again to eliminate as many as possible.

2 Reply to referee B

1. The model is at best contrived. The authors choose a very specific gauge group for dark matter namely G2 in order to my understanding to avoid having to deal with the notorious sign problem that arises in nonzero density studies of QCD-like theories. However there is not much theoretical justification and motivation for such a peculiar group. It is highly unlike that dark matter will have such a gauge interaction.

Answer: There is, at the moment, no a-priori justification for any dark matter model. Thus, we consider it important to exhaust as many options as possible. The present theory is the simplest non-Abelian gauge theory, which allows for fermionic dark matter, for which a first-principles calculation of the impact on neutron stars is possible at all. We consider this to be an important check for estimating the range of impact different dark matter scenarios can have on neutron stars. Nothing comparable is likely feasible for the foreseeable future for any other non-Abelian gauge theory scenario, making this investigation unique in exploring a range of parameter space inaccessible otherwise.

1. In addition contrary to what the authors claim, that this model produces unique mass-radius relations, this is hard to believe. One could easily mimic similar mass-radius relations by having dark matter particles exhibit some repulsive or attractive self-interactions via mediation of scalars or vectors with appropriate couplings. It is well known in neutron star physics that lots of EoS end up producing the same mass-radius relations and therefore it is hard to pinpoint which particular EoS is correct based on sole observations of mass and radius. The authors have to explain better in what sense they consider unique these EoS or they should simply tune down their statement.

Answer: As we see that the statement can be misunderstood, we have adjusted it. It is, of course, true that any mass-radius-relation can be reproduced with different models, provided they become sufficiently baroque. This is especially true, as long as the mass-radius relation is not very precisely measured. However, together with the tidal deformability, there are exactly three measurements to be compared to three parameters for the dark matter sector, which, due to the first-principle calculations, can fix each other within uncertainties.

1. Another relevant issue within the context of the studied model is how dark matter abundance is produced in the first place. The dark sector will have to go through a stage of confinement as it happened with QCD. The authors should comment on how their model produces the observed amount of dark matter abundance. Related to that is also the issue of dark matter self-interactions. If this component composes 100% of dark matter, they are constrained by limits from the bullet cluster and the ellipticity of galaxies. If not they should state more clearly that this is a small component of dark matter that must be less than 5% in order to avoid the aforementioned constraints.

Answer: Generically, such dark matter scenarios basically only need very simple constraints to satisfy these bounds from the 2 → 3 process, which to leading order is topologically fixed, and dominates. This has been discussed explicitly already for the Sp(4) case, and similar theories extensively in the literature, which we highlight more now. Due to the topological nature of the relevant interaction, it is unlikely that will substantially change for the present case, and an explicit calculation will be a project on its own. At any rate, the constraints on theories of this type turn out to be fairly mild so far, and will not suffice to exclude this theory.

1. Finally although the authors talk about accretion, there is practically not so much time to accrete such a huge component of mass in the lifetime of a neutron star. Therefore accretion is not the main mechanism of this admixed star formation. The authors should clarify this point and present or comment on a reasonable formation scenario.

Answer: This is certainly both beyond our expertise and the scope of this paper. However, in the literature partly even larger numbers are cited (see new reference [20] and [21]), and around 1% is assumed to be feasible [17,48,49], or even plausible. In addition, we have altered the way how results are shown in the plot, such that if people prefer a different amount, the result can be easily read off. Finally, we will make the data available upon publication, such that even a very precise preferred number can be obtained from our primary data, if anyone so desires. We have also added a footnote that we assume that, if dark matter should contain more components, these are assumed tacitly to be not significantly present in the neutron star considered in this study. Otherwise, its equation fo state could be added, for a treatment using a three-fluid approach straightforwardly, anyhow. But justification for concrete multiple components is right now not such that this would warrant a study as the one presented.

1. I also see a couple of flaws not related to physics but rather to the way the paper is written. Firstly, there is a vast literature on dark matter effects on neutron stars ranging from cooling effects, black hole formation, admixed/dark stars etc that the authors ignore. They should cite relevant work on dark matter effects on neutron stars. Furthermore the abstract has to be rewritten so it reflects in a more precise way what has been done in this study. Accumulation as I mentioned does not seem to be valid. Moreover it should be clarified in the abstract if they refer to a 100% component of dark matter or a subdominant one, clarifying also that they choose a particular gauge group to model the dark matter interactions. At first read I thought that G2 is implemented to deal with the QCD baryonic part of the star.

Answer: It is impossible to provide here a full review of what has been done in the literature. Such a complete overview would be rather a review article, which is certainly not within the scope of SciPost Core. The way we have approached this has been commensurate what other publications on the same subject, see especially [17-19,22-45], with other dark matter candidate theories, have done, and our work includes already almost 50 references to work on dark matter in neutron stars. We have rewritten the abstract to make it more clear what the scope is: We derive the impact of varying amounts of strongly-interacting dark matter on neutron stars in a unique way, which has never done before and unlikely to be happening anytime soon again, using a first-principles description of a non-Abelian strongly interactingdark matter theory including fully backcoupled dark gluodynamics.