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.
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Adler, R.J.; Bjorken, J.D.; Chen, P.; Liu, J.S.: Simple analytic models of gravitational collapse. Am. J. Phys. 73, 1148–1159 (2005)
Annual Review of Astronomy and Astrophysics, vol. 17. Annual Reviews, Palo Alto (1979)
Beesham, A.; Ghosh, S.G.: Naked singularities in the charged Vaidya–de Sitter space-time. Int. J. Mod. Phys. D 12, 801 (2003)
Bharadwaj, S.; Kar, S.: Modeling galaxy halos using dark matter with pressure. Phys. Rev. D 68, 023516 (2003)
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]
Binney, J.: ApJ 215, 483–491 (1977)
Birnboim, Y.; Dekel, A.: Virial shocks in galactic haloes. Mon. Not. R. Astron. Soc. 345, 349–364 (2003)
Bizon, P.; Malec, E.; O’Murchadha, N.: Trapped surfaces in spherical stars. Phys. Rev. Lett. 61, 1147–1450 (1988)
Bondi, H.: Mon. Not. Astron. Soc. 107, 343 (1947)
Bondi, H.: Spherically symmetrical models in general relativity. Mon. Not. R. Astron. Soc. 107, 410–425 (1947)
Bullock, J.S.; Boylan-Kolchin, M.: Small-scale challenges to the \(\Lambda \)CDM paradigm. Annu. Rev. Astron. Astrophys. 55, 343 (2017)
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)
Chandrasekhar, S.: Stellar configurations with degenerate cores. The Observatory 57, 373–377 (1934)
Christodoulou, D.: Violation of cosmic censorship in the gravitational collapse of a dust cloud. Commun. Math. Phys. 93, 171 (1984)
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)
Cooray, A.; Sheth, R.K.: Halo models of large scale structure. Phys. Rep. 372, 1–129 (2002)
Darmois, G.: Les équations de la gravitation einsteinienne. Mémorial Sci. Math. 25, 1–48 (1927)
Datt, B.: Über eine Klasse von Lösungen der Gravitationsgleichungen der Relativität. Z. Phys. 108, 314–321 (1938)
Del Popolo, A.; Le Delliou, M.: Small scale problems of the \(\Lambda \)CDM model: a short review. Galaxies 5(1), 17 (2017)
Ellis, G.F.R.: Closed trapped surfaces in cosmology. Gen. Relativ. Gravit. 35, 1309–1319 (2003)
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)
Frenk, C.S.; White, S.D.M.: Dark matter and cosmic structure. Annalen der Physik. 524, 507–534 (2012)
Friedman, A.: On the curvature of space. Z. Phys. 10, 377 (1922)
Friedman, A.: On the curvature of space. Gen. Relativ. Gravit. 31, 1991 (1999)
Goncalves, S.M.C.V.: Naked singularities in Tolman–Bondi–de Sitter collapse. Phys. Rev. D 63, 064017 (2001)
Goswami, R.; Joshi, P.S.: What role pressures play to determine the final end state of gravitational collapse? Class. Quantum Gravity 19, 5229 (2002)
Goswami, R.; Joshi, P.S.: Black hole formation in perfect fluid collapse. Phys. Rev. D 69, 027502 (2004)
Goswami, R.; Joshi, P.S.; Malafarina, D.: Scalar field collapse and cosmic censorship. arXiv:1202.6218 [gr-qc]
Gundlach, C.: Critical phenomena in gravitational collapse. Phys. Rep. 376, 339 (2003)
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)
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)
Harada, T.; Iguchi, H.; Nakao, K.: Physical processes in naked singularity formation. Prog. Theor. Phys. 107, 449 (2002)
Hawking, S.W.; Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973)
Hellaby, C.; Lake, K.: Shell crossings and the Tolman model. Astrophys. J. 290, 381 (1985)
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)
Israel, W.: Singular hypersurfaces and thin shells in general relativity. Nuovo Cim. B 44S10, 1 (1966)
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)
Jhingan, S.; Magli, G.: Gravitational collapse of spherically symmetric clusters of rotating particles. arXiv:gr-qc/9902041
Jing, Y.: The density profile of equilibrium and nonequilibrium dark matter halos. Astrophys. J. 535, 30 (2000)
Joshi, P.S.: Visibility of a spacetime singularity. Phys. Rev. D 75, 044005 (2007)
Joshi, P.S.: On the genericity of spacetime singularities. Pramana 69, 119 (2007)
Joshi, P.S.: Gravitational Collapse and Spacetime Singularities. Cambridge University Press, Cambridge (2007)
Joshi, P.S.; Dwivedi, I.H.: Naked singularities in spherically symmetric inhomogeneous Tolman–Bondi dust cloud collapse. Phys. Rev. D 47, 5357 (1993)
Joshi, P.S.; Malafarina, D.: Recent developments in gravitational collapse and spacetime singularities. Int. J. Mod. Phys. D 20, 2641 (2011)
Joshi, P.S.; Malafarina, D.: Instability of black hole formation under small pressure perturbations. Gen. Relativ. Gravit. 45, 305 (2013)
Joshi, P.S.; Saraykar, R.V.: Shell-crossings in gravitational collapse. Int. J. Mod. Phys. D 22, 1350027 (2013)
Joshi, P.S.; Goswami, R.; Dadhich, N.: Why do naked singularities form in gravitational collapse? 2. Phys. Rev. D 70, 087502 (2004)
Joshi, P.S.; Malafarina, D.; Narayan, R.: Equilibrium configurations from gravitational collapse. Class. Quantum Gravity 28, 235018 (2011)
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)
Joshi, P.S.; Malafarina, D.; Narayan, R.: Distinguishing black holes from naked singularities through their accretion disc properties. Class. Quantum Gravity 31, 015002 (2014)
Kong, L.; Malafarina, D.; Bambi, C.: Can we observationally test the weak cosmic censorship conjecture? Eur. Phys. J. C 74, 2983 (2014)
Kuroda, Y.: Naked singularity in Vaidya space-time. Prog. Theor. Phys. 72, 63–72 (1984)
Lake, K.: Collapse of radiating imperfect fluid spheres. Phys. Rev. D 26, 518 (1982)
Lake, K.; Hellaby, C.: Collapse of radiating fluid spheres. Phys. Rev. D 24, 3019 (1981)
Lasky, P.D.; Lun, A.W.C.: Generalized Lemaitre–Tolman–Bondi solutions with pressure. Phys. Rev. D 74, 084013 (2006)
Lasky, P.D.; Lun, A.W.C.; Burston, R.B.: Initial value formalism for Lemaître-Tolman-Bondi collapse. ANZIAM J. 49, 53–73 (2007)
Lattimer, J.M.; Prakash, M.: The physics of neutron stars. Science 304, 536 (2004)
Lemaître, G.: L’Univers en expansion. Ann. Soc. Sci. Brux. I A 53, 51–85 (1933)
Liddle, A.R.; Lyth, D.H.: The cold dark matter density perturbation. Phys. Rep. 231, 1 (1993)
Lieb, E.H.; Yau, H.T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112, 147 (1987)
Lynden-Bell, D.: Statistical mechanics of violent relaxation in stellar systems. Mon. Not. R. Astron. Soc. 136, 101–121 (1967)
May, M.M.; White, R.H.: Hydrodynamic calculations of general-relativistic collapse. Phys. Rev. 141, 1232–1241 (1966)
Merritt, D.: Elliptical galaxy dynamics. Publ. Astron. Soc. Pac. 111, 129 (1999)
Misner, C.W.; Sharp, D.H.: Relativistic equations for adiabatic, spherically symmetric gravitational collapse. Phys. Rev. 136, B571–B576 (1964)
Navarro, J.F.; Frenk, C.S.; White, S.D.M.: The structure of cold dark matter Halos. Astrophys. J. 462, 563–575 (1996)
Navarro, J.F.; Frenk, C.S.; White, S.D.M.: A universal density profile from hierarchical clustering. Astrophys. J. 490, 493–508 (1997)
Oppenheimer, J.R.; Snyder, H.: On continued gravitational contraction. Phys. Rev. 56, 455 (1939)
Padmanabhan, T.: Structure Formation in the Universe. Cambridge University Press, Cambridge (1993)
Penrose, R.: Gravitational collapse: the role of general relativity. Riv. Nuovo Cim. 1, 252 (1969)
Penrose, R.: The question of cosmic censorship. J. Astrophys. Astron. 20, 233 (1999)
Poisson, E.: A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge University Press, Cambridge (2004)
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)
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)
Sarwe, S.; Saraykar, R.V.; Joshi, P.S.: Gravitational collapse with equation of state. arXiv:1207.3200 [gr-qc]
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)
Saxton, C.J.: Galaxy stability within a self-interacting dark matter halo. Mon. Not. R. Astron. Soc. 430, 1578 (2013)
Sengor, G.: Cosmological Perturbations in the Early Universe. arXiv:1807.08007
Senovilla, J.M.M.; Garfinkle, D.: The 1965 Penrose singularity theorem. Class. Quantum Gravity 32(12), 124008 (2015)
Silk, J.: Galaxy Formation and Large Scale Structure, pp. 277–378. Springer, New York (1987)
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)
Singh, T.P.; Joshi, P.S.: The final fate of spherical inhomogeneous dust collapse. Class. Quantum Gravity 13, 559 (1996)
Spergel, D.N.; Steinhardt, P.J.: Observational evidence for selfinteracting cold dark matter. Phys. Rev. Lett. 84, 3760 (2000)
Szekeres, P.; Lun, A.: What is a shell-crossing singularity? J. Aust. Math. Soc. Ser. B Appl. Math. 41, 167–179 (1999)
Tolman, R.C.: Effect of inhomogeneity on cosmological models. Proc. Natl. Acad. Sci. USA 20, 410 (1934)
Tulin, S.: Dark matter self-interactions and small scale structure. Phys. Rep. 730, 1–57 (2018)
Vaz, C.; Witten, L.: Do naked singularities form? Class. Quantum Gravity 13, L59 (1996)
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)
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)
Weinberg, S.: Gravitation and Cosmology, p. 482. Wiley, New York (1972)
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)
White, S.D.M.: arXiv:astro-ph/9410043
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)
White, S.D.M.; Rees, M.J.: Mon. Not. R. Astron. Soc. 183, 341 (1978)
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Dey, D., Joshi, P.S. Gravitational collapse of baryonic and dark matter. Arab. J. Math. 8, 269–292 (2019). https://doi.org/10.1007/s40065-019-0252-x
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DOI: https://doi.org/10.1007/s40065-019-0252-x