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Gravitational collapse of baryonic and dark matter

  • Dipanjan Dey
  • Pankaj S. JoshiEmail author
Open Access
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  • 65 Downloads

Abstract

A massive star undergoes a continual gravitational collapse when the pressures inside the collapsing star become insufficient to balance the pull of gravity. The Physics of gravitational collapse of stars is well studied. Using general relativistic techniques, one can show that the final fate of such a catastrophic collapse can be a black hole or a naked singularity, depending on the initial conditions of gravitational collapse. While stars are made of baryonic matter whose collapse is well studied, there is good indirect evidence that another type of matter, known as dark matter, plays an important role in the formation of large-scale structures in the universe, such as galaxies. It is estimated that some 85% of the total matter in the universe is dark matter. Since the particle constituent of dark matter is not known yet, the gravitational collapse of dark matter is less explored. Here, we consider first some basic properties of baryonic matter and dark matter collapse. Then, we discuss the final fate of gravitational collapse for different types of matter fields and the nature of the singularity which can be formed as an endstate of gravitational collapse. We then present a general relativistic technique to form equilibrium configurations, and argue that this can be thought of as a general relativistic analog of the standard virialization process. We suggest a modification, where the top-hat collapse model of primordial dark-matter halo formation is modified using the general relativistic technique of equilibrium. We also explain why this type of collapse process is more likely to happen in the dark-matter fields.

Mathematics Subject Classification

83C05 83C57 83C75 83F05 85A15 

Notes

References

  1. 1.
    Adler, R.J.; Bjorken, J.D.; Chen, P.; Liu, J.S.: Simple analytic models of gravitational collapse. Am. J. Phys. 73, 1148–1159 (2005)CrossRefGoogle Scholar
  2. 2.
    Annual Review of Astronomy and Astrophysics, vol. 17. Annual Reviews, Palo Alto (1979)Google Scholar
  3. 3.
    Beesham, A.; Ghosh, S.G.: Naked singularities in the charged Vaidya–de Sitter space-time. Int. J. Mod. Phys. D 12, 801 (2003)CrossRefGoogle Scholar
  4. 4.
    Bharadwaj, S.; Kar, S.: Modeling galaxy halos using dark matter with pressure. Phys. Rev. D 68, 023516 (2003)CrossRefGoogle Scholar
  5. 5.
    Bhattacharya, K.; Dey, D.; Mazumdar, A.; Sarkar, T.: On the end stage of spherical gravitational collapse in a cosmological scenario. arXiv:1709.03798 [gr-qc]
  6. 6.
    Binney, J.: ApJ 215, 483–491 (1977)CrossRefGoogle Scholar
  7. 7.
    Birnboim, Y.; Dekel, A.: Virial shocks in galactic haloes. Mon. Not. R. Astron. Soc. 345, 349–364 (2003)CrossRefGoogle Scholar
  8. 8.
    Bizon, P.; Malec, E.; O’Murchadha, N.: Trapped surfaces in spherical stars. Phys. Rev. Lett. 61, 1147–1450 (1988)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bondi, H.: Mon. Not. Astron. Soc. 107, 343 (1947)Google Scholar
  10. 10.
    Bondi, H.: Spherically symmetrical models in general relativity. Mon. Not. R. Astron. Soc. 107, 410–425 (1947)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bullock, J.S.; Boylan-Kolchin, M.: Small-scale challenges to the \(\Lambda \)CDM paradigm. Annu. Rev. Astron. Astrophys. 55, 343 (2017)CrossRefGoogle Scholar
  12. 12.
    Bullock, J.S.; Kolatt, T.S.; Sigad, Y.; Somerville, R.S.; Kravtsov, A.V.; Klypin, A.A.; et al.: Profiles of dark haloes. Evolution, scatter, and environment. Mon. Not. R. Astron. Soc. 321, 559–575 (2001)CrossRefGoogle Scholar
  13. 13.
    Chandrasekhar, S.: Stellar configurations with degenerate cores. The Observatory 57, 373–377 (1934)Google Scholar
  14. 14.
    Christodoulou, D.: Violation of cosmic censorship in the gravitational collapse of a dust cloud. Commun. Math. Phys. 93, 171 (1984)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Cooperstock, F.I.; Jhingan, S.; Joshi, P.S.; Singh, T.P.: Negative pressure and naked singularities in spherical gravitational collapse. Class. Quantum Gravity 14, 2195 (1997)CrossRefGoogle Scholar
  16. 16.
    Cooray, A.; Sheth, R.K.: Halo models of large scale structure. Phys. Rep. 372, 1–129 (2002)CrossRefGoogle Scholar
  17. 17.
    Darmois, G.: Les équations de la gravitation einsteinienne. Mémorial Sci. Math. 25, 1–48 (1927)zbMATHGoogle Scholar
  18. 18.
    Datt, B.: Über eine Klasse von Lösungen der Gravitationsgleichungen der Relativität. Z. Phys. 108, 314–321 (1938)CrossRefGoogle Scholar
  19. 19.
    Del Popolo, A.; Le Delliou, M.: Small scale problems of the \(\Lambda \)CDM model: a short review. Galaxies 5(1), 17 (2017)CrossRefGoogle Scholar
  20. 20.
    Ellis, G.F.R.: Closed trapped surfaces in cosmology. Gen. Relativ. Gravit. 35, 1309–1319 (2003)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Florides, P.S.: A new interior schwarzschild solution. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 337, pp. 529–535. The Royal Society, London (1974)Google Scholar
  22. 22.
    Frenk, C.S.; White, S.D.M.: Dark matter and cosmic structure. Annalen der Physik. 524, 507–534 (2012)Google Scholar
  23. 23.
    Friedman, A.: On the curvature of space. Z. Phys. 10, 377 (1922)CrossRefGoogle Scholar
  24. 24.
    Friedman, A.: On the curvature of space. Gen. Relativ. Gravit. 31, 1991 (1999)CrossRefGoogle Scholar
  25. 25.
    Goncalves, S.M.C.V.: Naked singularities in Tolman–Bondi–de Sitter collapse. Phys. Rev. D 63, 064017 (2001)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Goswami, R.; Joshi, P.S.: What role pressures play to determine the final end state of gravitational collapse? Class. Quantum Gravity 19, 5229 (2002)CrossRefGoogle Scholar
  27. 27.
    Goswami, R.; Joshi, P.S.: Black hole formation in perfect fluid collapse. Phys. Rev. D 69, 027502 (2004)CrossRefGoogle Scholar
  28. 28.
    Goswami, R.; Joshi, P.S.; Malafarina, D.: Scalar field collapse and cosmic censorship. arXiv:1202.6218 [gr-qc]
  29. 29.
    Gundlach, C.: Critical phenomena in gravitational collapse. Phys. Rep. 376, 339 (2003)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Gunn, J.E.; Gott III, J.R.: On the infall of matter into clusters of galaxies and some effects on their evolution. Astrophys. J. 176, 1–19 (1972)CrossRefGoogle Scholar
  31. 31.
    Harada, T.; Iguchi, H.; Nakao, K.: Naked singularity formation in the collapse of a spherical cloud of counter rotating particles. Phys. Rev. D 58, 041502 (1998)CrossRefGoogle Scholar
  32. 32.
    Harada, T.; Iguchi, H.; Nakao, K.: Physical processes in naked singularity formation. Prog. Theor. Phys. 107, 449 (2002)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Hawking, S.W.; Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973)CrossRefGoogle Scholar
  34. 34.
    Hellaby, C.; Lake, K.: Shell crossings and the Tolman model. Astrophys. J. 290, 381 (1985)CrossRefGoogle Scholar
  35. 35.
    Hogan, C.J.; Kaiser, N.; Turner, M.S.; Vittorio, N.; White, S.D.M.: The formation of structure in the universe. FERMILAB-CONF-85-057-A (1985)Google Scholar
  36. 36.
    Israel, W.: Singular hypersurfaces and thin shells in general relativity. Nuovo Cim. B 44S10, 1 (1966)CrossRefGoogle Scholar
  37. 37.
    Jenkins, A.; Frenk, C.S.; White, S.D.M.; Colberg, J.M.; Cole, S.; Evrard, A.E.; et al.: The mass function of dark matter halos. Mon. Not. R. Astron. Soc. 321, 372 (2001)CrossRefGoogle Scholar
  38. 38.
    Jhingan, S.; Magli, G.: Gravitational collapse of spherically symmetric clusters of rotating particles. arXiv:gr-qc/9902041
  39. 39.
    Jing, Y.: The density profile of equilibrium and nonequilibrium dark matter halos. Astrophys. J. 535, 30 (2000)CrossRefGoogle Scholar
  40. 40.
    Joshi, P.S.: Visibility of a spacetime singularity. Phys. Rev. D 75, 044005 (2007)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Joshi, P.S.: On the genericity of spacetime singularities. Pramana 69, 119 (2007)CrossRefGoogle Scholar
  42. 42.
    Joshi, P.S.: Gravitational Collapse and Spacetime Singularities. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  43. 43.
    Joshi, P.S.; Dwivedi, I.H.: Naked singularities in spherically symmetric inhomogeneous Tolman–Bondi dust cloud collapse. Phys. Rev. D 47, 5357 (1993)CrossRefGoogle Scholar
  44. 44.
    Joshi, P.S.; Malafarina, D.: Recent developments in gravitational collapse and spacetime singularities. Int. J. Mod. Phys. D 20, 2641 (2011)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Joshi, P.S.; Malafarina, D.: Instability of black hole formation under small pressure perturbations. Gen. Relativ. Gravit. 45, 305 (2013)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Joshi, P.S.; Saraykar, R.V.: Shell-crossings in gravitational collapse. Int. J. Mod. Phys. D 22, 1350027 (2013)CrossRefGoogle Scholar
  47. 47.
    Joshi, P.S.; Goswami, R.; Dadhich, N.: Why do naked singularities form in gravitational collapse? 2. Phys. Rev. D 70, 087502 (2004)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Joshi, P.S.; Malafarina, D.; Narayan, R.: Equilibrium configurations from gravitational collapse. Class. Quantum Gravity 28, 235018 (2011)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Joshi, P.S.; Malafarina, D.; Saraykar, R.V.: Genericity aspects in gravitational collapse to black holes and naked singularities. Int. J. Mod. Phys. D 21, 1250066 (2012)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Joshi, P.S.; Malafarina, D.; Narayan, R.: Distinguishing black holes from naked singularities through their accretion disc properties. Class. Quantum Gravity 31, 015002 (2014)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Kong, L.; Malafarina, D.; Bambi, C.: Can we observationally test the weak cosmic censorship conjecture? Eur. Phys. J. C 74, 2983 (2014)CrossRefGoogle Scholar
  52. 52.
    Kuroda, Y.: Naked singularity in Vaidya space-time. Prog. Theor. Phys. 72, 63–72 (1984)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Lake, K.: Collapse of radiating imperfect fluid spheres. Phys. Rev. D 26, 518 (1982)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Lake, K.; Hellaby, C.: Collapse of radiating fluid spheres. Phys. Rev. D 24, 3019 (1981)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Lasky, P.D.; Lun, A.W.C.: Generalized Lemaitre–Tolman–Bondi solutions with pressure. Phys. Rev. D 74, 084013 (2006)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Lasky, P.D.; Lun, A.W.C.; Burston, R.B.: Initial value formalism for Lemaître-Tolman-Bondi collapse. ANZIAM J. 49, 53–73 (2007)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Lattimer, J.M.; Prakash, M.: The physics of neutron stars. Science 304, 536 (2004)CrossRefGoogle Scholar
  58. 58.
    Lemaître, G.: L’Univers en expansion. Ann. Soc. Sci. Brux. I A 53, 51–85 (1933)zbMATHGoogle Scholar
  59. 59.
    Liddle, A.R.; Lyth, D.H.: The cold dark matter density perturbation. Phys. Rep. 231, 1 (1993)CrossRefGoogle Scholar
  60. 60.
    Lieb, E.H.; Yau, H.T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112, 147 (1987)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Lynden-Bell, D.: Statistical mechanics of violent relaxation in stellar systems. Mon. Not. R. Astron. Soc. 136, 101–121 (1967)CrossRefGoogle Scholar
  62. 62.
    May, M.M.; White, R.H.: Hydrodynamic calculations of general-relativistic collapse. Phys. Rev. 141, 1232–1241 (1966)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Merritt, D.: Elliptical galaxy dynamics. Publ. Astron. Soc. Pac. 111, 129 (1999)CrossRefGoogle Scholar
  64. 64.
    Misner, C.W.; Sharp, D.H.: Relativistic equations for adiabatic, spherically symmetric gravitational collapse. Phys. Rev. 136, B571–B576 (1964)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Navarro, J.F.; Frenk, C.S.; White, S.D.M.: The structure of cold dark matter Halos. Astrophys. J. 462, 563–575 (1996)CrossRefGoogle Scholar
  66. 66.
    Navarro, J.F.; Frenk, C.S.; White, S.D.M.: A universal density profile from hierarchical clustering. Astrophys. J. 490, 493–508 (1997)CrossRefGoogle Scholar
  67. 67.
    Oppenheimer, J.R.; Snyder, H.: On continued gravitational contraction. Phys. Rev. 56, 455 (1939)CrossRefGoogle Scholar
  68. 68.
    Padmanabhan, T.: Structure Formation in the Universe. Cambridge University Press, Cambridge (1993)Google Scholar
  69. 69.
    Penrose, R.: Gravitational collapse: the role of general relativity. Riv. Nuovo Cim. 1, 252 (1969)Google Scholar
  70. 70.
    Penrose, R.: The question of cosmic censorship. J. Astrophys. Astron. 20, 233 (1999)CrossRefGoogle Scholar
  71. 71.
    Poisson, E.: A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  72. 72.
    Rubin, V.C.; Ford Jr., W.K.; Thonnard, N.: Extended rotation curves of high-luminosity spiral galaxies. IV. Systematic dynamical properties, Sa through Sc. Astrophys. J. 225, L107 (1978)CrossRefGoogle Scholar
  73. 73.
    Rubin, V.C.; Thonnard, N.; Ford Jr., W.K.: Rotational properties of 21 SC galaxies with a large range of luminosities and radii, from NGC 4605 (R = 4 kpc) to UGC 2885 (R = 122 kpc). Astrophys. J. 238, 471 (1980)CrossRefGoogle Scholar
  74. 74.
    Sarwe, S.; Saraykar, R.V.; Joshi, P.S.: Gravitational collapse with equation of state. arXiv:1207.3200 [gr-qc]
  75. 75.
    Satin, S.; Malafarina, D.; Joshi, P.S.: Genericity aspects of black hole formation in the collapse of spherically symmetric slightly inhomogeneous perfect fluids. Int. J. Mod. Phys. D 25, 1650023 (2016)MathSciNetCrossRefGoogle Scholar
  76. 76.
    Saxton, C.J.: Galaxy stability within a self-interacting dark matter halo. Mon. Not. R. Astron. Soc. 430, 1578 (2013)CrossRefGoogle Scholar
  77. 77.
    Sengor, G.: Cosmological Perturbations in the Early Universe. arXiv:1807.08007
  78. 78.
    Senovilla, J.M.M.; Garfinkle, D.: The 1965 Penrose singularity theorem. Class. Quantum Gravity 32(12), 124008 (2015)MathSciNetCrossRefGoogle Scholar
  79. 79.
    Silk, J.: Galaxy Formation and Large Scale Structure, pp. 277–378. Springer, New York (1987)Google Scholar
  80. 80.
    Simon, J.D.; Bolatto, A.D.; Leroy, A.; Blitz, L.; Gates, E.L.: High-resolution measurements of the halos of four dark matter-dominated galaxies: deviations from a universal density profile. Astrophys. J. 621, 757–776 (2005)CrossRefGoogle Scholar
  81. 81.
    Singh, T.P.; Joshi, P.S.: The final fate of spherical inhomogeneous dust collapse. Class. Quantum Gravity 13, 559 (1996)MathSciNetCrossRefGoogle Scholar
  82. 82.
    Spergel, D.N.; Steinhardt, P.J.: Observational evidence for selfinteracting cold dark matter. Phys. Rev. Lett. 84, 3760 (2000)CrossRefGoogle Scholar
  83. 83.
    Szekeres, P.; Lun, A.: What is a shell-crossing singularity? J. Aust. Math. Soc. Ser. B Appl. Math. 41, 167–179 (1999)MathSciNetCrossRefGoogle Scholar
  84. 84.
    Tolman, R.C.: Effect of inhomogeneity on cosmological models. Proc. Natl. Acad. Sci. USA 20, 410 (1934)zbMATHGoogle Scholar
  85. 85.
    Tulin, S.: Dark matter self-interactions and small scale structure. Phys. Rep. 730, 1–57 (2018)MathSciNetCrossRefGoogle Scholar
  86. 86.
    Vaz, C.; Witten, L.: Do naked singularities form? Class. Quantum Gravity 13, L59 (1996)MathSciNetCrossRefGoogle Scholar
  87. 87.
    VIRGO Consortium Collaboration; Smith, R.E.; Peacock, J.A.; Jenkins, A.; White, S.D.M.; Frenk, C.S.; Pearce, F.R.; et al.: Stable clustering, the halo model and nonlinear cosmological power spectra. Mon. Not. R. Astron. Soc. 341, 1311 (2003)Google Scholar
  88. 88.
    Wagh, S.M.; Maharaj, S.D.: Naked singularity of the Vaidya–de Sitter space-time and cosmic censorship conjecture. Gen. Relativ. Gravit. 31, 975 (1999)CrossRefGoogle Scholar
  89. 89.
    Weinberg, S.: Gravitation and Cosmology, p. 482. Wiley, New York (1972)Google Scholar
  90. 90.
    Weinberg, D.H.; Bullock, J.S.; Governato, F.; Kuzio de Naray, R.; Peter, A.H.G.: Cold dark matter: controversies on small scales. Proc. Natl. Acad. Sci. 112, 12249 (2015)CrossRefGoogle Scholar
  91. 91.
    White, S.D.M.: arXiv:astro-ph/9410043
  92. 92.
    White, S.D.M.; Liu, D.Q.: The origin and evolution of structures in a universe dominated by cold dark matter. In: Origin, Structure and Evolution of Galaxies, Proceedings of the Guo Shoujing Summer School of Astrophysics, Tunxi, China, pp. 281–317. World Scientific, Singapore and Teaneck, NJ (1988)Google Scholar
  93. 93.
    White, S.D.M.; Rees, M.J.: Mon. Not. R. Astron. Soc. 183, 341 (1978)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.International Center for CosmologyCharusat UniversityAnandIndia

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