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Universal relations for elastic hybrid stars and quark stars

Chun-Ming Yip1,2, Shu Yan Lau3, and Kent Yagi4

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Phys. Rev. D 114, 043024 – Published 13 August, 2026

DOI: https://doi.org/10.1103/cflw-5fps

Abstract

Some compact stars may contain deconfined quark matter, forming hybrid stars or quark stars. If the quark matter forms an inhomogeneous condensate in the crystalline color superconducting phase, its rigidity may be high enough to noticeably alter the stellar properties. In this paper, we investigate whether these elastic stars follow the universal relations, i.e., relations insensitive to equations of state, that have been well established for fluid stars. We improve upon previous studies by allowing quark matter in the background, static, and spherically symmetric configuration to be sheared. Such background shear can be treated in the form of an effective pressure anisotropy. We then calculate the moment of inertia I, tidal deformability λ2, and spin-induced quadrupole moment Q of these models with pressure anisotropy. The I−λ2−Q universal relations for the elastic hybrid (quark) star models are valid up to a variation of ≈3%(4%), larger than that for typical fluid star models, when the maximal magnitude of quark matter shear modulus is considered in the crystalline color superconducting phase from realistic calculations. The uncertainty in universal relations related to the stellar compactness for these elastic star models, on the other hand, remain comparable to those for typical fluid star models. Our results demonstrate the validity of universal relations for hybrid stars and quark stars with a realistic degree of pressure anisotropy due to the crystalline color superconducting quark matter.

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References (92)

  1. M. Alford, M. Braby, M. W. Paris, and S. Reddy, Hybrid stars that masquerade as neutron stars, Astrophys. J. 629, 969 (2005).
  2. M. G. Alford, S. Han, and M. Prakash, Generic conditions for stable hybrid stars, Phys. Rev. D 88, 083013 (2013).
  3. E. Witten, Cosmic separation of phases, Phys. Rev. D 30, 272 (1984).
  4. M. G. Alford, K. Rajagopal, and F. Wilczek, QCD at finite baryon density: Nucleon droplets and color superconductivity, Phys. Lett. B 422, 247 (1998).
  5. M. G. Alford, K. Rajagopal, and F. Wilczek, Color flavor locking and chiral symmetry breaking in high density QCD, Nucl. Phys. B537, 443 (1999).
  6. M. G. Alford, J. A. Bowers, and K. Rajagopal, Crystalline color superconductivity, Phys. Rev. D 63, 074016 (2001).
  7. K. Rajagopal and R. Sharma, The crystallography of three-flavor quark matter, Phys. Rev. D 74, 094019 (2006).
  8. R. Anglani, R. Casalbuoni, M. Ciminale, N. Ippolito, R. Gatto, M. Mannarelli, and M. Ruggieri, Crystalline color superconductors, Rev. Mod. Phys. 86, 509 (2014).
  9. M. G. Alford, Color superconducting quark matter, Annu. Rev. Nucl. Part. Sci. 51, 131 (2001).
  10. M. G. Alford, A. Schmitt, K. Rajagopal, and T. Schäfer, Color superconductivity in dense quark matter, Rev. Mod. Phys. 80, 1455 (2008).
  11. G. Pagliara, Formation of quark phases in protoneutron stars: The transition from the 2SC to the normal quark phase, Phys. Rev. D 83, 125013 (2011).
  12. M. Mannarelli, K. Rajagopal, and R. Sharma, The Rigidity of crystalline color superconducting quark matter, Phys. Rev. D 76, 074026 (2007).
  13. M. Karlovini and L. Samuelsson, Elastic stars in general relativity. 1. Foundations and equilibrium models, Classical Quantum Gravity 20, 3613 (2003).
  14. M. Karlovini, L. Samuelsson, and M. Zarroug, Elastic stars in general relativity. 2. Radial perturbations, Classical Quantum Gravity 21, 1559 (2004).
  15. R. L. Bowers and E. P. T. Liang, Anisotropic spheres in general relativity, Astrophys. J. 188, 657 (1974).
  16. D. Horvat, S. Ilijic, and A. Marunovic, Radial pulsations and stability of anisotropic stars with quasi-local equation of state, Classical Quantum Gravity 28, 025009 (2011).
  17. X.-G. Huang, M. Huang, D. H. Rischke, and A. Sedrakian, Anisotropic hydrodynamics, bulk viscosities and R-modes of strange quark stars with strong magnetic Fields, Phys. Rev. D 81, 045015 (2010).
  18. E. J. Ferrer, V. de la Incera, J. P. Keith, I. Portillo, and P. L. Springsteen, Equation of state of a dense and magnetized Fermion system, Phys. Rev. C 82, 065802 (2010).
  19. S. Yazadjiev, Relativistic models of magnetars: Nonperturbative analytical approach, Phys. Rev. D 85, 044030 (2012).
  20. L. M. Becerra, E. A. Becerra-Vergara, and F. D. Lora-Clavijo, Realistic anisotropic neutron stars: Pressure effects, Phys. Rev. D 109, 043025 (2024).
  21. R. F. Sawyer and D. J. Scalapino, Pion condensation in superdense nuclear matter, Phys. Rev. D 7, 953 (1973).
  22. W. H. Dickhoff, A. Faessler, J. Meyer-Ter-Vehn, and H. Muther, Pion condensation and realistic interactions, Phys. Rev. C 23, 1154 (1981).
  23. A. B. Migdal, E. E. Saperstein, M. A. Troitsky, and D. N. Voskresensky, Pion degrees of freedom in nuclear matter, Phys. Rep. 192, 179 (1990).
  24. P. H. R. S. Moraes, G. Panotopoulos, and I. Lopes, Anisotropic dark matter stars, Phys. Rev. D 103, 084023 (2021).
  25. J. Bramante and N. Raj, Dark matter in compact stars, Phys. Rep. 1052, 1 (2024).
  26. S. K. Maurya, K. N. Singh, A. Aziz, S. Ray, and G. Mustafa, Compact stars with dark matter induced anisotropy in complexity-free background and effect of dark matter on GW echoes, Mon. Not. R. Astron. Soc. 527, 5192 (2023).
  27. C. R. Ghezzi, Anisotropic dark energy stars, Astrophys. Space Sci. 333, 437 (2011).
  28. K. Das and N. Ali, Anisotropic charged dark energy star, Astrophys. Space Sci. 356, 57 (2015).
  29. T. Cadogan and E. Poisson, Self-gravitating anisotropic fluids. I: Context and overview, Gen. Relativ. Gravit. 56, 118 (2024).
  30. T. Cadogan and E. Poisson, Self-gravitating anisotropic fluid. II: Newtonian theory, Gen. Relativ. Gravit. 56, 119 (2024).
  31. T. Cadogan and E. Poisson, Self-gravitating anisotropic fluid. III: Relativistic theory, Gen. Relativ. Gravit. 56, 120 (2024).
  32. A. Alho, J. Natário, P. Pani, and G. Raposo, Compact elastic objects in general relativity, Phys. Rev. D 105, 044025 (2022); 105, 129903(E) (2022).
  33. Z. Z. Dong, S. Y. Lau, and K. Yagi, New modeling for hybrid stars with an elastic quark core, Phys. Rev. D 111, 123043 (2025).
  34. A. Alho, J. Natário, P. Pani, and G. Raposo, Compactness bounds in general relativity, Phys. Rev. D 106, L041502 (2022).
  35. J. B. Hartle, Slowly rotating relativistic stars. 1. Equations of structure, Astrophys. J. 150, 1005 (1967).
  36. J. B. Hartle and K. S. Thorne, Slowly rotating relativistic stars. II. Models for neutron stars and supermassive stars, Astrophys. J. 153, 807 (1968).
  37. D. G. Ravenhall and C. J. Pethick, Neutron star moments of inertia, Astrophys. J. 424, 846 (1994).
  38. M. Bejger and P. Haensel, Moments of inertia for neutron and strange stars: Limits derived for the Crab pulsar, Astron. Astrophys. 396, 917 (2002).
  39. J. M. Lattimer and B. F. Schutz, Constraining the equation of state with moment of inertia measurements, Astrophys. J. 629, 979 (2005).
  40. M. Urbanec, J. C. Miller, and Z. Stuchlík, Quadrupole moments of rotating neutron stars and strange stars, Mon. Not. R. Astron. Soc. 433, 1903 (2013).
  41. M. Bauböck, E. Berti, D. Psaltis, and F. Özel, Relations between neutron-star parameters in the Hartle-Thorne approximation, Astrophys. J. 777, 68 (2013).
  42. C. Breu and L. Rezzolla, Maximum mass, moment of inertia and compactness of relativistic stars, Mon. Not. R. Astron. Soc. 459, 646 (2016).
  43. N. Jiang and K. Yagi, Analytic I-Love-C relations for realistic neutron stars, Phys. Rev. D 101, 124006 (2020).
  44. K. Yagi and N. Yunes, Approximate universal relations for neutron stars and quark stars, Phys. Rep. 681, 1 (2017).
  45. K. Yagi and N. Yunes, I-Love-Q, Science 341, 365 (2013).
  46. K. Yagi and N. Yunes, I-Love-Q relations in neutron stars and their applications to astrophysics, gravitational waves and fundamental physics, Phys. Rev. D 88, 023009 (2013).
  47. D. D. Doneva, S. S. Yazadjiev, N. Stergioulas, and K. D. Kokkotas, Breakdown of I-Love-Q universality in rapidly rotating relativistic stars, Astrophys. J. Lett. 781, L6 (2013).
  48. G. Pappas and T. A. Apostolatos, Effectively universal behavior of rotating neutron stars in general relativity makes them even simpler than their Newtonian counterparts, Phys. Rev. Lett. 112, 121101 (2014).
  49. S. Chakrabarti, T. Delsate, N. Gürlebeck, and J. Steinhoff, I-Q relation for rapidly rotating neutron stars, Phys. Rev. Lett. 112, 201102 (2014).
  50. K. Yagi, K. Kyutoku, G. Pappas, N. Yunes, and T. A. Apostolatos, Effective no-hair relations for neutron stars and quark stars: Relativistic results, Phys. Rev. D 89, 124013 (2014).
  51. V. Paschalidis, K. Yagi, D. Alvarez-Castillo, D. B. Blaschke, and A. Sedrakian, Implications from GW170817 and I-Love-Q relations for relativistic hybrid stars, Phys. Rev. D 97, 084038 (2018).
  52. N. K. Largani, T. Fischer, A. Sedrakian, M. Cierniak, D. E. Alvarez-Castillo, and D. B. Blaschke, Universal relations for rapidly rotating cold and hot hybrid stars, Mon. Not. R. Astron. Soc. 515, 3539 (2022).
  53. G. Papigkiotis and G. Pappas, Universal relations for rapidly rotating neutron stars using supervised machine-learning techniques, Phys. Rev. D 107, 103050 (2023).
  54. C. J. Krüger and S. H. Völkel, Rapidly rotating neutron stars: Universal relations and EOS inference, Phys. Rev. D 108, 124056 (2023).
  55. S. Y. Lau, P. T. Leung, and L. M. Lin, Tidal deformations of compact stars with crystalline quark matter, Phys. Rev. D 95, 101302 (2017).
  56. S. Y. Lau, P. T. Leung, and L. M. Lin, Two-layer compact stars with crystalline quark matter: Screening effect on the tidal deformability, Phys. Rev. D 99, 023018 (2019).
  57. F. Gittins, N. Andersson, and J. P. Pereira, Tidal deformations of neutron stars with elastic crusts, Phys. Rev. D 101, 103025 (2020).
  58. Z. Z. Dong, J. C. Faggert, S. Y. Lau, and K. Yagi, Love–C relations for elastic hybrid stars, Gen. Relativ. Gravit. 56, 113 (2024).
  59. K. Yagi and N. Yunes, I-Love-Q anisotropically: Universal relations for compact stars with scalar pressure anisotropy, Phys. Rev. D 91, 123008 (2015).
  60. H. C. Das, I-Love-C relation for an anisotropic neutron star, Phys. Rev. D 106, 103518 (2022).
  61. S. R. Mohanty, S. Ghosh, P. Routaray, H. C. Das, and B. Kumar, The impact of anisotropy on neutron star properties: Insights from I–f–C universal relations, J. Cosmol. Astropart. Phys. 03 (2024) 054.
  62. J. M. Z. Pretel and C. Zhang, Universal relations for anisotropic interacting quark stars, J. Cosmol. Astropart. Phys. 10 (2024) 032.
  63. A. Akmal, V. R. Pandharipande, and D. G. Ravenhall, The Equation of state of nucleon matter and neutron star structure, Phys. Rev. C 58, 1804 (1998).
  64. P. Haensel, A. Y. Potekhin, and D. G. Yakovlev, Neutron Stars 1: Equation of State and Structure (Springer, New York, 2007), Vol. 326.
  65. P. J. Davis, H. D. Thi, A. F. Fantina, F. Gulminelli, M. Oertel, and L. Suleiman, Inference of neutron-star properties with unified crust-core equations of state for parameter estimation, Astron. Astrophys. 687, A44 (2024).
  66. P. J. Davis, H. Dinh Thi, A. F. Fantina, F. Gulminelli, M. Oertel, and L. Suleiman, Crust (unified) tool for equation-of-state reconstruction (CUTER) v2, Eur. Phys. J. A 61, 120 (2025).
  67. F. Douchin and P. Haensel, A unified equation of state of dense matter and neutron star structure, Astron. Astrophys. 380, 151 (2001).
  68. I. Bombaci and D. Logoteta, Equation of state of dense nuclear matter and neutron star structure from nuclear chiral interactions, Astron. Astrophys. 609, A128 (2018).
  69. H. Shen, H. Toki, K. Oyamatsu, and K. Sumiyoshi, Relativistic equation of state of nuclear matter for supernova explosion, Prog. Theor. Phys. 100, 1013 (1998).
  70. H. Shen, H. Toki, K. Oyamatsu, and K. Sumiyoshi, Relativistic equation of state of nuclear matter for supernova and neutron star, Nucl. Phys. A637, 435 (1998).
  71. Y. Sugahara and H. Toki, Relativistic mean field theory for unstable nuclei with nonlinear sigma and omega terms, Nucl. Phys. A579, 557 (1994).
  72. S. Typel, M. Oertel, and T. Klähn, CompOSE CompStar online supernova equations of state harmonising the concert of nuclear physics and astrophysics compose.obspm.fr, Phys. Part. Nucl. 46, 633 (2015).
  73. M. Oertel, M. Hempel, T. Klähn, and S. Typel, Equations of state for supernovae and compact stars, Rev. Mod. Phys. 89, 015007 (2017).
  74. S. Typel et al. (CompOSE Core Team), CompOSE reference manual, Eur. Phys. J. A 58, 221 (2022).
  75. J. C. Faggert, Probing Crystalline Color Superconducting Phase of Quark Matter through Universal Relations for Hybrid Stars, Virginia U. 10.18130/njvn-1964 (2023), 10.18130/njvn-1964.
  76. B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), GW170817: Measurements of neutron star radii and equation of state, Phys. Rev. Lett. 121, 161101 (2018).
  77. M. C. Miller et al., PSR J0030+0451 mass and radius from NICER data and implications for the properties of neutron star matter, Astrophys. J. Lett. 887, L24 (2019).
  78. T. Salmi et al., The radius of the high-mass pulsar PSR J0740+6620 with 3.6 yr of NICER data, Astrophys. J. 974, 294 (2024).
  79. V. Doroshenko, V. Suleimanov, G. Pühlhofer, and A. Santangelo, A strangely light neutron star within a supernova remnant, Nat. Astron. 6, 1444 (2022).
  80. I. A. Rather, G. Panotopoulos, and I. Lopes, Quark models and radial oscillations: Decoding the HESS J1731-347 compact object’s equation of state, Eur. Phys. J. C 83, 1065 (2023).
  81. F. Di Clemente, A. Drago, and G. Pagliara, Is the compact object associated with HESS J1731-347 a strange quark star? A possible astrophysical scenario for its formation, Astrophys. J. 967, 159 (2024).
  82. Z. F. Seidov, The stability of a star with a phase change in general relativity theory, Sov. Astron. 15, 347 (1971).
  83. S. Vinciguerra et al., An updated mass–radius analysis of the 2017–2018 NICER data set of PSR J0030+0451, Astrophys. J. 961, 62 (2024).
  84. T. K. Chan, A. P. O. Chan, and P. T. Leung, I-Love relations for incompressible stars and realistic stars, Phys. Rev. D 91, 044017 (2015).
  85. T. K. Chan, A. P. O. Chan, and P. T. Leung, Universality and stationarity of the I-Love relation for self-bound stars, Phys. Rev. D 93, 024033 (2016).
  86. M. Karlovini and L. Samuelsson, Elastic stars in general relativity: IV. Axial perturbations, Classical Quantum Gravity 24, 3171 (2007).
  87. B. Carter, Elastic perturbation theory in general relativity and a variation principle for a rotating solid star, Commun. Math. Phys. 30, 261 (1973).
  88. K. Yagi, L. C. Stein, G. Pappas, N. Yunes, and T. A. Apostolatos, Why I-Love-Q: Explaining why universality emerges in compact objects, Phys. Rev. D 90, 063010 (2014).
  89. T. K. Chan, Y. H. Sham, P. T. Leung, and L. M. Lin, Multipolar universal relations between f-mode frequency and tidal deformability of compact stars, Phys. Rev. D 90, 124023 (2014).
  90. Y. H. Sham, T. K. Chan, L. M. Lin, and P. T. Leung, Unveiling the universality of I-Love-Q relations, Astrophys. J. 798, 121 (2015).
  91. K. Kyutoku, Assessment of universal relations among second-order moments of relativistic stars via reformulated perturbation equations, Phys. Rev. D 112, 024059 (2025).
  92. T. Katagiri, G. R. Mukkamala, and K. Yagi, Theoretical modeling of approximate universality of tidally deformed neutron stars, Phys. Rev. D 112, 023030 (2025).

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Phys. Rev. D 114, 043024– Published 13 August, 2026
  • Received 15 October 2025
  • Accepted 9 July 2026
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