http://lib.uliege.be https://matheo.uliege.be Non-standard formation processes of low-mass black holes Auteur : Kumar, Shami Promoteur(s) : Cudell, Jean-Rene Faculté : Faculté des Sciences Diplôme : Master en sciences spatiales, à finalité approfondie Année académique : 2018-2019 URI/URL : http://hdl.handle.net/2268.2/6992 Avertissement à l'attention des usagers : Tous les documents placés en accès ouvert sur le site le site MatheO sont protégés par le droit d'auteur. Conformément aux principes énoncés par la "Budapest Open Access Initiative"(BOAI, 2002), l'utilisateur du site peut lire, télécharger, copier, transmettre, imprimer, chercher ou faire un lien vers le texte intégral de ces documents, les disséquer pour les indexer, s'en servir de données pour un logiciel, ou s'en servir à toute autre fin légale (ou prévue par la réglementation relative au droit d'auteur). Toute utilisation du document à des fins commerciales est strictement interdite. 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Master thesis (research focus) University of Liège, AGO department Non-standard formation processes of low-mass black holes KUMAR Shami Supervisor: Jean-René Cudell Master in Space Sciences Academic year 2018-2019 Contents Introduction 1 1 The detection of gravitational waves by LIGO and Virgo . .2 2 The observed gravitational waves and their progenitors . .3 3 The first image of a black hole . .4 4 The dark matter mystery . .6 5 The tests of general relativity . .6 1 The maximum mass of a white dwarf 9 1.1 Stellar evolution . .9 1.2 The pressure of a relativistic degenerate gas of fermions . .9 1.3 The pressure at the center of a star . 11 1.4 The Chandrasekhar mass . 11 2 The maximum mass of a neutron star 13 2.1 The maximum mass of a neutron star and the Oppenheimer-Volkoff limit . 13 2.2 Modern computations . 17 2.3 Observational determination of the mass of a neutron star . 17 2.3.1 X-ray binaries . 17 2.3.2 Binaries with two neutron stars and relativistic effects . 19 2.4 Measured neutron star masses . 20 3 The Kerr metric properties 23 3.1 The Schwarzschild metric . 23 3.2 The Kerr metric properties . 24 3.2.1 The curvature singularity geometry . 26 3.2.2 The event horizons and the ergosphere . 27 3.2.3 The maximum spin of a Kerr black hole . 31 3.3 Measured black holes and neutron stars masses . 32 4 The formation of black holes by accretion of dark matter onto neutron stars 33 4.1 The accretion of fermionic dark matter . 33 4.1.1 The number of accreted dark matter particles . 34 4.1.2 Thermalization, start of the collapse and value of Ncoll .............. 35 4.1.3 Formation of the black hole and value of Ncrit ................... 36 4.2 The accretion of bosonic dark matter . 38 4.2.1 The number of accreted dark matter particles . 38 4.2.2 Start of the collapse and formation of the black hole . 40 4.2.3 The effects of a dark matter Bose-Einstein condensate . 40 4.3 The fraction of neutron stars affected by these phenomena . 41 5 The primordial black holes and their constraints 43 5.1 The inflation models and the origin of the primordial black holes . 43 5.2 The constraints derived from the observations . 44 5.2.1 The constraints from the gravitational lensing . 45 5.2.1.1 Theoretical elements of gravitational lensing . 45 5.2.1.2 The magnification due to the microlensing . 46 5.2.1.3 Studies of gravitational lensing . 46 5.2.2 The impact of the gravitational field of a (primordial) black hole on some celestial objects . 47 5.2.3 Other sources of constraints . 49 5.3 The future constraints . 50 Conclusion 55 ii Introduction Black holes are the densest objects known in the Universe. The classical theory for their formation is that they result from the death of a very massive star during a supernova explosion. The core of the star is then so dense that the pressure from degenerate matter is not enough to prevent the gravitational collapse. The topic of the black holes regained interest in the last years following the detection of gravitational waves due to black holes merging (and one neutron stars merging event) by the LIGO-Virgo collaboration and more recently the first close-up image of a black hole, namely the supermassive black hole at the centre of the M87 galaxy. It also motivated again the investigations on the black hole formation processes, from a better understanding of the classical stellar evolution theory to less traditional ways that could explain some anomalies and less understood points. The modern point of view of what a black hole is comes from the theory of general relativity of A. Einstein. In this theory, where the gravity is no longer viewed as an instantaneous remote force but as a deformation of a structure called space-time, some properties and observations not predicted by the Newtonian theory of gravitation appear and are famous nowadays: the precession of the perihelion of Mercury, the deviation of light in a gravity field, the gravitational redshifting and of course the black holes, objects so dense that they push our understanding of the physics to its limits. Describing the space-time can be done by developing the metric ds2, giving an expression of the element of length in a quadri-dimensionnal space-time and in a given set of coordinates, depending on the energy content of the medium considered. This metric can be expressed as a function of the metric tensor elements, which appear in the fundamental equations of general relativity: the Einstein equations. The black holes, while being an extreme environment, are also paradoxically "simple" to characterise. Indeed three quantities are sufficient to categorise a black hole: its mass M, its angular momentum J and its electric charge Q. Following that, the metrics describing the environment around a black hole are limited as briefly resumed in Table 1. Q J Metric = 0 = 0 Schwarzschild = 0 6= 0 Kerr 6= 0 = 0 Reissner-Nordström 6= 0 6= 0 Kerr-Newman Table 1: Table summarising the different metrics used to describe the space-time around a black hole according to its properties. All these metrics have the peculiarity of being analytical solutions of the Einstein field equations. While the Schwarzschild solution has been found very rapidly the next year after the publication of general relativity by Albert Einstein in 1915 [1], and the Reissner-Nordström solution in the following years [2], about fifty years passed before R. Kerr found an analytical solution for the case of a rotation non-electrically charged black hole in 1963 [3], followed soon after by the more general Kerr-Newman solution for a electrically charged black hole in rotation in 1965 [4]. 1 1 The detection of gravitational waves by LIGO and Virgo Gravitational waves are one among other predictions of the theory of general relativity, and therefore their (non)detection can confirm or infirm it. There was already indirect evidence of the existence of the gravitational waves from Hulse and Taylor in 1974 who studied a binary neutron stars system, that had a decreasing period of revolution which could be explained by the emission of gravitational waves. However there still wasn’t a direct proof of their existence until September 14th 2015, when the collaborations LIGO (Laser Interferometer Gravitational-wave Observatory) and Virgo succeeded and measured them and some of their effects directly [5]. They did this, as the LIGO acronym suggests, by using interferometers, with kilometer-long arm lengths, as shown on the images of the different observatories below. Figure 1: Aerial pictures of the different gravitational waves observatories. Left: LIGO Hanford Observatory in Washington state, USA. Middle: LIGO Livingston Observatory in Louisiana state, USA. Right: Virgo detector in Cascina, Italy [6]. In these interferometers a laser goes through by a beam splitter, and its light travels in each arm before going back, creating a destructive interference. If a gravitational wave goes through the arm, it will modify its length vert slightly (by an amount of the order of 1/1000th of the proton size) so that the interference pattern will not be totally destructive as previously. From that, it is possible to acquire information on the gravitational waves and on their sources. The precision needed to measure such deformations is extremely high, thus lots of different environmental issues must be very well controlled or compensated: thermal dilation, tidal effects, ground vibrations and so on. To be sure that the signal received by a detector is indeed a gravitational wave and not something else, several observatories are used. Since there will be a slight time delay on the reception of the wave between the observatories (about 10 milliseconds between the two LIGO facilities) and the waves propagate at the speed of light, one checks that the same perturbation has also been felt by the other detector, which shows that one has detected a gravitational wave. Moreover, having several detectors helps to narrow down the position in the sky from where the gravitational wave comes from.