Non-Standard Formation Processes of Low-Mass Black Holes

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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

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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. To measure such tiny length variations, one needs to have a large arm length and a large resolution [7]. The first condition cannot be in fact be achieved with "only" 4 km arms (the length of the arm of the LIGO observatory). To increase the effective length, Fabry-Perot cavities are placed (see Fig . 2), i.e. a second mirror in each arm will reflect several times the light before it goes back to the beam splitter region. The resolution condition is also achieved by doing these multiple reflections. By this method, the effective power of the laser at the end can be equal to about 750 kW, compared to the initial 200 W of the laser source.

2 Figure 2: Schematic and simplified representation of the principle of the LIGO interferometer. A laser source sends a light which is separated in two by a beam splitter mirror. At the end of each arm, a mirror reflects back the laser to another mirror in this arm, back and forth to increase the effective length travelled by the light and the power of the light (also increased by a power-recycling mirror between the source and the beam splitter). Finally, the laser is sent to the detector. A variation in the interference pattern can potentially indicate the passage of a gravitational wave which has extremely slightly distorted the length of the arms. Figure modified from [7].

The gravitational waves can lead to a whole new kind of observations. The current messengers used to observe the Universe and its components are mainly the photons, and in some cases other elementary particles such as neutrinos or muons. However these methods have some drawbacks: photons can be easily absorbed by a medium depending on the wavelength ones studies, and neutrinos interact very little with matter. Gravitational waves are not absorbed and because of this, they could help us to look at places otherwise very hard or even impossible to reach. One of the examples would be the very early Universe. While we cannot go back further than about 380 000 years after the Big Bang (the moment when the Universe was suﬃciently "transparent" to let the photons freely propagate), corresponding to the cosmic microwave background emission, the most ancient light we can observe, gravitational waves are not so restricted and thus we could gain access to information beyond the cosmic microwave background and earlier in the history of our Universe.

2 The observed gravitational waves and their progenitors

Until now, the only gravitational waves we have been able to detect have been produced by the merging of compact objects (the merging of two black holes except for one case where it was two neutron stars). The two compact objects orbited around each other closer and closer while emitting gravitational waves (evacuating some energy from the binary system in the process) until the ﬁnal merging event of the objects which, in the case of two black holes, creates another black hole heavier than the individual progenitors. There have been two observations runs performed in total by LIGO and Virgo (abridged Ox): O1 (from September 12th 2015 to January 19th 2016) and O2 (from November 30th 2016 to Augustus 25th 2017). During the course of these two runs, and after analysis and treatment of the data, 11 gravitational waves events have been conﬁrmed. Some of the parameters linked to each of these events are shown in Table 2.

3 Event m1(M ) m2(M ) Mf (M ) Erad (M ) z Run +4.8 +3.0 +3.3 +0.4 +0.03 GW150914 35.6−3.0 30.6−4.4 63.1−3.0 3.1−0.4 0.09−0.03 O1 +14.0 +4.1 +9.9 +0.5 +0.09 GW151012 23.3−5.5 13.6−4.8 35.7−3.8 1.5−0.5 0.21−0.09 O1 +8.8 +2.2 +6.4 +0.1 +0.04 GW151226 13.7−3.2 7.7−2.6 20.5−1.5 1.0−0.2 0.09−0.04 O1 +7.2 +4.9 +5.2 +0.5 +0.07 GW170104 31.0−5.6 20.1−4.5 49.1−3.9 2.2−0.5 0.19−0.08 O2 +5.3 +1.3 +3.2 +0.05 +0.02 GW170608 10.9−1.7 7.6−2.1 17.8−0.7 0.9−0.1 0.07−0.02 O2 +16.6 +9.1 +14.6 +1.7 +0.19 GW170729 50.6−10.2 34.3−10.1 80.3−10.2 4.8−1.7 0.48−0.20 O2 +8.3 +5.2 +5.2 +0.6 +0.05 GW170809 35.2−6.0 23.8−5.1 56.4−3.7 2.7−0.6 0.20−0.07 O2 −5.7 +2.9 +3.2 +0.4 +0.03 GW170814 30.7−3.0 25.3−4.1 53.4−2.4 2.7−0.3 0.12−0.04 O2 +0.12 +0.09 +0.00 GW170817 1.46−0.10 1.27−0.09 ≤ 2.8 ≥ 0.04 0.01−0.00 O2 +7.5 +4.3 +4.8 +0.5 +0.07 GW170818 35.5−4.7 26.8−5.2 59.8−3.8 2.7−0.5 0.20−0.07 O2 +10.0 +6.3 +9.4 +0.9 +0.13 GW170823 39.6−6.6 29.4−7.1 65.6−6.6 3.3−0.8 0.34−0.14 O2 Table 2: Table listing some of the properties of the compact objects involved in the different gravitational wave events detected (listed by chronological order of detection) by the LIGO-Virgo collaboration between September 12th 2015 and Augustus 25th 2017. m1 is the mass of the heavier component, m2 the mass of the lighter one, Mf the mass of the object resulting from the merger of the two components and z is the redshift. All masses are expressed in solar mass. Erad is the total radiated energy by gravitational waves, corresponding roughly to the energy mass difference between Mf and m1 + m2 [8]. GW170817 is the only gravitational wave event involving two neutron stars (as suggested by the low masses of the components compared to the other events) in the first two runs. Moreover, since this is not a black hole merger, there should be a electromagnetic signal in addition to the gravitational one, which has been confirmed by Fermi, within a two seconds delay. The object resulting from the merging of the neutron stars is also interesting, since it could be a high-mass neutron star or a very low-mass black hole. As a last remark, the events GW170729, GW170809, GW170818 and GW170823 have not been discovered initially from the data, but they were validated as gravitational wave events after a second analysis of the data. As said previously, with several detectors it is possible to localise to some extent the position of the gravitational wave events in the sky. The results of theses localisations are given in Fig. 3. The confidence region of the position is much smaller when the three detectors (LIGO and Virgo) detect the same event. It was the case for GW170818, as one can see the localisation is much better than for the other events.

Other runs of observations of gravitational wave events are planned in the future. Thanks to the upgrade in the technologies and facilities of the diﬀerent observatories and a better understanding of the phenomena, the expected number of events is higher than for the ﬁrst two runs. The O3 run at the date of this writing has already begun since the 1st April 2019 and should continue for about 12 months. The latest detected events can be seen in reference [9].

3 The ﬁrst image of a black hole

Between the 5th and the 11th of April 2017, the EHT (Event Horizon Telescope) has observed the supermassive black hole at the center of the M87 galaxy (named hereafter M87∗), an elliptical galaxy at about 55 millions light-years from us, while the processed images have been released to the public on the 10th of April 2019. This is a ﬁrst in history, and to achieve such a feat, a large collaboration between diﬀerent institutions and telescopes was necessary. M87∗ has been chosen due to the fact that it is the supermassive black hole with the largest apparent size (with our own, Sagitarrius A∗). The apparent angular size of M87∗ is of the order of a few tenths of micro-arcseconds, and as such it required a very high resolution. To achieve this, a network of eight telescopes around the world worked together to simulate an Earth-size radio telescope (at a wavelength of 1.3 mm), with a resolution of about 20 micro-arcseconds. Following this study, the mass of M87∗ has been estimated to be 6.5 ± 0.7 billions solar masses [10].

4 Figure 3: Positions in the sky of the sources of the gravitational wave events of the O1 and O2 runs with the 50 % and 90 % conﬁdence regions. The upper image corresponds to the events of the second run for which an alert to electromagnetic detectors have been sent, while the lower image corresponds to all the other events of the ﬁrst two runs of the LIGO-Virgo collaboration. Both images are in equatorial coordinates [8].

The resulting images are shown in Fig. 4. The interpretation of the image is not straightforward. The bright ring corresponds to the light deviated in the gravitational ﬁeld of the supermassive black hole and coming originally from the accretion disk emission around M87∗. However the central blackened region in the center is not just the black hole itself, with its event horizon delimiting the darker inner region and the bright ring, but a larger region called the shadow of the black hole. The actual event horizon is about 2.5 times smaller. Another striking feature of the images is the asymmetric intensity of the ring, with a clear brighter region to the south-est. This eﬀect is due to the relativistic beaming of the light from the material rotating with M87∗.

Figure 4: Images of the supermassive black hole at the center of the M87 galaxy in the Virgo cluster on diﬀerent days of observation. The white circle corresponds to the angular resolution of the EHT (∼ 20 micro-arcseconds). The inner dark region is the shadow of the black hole and the asymmetry of the ring is due to the relativistic beaming coming from the emission of the matter going towards the Earth. The North is the top of the images and the East is the left [10].

5 4 The dark matter mystery

The enigma of dark matter is one of the big issues in today’s astrophysics. The idea of its possible existence is notably supported when one analyses the rotation curves of galaxies. Even though the rotational speed should globally decrease when far from the center, it is not what is observed. Instead the speed is more or less constant when sufficiently far. These observations, as well as the study of galaxy clusters and their collisions, lead to the introduction of some missing mass which could explain these curves and which is dominant compared to the luminous matter we can see. However, even if this hypothesis has been made in the last century, the nature of dark matter is not known. Lots of different models and constraints have been developed, without to this day a clear direct signature (provided it is possible). One possible explanation would be that the dark matter is composed of new particles, which don’t interact a lot with the ordinary matter. Another explanation could be that we missed some celestial objects during the observations, for example black holes, planets or in general all sorts of compact objects. However several studies have already been performed on this subject, and these compact objects cannot make the majority of the dark matter for a large range of masses (from a fraction of solar mass to several tenth of solar masses). Finally, one straightforward idea in spirit would be that the theory itself is wrong. Other models of gravity have been developed (like the MOdified Newtonian Dynamics), but to this day there aren’t any which really explain all the observations. Even if dark matter interacts very little with baryonic matter, we know it should interact at least via gravity. Moreover, if dark matter is indeed composed of black holes (or even primordial black holes if they originated from the very early Universe), one could use the gravitational waves to study these black holes.

5 The tests of general relativity

Since the gravitational waves are a consequence of general relativity, they can be used to test this theory and to see if there are any deviation. If one supposes that the origin of the gravitational waves is indeed the coalescence of two black holes, one can use general relativity to infer parameters such as the final mass and spin from the different parts of the signal and see if they overlap or not. Abbott et al. [11] studied GW150914 to test any deviation from theory. To do so, they inferred the mass and the spin of the final object from the inspiral phase (before the actual merger) and the post-inspiral phase. Their results are shown in Fig 5. As one can see the 90 % confidence regions overlap, consistent with theory. Another way to see the agreement is to look at the difference between the two models (inspiral and post-inspiral) as illustrated in Fig 6.

This Master thesis investigates different mechanisms of black hole formation. First, before beginning the subject itself, a review of the maximum mass of less compact objects, white dwarfs and neutron stars, is given. In particular, the mass limit on neutron stars, called the Oppenheimer-Volkoff mass, is important due to the fact that beyond it the neutron star should collapse into a black hole. Second, the rotating black holes, or Kerr black holes, and their properties are reviewed. Most notably, the two event horizons and the notion of ergosphere will be investigated. Then different mechanisms other than the classical stellar evolution are investigated, notably the accretion of dark matter, bosonic or fermionic, and the possibility of primordial black holes.

The fundamental constants c = ~ = G = kb = 1 in this Master thesis, where c is the speed of light in vacuum, ~ the reduced Planck constant, G the gravitational constant and kb the Boltzmann constant.

6 Figure 5: 90 % confidence regions of the final mass (in solar mass) and dimensionless spin of GW150914 deduced from the inspiral phase and the post-inspiral phase. In agreement with general relativity, the two regions overlap. The 90 % confidence region deduced from all the phases is also shown in black, and is located within the joint area of the two other regions [11].

Figure 6: Graphics of the relative difference (between the value deduced from the inspiral phase and from the post-inspiral phase) for the dimensionless spin of the final object and od the relative difference for its mass. The 90 % confidence region is shown. The expected value is at the center (plus symbol) [11].

7 8 Chapter 1

The maximum mass of a white dwarf

1.1 Stellar evolution

Depending on the core mass at the end of its life, a star will die differently. If the core possesses a mass below about 1.4 solar masses (called the Chandrasekhar limit) the core of the star will become a white dwarf, the electron degeneracy pressure being sufficient to compensate its weight and sustain hydrostatic equilibrium. For more massive stars, the electron degeneracy pressure is not enough to balance the gravitational contraction and the star will undergo a supernova event. The fate of the core will then depend again on its mass. If it is over the Chandrasekhar limit but below the Oppenheimer- Volkoff limit, then the neutron degeneracy pressure will be sufficient to balance the gravitational force and it will become a neutron star. But if it’s too massive this neutron degeneracy pressure won’t be enough and it will collapse into a black hole.

The Chandrasekhar limit can be estimated from a theoretical and analytical point of view. The following demonstration is taken from reference [12] where more details can be found.

1.2 The pressure of a relativistic degenerate gas of fermions

We consider here the general case of an electron gas, where the fact that electrons can become relativistic is taken into account. In that case, the relativistic expression of the energy E has to be taken:

E2 = m2 + p2 (1.1) where m is the mass of the electron and p its momentum. First one needs to determine what the pressure due to an ideal degenerate gas of relativistic electrons is. The internal energy U can be expressed as follows:

Z +∞ U = E f(E) g(p)dp (1.2) 0 where f(E) in the case of the electrons is the Fermi-Dirac distribution and g(p)dp the density of states expressed as

p 2 g(p)dp = V dp (1.3) π where V is the volume. The variation of this internal energy can be linked to the other properties of the gas:

dU = T dS − P dV + µ dN (1.4) where T is the temperature, dS the entropy variation, P the pressure, dV the volume variation, µ the chemical potential and dN the variation of the number of particles.

9 By looking at Eq. (1.4), one immediately sees that the pressure of the gas corresponds in absolute value to the derivative of the internal energy with respect to volume, considering all the other properties constant, in particular the number of particles for each state. Thus, using Eq. (1.2), one has:

∂U Z +∞ dE dp P = − = − f(E) g(p)dp (1.5) ∂V 0 dp dV The term dE/dp can be directly computed from Eq. (1.1) and since the momentum is proportional −1 − 1 to the wave vector, and the latter is itself proportional to the characteristic length L = V 3 , we can get an expression for the factor dp/dV .

− 1 dp d(αV 3 ) −α −p = = 4 = (1.6) dV dV 3V 3 3V with α the proportionality factor. Finally Eq. (1.5) becomes

1 Z +∞ p2 P = f(E) g(p)dp (1.7) 3V 0 E If we consider that the gas of electrons is fully degenerate, there is no occupied state beyond the Fermi momentum pF and the Fermi-Dirac distribution is taken equal to 1 as Eq. (1.7) is integrated p up to the Fermi momentum. Lastly, by deﬁning a new variable x = m , Eq. (1.7) becomes

m4 Z xF x4 P = 2 1 dx (1.8) 3π 0 (1 + x2) 2 where xF is the dimensionless Fermi pressure, given by

p 1 1 x = F = 3π2n 3 (1.9) F m m e where ne is the electron density. The second equality in Eq. (1.9) can be directly derived from Eq. (1.3). Indeed, the total number N of states for a degenerate gas of electrons is

Z pF 1 1 3 2 3 N = g(p)dp = 2 V pF ⇐⇒ pF = 3π ne (1.10) 0 3π Eq. (1.8) is then equal to

4 3 P = K ne I(xF ) (1.11) The diﬀerent terms of Eq. (1.11) are given by

1 π 3 3 K = (1.12) 2 8π

Ye ρc ne = (1.13) mH

2 3 2 1 2x h 2 1 i I(x) = x(1 + x ) 2 − 1 + ln x + (1 + x ) 2 (1.14) 2x4 3 where Ye is the number of electrons per nucleon, ρc the density in the core and mH the mass of a hydrogen atom.

10 1.3 The pressure at the center of a star

Let us consider in this section the separate case of a star of uniform chemical composition. Let us also consider that the star is in hydrostatic equilibrium. Then, the pressure gradient can be expressed as

dP m(r)ρ(r) = − (1.15) dr r2 where ρ(r) is the density and m(r) the mass inside a sphere of radius r with its radial variation given by

dm = 4πr2ρ(r) (1.16) dr To ﬁnd an expression for the pressure inside this star as a function of the radial distance r, we will use the Clayton model [13]. The idea of the model is to model the pressure gradient by a simple expression depending only on r. To do so, an exponential factor is used:

2 dP 4πρ 2 2 = − c e−r /a (1.17) dr 3 where ρc is the density at the center and a a characteristic length small compared to the radius of the star R. Eq. (1.17) is a good approximation at small r but also at large r if a is small compared to R. Integrating Eq. (1.17), we get the expression of the pressure:

2π h 2 2 2 2 i P (r) = ρ2a2 e−r /a − e−R /a (1.18) 3 c Using Eq. (1.18), one gets the pressure at the center (by neglecting the second exponential):

2π P = P (r = 0) = ρ2a2 (1.19) c 3 c Also, the evolution of the mass m(r) is given by

2 Z r 4 m (r) 04 dP 0 m(r)dm = −4πr dP ⇐⇒ = −4π r 0 dr (1.20) 2 0 dr where dm is an inﬁnitesimal variation of mass and dP an inﬁnitesimal variation of pressure. By deriving with respect to r Eq. (1.18), using Eq. (1.20) and supposing a small compared to R, one can get an approximate expression for the parameter a:

1 3M 3 a ' √ (1.21) 4πρc 6 where M is the total mass of the star. Finally, substituting Eq. (1.21) into Eq. (1.19), one obtains a relation between the central pressure and the mass of the star:

1 4 π 3 2 P ' M 3 ρ 3 (1.22) c 36 c 1.4 The Chandrasekhar mass

Eq. (1.22) corresponds roughly to the pressure needed to support the star. Finally, if we consider the white dwarf as a star in which the pressure comes from the degenerate gas of electrons, we can equate Eqs. (1.11) and (1.22).

1 4 1 3 3 2 4 π 3 Yeρc π 3 3 I(xF ) ' M 3 ρc (1.23) 2 8π mH 36

11 Finally, by rearranging the terms in Eq. (1.23), we can obtain an approximation of the mass of the white dwarf where the pressure needed comes from the degeneracy pressure of the electrons:

3 2 6π 1 2 3 2 3 M ' Y I(x ) 2 = 4.3 Y I(x ) 2 M (1.24) 1/3 2 e F e F 128 mH π 30 where M = 1.98 10 kg is the mass of the Sun

Following stellar evolution theory, a white dwarf is mainly composed of 12C and 16O. Thus typically Ye ' 1/2. As for the value of I(xF ), one needs to use the value of Eq. (1.9) in Eq. (1.14). When plotting the central density versus the mass of the white dwarf, one can see that the density of the 2 white dwarf tends to inﬁnity when M tends to 4.3 Ye M = 1.1 M , meaning that other aspects have to be considered (for example the degeneracy pressure of the neutrons in the case of neutron stars). This limit value corresponds to the famous Chandrasekhar mass. A more accurate computation of the Chandrasekhar mass can be considered numerically if instead of Eq. (1.18) a polytropic model is used [14]. In this kind of model, the relation between the pressure and the density ρ is deﬁned by the following relation:

1+ 1 P = Cρ( n ) (1.25) where C is a constant and n a strictly positive real number called the polytropic index, their values depending on the nature of the medium considered. Moreover, if we derive with respect to r Eq. (1.15), use Eq. (1.16) and ﬁnally use Eq.(1.25) to get rid of the pressure, we can get a second order diﬀerential equation for the density:

2 1 d r d 1+ 1 Cρ( n ) = −4πρ (1.26) r2 dr ρ dr If boundary conditions are used, Eq. (1.26) can be solved numerically to get the density proﬁle. 4 For a degenerate gas of electrons n = 3 and so P (r) ∝ ρ(r) 3 . In that case the numerical factor in 1/3 Eq. (1.22) is no longer equal to (π/36) ' 0.44 but to ' 0.36. Then the Chandrasekhar mass MCh is given by Eq. (1.27) (considering Ye ' 1/2).

2 MCh = 5.8 Ye M ' 1.45 M (1.27) Finally, it is important to keep in mind that these models are still simpliﬁed compared to reality. The chemical composition is not rigorously uniform and one also needs to know the density proﬁle, which can be tricky.

12 Chapter 2

The maximum mass of a neutron star

2.1 The maximum mass of a neutron star and the Oppenheimer- Volkoﬀ limit

One interesting question about neutron stars is their maximum mass. Indeed, like the Chandrasekhar limit for white dwarfs, we can go further and try to estimate what would be the limiting mass for a neutron star, beyond which the mass of the neutron star would be too large for the degeneracy pressure of the neutrons (and would become a black hole). One of the ﬁrst attempts to compute this limit has been made in 1939 by Oppenheimer and Volkoﬀ [15]. In their paper they considered the metric for a spherically symmetric object:

ds2 = eνdt2 − eλdr2 − r2dθ2 − r2 sin2(θ)dφ2 (2.1) where, in the case of an empty space around the object, λ and ν are functions of r given by

2m−1 eλ = 1 − (2.2) r 2m eν = 1 − (2.3) r where m is the total mass of the neutron star from the point of view of a distant observer, and if the object doesn’t rotate too fast and is electrically neutral. By supposing no mass motion, we can rewrite the Einstein equations (2.4) inside the neutron star, which requires the expression of the energy-momentum tensor Tµν and the Einstein tensor Gµν.

µ µ µ Gµν + Λgµν = 8πTµν ⇐⇒ Gν + Λδν = 8πTν (2.4) µ where Λ is the cosmological constant, δν the Kronecker symbol and gµν the element (µ, ν) of the metric tensor. An important simplification is the fact that we consider the case of a perfect fluid. In that situation the energy-momentum tensor simplifies a lot and is diagonal such as one has

µ (Tν ) = diag(ρ, −p, −p, −p) (2.5) where p is the pressure and ρ the energy density in proper coordinates, with an equation of state p(ρ) linking both, to be deﬁned depending on the medium considered. Thus it is only necessary to compute the diagonal elements of the Einstein tensor. This tensor is deﬁned as follows:

R R G = R − g ⇐⇒ Gµ = Rµ − δµ (2.6) µν µν 2 µν ν ν 2 ν µ where Rµν is the Ricci tensor and R the scalar curvature, itself deﬁned as R = Rµ. However the Ricci α ρ tensor is derived from the Riemann tensor Rβγδ since Rµν = Rµρν, which can itself be expressed in ρ function of the Christoﬀel symbols Γσξ:

13 α α α ξ α α ξ Rβγδ = Γβδ,γ + ΓγξΓβδ − Γβγ,δ − ΓδξΓβγ (2.7) where the comma corresponds to the ordinary derivative. Finally, the Christoﬀel symbols can themselves be expressed in function of the metric components 2 µ ν which are known thanks to Eq. (2.1) since ds = gµνdx dx :

gδα Γδ = (g + g − g ) (2.8) βµ 2 αβ,µ µα,β βµ,α The idea is then the following: computing the Christoffel symbols from the metric components, then the Riemann tensor then the Ricci tensor and finally the scalar curvature, all this to get the expression of the Einstein tensor. In the following, to avoid confusion, numbers will be attributed to the different variables when in indices (t = 0, r = 1, θ = 2 and φ = 3). Thus the first step is the derivation of the expression of the Christoffel symbols. Due to the fact that they are symmetric on the two lower indices, that no component of the metric depends explicitly on t and φ and that the metric is diagonal, it reduces drastically the number of non-zero symbols to be computed, which are the following:

g00 1 dν g e(ν−λ) dν g11 1 dλ Γ0 = g = ; Γ1 = − 11 g = ; Γ1 = g = 01 2 00,1 2 dr 00 2 00,1 2 dr 11 2 11,1 2 dr g11 g11 g22 1 Γ1 = g = −re−λ ; Γ1 = − g = −r sin2(θ)e−λ ; Γ2 = g = 22 2 22,1 33 2 33,1 12 2 22,1 r g22 g33 1 g33 Γ2 = − g = − sin(θ) cos(θ) ; Γ3 = g = ; Γ3 = g = cot(θ) 33 2 33,2 13 2 33,1 r 23 2 33,2 Using Eq. (2.7), the diﬀerent Ricci tensor components can be computed:

0 1 2 3 R00 = R000 + R010 + R020 + R030 " # e(ν−λ) d2ν 1 dν 2 1 dν dλ e(ν−λ) dν e(ν−λ) dν = 0 + + − + + 2 dr2 2 dr 2 dr dr 2r dr 2r dr " # e(ν−λ) d2ν 1 dν 2 1 dν dλ 2 dν = + − + 2 dr2 2 dr 2 dr dr r dr

0 1 2 3 R11 = R101 + R111 + R121 + R131 " # 1 d2ν 1 dν dλ 1 dν 2 1 dλ 1 dλ = − + − + 0 + + 2 dr2 4 dr dr 4 dr 2r dr 2r dr 1 d2ν 1 dν dλ 1 dν 2 1 dλ = − + − + 2 dr2 4 dr dr 4 dr r dr

0 1 2 3 R22 = R202 + R212 + R222 + R232 r dν r dλ h i = − e−λ + e−λ + 0 + 1 − e−λ 2 dr 2 dr r dλ dν = 1 − e−λ + − e−λ 2 dr dr

0 1 2 3 R33 = R303 + R313 + R323 + R333 1 dν r dλ = − r sin2(θ)e−λ + sin2(θ)e−λ + sin2(θ) 1 − e−λ + 0 2 dr 2 dr r dλ dν = sin2(θ) 1 − e−λ + − e−λ 2 dr dr

14 Then, using its deﬁnition, the scalar curvature can be obtained from the Ricci tensor components:

µ 0 1 2 3 00 11 22 33 R = Rµ = R0 + R1 + R2 + R3 = g R00 + g R11 + g R22 + g R33 " # " # e−λ d2ν 1 dν 2 1 dν dλ 2 dν 1 d2ν 1 dν dλ 1 dν 2 1 dλ = + − + − e−λ − + − + 2 dr2 2 dr 2 dr dr r dr 2 dr2 4 dr dr 4 dr r dr 1 r dλ dν 1 r dλ dν − 1 − e−λ + − e−λ − 1 − e−λ + − e−λ r2 2 dr dr r2 2 dr dr " # d2ν 1 dν 2 1 dν dλ 2 dν dλ 2 2 = e−λ + − + − + − dr2 2 dr 2 dr dr r dr dr r2 r2

After that, the Einstein tensor components can be determined:

R 1 1 dλ 1 G0 = R0 − = + e−λ − 0 0 2 r2 r dr r2

R 1 1 dν 1 G1 = R1 − = − e−λ + 1 1 2 r2 r dr r2

" # R 1 d2ν 1 dν 2 1 dν dλ 1 dν dλ G2 = R2 − = −e−λ + − + − = G3 2 2 2 2 dr2 4 dr 4 dr dr 2r dr dr 3 Finally, we can write the Einstein equations (taking Λ = 0):

1 dλ 1 1 G0 = 8πT 0 ⇐⇒ e−λ − + = 8πρ (2.9) 0 0 r dr r2 r2 1 dν 1 1 G1 = 8πT 1 ⇐⇒ e−λ + − = 8πp (2.10) 1 1 r dr r2 r2 " # 1 d2ν 1 dν 2 1 dν dλ 1 dν dλ G2 = 8πT 2 ⇐⇒ e−λ + − + − = 8πp (2.11) 2 2 2 dr2 4 dr 4 dr dr 2r dr dr

3 The Einstein equation for G3 is not written since it is the same that Eq. (2.11). Now that we have the expression of the Einstein equations, let us do the sum of Eqs. (2.9) and (2.10) to get a new relation: e−λ dλ dν 8π(ρ + p) = + (2.12) r dr dr Also let us derive Eq. (2.10) with respect to r:

dp dλ 1 dν 1 1 dν 1 d2ν 2 2 8π = −e−λ + + e−λ − + − + (2.13) dr dr r dr r2 r2 dr r dr2 r3 r3

Finally, equalising Eqs. (2.10) and (2.11) and implementing the expression of Eqs. (2.12) and (2.13), one obtains a last relation:

dp p + ρ dν = − (2.14) dr 2 dr

15 Now if we consider the speciﬁc case of a cold Fermi gas of one species, a parametric expression for the equation of state can be obtained [16]:

ρ = K(sinh(t) − t) (2.15)

K t p = sinh(t) − 8 sinh + sinh(3t) (2.16) 3 2 with the parameters K and t given by:

µ4 K = 0 (2.17) 32π2   " 2#1/2 pF pF t = 4 log  + 1 +  (2.18) µ0 µ0 where µ0 is the rest mass of the particles. Let us deﬁne a variable u as follows:

r(1 − e−λ) u = (2.19) 2 Using Eq. (2.14) and this variable u, Eqs. (2.9) and (2.10) become respectively:

du = 4πρr2 (2.20) dr dp p + ρ = − 4πpr3 + u (2.21) dr r(r − 2u) Combined with an equation of state, they allow to determine the structure of the neutron star. Using Eqs. (2.15) and (2.16) , Oppenheimer and Volkoﬀ obtained a speciﬁc form for Eqs. (2.20) and (2.21) respectively:

du = r2 (sinh(t) − t) (2.22) dr dt 4 sinh(t) − 2 sinh(t/2) r3 = − (sinh(t) + 8 sinh(t/2) + 3t) + u (2.23) dr r(r − 2u) cosh(t) − 4 cosh(t/2) + 3 3 Finally, considering only neutrons, they integrated the Eqs. (2.22) and (2.23) from diﬀerent values of t0 = t(r = 0) to tb = t(r = rb) = 0, where rb is the value of r for which the pressure p = 0.

If we look at Eq. (2.19) at r = rb and use Eq. (2.2), we can see that ub corresponds to the total mass m:

r (1 − e−λ(rb)) u = u(r = r ) = b = m (2.24) b b 2

By looking at the evolution of the core mass m with t0, one can see the presence of a maximum for t0 ∼ 3, corresponding to a mass m ∼ 0.75 solar mass, beyond which there is no static solutions. This value corresponds to the original Oppenheimer-Volkoﬀ limit.

16 2.2 Modern computations

The main issue compared to white dwarfs is that the equation of state of a neutron star is more difficult to estimate and evaluate, due to the very high density and its nature. Looking back at the original computation of the mass limit of neutron stars by Oppenheimer and Volkoff, one can see several strong approximations. Mainly, they considered only a cold gas of neutrons (without any other species). In addition, to rewrite the Einstein equations, the energy-momentum tensor is the simple one of a perfect fluid. The structure of a neutron star is more complicated than just being composed of neutrons [17]. In the outer layers, there is still a sizeable fraction of protons and electrons. It is while going deeper in the neutron star that the neutron fraction is more important. Moreover, near the center the density is extremely high and becomes bigger than the nuclear density. In these conditions it is very difficult to know the equation of state that is able to describe such conditions. It is theorised that at such densities, new states of matter could appear. Some hypotheses involve pions condensates or even quark matter, where there would be unconfined quarks. In consequence, many different equations can be used depending on what particles and temperature are considered. Bombaci (1995) [18] for example distinguishes "conventional" equations of states (where all negative charges are carried only by leptons) and "exotic" equations of state (in which negative charges can also be carried by hadrons). By taking into account more complex hypotheses, the main consequence of these different equations is the fact that the actual value of the maximum mass of a neutron star can be fairly different from the one computed by Oppenheimer and Volkoff. Typically, the theoretical values range between 2 and 3 solar masses.

2.3 Observational determination of the mass of a neutron star

Generally it is easier to determine the mass of a neutron star when it is in a binary system thanks to the constraints that the companion object and the neutron star put on each other [19]. The precise methods will depend on the type of binary [20].

2.3.1 X-ray binaries X-ray binaries consist of a neutron star and a companion star. Generally they form a compact system and can be described in ﬁrst approximation by classical Keplerian theory [21]. The position of the companion j can be expressed in the orbital plane (see Fig. 2.1):

xj = rj cos(ω + φj) (2.25)

yj = rj sin(ω + φj) (2.26) where ω is the periastron longitude, φj the phase of the component j (the angle on the orbital plane between the actual position of the star and the line crossing the periastron and the center of mass) and rj its radial coordinate. One can also link the phase with the eccentric anomaly E:

rj cos(φj) = aj(cos(E) − e) (2.27)

p 2 rj sin(φj) = aj 1 − e sin(E) (2.28) where aj is the semi-major axis of the component j and e the eccentricity. The eccentric anomaly is deﬁned according to the Kepler equation for elliptical orbits:

Ωb(t − t0) = E − e sin(E) (2.29) where Ωb is the angular velocity, t the time and t0 the moment of periastron passage.

17 Figure 2.1: Schematic view of some elements of the orbit which is centred on the origin corresponding here to the center-of-mass. The line of nodes containing N is the intersection of the orbital plane and the plane of the sky. P corresponds to the periastron and ω to its longitude, or its angle from the line of nodes in the revolution direction. The orbital plane is considered to be the X-Y plane. Modiﬁed from [19].

Thus Eqs. (2.25) and (2.26) can be rewritten to depend explicitly on the eccentric anomaly:

h p 2 i xj = aj (cos(E) − e) cos(ω) − 1 − e sin(E) sin(ω) (2.30)

h p 2 i yj = aj (cos(E) − e) sin(ω) + 1 − e cos(E) sin(E) (2.31)

As we can see, Ωb, aj, e, ω and t0 are elements required to deﬁne the orbit of the stars. It is possible to evaluate some of them thanks to the observations:

• By measuring the orbital variability of the radiation of one of the components, we can determine the orbital period Pb;

• Also, measuring the orbital evolution of the radial velocity vlj of the component j is helpful. The radial velocity is the component of the velocity along the line of sight. Using Eq. (2.26), we obtain its expression in the center-of-mass reference frame:

· · vlj = sin(i) rj sin(ω + φj) + rjφj cos(ω + φj) (2.32)

where i is the inclination of the orbital plane compared to the line of sight. The dot corresponds to the time derivative. Then we can use the ﬁrst and second Kepler laws for elliptical orbits:

2 aj(1 − e ) rj = (2.33) 1 + e cos(φj) · q 2 2 rj φj = ajM(1 − e ) (2.34)

where M is the total mass of the system.

18 One can easily rewrite Eq. (2.32) using Eqs. (2.33), (2.34) and trigonometric formulae:

vlj = Kj [cos(ω + φj) + e cos(ω)] (2.35)

s M Ω a sin(i) b√ j Kj = sin(i) 2 = (2.36) aj(1 − e ) 1 − e2

where Kj is the amplitude.

By ﬁtting Eq. (2.35) to the observed radial velocity, Kj, e, ω and aj sin(i) can be determined.

• With these parameters, one can determine the mass function fj of one of the components. Using the third Kepler law (2.37), we get Eq. (2.38) for the mass function (here for the companion of mass M2).

M Ω2 = (2.37) b a3

(M sin(i))3 f = 1 = (a sin(i))3 Ω2 (2.38) 2 M 2 2 b

where a = a1 + a2. We have two relations with Eqs. (2.36) and (2.38). However there are four unknowns: the masses of each component, a and sin(i).

• A third equation can come from the mass ratio q. Indeed, by deﬁnition we have:

M1a1 = M2a2 (2.39)

By measuring the radial velocity of the other component, we can measure the ratio of the radial velocities and by using Eq. (2.36) we have:

M K q = 1 = 2 (2.40) M2 K1 • A fourth equation is still needed. Additional information can be obtained by the study of eclipses in the binary for example.

However, it is important to keep in mind that in reality the Keplerian motions can be perturbed by other eﬀects such as accretion or tidal interactions, which can lead to higher uncertainties.

2.3.2 Binaries with two neutron stars and relativistic effects If a binary system is a close system of neutron stars, it can be treated as two point masses. Moreover, relativistic corrections might be needed to get more accurate measurements. Notably, in the particular case of two neutrons stars, gravitational waves can be emitted, coming from the loss of energy and angular momentum of the system. The relativistic effects also lead to long-term variations of the orbital parameters such as a, e, Ωb or ω. The determination of the parameters in this case is at first similar to that for X-ray binaries. The measurements of the radial velocity and its amplitude Kj allow to determine some of the parameters, which allow to determine the mass function fj of one of the components. As in the case of X-ray binaries, the expressions of Kj and fj give two equations, and two others are still needed.

19 Here we can take into account the fact that relativistic eﬀects become important. For example, the expression of the periastron advance is given by Eq. (2.41) and can help to put constraints on the masses of the neutron stars by measuring it.

5 2 3 · 3Ω M 3 ω = b (2.41) (1 − e2) Another consequence of relativistic effects is the modification on the Doppler effect and the gravitational redshift, shifting the pulse arrival time. The delay ∆E due to these effects is called the Einstein delay. This delay can be expressed in function of the Keplerian parameters [22]. As previously said, the delay of the pulse (of the pulsar of mass M1) is a combination of two effects, the gravitational redshift and the Doppler effect. These effects can be expressed respectively as follows (taking only the terms in 1/c2):

M2 γgrav = − (2.42) r12 v2 γ = 1 (2.43) Doppler 2 where r12 is the distance between the two objects of the binary and v1 the orbital velocity of M1 Combining Eqs. (2.42) and (2.43) and expressing v1 in function of the masses M1 and M2, the proper time dτ of the pulsar can be expressed by M2 M2 M2 dτ ' dt [1 + γgrav − γDoppler] = dt 1 − − (2.44) r12 M r12 where dt is the temporal element of the metric. Moreover, by summing the squares of Eqs. (2.27) and (2.28), one can get an expression for rj = r12 if aj = a:

r12 = a(1 − e cos(E)) (2.45) The variation dt can be expressed in function of the variation dE of the eccentric anomaly by taking the diﬀerential of Eq. (2.29). Doing this, integrating Eq. (2.44) and using (2.45), one can ﬁnally get the expression of the Einstein delay:

∆E = τ − t = γ sin(E) (2.46) where the parameter γ is given by

eM (M + 2M ) γ = 2 1 2 (2.47) ΩbaM By measuring the shift of the pulses arrival times, one can get information on the value of γ, providing another relation.

2.4 Measured neutron star masses

Measured masses for neutron stars can help to put constraints on the actual value of the Oppenheimer- Volkoﬀ limit, but also on the nature of small black holes. Indeed, if we follow the stellar evolution models, a black hole is a possible result of a supernova event as long as the star had a suﬃcient mass. Moreover, these black holes should themselves have a mass larger than the maximum mass of a neutron star. So, if we detect a black hole with a mass lower than this limit (and so lower than some of the detected neutron stars), one should consider other possible ways to create black holes. A list of the four heaviest neutron stars detected is shown in Table 2.1. The heaviest detected neutron stars are around 2 solar masses. In consequence any black hole below this limit should be considered. More comprehensive lists can be found in references [23], [24] and [25].

20 System Mass of the neutron star (in solar mass) Study J1614-2230 1.97 ± 0.04 Demorest et al. (2010) [26] J0348+0432 2.01 ± 0.04 Antoniadis et al. (2013) [27] +0.17 PSR B1516+02B 1.94−0.19 Freire (2008) [28] +0.17 PSR J2215+5135 2.27−0.15 Linares, Shahbaz and Casares (2018) [29] Table 2.1: Table listing some of the most massive neutron stars.

21 22 Chapter 3

The Kerr metric properties

3.1 The Schwarzschild metric

The Schwarzschild metric is an exact solution of the Einstein equations which is valid if the isolated object curving the spacetime has a spherical symmetry, doesn’t spin and is electrically neutral. In that case the line element ds2 outside the object (so where the energy-momentum tensor is identically equal to zero) can be expressed as follows [1]:

2m dr2 ds2 = − 1 − dt2 + + r2(dθ2 + sin2(θ)dφ2) (3.1) r 2m 1 − r where r, θ and φ are the spherical coordinates and m the total mass of the object. One may immediately see what seems to be a singularity when r tends to the Schwarzschild radius RS = 2m and that the gtt term of Eq. (3.1) becomes positive when r < 2m. However this apparent singularity can be removed by choosing other sets of coordinates [30]. To construct these, we start by deﬁning the Regge-Wheeler tortoise coordinate r∗ as follows:

2 2 dr r dr∗ = ⇐⇒ r∗ = r + 2m ln − 1 (3.2) 2m2 2m 1 − r With this new coordinate in a two-dimensions situation (dθ2 = dφ2 = 0), for a null line element (which 2 2 is the case of the light), i.e. when Eq. (3.1) is equal to zero, dr∗ = dt . From there we can deﬁne two other coordinates u = t − r∗ and v = t + r∗, which will lead to two systems of coordinates [30]. The constants u and v correspond respectively to outgoing and ingoing null geodesics, or in other words the wordlines with theses coordinates constant correspond to outgoing or ingoing null geodesics.

The ingoing Eddington-Finkelstein coordinates are the same as the ordinary spherical coordinates (t, r, θ, φ) but in which the temporal coordinate t is replaced by v. Finally, doing this change of coordinate in Eq. (3.1), one obtains the Schwarzschild metric in the ingoing Eddington-Finkelstein coordinates:

2m dr2 ds2 = − 1 − (dv − dr )2 + + r2 dθ2 + sin2(θ)dφ2 (3.3) r ∗ 2m 1 − r 2m = − 1 − dv2 + 2dvdr + r2 dθ2 + sin2(θ)dφ2 (3.4) r

One sees that there is no longer a singularity at r = RS. However there is still one at r = 0. As it will be shown in the following, there is a similar singularity but with a diﬀerent geometry in the Kerr metric. The usefulness of the ingoing Eddington-Finkelstein coordinates for the description of black holes can be seen in Fig. 3.1.

23 Figure 3.1: Two-dimensions space-time diagram in the ingoing Eddington-Finkelstein coordinates (t, r) with the radial coordinate expressed in units of mass of the object considered. The blue lines correspond to the outgoing null geodesics (constant u) and the red lines to the ingoing ones (constant v). Constant u and v representing the coordinate lines of the null geodesics, they define the limits of the lightcones. As one can see by observing the figure, wherever something is within the Schwarzschild radius (black vertical line), it will remain within this limit. Modified from [30].

The outgoing Eddington-Finkelstein coordinates are the same as the ordinary spherical coordinates (t, r, θ, φ) but in which the temporal coordinate t is replaced by u. Following the same reasoning as for the ingoing Eddington-Finkelstein coordinates, one can get the Schwarzschild metric in the outgoing Eddington-Finkelstein coordinates:

2m ds2 = 1 − du2 − 2dudr + r2 dθ2 + sin2(θ)dφ2 (3.5) r The reason these coordinates will not be considered in the following is shown in Fig. 3.2, where one can see that the outgoing coordinates describe the hypothetical object which is a white hole.

Figure 3.2: Two-dimensions space-time diagram in the outgoing Eddington-Finkelstein coordinates (t, r) with the radial coordinate expressed in units of mass of the object considered. The blue lines correspond to the outgoing null geodesics (constant u) and the red lines to the ingoing ones (constant v). Constant u and v representing the coordinate lines of the null geodesics, they define the limits of the lightcones. As one can see by observing the figure, wherever something is within the "Schwarzschild radius" (black vertical line), it will be ejected towards this limit. Modified from [30].

3.2 The Kerr metric properties

When a black hole rotation is considered, we cannot use any longer the Schwarzchild metric, and have to use a more general metric, called the Kerr metric (named after its discoverer Roy Kerr [3]).

24 Like the Schwarzchild metric, the Kerr metric is an exact analytical solution of the Einstein equations and can be expressed in the ingoing Eddington-Finkelstein coordinates, which is the form used in the original paper of Kerr:

2mr ds2 = − 1 − (dv − a sin2(θ)dφ)2 (3.6) r2 + a2 cos2(θ) + 2(dv − a sin2(θ)dφ)(dr − a sin2(θ)dφ) (3.7) + (r2 + a2 cos2(θ))(dθ2 + sin2(θ)dφ2) (3.8) where a is the angular momentum of the black hole divided by its mass. One may notice that when a is taken equal to zero, the Schwarzchild metric (3.4) is recovered. However the Kerr metric in ingoing Eddington-Finkelstein coordinates, due to its several oﬀ- diagonal terms, is not always the most practical. Depending on the properties one wants to study, it is better to express the Kerr metric in other coordinate systems [31]. Two of them are listed below and will be useful for the rest of this chapter when looking at the properties themselves.

∼ ∼ • The Boyer-Lindquist coordinates (t, r, θ, φ) where the r and θ coordinates are the same as in ∼ ∼ the ingoing Eddington-Finkelstein coordinates and t and φ are deﬁned as follows [32]:

∼ Z r2 + a2 t = v − dr (3.9) ∆ ∼ Z a φ = φ − dr (3.10) ∆ where ∆ = r2 −2mr +a2. In these coordinates, the Kerr metric (3.8) can be expressed as follows:

2mr ∼2 4mar sin2(θ) ∼ ∼ Σ ds2 = − 1 − dt − dtdφ + dr2 + Σdθ2 (3.11) Σ Σ ∆ 2ma2r sin2(θ) ∼2 + r2 + a2 + sin2(θ)dφ (3.12) Σ where Σ = r2 + a2 cos2(θ). These coordinates highlight the fact that the Kerr metric is stationary and axisymmetric (since there is not direct dependence in t or φ). Moreover the Minkowski metric for a flat spacetime can be recovered by making r tend to infinity. The Boyer-Lindquist coordinates, having only one independent off-diagonal term, will be useful to discuss the event horizons and what is called the ergosphere of a Kerr black hole, due to the coordinate singularity ∆ = 0 appearing in this form which was not present in the ingoing Eddington-Finkelstein coordinates. It will also help to express the metric tensor in such a way that it will be easier to see the nature of the hypersurfaces according to the region where they are located.

• The Kerr-Schild coordinates (t, x, y, z) where x, y and z are "euclidean" coordinates. These coordinates can be linked to the ingoing Eddington-Finkelstein coordinates by the following transformations:

t = v − r (3.13)

x + iy = (r − ia)eiφ sin(θ) (3.14)

z = r cos(θ) (3.15) where i is the unit imaginary number.

25 In that case the Kerr metric can be expressed as follows [31]:

rx¯ + ay ry¯ − ax z 2 ds2 = −dt2 + dx2 + dy2 + dz2 + H dt + dx + dy + dz (3.16) a2 +r ¯2 a2 +r ¯2 r¯ where H and r¯ are deﬁned respectively as

2mr¯3 H = (3.17) r¯4 + a2z2 z2 x2 + y2 + z2 =r ¯2 + a2 1 − (3.18) r¯2

One of the advantages of this formulation of the Kerr metric is the fact that we recover "familiar" coordinates. Moreover, as one can see in Eq. (3.16), the Minkowski metric in cartesian coordinates appears, such that it can be rewritten in a more compact way [33]:

2 α β ds = (ηαβ + Hlαlβ) dx dx (3.19) where ηαβ is the Minkowski metric and lα is expressed as

rx¯ + ay ry¯ − ax z (l ) = 1, , , (3.20) α r¯2 + a2 r¯2 + a2 r¯

One may verify that lα is a vector of null norm with respect to the Minkoswki metric and the Kerr metric. As Eq. (3.19) shows, the metric element gαβ is the Minkowski metric element to which a "perturbation" is added. The Kerr-Schild coordinates will help to highlight the geometry of the curvature singularity of a Kerr black hole. Such metrics can be used to describe a rotating (but still electrically neutral) black hole. By including a non-zero angular momentum, diﬀerent properties appear, some of which will be detailed in the following sections.

3.2.1 The curvature singularity geometry Similarly to Eq. (3.4), there is a singularity which appears when r2 + a2 cos2(θ) = 0 in Eqs (3.8) and π (3.12). While it means that r = 0 and θ = 2 , it is easier to study the singularity in another set of coordinates called the Kerr-Schild coordinates. Looking at Eq. (3.14) and developing the exponential, one can identify the real and imaginary parts of the equation:

x = r sin(θ) cos(φ) + a sin(θ) sin(φ) (3.21)

y = r sin(θ) sin(φ) − a sin(θ) cos(φ) (3.22) Using the fundamental relation of trigonometry and Eq. (3.15), one can regroup x, y and z together:

x2 + y2 z2 x2 + y2 = r2 + a2 sin2(θ) ⇐⇒ + = 1 (3.23) r2 + a2 r2 It is also possible to express the left side of Eq. (3.23) in another way, using again Eq. (3.15):

x2 + y2 z2 x2 + y2 z2 − = r2 + a2 − r2 ⇐⇒ − = 1 (3.24) sin2(θ) cos2(θ) a2 sin2(θ) a2 cos2(θ) π Since the singularity we are interested in corresponds to r = 0 and θ = 2 , it is interesting to look at the shape of the surfaces of constant r but also of constant θ. Eq. (3.23) corresponds to the equation of an ellipsoid (more precisely an oblate spheroid) while Eq. (3.24) corresponds to the equation of a hyperboloid of one sheet. The corresponding plots are showed in Fig. 3.3.

26 Figure 3.3: Visual representation of a cut along the z-axis of oblate spheroids of constant r (top) and hyperboloids of one sheet of constant θ (bottom) in the Kerr-Schild coordinates [32].

The ellipsoid corresponding to r = 0 is a degenerate case, reducing to a disk in z = 0. The additional π condition for the singularity θ = 2 will restrain the singularity to a radius a, making it an annular singularity.

3.2.2 The event horizons and the ergosphere Looking again at Eq. (3.12), Σ = r2 + a2 cos2(θ) = 0 is not the only singular value. Indeed, it is also the case of ∆ = r2 − 2mr + a2 = 0. However the situation is diﬀerent. Σ = 0 is a curvature singularity in the sense that a change of the system of coordinates will not get rid of it. But for ∆ = 0, this is the case. One can simply look at Eq. (3.8) to see that this singularity is not there in the ingoing Eddington-Finkelstein coordinates. For this reason, ∆ = 0 is called a coordinate singularity. This is very similar of the Schwarzschild case when r = RS, to which corresponds an event horizon. In the Kerr case, ∆ = 0 will also corresponds to event horizons, but because this coordinate singularity is a quadratic function of r means there will be two solutions. It is straightforward to get their expressions, by resolving a simple algebraic equation of the second order in r:

2 2 p 2 2 ∆ = r − 2mr + a = 0 ⇐⇒ r± = m ± m − a (3.25)

This is an important diﬀerence compared to the Schwarzschild metric: there is an inner horizon r = r− and an outer horizon r = r+ which have in general r 6= 0. As expected, a = 0 recovers r = RS = 2m. These two hypersurfaces are null hypersurfaces, i.e. the norm of the normal vector to them is equal to zero. To demonstrate this, one needs to obtain the expression of the inverse metric tensor gµν. First, the metric tensor in the Boyer-Lindquist coordinates can be obtained using Eq. (3.12) and the general deﬁnition of the line element:

2 µ ν ds = gµνdx dx (3.26)

27  2mr 2mra  − 1 − 0 0 − sin2(θ) g 0 0 g   Σ Σ  tt tφ    Σ   0 grr 0 0   0 0 0  gµν =   =  ∆   0 0 gθθ 0   0 0 Σ 0    gφt 0 0 gφφ  2mra 2mra2  − sin2(θ) 0 0 sin2(θ) r2 + a2 + sin2(θ) Σ Σ

Then after computing the inverse of the metric tensor one obtains [32]:

 2mra2  r2 + a2 + sin2(θ)  Σ 2mra   tt tφ  − 0 0 −  g 0 0 g  ∆ Σ∆  rr  ∆  µν  0 g 0 0   0 0 0  g =   =  Σ   0 0 gθθ 0   1  φt φφ  0 0 0  g 0 0 g  Σ   2mra 1 a2   − 0 0 −  Σ∆ Σ sin2(θ) Σ∆

A hypersurface which has for equation r = constant has by deﬁnition a normal (nµ) = (nt, nr, nθ, nφ) = µν (0, 1, 0, 0). The norm of the normal with respect to the metric gµν is given by g nµnν:

∆ n nα = gµνn n = grrn2 = (3.27) α µ ν r Σ In consequence when ∆ = 0, which is the case on the event horizons (which are hypersurfaces of constant r in the sense that their equations don’t depend on the other coordinates as one can see by looking at Eq. (3.25)), the norm of the normal to these hypersurfaces is equal to zero, conﬁrming the null nature of these hypersurfaces r = r±. Since r+ and r− are both roots of ∆, it allows to determine if a given hypersurface with r constant α α is spacelike (timelike normal vector: nαn < 0) or timelike (spacelike normal vector: nαn > 0).

∆ = (r − r−)(r − r+) > 0 if r > r+ or r < r− → timelike hypersurface (3.28)

= 0 if r = r± → null hypersurface (3.29)

< 0 if r+ > r > r− → spacelike hypersurface (3.30)

Finally to introduce the ergosphere, it is interesting to look at the gtt term from Eq. (3.12). In the case of the Schwarzschild metric, it is when r = RS that this term changes sign. However the situation is diﬀerent in the case of the Kerr metric. The following reasoning is taken from reference [32] where additional information can be found. First one needs to look at the values nullifying the gtt term of the Kerr metric in the Boyer-Lindquist coordinates:

2mr g = − 1 − = 0 ⇐⇒ 2mr − Σ = 0 (3.31) tt Σ p 2 2 2 ⇐⇒ r = RK± = m ± m − a cos (θ) (3.32)

Subsequently, gtt is positive only when RK+ > r > RK−. However, contrarily to the non-rotating case, RK± is diﬀerent from the values of r = r± giving rise to the coordinate singularity. In summary, the event horizons are between the zeroes of the gtt term such as RK+ > r+ > r− > RK−. The upper value RK+ corresponds to the outer boundary of what is called the ergosphere (see Fig. 3.4), the latter corresponding to the region between RK + and r+.

28 Figure 3.4: Schematic representation of the ergosphere compared to the outer event horizon. The arrow corresponds to the spin axis [32].

To understand what happens to an observer in the ergosphere, it is necessary beforehand to introduce the notion of Killing vector ﬁeld. By deﬁnition, a Killing vector is a vector which preserves the metric, or in other words the distances are conserved if all the points are equally moved in the direction of this vector. A given Killing vector X is a solution of the Killing equation:

Xµ;ν + Xν;µ = 0 (3.33) where the semicolon corresponds to the covariant derivative. If the diﬀerent components of a metric are independent of some variable, it indicates directly a Killing vector. For example, since the Schwarzschild metric is independent of t and φ, one can deduce two Killing vectors from that observation which are the following:

X1 = ∂t = (gtt, 0, 0, 0) (3.34)

X2 = ∂φ = (0, 0, 0, gφφ) (3.35) A transformation along these vectors won’t change the metric since it is independent of these variables. Going back to Eq. (3.12), one can see that none of the components depends explicitly on t. As such, there is a Killing vector (Xν) = (gtt, 0, 0, 0), expressed in contravariant indices (using the matrices of the Eq. (3.26)):

µ µν t tt X = g Xν ⇐⇒ X = g Xt = 1 (3.36)

But since in the ergosphere gtt is positive, the nature of the Killing vector is spacelike:

α µ ν t 2 XαX = gµνX X = gtt(X ) = gtt (3.37) This Killing vector can then help to define what is a static observer and a stationary one [32]. Firstly, a stationary observer is defined as an observer for whom the metric is constant during its motion. On the hand, because of this definition, it means the tangent vector to the wordline of this observer must be a Killing vector, which by definition preserves the metric. On the other hand, the four-velocity of an observer is always tangent to its wordline too. One can then deduce that the four-velocity is at least proportional to a Killing vector, which in a general point of view is a combination of the two Killing vectors X1 = ∂t and X2 = ∂φ since the elements of the metric don’t depend on these coordinates as one can see by looking at Eq. (3.12). One has then for the four-velocity a vector with only two non-zero components since the Killing vectors have only the t and φ components which are different from zero:

(X )µ + ω(X2)µ uµ = 1 = (ut, 0, 0, uφ) = ut(1, 0, 0, ω) (3.38) ||X1 + ωX2|| To keep the consistency in the numerator of Eq. (3.38), one introduces a factor ω = dφ/dt = uφ/ut which in fact is the angular velocity of the observer.

29 It will set conditions on which situations and in which regions of a Kerr black hole a stationary observer can be allowed. Since the four-velocity of an observer is a timelike vector of norm -1, one can get a quadratic inequality for ω:

α µ ν t 2 φ 2 t φ t 2 2 uαu = gµνu u = gtt(u ) + gφφ(u ) + 2gtφu u = (u ) gtt + ω gφφ + 2ωgtφ < 0 (3.39)

To look at this inequality, let us solve the equation between brackets in Eq. (3.39):

q 2 −gtφ ± g − gφφgtt 2 tφ gtt + ω gφφ + 2ωgtφ = 0 ⇐⇒ ω = ω± = (3.40) gφφ The discriminant needs to be greater than zero so one has from Eq. (3.12):

4m2a2r2 2mr 2mra2 sin2(θ) g2 − g g = sin4(θ) + 1 − a2 + r2 + sin2(θ) (3.41) tφ φφ tt Σ2 Σ Σ a2 sin2(θ) − (a2 + r2) = (a2 + r2) sin2(θ) + 2mr sin2(θ) (3.42) Σ = ∆ sin2(θ) (3.43)

Since the discriminant needs to be greater than zero, it is also the case for ∆. In consequence if ∆ < 0, there is no real solutions and the observer cannot be stationary. But as seen above, the only region where ∆ < 0 is between the two horizons of the Kerr black hole, so it is not possible for an observer to be stationary in that region. Moreover, Eq. (3.39) is valid if ∆ > 0 and ω+ > ω > ω−, which is the case in the exterior of the outer horizon (and so in the ergosphere). Finally, if ∆ = 0 (on the outer horizon for example), the inequality cannot be solved. However, there is a unique solution for Eq. (3.40) which corresponds to a null four-velocity. Secondly, a static observer is an observer who seems at rest for an asymptotic distant another observer. Thus it is also an observer for whom the tangent vector to its wordline is proportional to the Killing vector (Xν) = ∂t. Let us suppose that this observer is located in the ergosphere. In that case, to be static, the tangent vector would need to be spacelike since the Killing vector (Xν) = ∂t there is spacelike as shown in Eq. (3.37). But this is not possible since the observer should have a timelike wordline (and so a timelike tangent vector) to respect causality. One concludes that an observer located in the ergosphere cannot be static. Concerning the ergosphere, on its outer limit by deﬁnition gtt = 0 and ω− = 0. According to whether or not the observer is located outside or inside the ergosphere it will change the sign of ω−. If outside, it will be negative and so it is possible to have an observer which has a zero angular velocity (and so is static) which respects Eq. (3.39). However, if inside the ergosphere, ω− is positive, which means it is not possible to have a static observer inside the ergosphere as already explained before. To put all this in a nutshell, here a summary of the diﬀerent situations according to the region concerned:

• r > RK+: It is outside of the ergosphere and the black hole, so it is possible to have a static or a stationary observer there.

• RK+ > r > r+: It is the ergosphere, there cannot be a static observer but it is possible to have a stationary observer as long its angular velocity is between ω− and ω+.

• r+ > r > r−: In this region there cannot be nor a static observer nor a stationary one (since ∆ < 0 there).

• r− > r > RK−: In that region there still cannot be a static observer but a stationary one is again possible since ∆ > 0.

• r < RK−: In the innermost region, there can be both a static or a stationary observer since ω− changes sign again and ∆ > 0.

30 3.2.3 The maximum spin of a Kerr black hole Returning to the subject of the spin of a black hole, one may wonder if there is a limit to its value. To answer that, it is interesting to take a closer look at Eq. (3.25). As already said, if a = 0, one of the 2 2 horizons has for value r = RS and the other is rejected to r = 0. However m − a has to be positive to have a solution to ∆ = 0. It means that the maximum value for the angular momentum per unit mass of the black hole is a = m. A black hole with such a high spin is called an extremal Kerr black hole, and in that situation the black hole would have a unique event horizon since r = r+ = r− = m, half of the Schwarzschild radius of a non-rotating black hole with the same mass. If a > m, then it means that ∆ cannot be equal to zero, or in other words that there is no event horizon. However the curvature singularity Σ = 0 still exists. That would mean that when a > m, the singularity of the Kerr black hole would be naked, i.e. not hidden by an event horizon. This would be in contradiction with the weak cosmic censorship hypothesis. This hypothesis states that there cannot exist a naked singularity and will always be hidden by an event horizon (with the exception of the Big Bang singularity). It has been proposed to put aside the problem as what is inside an event horizon is not causally connected to us. Table 3.1 and Fig. 3.5 indicate the final dimensionless spins (the spins of the final objects) of the different gravitational waves events detected by the LIGO-Virgo collaboration. As these data illustrates, all of them are in agreement with the weak cosmic censorship hypothesis.

Events Final dimensionless spin a/m +0.05 GW150914 0.69−0.04 +0.13 GW151012 0.67−0.11 +0.07 GW151226 0.74−0.05 +0.08 GW170104 0.66−0.10 +0.04 GW170608 0.69−0.04 +0.07 GW170729 0.81−0.13 +0.08 GW170809 0.70−0.09 +0.07 GW170814 0.72−0.05 GW170817 6 0.89 +0.07 GW170818 0.67−0.08 +0.08 GW170823 0.71−0.10 Table 3.1: Table listing the ﬁnal dimensionless spins of the diﬀerent gravitational waves events detected by the LIGO-Virgo collaboration [8].

Figure 3.5: Plot of the final dimensionless spins of the different gravitational waves events dectected by the LIGO-Virgo collaboration in function of the final mass (mass of the objets resulting of the merge) expressed in solar masses. The contours define the 90% confidence regions for each event [8].

31 3.3 Measured black holes and neutron stars masses

Neutron stars and black holes masses have already been measured by electromagnetic messengers (pulsar, accretion disks...). However the direct detection of gravitational waves in the last few years provided an alternative way of detecting systems of binary black holes. Fig. 3.6 shows the masses of different neutron stars and black holes. As one can directly see by looking at this figure there is the presence of a gap between around 2 and 5 solar masses, separating the neutron stars and the black holes. It also suggests as discussed in the previous chapters that there is a maximum limit on the mass of a neutron star (the Oppenheimer-Volkoff limit) and that black holes formation due to supernova events should not have a low mass, and the discovery of a solar-mass black hole would trigger many investigations on its origin.

Figure 3.6: Visual representation of the masses of diﬀerent black holes and neutron stars (in solar masses), including those detected by the LIGO-Virgo collaboration by gravitational waves, with the error bars. EM black holes/neutron stars means they have been detected by electromagnetic means. When two objects are linked together by an arrow, it means they were part of a binary system and the circle at the end of the arrow is the total mass of the ﬁnal object resulting from the merge. The total mass of the neutron stars binary has a lot of incertitude and as such is labelled by an interrogation mark. One can clearly see the mass gap separating the black holes and the neutron stars in the region between around 2 and 5 solar masses. The black holes detected by the LIGO-Virgo collaboration are mainly heavy compared to the ones detected by X-rays [34].

32 Chapter 4

The formation of black holes by accretion of dark matter onto neutron stars

The classical stellar evolution models predict that black holes are the ﬁnal state of only the most massive stars. In consequence, the mass of the black hole is at least several solar masses. One can then wonder if low-mass black holes could exist (around the mass of the Sun or below). If this is the case, other mechanisms have to be considered to make them possible. One of the possible ways is by using the accretion of dark matter. Even if the accretion of ordinary matter is more known , its interactions with compact objects and their magnetic ﬁeld (which is for a white dwarf or a neutron star very large) complicate a lot the understanding of the process and it is more likely to cause the explosion of the object (type Ia supernova of a white dwarf for example) or to make an accretion disk around them [35]. The dark matter is another possible way to make a black hole by accretion onto a compact object. Even if progress has been made in the last decades on the subject, the dark matter puzzle is still one of the big mysteries in astrophysics to this day. The formation of black holes by dark matter accretion would help to put constraints on its characteristics such as the type of particles it is made of, their mass and so on. In the following chapter two main categories will be considered: the case where the dark matter particles are fermions and the one where they are bosons.

4.1 The accretion of fermionic dark matter

The idea is straightforward [36]: during the course of time dark matter would accumulate into a neutron star until the moment where the number of dark matter particles is too high and they overcome their Fermi degeneracy pressure. The neutron star would then collapse into an initial black hole which will feed on the rest of the neutron star. Since a neutron star has a mass of the order of the solar mass, the black hole wouldn’t have as high a mass as one resulting from a supernova event. Two elements concerning the number of particles are interesting here: the number Ncoll of particles over which the collapse of the dark matter cloud begins and the number Ncrit of particles over which the attraction of the particles exceeds the degeneracy pressure of the dark matter. Some hypothesis on the nature of the fermionic dark matter particles will be made:

• The dark matter must be able to interact with the ordinary matter. It will be necessary to the capture of the particles and their subsequent thermalization.

• The dark matter particles are non-annihilating, in other words there are more particles than anti-particles. This hypothesis is also necessary to allow the accumulation of the dark matter.

• Finally, the dark matter particles interact among themselves, speciﬁcally they can attract each other, which will help to initiate more easily the collapse.

33 4.1.1 The number of accreted dark matter particles

It is possible to obtain an expression of the number of dark matter particles accumulated Nacc as a function of time [37, 38]. In order to do this, one ﬁrst assumes that the dark matter particles velocities v follow a Maxwell-Boltzmann distribution f(v):

m 3 mv2 f(v)dv = n 2 4πv2 exp − dv (4.1) 0 2πT 2T where n0 is the number density of dark matter particles around the neutron star, m the mass of mv¯2 3T the particles and T their temperature. By using the equipartition theorem, one gets 2 = 2 , q 2 2 2 where v¯ = hvx + vy + vz i is the root-mean-squared velocity of the dark matter. Implementing this expression in Eq. (4.1) to get rid of the temperature, one has

3 3 2 3v2 f(v)dv = n 4πv2 exp − dv (4.2) 0 2πv¯2 2¯v2 The diﬀerential accretion rate dF will correspond to the ﬂux of dark matter particles through a surface element on a sphere centered onto the neutron star. If ones considers only the particles with a velocity in the interval [v, v + dv] and an angular interval [θ, θ + dθ] from the normal to the surface element, taking into account the projection factor v cos(θ), one obtains

sin(θ)dθ v dF = −f(v)dv v cos(θ) = f(v) dv d(cos2(θ)) (4.3) 2 4 3 3v2 = n πv3 exp − dv d(cos2(θ)) (4.4) 0 2πv¯2 2¯v2 v2 Let us then express Eq. (4.4)in terms of the kinetic energy E = 2 and the angular momentum J = vR sin(θ), both per unit mass, where R is the radius of the sphere on which the surface element located the surface element. One gets the diﬀerential accretion rate dFacc:

3 3 2 3E dF = 4πR2dF = 4π2n exp − dE dJ 2 (4.5) acc 0 2πv¯2 v¯2 3 3 2 ' 4π2n dE dJ 2 (4.6) 0 2πv¯2 The approximation made in Eq. (4.6) is that the energy can only vary between zero and the value v¯2/3 by which Eq. (4.5) decreases by a factor e, such that the exponential in Eq. (4.5) is considered of the order of 1. One also deﬁnes E0 as a parameter corresponding to the maximum kinetic energy allowing the capture by the neutron star. It is also possible to express the periastron of the trajectories of the dark matter particles around the neutron star with respect to E and J [38]:

J 2/M J 2 rperi = r ! ' (4.7) J 2E 2M 1 + 1 + 2 M 2 Here we suppose that M 2 is much larger that J 2E, with M the mass of the neutron star. For a particle to be accreted onto the neutron star, its periastron needs to be smaller than the radius of the neutron star.√ Looking at Eq. (4.7), one deduces that rperi is equal to the radius of the neutron star R when J = 2MR. Finally, integrating Eq. (4.6), one gets

3 v¯2 3 2 Z min ,E0 Z 2MR 2 2 2 3 3 2 2 3 v¯ Facc = 4π n0 2 dE dJ = 8π MRn0 2 min ,E0 (4.8) 2πv¯ 0 0 2πv¯ 3 The last step consists in taking into account the relativistic eﬀects, which are non negligible when a neutron star is considered.

34 One can express the trajectory of the dark matter particles as follows [39]:

du2 2M 2E = u 2Mu2 − u + + (4.9) dφ J 2 J 2 1 where u = r , r is the radial coordinate and φ the azimutal angle. There is still the condition that the periastron needs to be equal to or smaller than R. Replacing r by R in Eq. (4.9) and noting the fact that at the periastron u is independant of φ, one has the relation between the kinetic energy and the angular momentum of the dark matter particles needed to respect this condition:

J 2 2M M E = 1 − − (4.10) 2R2 R R In the classical case, when E was supposed small, J 2 = 2MR. If one takes also E very small in Eq. 2M −1 (4.10), one has J 2 = 2MR 1 − . Thus there is a corrective factor which needs to be taken R 2 into account when integrating Eq. (4.8). Also, in practise, E0 is bigger than v¯ /3 for a neutron star [38]. Finally one has to take into account the fact that even if the particle goes trough the neutron star, it will not necessarily mean that it will accrete. This will depend notably on its interaction cross- section σ with the baryons constituting the neutron star. As such, one introduces an efficiency factor −45 2 f = σ/σcrit, with σcrit some critical value (of the order of 10 cm ). If the cross-section is bigger than this critical value, f = 1 and every particle will be captured on the first passage on average. With all this information, one finally gets the number of accreted particles over some time t:   3 2M MR 3 2 v¯2 N = F 1 − f t = 8π2n   f t (4.11) acc acc R 0  2M  2πv¯2 3 1 − R √   6π ρ R R σt = DM  S  (4.12) v¯ m  RS  σ 1 − crit R where ρDM is the dark matter massive density. Some values are given in Table 4.1.

t (years) f Nacc (TeV/m) mtot (solar masses) 106 0.1 1.7 1035 1.5 10−19 109 0.5 8.5 1038 7.6 10−16 1012 1 1.7 1042 1.5 10−12

Table 4.1: Number of accreted particles Nacc for some values of f and t. Here the typical values −1 −3 4 3 v¯ = 220 km s , ρDM = 0.3 GeV cm , R = 10 m and RS = 5 10 m are ﬁxed. mtot = mNacc is the total mass of the dark matter accumulated in the neutron star.

4.1.2 Thermalization, start of the collapse and value of Ncoll After that the dark matter particles are captured by the neutron star, they will thermalize with the baryons in it and all the particles will go inside some radius called the thermal radius. After enough particles are concentrated in this thermal radius, the collapse of this dark matter cloud can start. The attraction between the dark matter particles will speed up this process, lowering the number of particles needed. The ﬁrst problem is now to determine the value of Ncoll, assuming self-interaction of the dark matter particles.

35 It is assumed that the attractive potential between two particles is a Yukawa potential VY (r) given as follows [36]:

a e−br V (r) = (4.13) Y r where a is a parameter to be determined and b the mass of the mediator particle of the self-attraction interaction. The value of Ncoll can then be determined using the virial theorem of a homogeneous spherical dark matter cloud of radius Rc:

2hKi = −hV i = −hVgrav + Vext + VY,toti (4.14) where hKi and hV i are respectively the mean kinetic and potential energy, Vgrav the gravitational potential energy of the dark matter cloud, Vext the potential energy due to the rest of the neutron star and VY,tot the total potential energy coming from the Yukawa interactions between all the dark matter particles. One can show that VY,tot for the dark matter cloud can be expressed as follows [40]:

2 3N 2 2 3 3 −bRc 2 VY,tot = − 5 6 3 − 3b Rc + 2b Rc − 3e (1 + bRc) (4.15) 4b Rc where N is the number of dark matter particles. Developing each potential energy term in Eq. (4.14), one has

2 2 2 2 3m 8πρcmRc 3N 2 2 3 3 −bRc 2 2hKi = N + N + 5 6 3 − 3b Rc + 2b Rc − 3e (1 + bRc) (4.16) 5Rc 5 4b Rc where ρc is the core density of the neutron star. Moreover, Eq. (4.16) allows to get an approximate expression for the thermal radius rth discussed above. Indeed, if the system is in thermal equilibrium, using the equipartition theorem and taking Rc large, one ﬁnds the expression of rth (for one particle):

1 2 2 8πρcmRc 3T 15T hKi ' = ⇔ rth = (4.17) 10 2 8πρcm Some values of the thermal radius for diﬀerent masses are given in Table 4.2.

m (TeV) rth (m) 10−6 88 10−3 2.8 1 8.8 10−2 103 2.8 10−3

Table 4.2: Values of the thermal radius rth for diﬀerent masses m of the dark matter particles. Here 5 18 −3 the typical values T = 10 K and ρC = 10 kg m are ﬁxed.

Finally to determine the value of Ncoll, one needs to take into account all the terms of Eq. (4.16). As long as the parameters a and b are small, one can see an unstable solution for r → 0 for N suﬃciently small. The value of Ncoll is the value of N (suﬃciently large) for which the two solutions disappear.

4.1.3 Formation of the black hole and value of Ncrit The last step after the beginning of the collapse to get a black hole is to have a number of particles larger than Ncrit, such that the degeneracy pressure won’t be enough to stop the collapse (the situation is very similar to the Chandrasekhar limit for the white dwarfs or the Oppenheimer-Volkoﬀ one for the neutron stars). It is possible to get a simpliﬁed expression for Ncrit which depends only on the parameters a and b introduced in the Yukawa potential (4.13) and the mass of the dark matter particles [40].

36 One ﬁrst develops the expression of the total energy Etot of the dark matter cloud, which is the sum of its kinetic energy K, its gravitational potential energy Vgrav and the self-interaction potential energy VY . The latter two have already been expressed in Eq. (4.16). The last term needed is thus the kinetic energy. To get its expression, we follow the same reasoning as for the pressure of a relativistic degenerate gas of fermions in Chapter 1. The only diﬀerence is the fact that here instead of the pressure we compute the kinetic energy by the following relation:

Z pF 4 3 V 2 4πm Rc K = V ρ = 3 4πp Etot(p)dp = J(xF ) (4.18) 4π 0 3 pF where ρ is the kinetic energy density of the dark matter cloud, V the volume of the cloud, xF = m and J(x) is given by

1 h 1 h 1 ii J(x) = x 1 + x2 2 1 + 2x2 − ln x + 1 + x2 2 (4.19) 8π2 p where x = m . Applying the same reasoning to obtain the number of particles N, one has:

Z pF 3 3 V 2 m x N = V n = 3 4πp dp = 2 (4.20) 4π 0 3π where n is the number density of particles. Then we make the non-relativistic short range approximation, i.e. we suppose respectively m much larger than pF and bRc much larger than 1. The non-relativistic approximation m >> pF simpliﬁes the expression of J(xF ) (4.19) since it means that xF is small by deﬁnition. In this approximation one has for the kinetic energy:

4 3 5 5 4πm R 2 9π 3 N 3 K = J(xF ) ' mN + 2 (4.21) 3 15π 4 mRc

The short range approximation bRc >> 1 simpliﬁes the expression of Eq. (4.15) as follows:

3aN 2 1 VY ' − 2 3 (4.22) 2b Rc

Combining Eqs. (4.21), (4.22) and the expression of Vgrav from Eq. (4.16), one gets an expression for Etot:

5 5 3 2 2 9π N 3 2 3a 1 3m Etot = mN + 2 − N 2 3 + (4.23) 15π 4 mRc 2b Rc 5Rc

Let us ﬁnd the extrema of Eq. (4.23) by determining the roots of its derivative with respect to Rc: s 5 − 1 10 − 2 dEtot 2 9π 3 N 3 4 9π 3 N 3 15a = 0 ⇔ Rc = 3 ± 2 6 − 2 2 (4.24) dRc 9π 4 m 81π 4 m 2b m

There are two diﬀerent solutions for Rc. However one can determine the value of N for which the solutions are equal (i.e. the discriminant is equal to zero), corresponding to the unstable situation of the collapse, which gives the value of Ncrit:

10 − 2 3 3 3 5 2 3 3 4 9π Ncrit 15a 2 9 8 b 1 b 1 2 6 − 2 2 = 0 ⇔ Ncrit = π 1 6 ' 0.3 1 6 81π 4 m 2b m 4 1215 a 2 m a 2 m

(4.25)

Depending on the values of a and b, one can then compute Mcrit = mNcrit. A graphical illustration is given in Fig. 4.2.

37 Figure 4.1: Value of Mcrit (in solar mass) with respect to the mass of the dark matter particle m, according to Eq. (4.25), in the case of the accretion of fermionic dark matter.

4.2 The accretion of bosonic dark matter

The accretion of bosonic dark matter is similar to the fermionic case discussed above. We suppose an asymmetric non-annihilating dark mater which will accrete in a neutron star until the number of particles is suﬃciently high to start a collapse and the formation of a black hole. The key point here is the fact that in the bosonic case there is no Fermi pressure which would oppose the collapse. However it doesn’t mean that there are no processes that hinder or accelerate the formation of the black hole, in this section mainly the pressure due to the zero point energy and the possible formation of a Bose-Einstein condensate will be considered [41].

4.2.1 The number of accreted dark matter particles The reasoning explained in the previous section could be used whatever the dark matter particles are fermions or bosons since the differences between them are not involved in the computations. However for this section we will follow this time a slighty different approach [41] and compare it to the approach used for the fermionic case, neglecting this time the self-attraction of the dark matter and considering still at first that the dark matter particles follow a Maxwell-Boltzmann velocities distributions. It is possible to show that in that case that the accretion rate is given by the following relation [42]:

Z R dFacc 2 Facc = 4π r dr (4.26) 0 dV The diﬀerential term is given by

1 " 2 # 2 2 −B dFacc 6 vesc(r) 1 − e ∆p = σn0(r)nB(r) 1 − 2 min , 1 (4.27) dV π v¯ B pF where n0 is now locally deﬁned, nB is the local baryon number density, vesc the local escape velocity, pF the Fermi momentum of the degenerated baryons.

38 The B term is given by the following expression:

2 2 vesc(r) mmB B = 6 2 (4.28) v¯ (m − mB) where mB is the mean mass of the ordinary baryonic particles of the neutron star. The minimum factor in Eq. (4.27) comes from the possible case where the momentum transfer when a dark matter particle is scattered by the baryons is inferior to the Fermi momentum. If we consider n0, nB and vesc independent of the radial coordinates, Eq. (4.26) becomes

1 " 2 # 2 2 −B 6 σNBρDM vesc 1 − e mvesc Facc ' 1 − 2 min , 1 (4.29) π mv¯ B pF One can look for the value of m for which the momentum transfer will be lower than the Fermi momentum. The latter is given by Eq. (1.10) where the electronic density is replaced by the baryonic density :

1 1 3π2ρ 3 2 3 B pF = 3π nB = ' 0.58 GeV (4.30) mB 15 −3 where ρB is the baryonic mass per unit volume and has been taken equal to 1.4 10 g cm . As such, one can determine the mass for which the minimum factor equal to 1: mv p esc > 1 ⇔ m > F ' 1 GeV (4.31) pF vesc 5 −1 where vesc = 1.8 10 km s . Finally one has the number of particles accreted over time Nacc = Facct:

• If m is over 1 GeV ρ N = 2.3 1030 DM f t (4.32) acc m • If m less than 1 GeV 32 Nacc = 3.4 10 ρDM f t (4.33)

−1 57 where the typical values v¯ = 220 km s and NB = 1.7 10 are ﬁxed, and ρDM is expressed in −3 GeV cm , t in years and m in TeV. The order of magnitude of Nacc for the same parameters that in Table 4.1 are indicated in Table 4.3.

t (years) f Nacc (TeV/m) mtot (solar masses) 106 0.1 6.8 1034 6.0 10−20 109 0.5 3.4 1038 3.0 10−16 1012 1 6.8 1041 6.0 10−13

Table 4.3: Orders of magnitude of the number of accreted particles Nacc for some values of f and t. −1 −3 Here the typical values v¯ = 220 km s and ρDM = 0.3 GeV cm are ﬁxed. mtot = mNacc is the total mass of the dark matter accumulated in the neutron star. Eq. (4.32) is used.

As one can directly see by looking at Table 4.3, the number of accreted particles over time is of the same order of magnitude (by a factor ∼ 2) than that for the fermionic case, which is in agreement with the fact that for the computation of Nacc no hypothesis has been made on the exact nature of the dark matter particles. The slight diﬀerence comes from several factors: even if in both cases an asymmetric dark matter interacting with baryons is assumed, for the bosonic dark matter the possible self-interaction between the particles is neglected. Moreover, from a computational point of view, some factors are taken into account in one case and not in the other and vice versa (for example, the numeric density of baryons and the escape velocity are directly considered for the bosonic dark matter and not for the fermionic dark matter, and on the opposite the radius of the dark matter cloud is not involved in the bosonic case).

39 4.2.2 Start of the collapse and formation of the black hole

After the thermalization of the dark matter particles within the thermal radius rth given by Eq. (4.17), the number of particles will become high enough to initiate the start of the collapse of the dark matter cloud. As a ﬁrst approximation, it is still possible to estimate the number of particles over which the cloud becomes self-gravitating [41]. Indeed, if one considers that the collapse will begin when the density of dark matter particles is higher than the baryonic density within the thermal radius, one has:

3 5 3 2 2 2 Nm 3mN 4πρBm 42 100GeV T ρDM > ρB ⇔ 3 > ρB ⇔ > ρB ⇔ N & 10 5 (4.34) 4πrth 4π 15T m 10 K 3

Finally, one can also estimate the value of Ncrit. However, contrarily to the accretion of fermionic dark matter, there is no Fermi pressure to oppose the gravitational collapse. The pressure that will play this role comes from the zero-point energy. Since there is no exclusion principle, the average distance between two bosons inside the spherical cloud is of the order of the radius of the cloud. However, looking at the Heinsenberg incertitude principle, it means that their zero-point energy is of the order of the inverse of the radius of the cloud. Then the average energy hEi of a boson inside the cloud is

Nm2 1 hEi = V + K ' − + (4.35) Rc Rc

Deriving Eq. (4.35) with respect to Rc, one ﬁnds an approximation of the critical value over which the gravitational term is larger than the kinetic one :

dhEi 1 = 0 ⇔ Ncrit ' 2 (4.36) dRc m

Some values of Ncrit in function of m are indicated in Table 4.4.

m (TeV) Ncrit 10−2 1.5 1036 0.2 3.8 1033 1 1.5 1032 102 1.5 1028

Table 4.4: Values of Ncrit for diﬀerent values of the mass of the dark matter particles m, in the case where the particles are bosonic.

Comparing Tables 4.2 and 4.4, one sees that due to the more constraining constraint imposed by the Pauli principle for the fermions, a lot fewer particles are needed to create a black hole, for the dark matter particles of the same mass.

4.2.3 The eﬀects of a dark matter Bose-Einstein condensate A Bose-Einstein condensate is a state of matter which appears when the temperature of a system of bosons falls below some critical temperature Tc, which depends mainly of the mass of the bosons and their density, and given by the following relation (supposing that the interactions between the dark matter particles are negligible and that all the particles are within the thermal radius):

2 2π n0 3 Tc = 2 3 (4.37) ζ 3 (1.5) m 2 where ζ is the Riemann zeta function. When experimenting on Earth, we have to get to extremely low temperatures very close to the absolute zero to get below this critical temperature.

40 However, inside a neutron star, the density of the dark matter could be high enough to obtain a critical temperature above the actual temperature in the center of the neutron star (which is also the temperature of the dark matter particles within the thermal radius by deﬁnition). When such state is attained, the Maxwell-Boltzmann distribution is no longer a valid approximation, and one has to use instead the Bose-Einstein distribution. Inverting Eq. (4.37), one obtains the critical number of particles NBE over which a Bose-Einstein condensate forms:

2 3 ! 3 3 3 2 3 3N 4πrthm 2 Tc Tc Tc ' 3.3 3 ⇔ NBE = ' 0.32 √ (4.38) 3 2 3 3.3 ρB 4πrthm

In an analogous way to the thermal radius, there is a radius rBE under which theses particles in the condensate state are located given by [41]

1 3 4 rBE = 2 (4.39) 8πρBm The number of particles over which the ones in the Bose-Einstein condensate begin to self-gravitate can be determined by using Eq. (4.34), replacing the thermal radius rth by rBE:

5 100GeV 2 N 1023 (4.40) & m Comparing Eqs. (4.34) and (4.40), one can see that for the same mass fewer particles are needed to start the self-gravitation when there is a Bose-Einstein condensate.

4.3 The fraction of neutron stars aﬀected by these phenomena

One can see when looking at Eqs. (4.12), (4.32) and (4.33) that given enough time, a neutron star would eventually accumulate enough dark matter particles that it will collapse into a black hole. However it is pretty obvious that the value of the interaction dark matter-baryon cross-section will play a big role in the actual time needed. Since we observe neutron stars, it means these still didn’t collapse. Thus the population of neutron stars could be used to help estimate the fraction of neutron stars that have collapsed into a black hole depending on the value of the interaction cross-section. Whatever the dark matter particles are fermions or bosons, the number of particles accreted depends in particular on three main factors: the dark matter density ρDM , the root-mean-squared velocity v¯ and the interaction cross-section σ. The ﬁrst two are not equally distributed in a same galaxy (the Milky Way being a good example). But this is also the case for the distribution of the neutron stars, which we can suppose at ﬁrst that it follows roughly the stellar distribution to some extent [36]. One needs to take into account all these elements to get the impact of the cross-section. The idea to compute more precisely the fraction of collapsed neutron stars is the following [36]. First of all, one needs a criteria to know when a given value of the cross-section will initiate the collapse of the neutron star. It can be achieved by looking at when the number of accreted particles Nacc is larger than the critical number of particles Ncrit, for example in the fermionic case:

 RS  ρ 0.3 σ 1 − b 3 1 1 DM √ crit  R  √ Nacc > Ncrit ⇐⇒ > 5 (4.41) v¯ 6π  RSR  a m σt

By looking at Eq. (4.41), one can see that the ratio of the dark matter density and the root-mean- squared velocity at the neutron star needs to be higher that some value, depending notably on the parameters involved in the Yukawa interaction, the time period studied and of course the interaction cross-section. This equation also shows explicitly that if we consider a smaller period or a smaller cross-section, less neutron stars will be converted into black holes, since the radio ρDM /v¯ needs to be higher to do so and thus less regions satisfy this condition. The same logic can also be applied to bosonic dark matter, using instead Eqs. (4.32), (4.33) and (4.34).

41 An example of ﬁgure for the fraction of collapsed neutron stars versus the cross-section is shown below, where the dark matter distribution follows a Burkert proﬁle described as follows:

" !#−1 r r 2 ρDM (r) = ρs 1 + 1 + (4.42) rs rs

−3 where r is the radial distance, ρs = 3.15 Gev cm and rs = 5 kpc.

Figure 4.2: Fraction of neutron stars which should have collapsed in t = 5 109 years with respect to the interaction dark matter-baryons cross-section in the fermionic case and in the Milky Way. The different curves correspond to different combinations of values of a, b (of the Yukawa interaction) and m, so that the curves have the same shape. The velocity distribution is supposed to increase linearly up to 220 km s−1 at 0.5 kpc from the center of the galaxy, then the velocity has constantly this value beyond 0.5 kpc. The dark matter distribution is a Burkert profile given by Eq. (4.42) [36].

42 Chapter 5

The primordial black holes and their constraints

One other class of low-mass black holes is constituted by the primordial black holes, which could have been created in the early Universe from the gravitational collapse of very dense regions. Some inﬂation- ary cosmological models can also explain where these regions come from. But more importantly is the fact that it is possible to put constraints on these primordial black holes, partly due to observational data. The primordial black holes are candidates to explain several phenomena:

• Dozens of quasars have been discovered at high redshifts, associated to supermassive black holes. However one cannot explain correctly how such massive objects could have been created so early in the Universe history in the classical models of accretion.

• Similarly, some very bright galaxies and dusty environments have also been discovered at high redshift and one does not understand how they could have existed so early.

• Even for present supermassive black holes, it is diﬃcult to justify how they acquired such mass even with more than ten billion additional years.

• The primordial black holes can also be candidates for solar mass and intermediate mass black 3 4 holes (∼ 10 − 10 M ).

5.1 The inﬂation models and the origin of the primordial black holes

The hot Big Bang model, which originated back in the first half of the 20th century, has been confirmed since then by several observations, most notably the famous cosmic microwave background (CMB) originally detected by serendipity by Penzias and Wilson in 1964 and later on refined by several satellites. However, despite its success, several elements are not explained by the classical Big Bang model. For example, the CMB has a quasi-uniform temperature of about 2.7 K and is a quasi-perfect black body spectrum. However all the regions seen on the CMB images were not initially causally connected, begging the question of how the temperature is so uniform. This problem, known as the horizon problem, is one among other issues that the classical Big Bang model doesn’t address. One can also cite the flatness problem (why the Universe is so flat) or the origin of the large-scale structures. Inflation (or preferably the inflation models) is one of the attempts to resolve these problems. The general idea is that in the very early history of the Universe, it experienced a short phase of exponential expansion, resolving some of the problems not addressed by the hot Big Bang model. To obtain such an expansion, one starts from a scalar field φ (or several ones depending on the model). From this, one can get an exponential increase of the time-dependent scale factor a(t).

43 To illustrate this, let us take as an example a simple model, with a scalar ﬁeld in the Friedmann- Lemaître-Robertson-Walker (FRLW) metric, used for a spatially homogeneous and isotropic spacetime, given by the following relation:

dr2 ds2 = dt2 − a2(t) + r2 dθ2 + sin2(θ)dφ2 (5.1) 1 − kr2 where k is the curvature and can be equal to -1, 0 or 1 and a(t) the scale-factor with a(t0) = 1, where t0 is the present time. Depending on the value of k, the geometry of the Universe is diﬀerent: • If k = −1, one has an open Universe • If k = 0, one has a ﬂat Universe • And if k = 1, one has a closed Universe Combining Eqs. (2.4), (2.5) and (5.1), one obtains the Friedmann-Lemaître equations (taking Λ = 0) [43, 44]:

a¨ 4π (ρ + 3p) = − (5.2) a 3 a˙ 2 8πρ k H2(t) = = − (5.3) a 3 a2 where H(t) is the Hubble parameter. Let us also suppose that there is a scalar ﬁeld φ associated to a potential V (φ). One can show using Eqs. (5.2) and (5.3), supposing an approximately ﬂat Universe (k = 0), φ independent of the spatial coordinates, the potential V much greater than the kinetic energy term φ˙2 and V » ∂V/∂φ:

V √ H2(t) ' ⇐⇒ a(t) = a(t )et V/3 (5.4) 3 0 As long as the condition V » φ˙2 is respected, there will be an exponential expansion of the Universe. The central point is the exact form of the potential V (φ). There are lots of possibilities for it, and each one of them corresponds to a different model of inflation, which goes well beyond the scope of this master thesis. A list of possible potentials can be found in reference [45]. Some of these inflation models lead to large density fluctuations and thus to regions with very high density compared to their neighbourhood. If the characteristic lengths of these regions are smaller than their corresponding gravitational radius, they undergo a gravitational collapse and create what is called a primordial black hole. It is important to keep in mind however that a priori there is not necesseraly a reason for which a primordial black hole should be in a particular range of masses. It could be a very light black hole or a massive one. But if the black hole was too light when it has been produced, it probably has disappeared due to the Hawking radiation. The critical mass Mcrit (in kg) under which a primordial black hole, in a time t, should have evaporated is given the following formula [46]:

1 1 α t 3 M ' 1012 3 (5.5) crit 4 10−4 13.8 109 where α is a coeﬃcient depending on the species emitted [47] and t is expressed in years. If one takes −4 12 t ' t0, α ' 4 10 and subsequently Mcrit ' 10 kg.

5.2 The constraints derived from the observations

Considering the primordial black holes suﬃciently massive not to have evaporated by now, the observations (electromagnetic waves and in the future the gravitational waves) can help put constraints on diﬀerent ranges of primordial black hole masses, but also on wheter or not primordial black holes could be candidates to be a sizeable fraction of the missing dark matter. In the following section, we will focus on some of these constraints [48].

44 5.2.1 The constraints from the gravitational lensing The primordial black holes being compact objects, one can learn information about them since these objects can act as lenses and deviate the light of background sources like stars. Among other things, depending on the mass of the black hole and its distance compared to us and the source, the deflection and the characteristics of the images won’t be the same and thus it allows to some extent to put constraints on the primordial black holes. One of the main way of studying gravitational lensing is by using the microlensing, i.e. the angular distance between the different images of the source is too small to be resolved such that the images (partially) superpose, resulting in a luminosity higher than the original source. Before the discussion on the different surveys of microlensing, let us remind the reader of some notions of gravitational lensing which will be useful later on [48, 49].

5.2.1.1 Theoretical elements of gravitational lensing An important notion in gravitational lensing is the lens equation, which is a relation between the diﬀerent angles and distances of interest. If one supposes that the lens is thin and the angles small (which is eﬀectively the case most of the time), one can get the expression of the distance S0Q in Fig. 5.1:

0 0 S Q = SQ + S S ⇐⇒ θdS = βdS + αdLS (5.6) where dLS and dS are the angular diameter distances between the plane of the lens and the plane of the source, and between the plane of the observer and the plane of the source respectively. The deﬂection angle α is given by

4M α = L (5.7) θdL where ML is the mass of the lens (here the mass of the primordial black hole) and dL the angular diameter distance between the plane of the observer and the plane of the lens. Fig. 5.1 illustrates these distances.

Figure 5.1: Schematic representation of the different planes and distances used commonly in gravitational lensing. O corresponds to the observer, M to the lens (or deflector), S to the source and S’ to the image of the source. Modified from [49].

45 Now that we have the lens equation, we can deﬁne two other quantities:

• The Einstein radius RE, deﬁned as follows:

r MLdLdLS RE = 2 (5.8) dS It corresponds to the radius of the Einstein ring (full ring of deﬂected light) which appears when the source is aligned with the lens and the observer

• From the Einstein radius, one can deﬁne the optical depth τ as "the probability that the light from the source passes inside the circle deﬁned by the Einstein radius on the lens plane (when it is much smaller than unity)" [48] and it can be expressed as follows:

Z dS Z dS dLdLS τ = nPBH(dL)σd(dL) = 4π ρPBH(dL) d(dL) (5.9) 0 0 dS

where nPBH is the number density of primordial black holes, σ the lensing cross-section (equal 2 here to πRE) and ρPBH the energy density of the primordial black hole lens.

5.2.1.2 The magnification due to the microlensing The magnification caused by the microlensing can be expressed in function of the angle β and the 2 Einstein radius. To show this, one first multiplies Eq. (5.6) by θdL/dS, using Eq. (5.7) and noting r = θdL and r0 = βdL:

q 2 2 r0 ± r0 + 4RE r2 − r r − R2 = 0 ⇔ r = (5.10) 0 E ± 2 In the thin-lens approximation, there are two images located at the positions given by Eq. (5.10). For a small source and supposing a circular symmetry of the lens, the magniﬁcation A of an image is given by [49]

θ dθ r dr A = = (5.11) β dβ r0 dr0

Noting u the ratio of r0 to the Einstein radius, one has using Eq. (5.11):

√ r r 1 u ±u u2 + 4 + u2 + 2 ± ± p 2 |A±| = = u ± u + 4 1 ± √ = √ (5.12) r0 dr0 4u u2 + 4 2u u2 + 4

The ratio of the two magniﬁcations in Eq. (5.12) thus gives √ A u2 + 2 + u u2 + 4 + R = = √ (5.13) A− u2 + 2 − u u2 + 4

5.2.1.3 Studies of gravitational lensing Now that we have reviewed some elements of gravitational lensing theory, we can take a closer look at which observations have been performed to put constraints on the primordial black holes. In particular, how much could the primordial black holes contribute to the elusive dark matter mass ? It is possible to link the optical depth and the fractional density fPBH of dark matter which is due to compact objects in our galaxy dark halo. For example, B. Paczynski demonstrated that there is a relation between the optical depth and fPBH in the simple case of an isothermal model [50] with sources in the Magellanic −6 clouds: τ ∼ 10 fPBH.

46 Following this, many observations have been performed to try and determine if primordial black holes could explain at least partially the "missing" mass needed to explain notably the velocity curve of our galaxy. Some famous projects are notably the MACHO project (MAssive Compact Halo Object) searching by microlensing objects like black holes, planets, brown dwarfs, etc in the dark matter halo, EROS (Expérience pour la Recherche d’Objets Sombres) or OGLE (Optical Gravitational Lensing Experiment). Some of them are grouped in Table 5.1. It is also important to remember that the results are for the fraction of compact objects, not necessarily primordial black holes. As such the results in the table should be taken as upper limit for the proportion of primordial black holes in all cases. As one can see when looking at Table 5.1, the duration of the surveys needs several years to attain a low number of microlensing events. Because of this, the incertitude on the optical depth can be very large if the number of events is very low (EROS for the Small Magellanic Cloud). It is also interesting to highlight the fact that even if no events have been detected (which is the case of EROS for the Large Magellanic Cloud)), it is still a piece of information that helps to put an upper limit on the fraction of dark matter constituted of compact objects.

−7 τ (10 ) fPBH (%) Number of events Period (years) Mass range (M ) Cloud Study +0.4 1.2−0.3 8 - 50 13 - 17 5.7 0.15 - 0.9 LMC MACHO [51] < 0.36 < 10 0 6.7 10−6 - 1 LMC EROS-2 [52] 0.085 to 8.0 1.8 - 100 1 6.7 > 10−2 SMC EROS-2 [52] 0.16 ± 0.12 < 4 2 ∼ 8 ∼ 0.1 LMC OGLE [53] 1.30 ± 1.01 < 20 3 ∼ 8 10−1 − 10−2 SMC OGLE [54]

Table 5.1: Determination of the optical length with diﬀerent surveys considering the sources in the Large Magellanic Cloud (LMC) and the Small Magellanic Cloud (SMC). The corresponding fraction of compact objects (value taken from the articles themselves), the number of microlensing events considered, the period of observation and the mass range of the compact objects concerned are also indicated.

Finally, in general the primordial black hole moves at some (tangential) speed vt with respect to the source-observer axis. Because of that the value of the brightness ampliﬁcation due to the microlensing will change over time. The charateristic time-scale (in years) between the ampliﬁcation maximum and its near-minimum is the ratio of the Einstein radius and the tangential speed of the lens:

s RE dLdLS ML(M ) dS(kpc) 200 tscale = ' 2 2 −1 (5.14) vt dS 10 100 vt(km s ) If one considers as lenses primordial black holes in the Milky Way dark matter halo and as sources p stars in the Magellanic clouds, tscale ' 2 ML(M )/10. For example, if one makes a ten years survey, one could theoretically follow the full ampliﬁcation variation due to black hole lenses as massive as a few hundred solar mass. It justiﬁes even more the need for years long surveys.

5.2.2 The impact of the gravitational ﬁeld of a (primordial) black hole on some celestial objects Constraints on primordial black holes can also be obtained by looking at the gravitational perturbations they produce on the celestial bodies or regions they encounter. In the case of a celestial body, let us take as an example a white dwarf [55]. If some circumstances make a primordial black hole encounter a white dwarf, provided that the black hole has a suﬃciently low mass (if it is too heavy, the black hole will directly destroy the white dwarf), it can go through the white dwarf, transferring some of its kinetic energy to the medium around its trajectory by dynamical interaction.

47 If the transfer of energy is large such as the temperature of the medium is high enough, it can start the carbon fusion which in turn causes a thermonuclear runaway (since by definition a white dwarf is composed of a state of degenerated electrons, which cannot really control the temperature to keep the white dwarf stable) if the thermal diffusion is slow enough and eventually a supernova explosion. The main difference with a classical type Ia supernova would be the fact that the white dwarf wouldn’t need to accrete material or/and to be close to the Chandrasekhar limit discussed in the first chapter.

First let us consider the characteristic length scale L over which the thermal diffusion will be slow enough compared to the fusion rate F to lead to a thermonuclear runaway and a supernova. Indeed the time-scale tdiff of the decrease in temperature of a hotter region of temperature T (which in our case would be at first the region around the trajectory of the primordial black hole inside the white dwarf) increases when L increases too. One can easily see it qualitatively starting from the one-dimensional heat equation:

∂T ∂2T κ ∂2T = α 2 = 2 (5.15) ∂t ∂x ρcp ∂x where α is the thermal diﬀusivity, κ the thermal conductivity, ρ the density of the medium considered and cp the speciﬁc heat of the medium. Reformulating Eq. (5.15) in term of dimensional analysis one has

T κ T ρcp 2 = 2 ⇐⇒ tdiff = L (5.16) tdiff ρcp L κ −1 Thus when L is bigger than some minimal value Lmin, tdiff is larger than F . In that case the fusion reactions will have enough time to develop before the excess thermal energy is diffused. As the medium is highly sensitive to the temperature, the energy freed by the fusion will ignite other fusions, triggering the thermonuclear runaway. The thermal conductivity can itself be expressed as a function of the temperature:

4aT 3 κ ' (5.17) ρκop where a is the radiation constant and κop is the opacity associated to the particles carrying the heat. The only thing left is to look at the diﬀerent particles one has to consider for the opacity, i.e. photons 8 −3 −2 8 and electrons. If ρ > 10 g cm , then the electrons are dominant and κop ÷ ρ , while when ρ < 10 −3 g cm , photons dominate and κop ÷ρ [55]. Combining these information, one ﬁnally has the following condition to have the thermonuclear runaway (considering F ÷ ρ):

3 2 κ T F tdiﬀ = 1 ⇔ L = ÷ 3 (5.18) R ρcp ρ κop 2 −0.5 ⇔ Lmin ÷ ρ (for electrons) (5.19) 2 −2 ⇔ Lmin ÷ ρ (for photons) (5.20) (5.21) One can also have an idea on the mass of the primordial black hole required to create a runaway [55]. It is the mass MPBH such as

s 4 R R ' 10 GMPBH log > Lmin (5.22) GMPBH A primordial black hole respecting this condition will heat the medium enough to lead to a supernova event. Since all black hole masses cannot make a white dwarf explode, and since some ranges of masses are already constrained by other means, one can also use the population of existing white dwarfs and the frequency of type Ia supernovae to put constraints on the importance of the primordial black holes in the dark matter content.

48 Finally one can also ﬁnd the mass of a white dwarf that a given black hole can destroy, using an equation of state to have the relation between the density of the medium and the mass of the white dwarf and Eq. (5.22) as illustrated in Fig. 5.2.

Several other objects perturbed by their gravitational interaction with potential primordial black holes can be used to put constraints, such as ultra-faint dwarf galaxies and the evolution of the half- light radius of a star cluster inside it [48], neutron stars, globular clusters or even the galatic disk itself by increasing the kinetic energy of the stars.

Figure 5.2: Minimum mass for a black hole to destroy a white dwarf of a given mass while going through it. The speed of the primordial black hole is supposed to be roughly equal to the escape velocity of the white dwarf, so it can go through it. The white dwarf mass is expressed in solar mass [55].

5.2.3 Other sources of constraints The few examples given in the previous subsections are far from being exhaustive. One can also cite [56]:

• The primordial black holes, as they are created in the early Universe, could had an impact on the cosmic microwave background, on its power spectrum or its anisotropies.

• Some models of primordial black holes lead to an increase of the metal fraction of some stars.

• The accretion on the primordial black holes should perturb the cosmic microwave background or more recently should emit, from their accretion disks, in X-rays and radio waves. Thus one can search for their signature in the CMB or in some domains of the electromagnetic spectrum.

A summary of all these constraints on the fraction density of the dark matter constituted of primordial black holes is presented in Fig. 5.3. However some of the constraints in Fig. 5.3 have been updated recently. On the one hand, the OGLE survey data have been updated. On the other hand, −13 the constraints on the very low-mass primordial black holes (below ∼ 10 M ), notably from the femtolensing with gamma-ray bursts is argued not to be reliable by Katz et al. [57], if the non-point like nature of the source and the lens is taken into account. The concerned mass ranges are shown in Fig. 5.4.

49 Figure 5.3: Graphics regrouping the diﬀerent methods of constraints on the fraction of primordial black holes in the dark matter depending on the mass of the primordial black holes (expressed in solar mass). All the curves should be seen as upper limits. WD, NS and WD stands for white dwarf, neutron star and wide binaries (binary systems with a large stellar separation) constraints respectively. The constraints derived from the study of gravitational lensing correspond to the blue lines for the general microlensing, and to black lines for the millilensing and femtolensing (the name comes from the size of the angular separation in arcsec). The red curves are the constraints derived from accretion considerations from X-rays and radio waves, and from the study of the cosmis microwave background. DF stands for dynamical friction and UFD for ultra-faint dwarf galaxy. The green curve associated to Eri-II corresponds to a particular dwarf galaxy, Eridanus II [58].

5.3 The future constraints

While many different ways to put constraints already exist, in the future one could further extend these constraints. Another example linked to gravitational lensing is the possible use in the future of the fast radio bursts [60], phenomena lasting a few milliseconds. Using a primordial black hole again as a lens (or to be more general any MACHO object), one could detect, if the mass of the black hole is sufficient, a time delay between the images of the burst. In the thin-lens approximation, the time delay ∆t between the two images can be expressed as follows [60]: " √ √ !# u u2 + 4 u2 + 4 + u ∆t = 4ML(1 + zL) + log √ (5.23) 2 u2 + 4 − u where zL is the redshift of the primordial black hole. Two conditions need to be fulfilled to have a sufficiently strong lensing:

• The observed time delay must be greater than some reference time delay (greater than the capability of the instruments at least). This condition ∆tobs > ∆tref imposes u > umin where umin(ML) is some lower bound corresponding to ∆tref.

50 • The diﬀerence of brightness between the two images must not be too large so both can be monitored. This condition Robs < Rref imposes u < umax where umax is some upper bound corresponding to Rref.

Figure 5.4: Top: Updated constraints on the fraction of primordial black holes in the dark matter from the OGLE 5-year survey. The red area is the region of 95 % confidence level in the case of no microlensing events by primordial black holes detected by the OGLE survey [59]. Bottom: Projections of the future constraints (in purple) depending on the number of correctly identified gamma-ray bursts and taking into account the extended size of the source. as is the actual transverse size of the emission region in the case of an extended source and corresponds to an upper limit under which constraints can be set. The cut-off for the HSC constraints is due to the limit of the validity of the geometrical approximation used in the HSC case. EGγ corresponds to the constraint from the extragalactic photons from PHH evaporation [57].

51 Like previously, one important thing left is the determination of the optical depth τ. Since u cannot take all values because of the above constraints, the cross-section won’t be a disk anymore but an annulus which inferior and superior radii are given by umin and umax respectively. Let us also introduce the comoving distance χ deﬁned as follows [44] and which can be linked to the angular diameter distance via the redshift (last equality):

Z r dr0 χ = a(t0)Θk(r) = a(t0) 02 = dP (1 + zP ) (5.24) 0 1 − kr where r corresponds here to the radial coordinate of Eq. (5.1) and zP is the redshift corresponding to dP . With these elements, one can formulate in another way Eq. (5.9) by integrating on the comoving distance:

Z dS Z dS 2 2 2 3 2 2 2 τ = nPBHπRE umax − umin d(dL) = NPBH (1 + zL) πRE umax − umin d(dL) (5.25) 0 0 Z zS 3 2 2 2 dχ = NPBH (1 + zL) πRE umax − umin (5.26) 0 (1 + zL) Z zS 2 = (1 + zL) NPBHσanndχ (5.27) 0

3 where NPBH is the number density per unit comoving volume (justifying the (1 + zL) term) and σann is the annulus cross-section. Then, from Eq. (5.27), one integrates on the redshift and express the result as fPBH:

Z zS Z zS 2 ΩM,0 2 dzL τ = (1 + zL) NPBHσanndχ = (1 + zL) NPBHσann (5.28) 0 0 ΩM,0 H(zl) Z zS 2 3H (t0) 2 dzL = ΩM,0 (1 + zL) NPBHσann (5.29) 0 8πρDM,0 H(zL) Z zS 2 3ΩM,0 H (t0) 2 NPBH dLdLS 2 2 = (1 + zL) 4πMPBH umax − umin dzL (5.30) 8π 0 H(zL) ρDM,0 dS Z zS 2 3ΩM,0 H (t0) 2 NPBHMPBH dLdLS 2 2 = (1 + zL) umax − umin dzL (5.31) 2 0 H(zL) ρDM,0 dS Z zS 2 3ΩM,0 H (t0) 2 dLdLS 2 2 = fPBH (1 + zL) umax − umin dzL (5.32) 2 0 H(zL) dS where H(zL) is the Hubble parameter at the time corresponding to the redshift zL, ΩM,0 the current matter density parameter and ρDM,0 the current matter density. Finally, taking into account that there is a distribution NFRB(z) of fast radio bursts, the integrated optical depth τint is given by Z τint = τNFRB(z)dz (5.33)

Having a relation between fPBH and ML (through ymin), one can then put constraints from it, depending on the value of ∆tref as illustrated in Fig. 5.5.

52 Figure 5.5: Constraints imposed by the fast radio bursts on the fraction fPBH of compacts objects in the dark matter with respect to the mass of the compact object acting as a lens (in solar mass). Depending on the condition on the time delay ∆tref = ∆¯t selected, the upper limit will change. It is supposed there have been 10 000 fast radio bursts and none of them have been lensed. The constraints from the MACHO and EROS projects are also presented, along the constraint from the study of wide binary systems. Modiﬁed ﬁgure from [60].

53 54 Conclusion

The modern concept of black hole goes back to the creation of the theory of general relativity, being one of the most extreme known object in the universe. Following the development of the knowledge on relativity along the 20th century, different metrics have been discovered in an attempt to describe the space-time around the black holes, which depend only on the mass, angular momentum and electric charge of the black hole. The Schwarzschild metric, when the black hole doesn’t spin and is electrically neutral, is a simpler metric than the other ones. However, in reality the black holes spin, and it is more appropriate in general to use at least the Kerr metric in that case. By taking into account a non-zero angular momentum, different properties are different from a Schwarzschild black hole, or even appear solely because of it. The curvature singularity has a toroidal geometry (in the appropriate coordinates), but also the coordinate singularity (the horizon) is no longer unique and doesn’t coincide with the curvature singularity. Because of this a region appears, called the ergosphere, in which there cannot be a static observer (who seems at rest for an asymptotic distant second observer). Moreover, according to the weak cosmic censorship hypothesis, the spin of a black hole cannot exceed a certain limit, which is its mass (in natural units). Until now, this hypothesis has not been violated.

Also, the recent years have been rich in discoveries: on one hand the direct detection with interfer- ometric methods of gravitational waves by the LIGO-Virgo collaboration produced by the coalescence of black holes, and even neutron stars. The third run is already ongoing this year. On the other hand the first close-up image of a (supermassive) black hole, M87∗, has been revealed in April 2019, showing the light emitted by the accretion disk bent by the gravitational field of M87∗ and the shadow of the black hole, within which resides the event horizon. These recent feats revived the interest in the subject of black holes. Among other things, the search for the low-mass black holes is an important topic. If one looks at the different measured neutron star and black hole masses measured, there seems to be a gap between around 2 and 5 solar masses. The (non)discovery of black holes in this range could bring precious information. Indeed, neutron stars have a maximum mass, the Oppenheimer-Volkoff limit, due to the fact than the neutron degeneracy pressure is not enough to counterbalance the large mass (which is analogous to the Chandrasekhar limit for the white dwarfs). Beyond that, it would collapse into a black hole. Even if this limit is not precisely known because of the extreme conditions and the uncertainty on the equation of state, the current consensus puts the limit around 2-3 solar masses. For the black holes, the classical way of producing them is to consider what remains after the supernova explosion of a very massive star. However, black holes produced that way cannot have a solar mass since the star they originate from is already very massive. As such, if solar-mass (or lighter) black holes are discovered one day, one should consider other mechanisms to explain their origin. Two interesting possibilities are the creation of solar-mass black holes by accretion of dark matter onto neutron stars and the primordial black holes.

Dark matter is one of the most puzzling mysteries in astrophysics. As such, lots of diﬀerent investigations have been and are currently done to attempt to resolve this problem. One option is to consider the dark matter as particles with unusual properties compared to the ordinary baryonic matter. This consideration has lead to the investigation of the possibility of creating black holes from dark matter. One of the ways to achieve this is by using a neutron star as a seed to accumulate dark matter until their gravitational collapse into a black hole, eating what is left of the neutron star. The idea is that the dark matter particles accumulate into the neutron star, thermalizing with the medium and going within a cloud of some thermal radius.

55 Two thresholds have to be crossed: the number Ncoll of particles over which the gravitational collapse begins and the number Ncrit over which the black hole is created (similarly to the Chandrasekhar and the Oppenheimer-Volkoﬀ limits). The values of these numbers will depend on whether the dark matter is composed of fermions or bosons. In the ﬁrst case, what will act against the gravitational collapse is the Fermi pressure, while in the second case the constraint of the Fermi pressure doesn’t exist, but a Bose-Einstein condensate can form. Globally the apparition of a Bose-Einstein condensate of dark matter will lower the number of particles required to the formation of the black hole. Moreover, if one considers the bosonic dark matter in general, due to the absence of Fermi pressure, the number of particles required, for a same mass of the particles, will also be lower compared to a fermionic dark matter. Combining the estimates of the value of the interaction cross-section between the dark matter and the baryonic matter, and the population of neutron stars, one can then set constraints on this cross-section.

The other way is to consider that these low-mass black holes don’t come from the stellar evolution but originate from the very early Universe. To resolve some issues not addressed by the hot Big Bang model, the inﬂation models have been developed. These models state there has been an exponential expansion of the Universe during a brief moment in the early Universe. In some of these models, primordial black holes are created from the gravitational collapse of very dense regions. Providing they were not too small they should still exist to this day (otherwise they evaporated by Hawking radiation). Due to their origin, a great range of masses is possible a priori, and they could account for at least a fraction of the dark matter mass. Many diﬀerent methods have been used to put constraints on the fraction of primordial black holes in dark matter (to be more precise the fraction of massive compact objects): gravitational lensing, celestial objects (neutron stars, dwarf galaxies, globular clusters...) perturbations, accretion process and so on. All these methods allowed to get constraints on a wide range of masses from 10−16 to 109 solar masses as shown in section 5.2.3. The coming years and decades will certainly bring a lot of new information, with the upgrade of existing instruments (which increases the expected number of detected events for LIGO and Virgo for example) and the creation of new ones. The gravitational waves direct detection can lead to a new era of observation to explore domains which were not accessible before. Combining this multi-messenger astronomy with the research on the dark matter, it could potentially lead to very interesting discoveries concerning its nature, wheter it is composed of particles or of a set of compact objects like primordial black holes.

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