Influence of EOS on compact star made of hidden sector nucleons

Shinji Maedan

REVIEW 2 major objections 4 minor 43 references

Influence of EOS on compact star made of hidden sector nucleons

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A hidden-sector chiral sigma model gives an analytic equation of state for compact stars of interacting dark nucleons, with maximum mass 2.1 times the free-gas value.

desk verdict Solid analytic EOS for hidden-sector nucleon stars, but the headline maximum mass rests on unproven stability and monotonicity conditions that need a numerical check. read the letter →

arxiv 1908.00711 v2 pith:Q2LESTW3 submitted 2019-08-02 hep-ph

Shinji Maedan This is my paper · ORCID

classification hep-ph

keywords hiddensectordarkmattercompactstarchiralsigmamodelequationofstateTolman-Oppenheimer-Volkoffequationsmeanfieldapproximationnucleonsmass-radiusrelation

topics Dark Matter

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies compact stars made of degenerate hidden-sector nucleons—candidates for cold dark matter—using a hidden SU(2) chiral $\sigma$ model with a vector meson in the mean-field approximation. Its central claim is that the equation of state is analytic: with the substitution $\theta = k'_F/y$, both pressure and energy density become explicit functions of $\theta$, so the EOS is fixed by just two dimensionless couplings $C'_\sigma$ and $C'_\omega$. Solving the TOV equations at $C'_\omega=(9\pi^2/4)(\sqrt{2}-\operatorname{arcsinh}1)\approx 11.83$ and $C'_\sigma=2C'_\omega$ gives a dimensionless maximum mass $M'_{\max}=0.567$ at radius $R'=2.24$, about 2.1 times the free degenerate-gas maximum. The paper also shows that a larger scalar coupling $C'_\sigma$ makes the maximum stable mass heavier. If correct, this provides an analytic template for interacting dark-matter stars whose mass-radius curves differ substantially from free-fermion stars.

What carries the argument

The load-bearing object is the variable $\theta=k'_F/y=k_F/m_f^*$, the hidden-nucleon Fermi momentum measured in units of the effective mass. Writing the mean-field equations in terms of $\theta$ turns the self-consistent condition for the scalar field into $f(\theta)=1/y^2$, where $f(\theta)$ is a known elementary function. That converts the energy density $\epsilon'$ and pressure $P'$ in Eqs. (23)--(24) into explicit analytic functions of a single variable, eliminating the numerical root-finding that usually accompanies mean-field EOS construction. A second ingredient, $g(\theta)\propto (f(\theta)-1)/\theta^3$, separates the pressure into free and interaction terms and locates the point $\theta=1$ where the EOS stops depending on $C'_\sigma$.

What would settle it

Evaluate $dP'/dn'_B$ and $d(\theta^2/f(\theta))/d\theta$ from Eqs. (23)--(25) over $0<k'_F<\infty$ for $C'_\omega=11.8326$ and $C'_\sigma=2C'_\omega$ (and at the upper limit $C'_\sigma=30$). If either quantity turns negative or changes monotonicity anywhere in this range, the EOS used in the TOV integration would not describe a homogeneous stable star, and the reported $M'_{\max}=0.567$ would not be a physical maximum mass. The paper contains no such numerical check.

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Extended reading notes

Core claim

The central claim is that the hidden-sector mean-field EOS can be written in closed form. Using $\theta = k'_F/y = k_F/m_f^*$, the scalar equation of motion becomes $f(\theta)=1/y^2$ with $f(\theta)=1+C'_\sigma\gamma[-\gamma C'_\omega \theta^6/(18\pi^4)+(\theta\sqrt{\theta^2+1}-\operatorname{arcsinh}\theta)/(2\pi^2)]$. Equations (23) and (24) then give $\epsilon'$ and $P'$ as explicit analytic functions of $\theta$ alone, so no self-consistent numerical solution for the effective mass is needed. For $C'_\omega=(9\pi^2/4)(\sqrt{2}-\operatorname{arcsinh}1)$, the TOV integrations yield $M'_{\max}=0.567$ at $R'_{\min}=2.24$ for $C'_\sigma=(6/3)C'_\omega$, a maximum mass 2.1 times the free-gas value; $M'_{\max}=0.550$ and $0.535$ for $C'_\sigma=(5/3)C'_\omega$ and $(4/3)C'_\omega$. At $\theta=1$ the EOS is independent of $C'_\sigma$, and for $k'_F\ll 1$ the interacting EOS is softer than the free gas while becoming stiffer at intermediate densities.

Load-bearing premise

The load-bearing premise is that for the chosen couplings ($C'_\omega\approx 11.83$, $C'_\sigma\lesssim 30$) the hidden-nucleon matter is a stable single phase—pressure rises when density rises, and energy per nucleon stays above the particle mass—and that $\theta$ rises smoothly with Fermi momentum. The paper asserts these conditions in Section 4 after Eq. (53) but does not prove or numerically verify them.

Editorial extensions

If this is right

With $C'_\sigma=(6/3)C'_\omega$, the maximum dimensionless mass is $M'_{\max}=0.567$, exactly 2.1 times the free-gas value $M'_{\max}=0.272$, while the radius at that maximum is only 0.94 times the free-gas radius.
Increasing $C'_\sigma$ from $(4/3)C'_\omega$ to $(6/3)C'_\omega$ raises $M'_{\max}$ from 0.535 to 0.567 and lowers $R'_{\min}$ from 2.28 to 2.24, so the scalar attraction makes the star heavier and more compact.
Dimensionful masses obey $M = 1.632\,M_\odot\, M' (1\,\mathrm{GeV}/m_f)^2$, so the same dimensionless sequence covers a wide range of astrophysical masses depending on the unknown hidden-nucleon mass $m_f$.
At low densities the interacting EOS is softer than a free gas when $C'_\sigma>C'_\omega$, so large-radius hidden-sector stars are lighter than free-gas stars; at very high densities $P'/\epsilon'$ approaches a constant close to 1/3.
The EOS curves for different $C'_\sigma$ all pass through the same point at $\theta=1$ ($k'_F=1$), where the pressure is stiffer than the free gas, because $g(1)=0$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

As an extension, the same analytic reduction should carry over to larger flavor numbers, since only $\gamma$ enters $f(\theta)$; the paper notes the $\gamma=6$ case but does not compute it.
As an observational extension, these mass-radius curves could be used to separate interacting from free dark-matter stars through their mass-radius relation; the paper does not make this comparison quantitative.
One obvious next calculation would be tidal deformability from the same EOS, since the factor-2.1 stiffness increase is exactly what such measurements probe; the paper stops at mass and radius.
Changing the calibration condition from $y(k'_F=1)=1$ to $y(k'_F=1/2)=1$ should move the $C'_\sigma$-independent point but preserve the qualitative story; the paper sketches this but gives no curves.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my read of Maedan's hidden-sector compact star paper.

The genuinely new thing is Eqs. (23)–(24): a closed-form, fully analytic EOS for the mean-field chiral sigma model with a vector meson, expressed in terms of theta = k'_F/y. The theta parametrization itself is not new (it comes from Wahidin et al. [19]), but the explicit closed-form EOS for this model and the scan of M'_max versus C'_sigma are not in the cited literature. That is a real, modest extension.

What the paper does well: the algebra is internally consistent; the theta = 1 identities check out; the EOS correctly reduces to the free Fermi gas at low k'_F; the comparison with the free gas is explicit; and the paper is honest about the effective-theory caveats and the unknown m_f. No circular fitting, no self-citation padding.

The soft spot is the one flagged by the stress test. In Sec. 4, after Eq. (53), the paper asserts that for C'_sigma ≲ 30, the conditions dP/dn_B > 0, epsilon/n_B - m_f > 0, and monotonicity of theta^2/f(theta) hold. These are not proven; the text just says one can 'ascertain' them. They are load-bearing: the TOV integration treats the EOS as a single-valued P(epsilon) curve, and the quoted M'_max = 0.567 and the 2.1x enhancement over the free gas come from that curve. If theta^2/f(theta) is not monotone, the EOS is multi-valued. A direct numerical check or a short proof of these inequalities would settle it. Fixable gap, not fatal.

Minor: the conclusion contains a self-contradictory sentence ('That is not the case.') that should be deleted. No code or data are provided, but the equations are explicit enough for someone to reproduce the TOV runs.

Who gets value: people working on dark matter compact stars, asymmetric dark matter, and mean-field EOS models. I'd send it to a referee: the core mathematics is self-contained and the missing verification is straightforward. With that added, it would be a solid contribution.

Referee Report

2 major / 4 minor

Summary. The paper constructs an equation of state (EOS) for degenerate hidden-sector nucleons using an SU(2) chiral sigma model with a dynamically massive vector meson, in the mean-field approximation. By introducing the variable theta = k'_F/y, the authors express the dimensionless energy density and pressure as explicit analytic functions of theta, Eqs. (23) and (24). The EOS is specified by two dimensionless couplings, C'_sigma and C'_omega. After fixing C'_omega by the normalization condition y=1 at k'_F=1, they vary C'_sigma and integrate the dimensionless TOV equations. For C'_sigma = (6/3)C'_omega they report a maximum mass M'_max = 0.567 at R'_min = 2.24, about 2.1 times the free-fermion value, and they study how the mass-radius relation depends on C'_sigma. The paper also derives a rough constraint on the hidden pion mass from degeneracy and BBN considerations.

Significance. The analytic parametrization of the mean-field EOS is a genuine technical simplification: Eqs. (23)-(24) give closed-form expressions and allow the crossing point theta = 1 to be understood analytically. The comparison with the independent free-fermion gas is a fair benchmark, and the conclusion that the interacting EOS is softer at low density and stiffer near k'_F ~ m_f is physically reasonable and clearly explained. I see no circular fitting: C'_omega is fixed by a normalization condition rather than by reproducing a target mass-radius curve. The main value is a tractable model for dark-matter compact stars and a clear demonstration that the hidden-sector scalar coupling can substantially raise the maximum mass. The TOV results are, however, contingent on the thermodynamic-stability and single-valuedness conditions discussed in the major comments, which need to be supplied before the quoted numbers can be fully accepted.

major comments (2)

[Sec. 4 (after Eq. (53))] The paper states that for C'_omega = 11.8326 and C'_sigma <= 30 the conditions dP/dn_B > 0 and epsilon/n_B - m_f > 0 hold, and that theta^2/f(theta) is a monotone increasing function of theta, but no proof or numerical verification is provided. These conditions are load-bearing: the TOV integration in Sec. 4.2 uses the EOS as a single-valued P'(epsilon') curve, the stellar surface is defined by P(R)=0, and the quoted maximum masses (0.567, 0.550, 0.535 for the three C'_sigma values) depend on the central density lying on the stable branch. If theta^2/f(theta) is not monotone, the map k'_F -> theta is multi-valued; if dP/dn_B <= 0 anywhere, the matter is thermodynamically unstable; if epsilon/n_B - m_f < 0, the matter is self-bound and the P(R)=0 boundary condition is inappropriate. Please add either an analytic argument or a numerical check (table or plot) covering the theta range actually used in the TOV integrations for C'_sigma = (4/3)C'_omega, (5/3)C'_omega, and (6/3)C'_omega.
[Eq. (8)] The equation of motion for sigma_h as printed appears to contain an error in the vector-meson term: it reads y^3 C_sigma C_omega k_F^6, whereas consistency with the dimensionless equation (13) and with Eq. (17) requires C_sigma C_omega k_F^6 / y^3 (with the m_f^{-2} prefactor unchanged). As written, Eq. (13) does not follow from Eq. (8), although Eqs. (13)-(24) are mutually consistent. Please correct Eq. (8) or explicitly state the corrected equation of motion, and check that no later equation relies on the wrong form.

minor comments (4)

[Appendix A] The proof that y < 1 for 0 < k'_F < 1 and y > 1 for k'_F > 1 assumes the unproven property that f(theta) > 1 for 0 < theta < 1, f(1) = 1, and f(theta) < 1 for 1 < theta < theta_f; this is the same gap noted in the major comment and should be justified explicitly.
[Sec. 4.2.2] The claim that the high-density limit of P'/epsilon' is 0.3333 for all three C'_sigma values is supported only by Eq. (48), but the corresponding values of theta_f (the positive solution of f(theta)=0) are not tabulated. Please provide theta_f and the limiting ratio for each parameter set so the reader can check this assertion.
[Sec. 1] The introduction misspells the name as 'Toleman-Oppenheimer-Volkoff'; it should be 'Tolman-Oppenheimer-Volkoff'.
[Eq. (12)] The equivalences C'_sigma = m_f^2 C_sigma = g_sigma^4/(2 lambda) and C'_omega = m_f^2 C_omega = g_sigma^2 are useful and should be stated explicitly, since they clarify that fixing C'_omega is a choice of the Yukawa coupling while varying C'_sigma changes the scalar self-coupling.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the C'_omega value is a normalization choice, not a fitted target, and the EOS/TOV derivation is self-contained.

full rationale

The central derivation is self-contained. The analytic EOS in Eqs. (23)-(24) follows from the mean-field equation of motion (17) by the substitution theta = k'_F/y, which is a change of variables rather than an input fitted to any output. The parameter C'_omega is fixed by Eq. (52) through the explicit normalization f(theta=1)=1, i.e., y=1 at k'_F=1, and the paper explicitly notes that other values such as f(1/2)=1 could equally be chosen; this is a model-parameter choice, not a fit to the predicted maximum mass or to any observable. The TOV integrations in Sec. 4 then use the resulting P'(eps') without adjusting parameters to reproduce the quoted M'_max values, and the comparison with the free hidden-sector-nucleon gas is an independent benchmark. No load-bearing step reduces to a self-citation: the model is introduced from Hur et al. [20] and mean-field EOS techniques from [41]-[42], but the analytic solution and TOV results are computed in this paper from the stated equations. The unproven assertions in Sec. 4 that the thermodynamic conditions and monotonicity of theta^2/f(theta) hold for C'_sigma <= 30 are a domain-of-validity and numerical-verification concern, not an indication of circularity.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central results depend on standard mean-field field theory and TOV equations plus model assumptions inherited from the hidden-sector literature. No observational data are used; the only numerical inputs are a hand-chosen value of C'_omega and three scanned values of C'_sigma. No new particles or forces are introduced beyond the chiral hidden-sector fields taken from Refs. [20, 22, 41, 42].

free parameters (3)

m_f, hidden sector nucleon vacuum mass
Stated in Sec. 2.1 as unknown; physical masses and radii scale as (1 GeV/m_f)^2 in Eq. (64), so absolute predictions are absent.
C'_omega = ~11.8326
Chosen by the normalization y(k'_F=1)=1 in Eq. (52); not fitted to data and not independently justified.
C'_sigma = (4/3)C'_omega, (5/3)C'_omega, (6/3)C'_omega, restricted to <~30
Scanned over three values and restricted to C'_sigma <~ 30 by asserted stability conditions; no empirical input.

assumptions (7)

standard math Tolman-Oppenheimer-Volkoff equations describe hydrostatic equilibrium of non-rotating compact stars
Used as the structural equations in Sec. 4 without modification.
domain assumption The hidden sector has a baryon asymmetry and contains stable hidden-sector nucleons and pions
Sec. 1 assumes hidden-sector baryogenesis satisfying Sakharov conditions; without it, hidden-sector compact stars would not form.
domain assumption The low-energy effective theory is the hidden SU(2)_L x SU(2)_R chiral sigma model plus a dynamically massive omega_h
Eq. (3), adopted from Refs. [20, 22, 41, 42].
domain assumption The mean-field approximation is valid for the degenerate hidden-sector nucleon gas
Sec. 2.1 uses the mean-field approximation for the sigma_h and omega_h fields.
domain assumption The small explicit chiral symmetry breaking term D*sigma_h is negligible
Sec. 2.1 states that the term D*sigma_h does not play an important role and is neglected.
domain assumption Hidden sector matter is isospin symmetric with equal numbers of p_h and n_h
Sec. 1 specifies equal numbers of p_h and n_h.
ad hoc to paper For C'_sigma <~ 30, thermodynamic stability and monotonicity conditions hold
Sec. 4 asserts dP/dn_B > 0 and epsilon/n_B - m_f > 0 and monotonicity of theta^2/f(theta) without proof; this restricts all parameter choices used in the TOV runs.

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Pith. "Pith review of Influence of EOS on compact star made of hidden sector nucleons." pith.science (2026). https://pith.science/paper/Q2LESTW3

@misc{pith2026190800711,
  author       = {Pith},
  title        = {Pith review of: Influence of EOS on compact star made of hidden sector nucleons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q2LESTW3}},
  note         = {Machine review of arXiv:1908.00711}
}

abstract

We study compact star made of degenerate hidden sector nucleons which will be a candidate for cold dark matter. A hidden sector like QCD is considered, and as the low energy effective theory we take (hidden sector) $ SU(2) $ chiral sigma model including hidden sector vector meson. With the mean field approximation, we find that one can treat the equation of state (EOS) of our model analytically by introducing a variable which depends on the Fermi momentum. The EOS is specified by the two parameters $ C'_{/sigma} $,$ C'_{/omega} $, and we discuss how these parameters affect on the mass-radius relation for compact star as well as on the EOS. The dependence of the maximum stable mass of compact stars on the parameter $ C'_{/sigma} $ will also be discussed.

Figures

Figures reproduced from arXiv: 1908.00711 by the authors.

**Figure 2.** The graphs of EOS (dimensionless energy density [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** This graph is the same with Fig.2 but magnified around [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: M′ − R′ relations, i.e., the relations between the dimensionless star mass M′ defined by Eq.(61) and the dimensionless star radius R′ defined by Eq.(61). The dotted line is the numerical result with C ′ σ = (6/3)C ′ ω and C ′ ω = (9π 2/4) √ 2 − arcsinh(1) , while the …

**Figure 5.** Figure 5: M′ − R′ relations with C ′ σ = (6/3)C ′ ω (dotted), C ′ σ = (5/3)C ′ ω (solid), C ′ σ = (4/3)C ′ ω (dashed), and a free gas case (dash-dotted). For all the cases except a free gas case, C ′ ω = (9π 2/4) √ 2 − arcsinh(1) . that for R′ ≫ 1, the (absolute value of) slop …

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

43 extracted references · 33 canonical work pages

[19]

A. B. Wahidin, A. Rahmansyah, and A. Sulaksono, Int. J. Mod. P hys. D 28, 1950071 (2019). 25

work page 2019
[1]

Bertone, D

For a review of dark matter, see G. Bertone, D. Hooper, and J. Silk, Phys. Rept. 405, 279 (2005)

work page 2005
[2]

R. C. Tolman, Phys. Rev. 55, 364 (1939)

1939
[3]

J. R. Oppenheimer and G. M. Volkoﬀ, Phys. Rev. 55, 374 (1939)

1939
[4]

A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967) [JETP Lett. 5, 24 (1967)]

work page 1967
[5]

M. Y. Khlopov, G. M. Beskin, N. E. Bochkarev, L. A. Pustylnik, an d S. A. Pustyl- nik, Astron. Zh. 68, 42 (1991)[Sov. Astron. 35, 21 (1991)]

work page 1991
[6]

Nakajima and M

T. Nakajima and M. Morikawa, [astro-ph/0506623]

arXiv
[7]

Narain, J

G. Narain, J. Schaﬀner-Bielich, and I. N. Mishustin, Phys. Rev. D 74, 063003 (2006)

work page 2006

Show all 43 references

[8]

Dietl, L

C. Dietl, L. Labun, and J. Rafelski, Phys. Lett. B 709, 123 (2012)

2012
[9]

Goldman, R

I. Goldman, R. N. Mohapatra, S. Nussinov, D. Rosenbaum, and V . Teplitz, Phys. Lett. B 725, 200 (2013)

2013
[10]

S. C. Leung, M. C. Chu, and L. M. Lin, Phys. Rev. D 84, 107301 (2011)

2011
[11]

A. Li, F. Huang, and R.-X. Xu, Astropart. Phys. 37, 70 (2012)

2012
[12]

Xiang, W.-Z

Q.-F. Xiang, W.-Z. Jiang, D.-R. Zhang, and R.-Y. Yang, Phys. Rev . C 89, 025803 (2014)

2014
[13]

Maselli, P

A. Maselli, P. Pnigouras, N. G. Nielsen, C. Kouvaris, and K. D. Kok kotas, Phys. Rev. D 96, 023005 (2017)

2017
[14]

Panotopoulos and I

G. Panotopoulos and I. Lopes, Phys. Rev. D 98, 083001 (2018)

2018
[15]

X. D. Wang, B. Qi, N. B. Zhang, and S. Y. Wang, [arXiv:1805.01314 [astro- ph.CO]]

arXiv
[16]

Barranco, A

J. Barranco, A. Bernal, and D. Delepine, [arXiv:1811.11125 [hep- ph]]
[17]

J. H. Chang, D. Egana-Ugrinovic, R. Essig, and C. Kouvaris, JC AP 1903, 036 (2019) [arXiv:1812.07000 [hep-ph]]

2019 arXiv
[18]

M. I. Gresham and K. M. Zurek, Phys. Rev. D 99, 083008 (2019)

2019
[20]

Hur, D.-W

T. Hur, D.-W. Jung, P. Ko, and J. Y. Lee, Phys. Lett. B 696, 262 (2011)

2011
[21]

Gell-Mann and M

M. Gell-Mann and M. Levy, Nuovo Cim. 16, 705 (1960)

1960
[22]

Boguta, Phys

J. Boguta, Phys. Lett. 120B, 34 (1983)

1983
[23]

Davoudiasl and R

H. Davoudiasl and R. N. Mohapatra, New J. Phys. 14, 095011 (2012)

2012
[24]

Petraki and R

K. Petraki and R. R. Volkas, Int. J. Mod. Phys. A 28, 1330028 (2013)

2013
[25]

K. M. Zurek, Phys. Rept. 537, 91 (2014)

2014
[26]

Foot and R

R. Foot and R. R. Volkas, Phys. Rev. D 68, 021304 (2003)

2003
[27]

Foot and R

R. Foot and R. R. Volkas, Phys. Rev. D 69, 123510 (2004)

2004
[28]

Berezhiani, [arXiv:hep-ph/0508233 [hep-ph]]

Z. Berezhiani, [arXiv:hep-ph/0508233 [hep-ph]]

arXiv
[29]

D. S. M. Alves, S. R. Behbahani, P. Schuster, and J. G. Wacker , Phys. Lett. B 692, 323 (2010)

2010
[30]

An, S.-L

H. An, S.-L. Chen, R. N. Mohapatra, and Y. Zhang, JHEP 03, 124 (2010)

2010
[31]

Spier Moreira Alves, S

D. Spier Moreira Alves, S. R. Behbahani, P. Schuster, and J. G. Wacker, JHEP 06, 113 (2010)

2010
[32]

Gu, Nucl

P.-H. Gu, Nucl. Phys. B 872, 38 (2013)

2013
[33]

M. R. Buckley and E. T. Neil, Phys. Rev. D 87, 043510 (2013)

2013
[34]

Detmold, M

W. Detmold, M. McCullough, and A. Pochinsky, Phys. Rev. D 90, 115013 (2014)

2014
[35]

Gu, JCAP 1412, 046 (2014)

P.-H. Gu, JCAP 1412, 046 (2014)

2014
[36]

S. J. Lonsdale and R. R. Volkas, Phys. Rev. D 97, 103510 (2018)

2018
[37]

M. Ibe, S. Kobayashi, R. Nagai, and W. Nakano, [arXiv:1907.114 64 [hep-ph]]

1907
[38]

J. I. Kapusta, Finite-temperature ﬁeld theory (Cambridge University Press, Cam- bridge, 1989), p. 182

1989
[39]

Prakash and T

M. Prakash and T. L. Ainsworth, Phys. Rev. C 36, 346 (1987)

1987
[40]

N. K. Glendenning, Nucl. Phys. A 480, 597 (1988)

1988
[41]

P. K. Sahu, R. Basu, and B. Datta, Astrophys. J. 416, 267 (1993)

1993
[42]

P. K. Sahu and A. Ohnishi, Prog. Theor. Phys. 104, 1163 (2000)

2000
[43]

L. E. Strigari, Phys. Rept. 530, 1 (2013). 26

2013

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [19]

A. B. Wahidin, A. Rahmansyah, and A. Sulaksono, Int. J. Mod. P hys. D 28, 1950071 (2019). 25

work page 2019

[2] [1]

Bertone, D

For a review of dark matter, see G. Bertone, D. Hooper, and J. Silk, Phys. Rept. 405, 279 (2005)

work page 2005

[3] [2]

R. C. Tolman, Phys. Rev. 55, 364 (1939)

1939

[4] [3]

J. R. Oppenheimer and G. M. Volkoﬀ, Phys. Rev. 55, 374 (1939)

1939

[5] [4]

A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967) [JETP Lett. 5, 24 (1967)]

work page 1967

[6] [5]

M. Y. Khlopov, G. M. Beskin, N. E. Bochkarev, L. A. Pustylnik, an d S. A. Pustyl- nik, Astron. Zh. 68, 42 (1991)[Sov. Astron. 35, 21 (1991)]

work page 1991

[7] [6]

Nakajima and M

T. Nakajima and M. Morikawa, [astro-ph/0506623]

arXiv

[8] [7]

Narain, J

G. Narain, J. Schaﬀner-Bielich, and I. N. Mishustin, Phys. Rev. D 74, 063003 (2006)

work page 2006