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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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Master thesis (research focus)
University of Liège, AGO department
Non-standard formation processes of low-massblack holes
KUMAR Shami Supervisor: Jean-René CudellMaster in Space Sciences Academic year 2018-2019
Contents
Introduction 11 The detection of gravitational waves by LIGO and Virgo . . . . . . . . . . . . . . . . . 22 The observed gravitational waves and their progenitors . . . . . . . . . . . . . . . . . . 33 The first image of a black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 The dark matter mystery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 The tests of general relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1 The maximum mass of a white dwarf 91.1 Stellar evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 The pressure of a relativistic degenerate gas of fermions . . . . . . . . . . . . . . . . . 91.3 The pressure at the center of a star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 The Chandrasekhar mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 The maximum mass of a neutron star 132.1 The maximum mass of a neutron star and the Oppenheimer-Volkoff limit . . . . . . . 132.2 Modern computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Observational determination of the mass of a neutron star . . . . . . . . . . . . . . . . 17
2.3.1 X-ray binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.2 Binaries with two neutron stars and relativistic effects . . . . . . . . . . . . . . 19
2.4 Measured neutron star masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 The Kerr metric properties 233.1 The Schwarzschild metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 The Kerr metric properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.1 The curvature singularity geometry . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.2 The event horizons and the ergosphere . . . . . . . . . . . . . . . . . . . . . . . 273.2.3 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 334.1 The accretion of fermionic dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1.1 The number of accreted dark matter particles . . . . . . . . . . . . . . . . . . . 344.1.2 Thermalization, start of the collapse and value of Ncoll . . . . . . . . . . . . . . 354.1.3 Formation of the black hole and value of Ncrit . . . . . . . . . . . . . . . . . . . 36
4.2 The accretion of bosonic dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2.1 The number of accreted dark matter particles . . . . . . . . . . . . . . . . . . . 384.2.2 Start of the collapse and formation of the black hole . . . . . . . . . . . . . . . 404.2.3 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 435.1 The inflation models and the origin of the primordial black holes . . . . . . . . . . . . 435.2 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 . . . . . . . . . . . . . . 455.2.1.2 The magnification due to the microlensing . . . . . . . . . . . . . . . . 465.2.1.3 Studies of gravitational lensing . . . . . . . . . . . . . . . . . . . . . . 46
5.2.2 The impact of the gravitational field of a (primordial) black hole on some celestialobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2.3 Other sources of constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 The future constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Conclusion 55
ii
Introduction
Black holes are the densest objects known in the Universe. The classical theory for their formationis that they result from the death of a very massive star during a supernova explosion. The coreof the star is then so dense that the pressure from degenerate matter is not enough to prevent thegravitational collapse. The topic of the black holes regained interest in the last years following thedetection of gravitational waves due to black holes merging (and one neutron stars merging event) bythe LIGO-Virgo collaboration and more recently the first close-up image of a black hole, namely thesupermassive black hole at the centre of the M87 galaxy. It also motivated again the investigations onthe black hole formation processes, from a better understanding of the classical stellar evolution theoryto less traditional ways that could explain some anomalies and less understood points. The modernpoint of view of what a black hole is comes from the theory of general relativity of A. Einstein. In thistheory, where the gravity is no longer viewed as an instantaneous remote force but as a deformationof a structure called space-time, some properties and observations not predicted by the Newtoniantheory 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 theelement of length in a quadri-dimensionnal space-time and in a given set of coordinates, dependingon the energy content of the medium considered. This metric can be expressed as a function of themetric tensor elements, which appear in the fundamental equations of general relativity: the Einsteinequations. The black holes, while being an extreme environment, are also paradoxically "simple" tocharacterise. Indeed three quantities are sufficient to categorise a black hole: its mass M , its angularmomentum J and its electric charge Q. Following that, the metrics describing the environment arounda black hole are limited as briefly resumed in Table 1.
Q J Metric= 0 = 0 Schwarzschild= 0 6= 0 Kerr6= 0 = 0 Reissner-Nordström6= 0 6= 0 Kerr-Newman
Table 1: Table summarising the different metrics used to describe the space-time around a black holeaccording 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 ofgeneral relativity by Albert Einstein in 1915 [1], and the Reissner-Nordström solution in the followingyears [2], about fifty years passed before R. Kerr found an analytical solution for the case of a rotationnon-electrically charged black hole in 1963 [3], followed soon after by the more general Kerr-Newmansolution 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 thereforetheir (non)detection can confirm or infirm it. There was already indirect evidence of the existence ofthe 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 gravitationalwaves. However there still wasn’t a direct proof of their existence until September 14th 2015, when thecollaborations LIGO (Laser Interferometer Gravitational-wave Observatory) and Virgo succeeded andmeasured 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 differentobservatories below.
Figure 1: Aerial pictures of the different gravitational waves observatories. Left: LIGO HanfordObservatory 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 armbefore 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) sothat the interference pattern will not be totally destructive as previously. From that, it is possible toacquire information on the gravitational waves and on their sources. The precision needed to measuresuch deformations is extremely high, thus lots of different environmental issues must be very wellcontrolled or compensated: thermal dilation, tidal effects, ground vibrations and so on. To be surethat the signal received by a detector is indeed a gravitational wave and not something else, severalobservatories are used. Since there will be a slight time delay on the reception of the wave between theobservatories (about 10 milliseconds between the two LIGO facilities) and the waves propagate at thespeed of light, one checks that the same perturbation has also been felt by the other detector, whichshows that one has detected a gravitational wave. Moreover, having several detectors helps to narrowdown 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 armof 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 beamsplitter region. The resolution condition is also achieved by doing these multiple reflections. By thismethod, the effective power of the laser at the end can be equal to about 750 kW, compared to theinitial 200 W of the laser source.
2
Figure 2: Schematic and simplified representation of the principle of the LIGO interferometer. A lasersource sends a light which is separated in two by a beam splitter mirror. At the end of each arm, amirror reflects back the laser to another mirror in this arm, back and forth to increase the effectivelength travelled by the light and the power of the light (also increased by a power-recycling mirrorbetween the source and the beam splitter). Finally, the laser is sent to the detector. A variation in theinterference pattern can potentially indicate the passage of a gravitational wave which has extremelyslightly 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 usedto observe the Universe and its components are mainly the photons, and in some cases other elementaryparticles such as neutrinos or muons. However these methods have some drawbacks: photons can beeasily absorbed by a medium depending on the wavelength ones studies, and neutrinos interact verylittle with matter. Gravitational waves are not absorbed and because of this, they could help us to lookat places otherwise very hard or even impossible to reach. One of the examples would be the very earlyUniverse. While we cannot go back further than about 380 000 years after the Big Bang (the momentwhen the Universe was sufficiently "transparent" to let the photons freely propagate), correspondingto the cosmic microwave background emission, the most ancient light we can observe, gravitationalwaves are not so restricted and thus we could gain access to information beyond the cosmic microwavebackground 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 mergingof 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 theobjects which, in the case of two black holes, creates another black hole heavier than the individualprogenitors.
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 Augustus25th 2017). During the course of these two runs, and after analysis and treatment of the data, 11gravitational waves events have been confirmed. Some of the parameters linked to each of these eventsare shown in Table 2.
3
Event m1(M) m2(M) Mf (M) Erad (M) z RunGW150914 35.6+4.8
−3.0 30.6+3.0−4.4 63.1+3.3
−3.0 3.1+0.4−0.4 0.09+0.03
−0.03 O1GW151012 23.3+14.0
−5.5 13.6+4.1−4.8 35.7+9.9
−3.8 1.5+0.5−0.5 0.21+0.09
−0.09 O1GW151226 13.7+8.8
−3.2 7.7+2.2−2.6 20.5+6.4
−1.5 1.0+0.1−0.2 0.09+0.04
−0.04 O1GW170104 31.0+7.2
−5.6 20.1+4.9−4.5 49.1+5.2
−3.9 2.2+0.5−0.5 0.19+0.07
−0.08 O2GW170608 10.9+5.3
−1.7 7.6+1.3−2.1 17.8+3.2
−0.7 0.9+0.05−0.1 0.07+0.02
−0.02 O2GW170729 50.6+16.6
−10.2 34.3+9.1−10.1 80.3+14.6
−10.2 4.8+1.7−1.7 0.48+0.19
−0.20 O2GW170809 35.2+8.3
−6.0 23.8+5.2−5.1 56.4+5.2
−3.7 2.7+0.6−0.6 0.20+0.05
−0.07 O2GW170814 30.7−5.7
−3.0 25.3+2.9−4.1 53.4+3.2
−2.4 2.7+0.4−0.3 0.12+0.03
−0.04 O2GW170817 1.46+0.12
−0.10 1.27+0.09−0.09 ≤ 2.8 ≥ 0.04 0.01+0.00
−0.00 O2GW170818 35.5+7.5
−4.7 26.8+4.3−5.2 59.8+4.8
−3.8 2.7+0.5−0.5 0.20+0.07
−0.07 O2GW170823 39.6+10.0
−6.6 29.4+6.3−7.1 65.6+9.4
−6.6 3.3+0.9−0.8 0.34+0.13
−0.14 O2
Table 2: Table listing some of the properties of the compact objects involved in the different gravita-tional wave events detected (listed by chronological order of detection) by the LIGO-Virgo collaborationbetween 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 componentsand z is the redshift. All masses are expressed in solar mass. Erad is the total radiated energy bygravitational 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 thelow masses of the components compared to the other events) in the first two runs. Moreover, since thisis not a black hole merger, there should be a electromagnetic signal in addition to the gravitationalone, which has been confirmed by Fermi, within a two seconds delay. The object resulting from themerging of the neutron stars is also interesting, since it could be a high-mass neutron star or a verylow-mass black hole. As a last remark, the events GW170729, GW170809, GW170818 and GW170823have not been discovered initially from the data, but they were validated as gravitational wave eventsafter a second analysis of the data.
As said previously, with several detectors it is possible to localise to some extent the position ofthe gravitational wave events in the sky. The results of theses localisations are given in Fig. 3. Theconfidence region of the position is much smaller when the three detectors (LIGO and Virgo) detectthe same event. It was the case for GW170818, as one can see the localisation is much better than forthe other events.
Other runs of observations of gravitational wave events are planned in the future. Thanks to theupgrade in the technologies and facilities of the different observatories and a better understanding ofthe phenomena, the expected number of events is higher than for the first two runs. The O3 run atthe date of this writing has already begun since the 1st April 2019 and should continue for about 12months. 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 thesupermassive black hole at the center of the M87 galaxy (named hereafter M87∗), an elliptical galaxyat about 55 millions light-years from us, while the processed images have been released to the publicon the 10th of April 2019. This is a first in history, and to achieve such a feat, a large collaborationbetween different institutions and telescopes was necessary. M87∗ has been chosen due to the factthat 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 itrequired a very high resolution. To achieve this, a network of eight telescopes around the world workedtogether to simulate an Earth-size radio telescope (at a wavelength of 1.3 mm), with a resolution ofabout 20 micro-arcseconds. Following this study, the mass of M87∗ has been estimated to be 6.5 ± 0.7billions solar masses [10].
4
Figure 3: Positions in the sky of the sources of the gravitational wave events of the O1 and O2 runswith the 50 % and 90 % confidence regions. The upper image corresponds to the events of the secondrun for which an alert to electromagnetic detectors have been sent, while the lower image correspondsto all the other events of the first two runs of the LIGO-Virgo collaboration. Both images are inequatorial 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 holeand coming originally from the accretion disk emission around M87∗. However the central blackenedregion in the center is not just the black hole itself, with its event horizon delimiting the darker innerregion and the bright ring, but a larger region called the shadow of the black hole. The actual eventhorizon is about 2.5 times smaller. Another striking feature of the images is the asymmetric intensityof the ring, with a clear brighter region to the south-est. This effect is due to the relativistic beamingof 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 clusteron 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 ofthe ring is due to the relativistic beaming coming from the emission of the matter going towards theEarth. 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 possibleexistence is notably supported when one analyses the rotation curves of galaxies. Even though therotational speed should globally decrease when far from the center, it is not what is observed. Insteadthe speed is more or less constant when sufficiently far. These observations, as well as the study ofgalaxy clusters and their collisions, lead to the introduction of some missing mass which could explainthese curves and which is dominant compared to the luminous matter we can see. However, evenif this hypothesis has been made in the last century, the nature of dark matter is not known. Lotsof 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 newparticles, which don’t interact a lot with the ordinary matter. Another explanation could be that wemissed some celestial objects during the observations, for example black holes, planets or in general allsorts of compact objects. However several studies have already been performed on this subject, andthese compact objects cannot make the majority of the dark matter for a large range of masses (from afraction of solar mass to several tenth of solar masses). Finally, one straightforward idea in spirit wouldbe that the theory itself is wrong. Other models of gravity have been developed (like the MOdifiedNewtonian 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 leastvia gravity. Moreover, if dark matter is indeed composed of black holes (or even primordial black holesif they originated from the very early Universe), one could use the gravitational waves to study theseblack holes.
5 The tests of general relativity
Since the gravitational waves are a consequence of general relativity, they can be used to test thistheory and to see if there are any deviation. If one supposes that the origin of the gravitational wavesis indeed the coalescence of two black holes, one can use general relativity to infer parameters such asthe final mass and spin from the different parts of the signal and see if they overlap or not. Abbott etal. [11] studied GW150914 to test any deviation from theory. To do so, they inferred the mass and thespin 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 withtheory. Another way to see the agreement is to look at the difference between the two models (inspiraland post-inspiral) as illustrated in Fig 6.
This Master thesis investigates different mechanisms of black hole formation. First, before beginningthe subject itself, a review of the maximum mass of less compact objects, white dwarfs and neutronstars, is given. In particular, the mass limit on neutron stars, called the Oppenheimer-Volkoff mass, isimportant 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. Mostnotably, the two event horizons and the notion of ergosphere will be investigated.
Then different mechanisms other than the classical stellar evolution are investigated, notably theaccretion 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 oflight in vacuum, ~ the reduced Planck constant, G the gravitational constant and kb the Boltzmannconstant.
6
Figure 5: 90 % confidence regions of the final mass (in solar mass) and dimensionless spin of GW150914deduced from the inspiral phase and the post-inspiral phase. In agreement with general relativity, thetwo 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 andfrom the post-inspiral phase) for the dimensionless spin of the final object and od the relative differencefor 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 amass below about 1.4 solar masses (called the Chandrasekhar limit) the core of the star will becomea white dwarf, the electron degeneracy pressure being sufficient to compensate its weight and sustainhydrostatic equilibrium. For more massive stars, the electron degeneracy pressure is not enough tobalance the gravitational contraction and the star will undergo a supernova event. The fate of the corewill 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 forceand it will become a neutron star. But if it’s too massive this neutron degeneracy pressure won’t beenough and it will collapse into a black hole.
The Chandrasekhar limit can be estimated from a theoretical and analytical point of view. Thefollowing 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 rel-ativistic is taken into account. In that case, the relativistic expression of the energy E has to betaken:
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:
U =
∫ +∞
0E f(E) g(p)dp (1.2)
where f(E) in the case of the electrons is the Fermi-Dirac distribution and g(p)dp the density of statesexpressed as
g(p)dp = V( pπ
)2dp (1.3)
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 thechemical 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 absolutevalue to the derivative of the internal energy with respect to volume, considering all the other propertiesconstant, in particular the number of particles for each state. Thus, using Eq. (1.2), one has:
P = −∂U∂V
= −∫ +∞
0
dE
dp
dp
dVf(E) g(p)dp (1.5)
The term dE/dp can be directly computed from Eq. (1.1) and since the momentum is proportionalto the wave vector, and the latter is itself proportional to the characteristic length L−1 = V −
13 , we can
get an expression for the factor dp/dV .
dp
dV=d(αV −
13 )
dV=−α3V
43
=−p3V
(1.6)
with α the proportionality factor.Finally Eq. (1.5) becomes
P =1
3V
∫ +∞
0
p2
Ef(E) g(p)dp (1.7)
If we consider that the gas of electrons is fully degenerate, there is no occupied state beyond theFermi momentum pF and the Fermi-Dirac distribution is taken equal to 1 as Eq. (1.7) is integratedup to the Fermi momentum. Lastly, by defining a new variable x = p
m , Eq. (1.7) becomes
P =m4
3π2
∫ xF
0
x4
(1 + x2)12
dx (1.8)
where xF is the dimensionless Fermi pressure, given by
xF =pFm
=1
m
(3π2ne
) 13 (1.9)
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
N =
∫ pF
0g(p)dp =
1
3π2V p3
F ⇐⇒ pF =(3π2ne
) 13 (1.10)
Eq. (1.8) is then equal to
P = K n43e I(xF ) (1.11)
The different terms of Eq. (1.11) are given by
K =π
2
(3
8π
) 13
(1.12)
ne =Ye ρcmH
(1.13)
I(x) =3
2x4
[x(1 + x2)
12
(2x2
3− 1
)+ ln
[x+ (1 + x2)
12
]](1.14)
where Ye is the number of electrons per nucleon, ρc the density in the core and mH the mass of ahydrogen 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 alsoconsider that the star is in hydrostatic equilibrium. Then, the pressure gradient can be expressed as
dP
dr= −m(r)ρ(r)
r2(1.15)
where ρ(r) is the density and m(r) the mass inside a sphere of radius r with its radial variation givenby
dm
dr= 4πr2ρ(r) (1.16)
To find an expression for the pressure inside this star as a function of the radial distance r, wewill use the Clayton model [13]. The idea of the model is to model the pressure gradient by a simpleexpression depending only on r. To do so, an exponential factor is used:
dP
dr= −4πρ2
c
3e−r
2/a2(1.17)
where ρc is the density at the center and a a characteristic length small compared to the radius of thestar 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:
P (r) =2π
3ρ2ca
2[e−r
2/a2 − e−R2/a2]
(1.18)
Using Eq. (1.18), one gets the pressure at the center (by neglecting the second exponential):
Pc = P (r = 0) =2π
3ρ2ca
2 (1.19)
Also, the evolution of the mass m(r) is given by
m(r)dm = −4πr4dP ⇐⇒ m2(r)
2= −4π
∫ r
0r′4dP
dr′dr
′(1.20)
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:
a '(
3M
4πρc√
6
) 13
(1.21)
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:
Pc '( π
36
) 13M
23 ρ
43c (1.22)
1.4 The Chandrasekhar mass
Eq. (1.22) corresponds roughly to the pressure needed to support the star. Finally, if we consider thewhite dwarf as a star in which the pressure comes from the degenerate gas of electrons, we can equateEqs. (1.11) and (1.22).
π
2
(3
8π
) 13(YeρcmH
) 43
I(xF ) '( π
36
) 13M
23 ρ
43c (1.23)
11
Finally, by rearranging the terms in Eq. (1.23), we can obtain an approximation of the mass of thewhite dwarf where the pressure needed comes from the degeneracy pressure of the electrons:
M '(
6π
1281/3
) 32 1
m2Hπ
Y 2e I(xF )
32 = 4.3 Y 2
e I(xF )32 M (1.24)
where M = 1.98 1030 kg is the mass of the Sun
Following stellar evolution theory, a white dwarf is mainly composed of 12C and 16O. Thus typicallyYe ' 1/2. As for the value of I(xF ), one needs to use the value of Eq. (1.9) in Eq. (1.14). Whenplotting the central density versus the mass of the white dwarf, one can see that the density of thewhite dwarf tends to infinity when M tends to 4.3 Y 2
e M = 1.1 M, meaning that other aspects haveto 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 insteadof Eq. (1.18) a polytropic model is used [14]. In this kind of model, the relation between the pressureand the density ρ is defined by the following relation:
P = Cρ(1+ 1n) (1.25)
where C is a constant and n a strictly positive real number called the polytropic index, their valuesdepending 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 getrid of the pressure, we can get a second order differential equation for the density:
1
r2
d
dr
[r2
ρ
d
dr
(Cρ(1+ 1
n))]
= −4πρ (1.26)
If boundary conditions are used, Eq. (1.26) can be solved numerically to get the density profile.For a degenerate gas of electrons n = 3 and so P (r) ∝ ρ(r)
43 . In that case the numerical factor in
Eq. (1.22) is no longer equal to (π/36)1/3 ' 0.44 but to ' 0.36. Then the Chandrasekhar mass MCh
is given by Eq. (1.27) (considering Ye ' 1/2).
MCh = 5.8 Y 2e M ' 1.45 M (1.27)
Finally, it is important to keep in mind that these models are still simplified compared to reality. Thechemical composition is not rigorously uniform and one also needs to know the density profile, whichcan 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 Chandrasekharlimit for white dwarfs, we can go further and try to estimate what would be the limiting mass for aneutron star, beyond which the mass of the neutron star would be too large for the degeneracy pressureof 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
eλ =
(1− 2m
r
)−1
(2.2)
eν = 1− 2m
r(2.3)
where m is the total mass of the neutron star from the point of view of a distant observer, andif the object doesn’t rotate too fast and is electrically neutral. By supposing no mass motion, wecan rewrite the Einstein equations (2.4) inside the neutron star, which requires the expression of theenergy-momentum tensor Tµν and the Einstein tensor Gµν .
Gµν + Λgµν = 8πTµν ⇐⇒ Gµν + Λδµν = 8πTµν (2.4)
where Λ is the cosmological constant, δµν the Kronecker symbol and gµν the element (µ, ν) of the metrictensor.
An important simplification is the fact that we consider the case of a perfect fluid. In that situationthe energy-momentum tensor simplifies a lot and is diagonal such as one has
(Tµν ) = 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 computethe diagonal elements of the Einstein tensor. This tensor is defined as follows:
Gµν = Rµν −R
2gµν ⇐⇒ Gµν = Rµν −
R
2δµν (2.6)
where Rµν is the Ricci tensor and R the scalar curvature, itself defined as R = Rµµ. However the Riccitensor is derived from the Riemann tensor Rαβγδ since Rµν = Rρµρν , which can itself be expressed infunction of the Christoffel symbols Γρσξ:
13
Rαβγδ = Γαβδ,γ + ΓαγξΓξβδ − Γαβγ,δ − ΓαδξΓ
ξβγ (2.7)
where the comma corresponds to the ordinary derivative.Finally, the Christoffel symbols can themselves be expressed in function of the metric components
which are known thanks to Eq. (2.1) since ds2 = gµνdxµdxν :
Γδβµ =gδα
2(gαβ,µ + gµα,β − gβµ,α) (2.8)
The idea is then the following: computing the Christoffel symbols from the metric components, thenthe Riemann tensor then the Ricci tensor and finally the scalar curvature, all this to get the expressionof the Einstein tensor. In the following, to avoid confusion, numbers will be attributed to the differentvariables 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 factthat they are symmetric on the two lower indices, that no component of the metric depends explicitlyon t and φ and that the metric is diagonal, it reduces drastically the number of non-zero symbols tobe computed, which are the following:
Γ001 =
g00
2g00,1 =
1
2
dν
dr; Γ1
00 = −g11
2g00,1 =
e(ν−λ)
2
dν
dr; Γ1
11 =g11
2g11,1 =
1
2
dλ
dr
Γ122 =
g11
2g22,1 = −re−λ ; Γ1
33 = −g11
2g33,1 = −r sin2(θ)e−λ ; Γ2
12 =g22
2g22,1 =
1
r
Γ233 = −g
22
2g33,2 = − sin(θ) cos(θ) ; Γ3
13 =g33
2g33,1 =
1
r; Γ3
23 =g33
2g33,2 = cot(θ)
Using Eq. (2.7), the different Ricci tensor components can be computed:
R00 = R0000 +R1
010 +R2020 +R3
030
= 0 +e(ν−λ)
2
[d2ν
dr2+
1
2
(dν
dr
)2
− 1
2
dν
dr
dλ
dr
]+e(ν−λ)
2r
dν
dr+e(ν−λ)
2r
dν
dr
=e(ν−λ)
2
[d2ν
dr2+
1
2
(dν
dr
)2
− 1
2
dν
dr
dλ
dr+
2
r
dν
dr
]
R11 = R0101 +R1
111 +R2121 +R3
131
=
[−1
2
d2ν
dr2+
1
4
dν
dr
dλ
dr− 1
4
(dν
dr
)2]
+ 0 +1
2r
dλ
dr+
1
2r
dλ
dr
= −1
2
d2ν
dr2+
1
4
dν
dr
dλ
dr− 1
4
(dν
dr
)2
+1
r
dλ
dr
R22 = R0202 +R1
212 +R2222 +R3
232
= −r2
dν
dre−λ +
r
2
dλ
dre−λ + 0 +
[1− e−λ
]= 1− e−λ +
r
2
(dλ
dr− dν
dr
)e−λ
R33 = R0303 +R1
313 +R2323 +R3
333
= −1
2
dν
drr sin2(θ)e−λ +
r
2sin2(θ)e−λ
dλ
dr+ sin2(θ)
(1− e−λ
)+ 0
= sin2(θ)
[1− e−λ +
r
2
(dλ
dr− dν
dr
)e−λ]
14
Then, using its definition, the scalar curvature can be obtained from the Ricci tensor components:
R = Rµµ = R00 +R1
1 +R22 +R3
3 = g00R00 + g11R11 + g22R22 + g33R33
=e−λ
2
[d2ν
dr2+
1
2
(dν
dr
)2
− 1
2
dν
dr
dλ
dr+
2
r
dν
dr
]− e−λ
[−1
2
d2ν
dr2+
1
4
dν
dr
dλ
dr− 1
4
(dν
dr
)2
+1
r
dλ
dr
]
− 1
r2
[1− e−λ +
r
2
(dλ
dr− dν
dr
)e−λ]− 1
r2
[1− e−λ +
r
2
(dλ
dr− dν
dr
)e−λ]
= e−λ
[d2ν
dr2+
1
2
(dν
dr
)2
− 1
2
dν
dr
dλ
dr+
2
r
(dν
dr− dλ
dr
)+
2
r2
]− 2
r2
After that, the Einstein tensor components can be determined:
G00 = R0
0 −R
2=
1
r2+ e−λ
(1
r
dλ
dr− 1
r2
)
G11 = R1
1 −R
2=
1
r2− e−λ
(1
r
dν
dr+
1
r2
)
G22 = R2
2 −R
2= −e−λ
[1
2
d2ν
dr2+
1
4
(dν
dr
)2
− 1
4
dν
dr
dλ
dr+
1
2r
(dν
dr− dλ
dr
)]= G3
3
Finally, we can write the Einstein equations (taking Λ = 0):
G00 = 8πT 0
0 ⇐⇒ e−λ(
1
r
dλ
dr− 1
r2
)+
1
r2= 8πρ (2.9)
G11 = 8πT 1
1 ⇐⇒ e−λ(
1
r
dν
dr+
1
r2
)− 1
r2= 8πp (2.10)
G22 = 8πT 2
2 ⇐⇒ e−λ
[1
2
d2ν
dr2+
1
4
(dν
dr
)2
− 1
4
dν
dr
dλ
dr+
1
2r
(dν
dr− dλ
dr
)]= 8πp (2.11)
The Einstein equation for G33 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 newrelation:
8π(ρ+ p) =e−λ
r
(dλ
dr+dν
dr
)(2.12)
Also let us derive Eq. (2.10) with respect to r:
8πdp
dr= −e−λdλ
dr
(1
r
dν
dr+
1
r2
)+ e−λ
(− 1
r2
dν
dr+
1
r
d2ν
dr2− 2
r3
)+
2
r3(2.13)
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
dr= −
(p+ ρ
2
)dν
dr(2.14)
15
Now if we consider the specific case of a cold Fermi gas of one species, a parametric expression forthe equation of state can be obtained [16]:
ρ = K(sinh(t)− t) (2.15)
p =K
3
(sinh(t)− 8 sinh
(t
2
)+ sinh(3t)
)(2.16)
with the parameters K and t given by:
K =µ4
0
32π2(2.17)
t = 4 log
pFµ0
+
[1 +
(pFµ0
)2]1/2
(2.18)
where µ0 is the rest mass of the particles.Let us define a variable u as follows:
u =r(1− e−λ)
2(2.19)
Using Eq. (2.14) and this variable u, Eqs. (2.9) and (2.10) become respectively:
du
dr= 4πρr2 (2.20)
dp
dr= − p+ ρ
r(r − 2u)
[4πpr3 + u
](2.21)
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
dr= r2 (sinh(t)− t) (2.22)
dt
dr= − 4
r(r − 2u)
sinh(t)− 2 sinh(t/2)
cosh(t)− 4 cosh(t/2) + 3
[r3
3(sinh(t) + 8 sinh(t/2) + 3t) + u
](2.23)
Finally, considering only neutrons, they integrated the Eqs. (2.22) and (2.23) from different valuesof 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 totalmass m:
ub = u(r = rb) =rb(1− e−λ(rb))
2= m (2.24)
By looking at the evolution of the core mass m with t0, one can see the presence of a maximum fort0 ∼ 3, corresponding to a mass m ∼ 0.75 solar mass, beyond which there is no static solutions. Thisvalue 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 difficultto estimate and evaluate, due to the very high density and its nature. Looking back at the originalcomputation of the mass limit of neutron stars by Oppenheimer and Volkoff, one can see several strongapproximations. Mainly, they considered only a cold gas of neutrons (without any other species). Inaddition, to rewrite the Einstein equations, the energy-momentum tensor is the simple one of a perfectfluid.
The structure of a neutron star is more complicated than just being composed of neutrons [17]. Inthe outer layers, there is still a sizeable fraction of protons and electrons. It is while going deeper inthe neutron star that the neutron fraction is more important. Moreover, near the center the density isextremely high and becomes bigger than the nuclear density. In these conditions it is very difficult toknow 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 temperatureare considered. Bombaci (1995) [18] for example distinguishes "conventional" equations of states (whereall negative charges are carried only by leptons) and "exotic" equations of state (in which negativecharges can also be carried by hadrons). By taking into account more complex hypotheses, the mainconsequence of these different equations is the fact that the actual value of the maximum mass of aneutron 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 tothe constraints that the companion object and the neutron star put on each other [19]. The precisemethods 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 systemand can be described in first approximation by classical Keplerian theory [21]. The position of thecompanion 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 planebetween 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)
rj sin(φj) = aj√
1− e2 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 correspondinghere to the center-of-mass. The line of nodes containing N is the intersection of the orbital plane andthe plane of the sky. P corresponds to the periastron and ω to its longitude, or its angle from the lineof nodes in the revolution direction. The orbital plane is considered to be the X-Y plane. Modifiedfrom [19].
Thus Eqs. (2.25) and (2.26) can be rewritten to depend explicitly on the eccentric anomaly:
xj = aj
[(cos(E)− e) cos(ω)−
√1− e2 sin(E) sin(ω)
](2.30)
yj = aj
[(cos(E)− e) sin(ω) +
√1− e2 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 possibleto evaluate some of them thanks to the observations:
• By measuring the orbital variability of the radiation of one of the components, we can determinethe 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), weobtain its expression in the center-of-mass reference frame:
vlj = sin(i)
[·rj sin(ω + φj) + rj
·φj cos(ω + φj)
](2.32)
where i is the inclination of the orbital plane compared to the line of sight. The dot correspondsto the time derivative.
Then we can use the first and second Kepler laws for elliptical orbits:
rj =aj(1− e2)
1 + e cos(φj)(2.33)
r2j
·φj =
√ajM(1− e2) (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)
Kj = sin(i)
√M
aj(1− e2)=
Ωb aj sin(i)√1− e2
(2.36)
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. Usingthe third Kepler law (2.37), we get Eq. (2.38) for the mass function (here for the companion ofmass M2).
Ω2b =
M
a3(2.37)
f2 =(M1 sin(i))3
M2= (a2 sin(i))3 Ω2
b (2.38)
where a = a1 + a2.
We have two relations with Eqs. (2.36) and (2.38). However there are four unknowns: the massesof 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 radialvelocities and by using Eq. (2.36) we have:
q =M1
M2=K2
K1(2.40)
• A fourth equation is still needed. Additional information can be obtained by the study of eclipsesin the binary for example.
However, it is important to keep in mind that in reality the Keplerian motions can be perturbedby 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 particularcase of two neutrons stars, gravitational waves can be emitted, coming from the loss of energy andangular momentum of the system. The relativistic effects also lead to long-term variations of theorbital 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. Themeasurements 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-raybinaries, 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 themasses of the neutron stars by measuring it.
·ω =
3Ω53b M
23
(1− e2)(2.41)
Another consequence of relativistic effects is the modification on the Doppler effect and the gravita-tional redshift, shifting the pulse arrival time. The delay ∆E due to these effects is called the Einsteindelay.
This delay can be expressed in function of the Keplerian parameters [22]. As previously said, thedelay of the pulse (of the pulsar of mass M1) is a combination of two effects, the gravitational redshiftand the Doppler effect. These effects can be expressed respectively as follows (taking only the termsin 1/c2):
γgrav = −M2
r12(2.42)
γDoppler =v2
1
2(2.43)
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, theproper time dτ of the pulsar can be expressed by
dτ ' dt [1 + γgrav − γDoppler] = dt
[1− M2
r12− M2
M
M2
r12
](2.44)
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 takingthe differential of Eq. (2.29). Doing this, integrating Eq. (2.44) and using (2.45), one can finally getthe expression of the Einstein delay:
∆E = τ − t = γ sin(E) (2.46)
where the parameter γ is given by
γ =eM2(M1 + 2M2)
ΩbaM(2.47)
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 evolutionmodels, 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 neutronstar. So, if we detect a black hole with a mass lower than this limit (and so lower than some of thedetected 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 detectedneutron stars are around 2 solar masses. In consequence any black hole below this limit should beconsidered. More comprehensive lists can be found in references [23], [24] and [25].
20
System Mass of the neutron star (in solar mass) StudyJ1614-2230 1.97 ± 0.04 Demorest et al. (2010) [26]J0348+0432 2.01 ± 0.04 Antoniadis et al. (2013) [27]
PSR B1516+02B 1.94+0.17−0.19 Freire (2008) [28]
PSR J2215+5135 2.27+0.17−0.15 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 isolatedobject curving the spacetime has a spherical symmetry, doesn’t spin and is electrically neutral. In thatcase the line element ds2 outside the object (so where the energy-momentum tensor is identically equalto zero) can be expressed as follows [1]:
ds2 = −(
1− 2m
r
)dt2 +
dr2(1− 2m
r
) + r2(dθ2 + sin2(θ)dφ2) (3.1)
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 apparentsingularity can be removed by choosing other sets of coordinates [30]. To construct these, we start bydefining the Regge-Wheeler tortoise coordinate r∗ as follows:
dr2∗ =
dr2(1− 2m
r
)2 ⇐⇒ r∗ = r + 2m ln( r
2m− 1)
(3.2)
With this new coordinate in a two-dimensions situation (dθ2 = dφ2 = 0), for a null line element (whichis the case of the light), i.e. when Eq. (3.1) is equal to zero, dr2
∗ = dt2. From there we can define twoother coordinates u = t − r∗ and v = t + r∗, which will lead to two systems of coordinates [30]. Theconstants u and v correspond respectively to outgoing and ingoing null geodesics, or in other wordsthe 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 ofcoordinate in Eq. (3.1), one obtains the Schwarzschild metric in the ingoing Eddington-Finkelsteincoordinates:
ds2 = −(
1− 2m
r
)(dv − dr∗)2 +
dr2(1− 2m
r
) + r2(dθ2 + sin2(θ)dφ2
)
= −(
1− 2m
r
)dv2 + 2dvdr + r2
(dθ2 + sin2(θ)dφ2
)(3.3)
(3.4)
One sees that there is no longer a singularity at r = RS . However there is still one at r = 0. As itwill be shown in the following, there is a similar singularity but with a different geometry in the Kerrmetric. The usefulness of the ingoing Eddington-Finkelstein coordinates for the description of blackholes 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 linescorrespond to the outgoing null geodesics (constant u) and the red lines to the ingoing ones (constantv). Constant u and v representing the coordinate lines of the null geodesics, they define the limits ofthe lightcones. As one can see by observing the figure, wherever something is within the Schwarzschildradius (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 forthe ingoing Eddington-Finkelstein coordinates, one can get the Schwarzschild metric in the outgoingEddington-Finkelstein coordinates:
ds2 =
(1− 2m
r
)du2 − 2dudr + r2
(dθ2 + sin2(θ)dφ2
)(3.5)
The reason these coordinates will not be considered in the following is shown in Fig. 3.2, whereone 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 linescorrespond to the outgoing null geodesics (constant u) and the red lines to the ingoing ones (constantv). Constant u and v representing the coordinate lines of the null geodesics, they define the limits ofthe lightcones. As one can see by observing the figure, wherever something is within the "Schwarzschildradius" (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 haveto 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 equa-tions and can be expressed in the ingoing Eddington-Finkelstein coordinates, which is the form usedin the original paper of Kerr:
ds2 = −(
1− 2mr
r2 + a2 cos2(θ)
)(dv − a sin2(θ)dφ)2
+ 2(dv − a sin2(θ)dφ)(dr − a sin2(θ)dφ)
+ (r2 + a2 cos2(θ))(dθ2 + sin2(θ)dφ2)
(3.6)
(3.7)
(3.8)
where a is the angular momentum of the black hole divided by its mass. One may notice that when ais 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 isbetter to express the Kerr metric in other coordinate systems [31]. Two of them are listed below andwill 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]:
∼t = v −
∫r2 + a2
∆dr (3.9)
∼φ = φ−
∫a
∆dr (3.10)
where ∆ = r2−2mr+a2. In these coordinates, the Kerr metric (3.8) can be expressed as follows:
ds2 = −(
1− 2mr
Σ
)d∼t
2− 4mar sin2(θ)
Σd∼td∼φ+
Σ
∆dr2 + Σdθ2
+
(r2 + a2 +
2ma2r sin2(θ)
Σ
)sin2(θ)d
∼φ
2
(3.11)
(3.12)
where Σ = r2 + a2 cos2(θ). These coordinates highlight the fact that the Kerr metric is station-ary and axisymmetric (since there is not direct dependence in t or φ). Moreover the Minkowskimetric for a flat spacetime can be recovered by making r tend to infinity. The Boyer-Lindquistcoordinates, having only one independent off-diagonal term, will be useful to discuss the eventhorizons 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 coor-dinates. It will also help to express the metric tensor in such a way that it will be easier to seethe 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. Thesecoordinates can be linked to the ingoing Eddington-Finkelstein coordinates by the followingtransformations:
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]:
ds2 = −dt2 + dx2 + dy2 + dz2 +H
[dt+
rx+ ay
a2 + r2dx+
ry − axa2 + r2
dy +z
rdz
]2
(3.16)
where H and r are defined respectively as
H =2mr3
r4 + a2z2(3.17)
x2 + y2 + z2 = r2 + a2
(1− z2
r2
)(3.18)
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 coordinatesappears, such that it can be rewritten in a more compact way [33]:
ds2 = (ηαβ +Hlαlβ) dxαdxβ (3.19)
where ηαβ is the Minkowski metric and lα is expressed as
(lα) =
(1,rx+ ay
r2 + a2,ry − axr2 + a2
,z
r
)(3.20)
One may verify that lα is a vector of null norm with respect to the Minkoswki metric and the Kerrmetric. As Eq. (3.19) shows, the metric element gαβ is the Minkowski metric element to whicha "perturbation" is added. The Kerr-Schild coordinates will help to highlight the geometry of thecurvature singularity of a Kerr black hole.
Such metrics can be used to describe a rotating (but still electrically neutral) black hole. Byincluding a non-zero angular momentum, different properties appear, some of which will be detailedin 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(3.12). While it means that r = 0 and θ = π
2 , it is easier to study the singularity in another set ofcoordinates called the Kerr-Schild coordinates.
Looking at Eq. (3.14) and developing the exponential, one can identify the real and imaginaryparts 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 =(r2 + a2
)sin2(θ)⇐⇒ x2 + y2
r2 + a2+z2
r2= 1 (3.23)
It is also possible to express the left side of Eq. (3.23) in another way, using again Eq. (3.15):
x2 + y2
sin2(θ)− z2
cos2(θ)=(r2 + a2
)− r2 ⇐⇒ x2 + y2
a2 sin2(θ)− z2
a2 cos2(θ)= 1 (3.24)
Since the singularity we are interested in corresponds to r = 0 and θ = π2 , 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 equationof an ellipsoid (more precisely an oblate spheroid) while Eq. (3.24) corresponds to the equation of ahyperboloid 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) andhyperboloids 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 additionalcondition for the singularity θ = π
2 will restrain the singularity to a radius a, making it an annularsingularity.
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 alsothe case of ∆ = r2− 2mr+ a2 = 0. However the situation is different. Σ = 0 is a curvature singularityin the sense that a change of the system of coordinates will not get rid of it. But for ∆ = 0, thisis the case. One can simply look at Eq. (3.8) to see that this singularity is not there in the ingoingEddington-Finkelstein coordinates. For this reason, ∆ = 0 is called a coordinate singularity. This isvery similar of the Schwarzschild case when r = RS , to which corresponds an event horizon. In theKerr case, ∆ = 0 will also corresponds to event horizons, but because this coordinate singularity is aquadratic 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:
∆ = r2 − 2mr + a2 = 0⇐⇒ r± = m±√m2 − a2 (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 tozero. 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 generaldefinition of the line element:
ds2 = gµνdxµdxν (3.26)
27
(gµν)
=
gtt 0 0 gtφ0 grr 0 00 0 gθθ 0gφt 0 0 gφφ
=
−(
1− 2mr
Σ
)0 0 −2mra
Σsin2(θ)
0Σ
∆0 0
0 0 Σ 0
−2mra
Σsin2(θ) 0 0 sin2(θ)
[r2 + a2 +
2mra2
Σsin2(θ)
]
Then after computing the inverse of the metric tensor one obtains [32]:
(gµν)
=
gtt 0 0 gtφ
0 grr 0 00 0 gθθ 0gφt 0 0 gφφ
=
−r2 + a2 +
2mra2
Σsin2(θ)
∆0 0 −2mra
Σ∆
0∆
Σ0 0
0 01
Σ0
−2mra
Σ∆0 0
1
Σ sin2(θ)− a2
Σ∆
A hypersurface which has for equation r = constant has by definition a normal (nµ) = (nt, nr, nθ, nφ) =(0, 1, 0, 0). The norm of the normal with respect to the metric gµν is given by gµνnµnν :
nαnα = gµνnµnν = grrn2
r =∆
Σ(3.27)
In consequence when ∆ = 0, which is the case on the event horizons (which are hypersurfaces ofconstant r in the sense that their equations don’t depend on the other coordinates as one can see bylooking at Eq. (3.25)), the norm of the normal to these hypersurfaces is equal to zero, confirming thenull nature of these hypersurfaces r = r±.
Since r+ and r− are both roots of ∆, it allows to determine if a given hypersurface with r constantis spacelike (timelike normal vector: nαnα < 0) or timelike (spacelike normal vector: nαnα > 0).
∆ = (r − r−)(r − r+) > 0 if r > r+ or r < r− → timelike hypersurface= 0 if r = r± → null hypersurface< 0 if r+ > r > r− → spacelike hypersurface
(3.28)(3.29)(3.30)
Finally to introduce the ergosphere, it is interesting to look at the gtt term from Eq. (3.12). In thecase of the Schwarzschild metric, it is when r = RS that this term changes sign. However the situationis different in the case of the Kerr metric. The following reasoning is taken from reference [32] whereadditional information can be found.
First one needs to look at the values nullifying the gtt term of the Kerr metric in the Boyer-Lindquistcoordinates:
gtt = −(
1− 2mr
Σ
)= 0⇐⇒ 2mr − Σ = 0
⇐⇒ r = RK± = m±√m2 − a2 cos2(θ)
(3.31)
(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, theevent horizons are between the zeroes of the gtt term such as RK+ > r+ > r− > RK−. The uppervalue RK+ corresponds to the outer boundary of what is called the ergosphere (see Fig. 3.4), the lattercorresponding to the region between RK+ and r+.
28
Figure 3.4: Schematic representation of the ergosphere compared to the outer event horizon. Thearrow corresponds to the spin axis [32].
To understand what happens to an observer in the ergosphere, it is necessary beforehand to introducethe 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 thisvector. 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 areindependent of some variable, it indicates directly a Killing vector. For example, since the Schwarzschildmetric is independent of t and φ, one can deduce two Killing vectors from that observation which arethe 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 ofthe Eq. (3.26)):
Xµ = gµνXν ⇐⇒ Xt = gttXt = 1 (3.36)
But since in the ergosphere gtt is positive, the nature of the Killing vector is spacelike:
XαXα = gµνX
µXν = gtt(Xt)2 = 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. Onthe hand, because of this definition, it means the tangent vector to the wordline of this observer mustbe a Killing vector, which by definition preserves the metric. On the other hand, the four-velocity ofan observer is always tangent to its wordline too. One can then deduce that the four-velocity is at leastproportional to a Killing vector, which in a general point of view is a combination of the two Killingvectors X1 = ∂t and X2 = ∂φ since the elements of the metric don’t depend on these coordinates as onecan see by looking at Eq. (3.12). One has then for the four-velocity a vector with only two non-zerocomponents since the Killing vectors have only the t and φ components which are different from zero:
uµ =(X1)µ + ω(X2)µ
||X1 + ωX2||= (ut, 0, 0, uφ) = ut(1, 0, 0, ω) (3.38)
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 stationaryobserver can be allowed. Since the four-velocity of an observer is a timelike vector of norm -1, one canget a quadratic inequality for ω:
uαuα = gµνu
µuν = gtt(ut)2 + gφφ(uφ)2 + 2gtφu
tuφ = (ut)2(gtt + ω2gφφ + 2ωgtφ
)< 0 (3.39)
To look at this inequality, let us solve the equation between brackets in Eq. (3.39):
gtt + ω2gφφ + 2ωgtφ = 0⇐⇒ ω = ω± =−gtφ ±
√g2tφ − gφφgtt
gφφ(3.40)
The discriminant needs to be greater than zero so one has from Eq. (3.12):
g2tφ − gφφgtt =
4m2a2r2
Σ2sin4(θ) +
(1− 2mr
Σ
)(a2 + r2 +
2mra2 sin2(θ)
Σ
)sin2(θ)
= (a2 + r2) sin2(θ) + 2mr sin2(θ)
(a2 sin2(θ)− (a2 + r2)
Σ
)= ∆ sin2(θ)
(3.41)
(3.42)
(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 regionwhere ∆ < 0 is between the two horizons of the Kerr black hole, so it is not possible for an observerto be stationary in that region. Moreover, Eq. (3.39) is valid if ∆ > 0 and ω+ > ω > ω−, whichis the case in the exterior of the outer horizon (and so in the ergosphere). Finally, if ∆ = 0 (on theouter horizon for example), the inequality cannot be solved. However, there is a unique solution forEq. (3.40) which corresponds to a null four-velocity.
Secondly, a static observer is an observer who seems at rest for an asymptotic distant anotherobserver. Thus it is also an observer for whom the tangent vector to its wordline is proportional tothe Killing vector (Xν) = ∂t. Let us suppose that this observer is located in the ergosphere. In thatcase, to be static, the tangent vector would need to be spacelike since the Killing vector (Xν) = ∂tthere is spacelike as shown in Eq. (3.37). But this is not possible since the observer should havea timelike wordline (and so a timelike tangent vector) to respect causality. One concludes that anobserver located in the ergosphere cannot be static.
Concerning the ergosphere, on its outer limit by definition gtt = 0 and ω− = 0. According towhether or not the observer is located outside or inside the ergosphere it will change the sign of ω−. Ifoutside, 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, whichmeans it is not possible to have a static observer inside the ergosphere as already explained before. Toput 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 ora stationary observer there.
• RK+ > r > r+: It is the ergosphere, there cannot be a static observer but it is possible to havea 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 isagain 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. Toanswer that, it is interesting to take a closer look at Eq. (3.25). As already said, if a = 0, one of thehorizons has for value r = RS and the other is rejected to r = 0. However m2 − a2 has to be positiveto have a solution to ∆ = 0. It means that the maximum value for the angular momentum per unitmass of the black hole is a = m. A black hole with such a high spin is called an extremal Kerr blackhole, 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 eventhorizon. However the curvature singularity Σ = 0 still exists. That would mean that when a > m, thesingularity of the Kerr black hole would be naked, i.e. not hidden by an event horizon. This would bein contradiction with the weak cosmic censorship hypothesis. This hypothesis states that there cannotexist a naked singularity and will always be hidden by an event horizon (with the exception of the BigBang singularity). It has been proposed to put aside the problem as what is inside an event horizon isnot causally connected to us.
Table 3.1 and Fig. 3.5 indicate the final dimensionless spins (the spins of the final objects) ofthe different gravitational waves events detected by the LIGO-Virgo collaboration. As these dataillustrates, all of them are in agreement with the weak cosmic censorship hypothesis.
Events Final dimensionless spin a/mGW150914 0.69+0.05
−0.04
GW151012 0.67+0.13−0.11
GW151226 0.74+0.07−0.05
GW170104 0.66+0.08−0.10
GW170608 0.69+0.04−0.04
GW170729 0.81+0.07−0.13
GW170809 0.70+0.08−0.09
GW170814 0.72+0.07−0.05
GW170817 6 0.89GW170818 0.67+0.07
−0.08
GW170823 0.71+0.08−0.10
Table 3.1: Table listing the final dimensionless spins of the different gravitational waves events detectedby the LIGO-Virgo collaboration [8].
Figure 3.5: Plot of the final dimensionless spins of the different gravitational waves events dectected bythe 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 yearsprovided an alternative way of detecting systems of binary black holes. Fig. 3.6 shows the masses ofdifferent neutron stars and black holes. As one can directly see by looking at this figure there is thepresence of a gap between around 2 and 5 solar masses, separating the neutron stars and the blackholes. It also suggests as discussed in the previous chapters that there is a maximum limit on the massof a neutron star (the Oppenheimer-Volkoff limit) and that black holes formation due to supernovaevents should not have a low mass, and the discovery of a solar-mass black hole would trigger manyinvestigations on its origin.
Figure 3.6: Visual representation of the masses of different black holes and neutron stars (in solarmasses), including those detected by the LIGO-Virgo collaboration by gravitational waves, with theerror 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 thecircle at the end of the arrow is the total mass of the final object resulting from the merge. The totalmass of the neutron stars binary has a lot of incertitude and as such is labelled by an interrogationmark. One can clearly see the mass gap separating the black holes and the neutron stars in the regionbetween around 2 and 5 solar masses. The black holes detected by the LIGO-Virgo collaboration aremainly heavy compared to the ones detected by X-rays [34].
32
Chapter 4
The formation of black holes by accretionof dark matter onto neutron stars
The classical stellar evolution models predict that black holes are the final state of only the mostmassive stars. In consequence, the mass of the black hole is at least several solar masses. One can thenwonder 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 usingthe accretion of dark matter. Even if the accretion of ordinary matter is more known , its interactionswith 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 theobject (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 oneof the big mysteries in astrophysics to this day. The formation of black holes by dark matter accretionwould help to put constraints on its characteristics such as the type of particles it is made of, theirmass and so on. In the following chapter two main categories will be considered: the case where thedark 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 neutronstar until the moment where the number of dark matter particles is too high and they overcome theirFermi degeneracy pressure. The neutron star would then collapse into an initial black hole which willfeed 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 elementsconcerning the number of particles are interesting here: the number Ncoll of particles over which thecollapse of the dark matter cloud begins and the number Ncrit of particles over which the attractionof the particles exceeds the degeneracy pressure of the dark matter. Some hypothesis on the nature ofthe fermionic dark matter particles will be made:
• The dark matter must be able to interact with the ordinary matter. It will be necessary to thecapture of the particles and their subsequent thermalization.
• The dark matter particles are non-annihilating, in other words there are more particles thananti-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 eachother, 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 afunction of time [37, 38]. In order to do this, one first assumes that the dark matter particles velocitiesv follow a Maxwell-Boltzmann distribution f(v):
f(v)dv = n0
( m
2πT
) 32
4πv2 exp
(−mv
2
2T
)dv (4.1)
where n0 is the number density of dark matter particles around the neutron star, m the mass ofthe particles and T their temperature. By using the equipartition theorem, one gets mv2
2 = 3T2 ,
where v =√〈v2x + v2
y + v2z〉 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
f(v)dv = n0
(3
2πv2
) 32
4πv2 exp
(−3v2
2v2
)dv (4.2)
The differential accretion rate dF will correspond to the flux of dark matter particles through asurface element on a sphere centered onto the neutron star. If ones considers only the particles witha velocity in the interval [v, v + dv] and an angular interval [θ, θ + dθ] from the normal to the surfaceelement, taking into account the projection factor v cos(θ), one obtains
dF = −f(v)dvsin(θ)dθ
2v cos(θ) =
v
4f(v) dv d(cos2(θ))
= n0πv3
(3
2πv2
)exp
(−3v2
2v2
)dv d(cos2(θ))
(4.3)
(4.4)
Let us then express Eq. (4.4)in terms of the kinetic energy E = v2
2 and the angular momentumJ = vR sin(θ), both per unit mass, where R is the radius of the sphere on which the surface elementlocated the surface element. One gets the differential accretion rate dFacc:
dFacc = 4πR2dF = 4π2n0
(3
2πv2
) 32
exp
(−3E
v2
)dE dJ2
' 4π2n0
(3
2πv2
) 32
dE dJ2
(4.5)
(4.6)
The approximation made in Eq. (4.6) is that the energy can only vary between zero and the value v2/3by which Eq. (4.5) decreases by a factor e, such that the exponential in Eq. (4.5) is considered of theorder of 1. One also defines E0 as a parameter corresponding to the maximum kinetic energy allowingthe capture by the neutron star. It is also possible to express the periastron of the trajectories of thedark matter particles around the neutron star with respect to E and J [38]:
rperi =J2/M(
1 +
√1 + 2
J2E
M2
) ' J2
2M(4.7)
Here we suppose thatM2 is much larger that J2E, withM the mass of the neutron star. For a particleto be accreted onto the neutron star, its periastron needs to be smaller than the radius of the neutronstar. Looking at Eq. (4.7), one deduces that rperi is equal to the radius of the neutron star R whenJ =√
2MR. Finally, integrating Eq. (4.6), one gets
Facc = 4π2n0
(3
2πv2
) 32∫ min
(v2
3,E0
)0
dE
∫ 2MR
0dJ2 = 8π2MRn0
(3
2πv2
) 32
min
(v2
3, E0
)(4.8)
The last step consists in taking into account the relativistic effects, which are non negligible when aneutron star is considered.
34
One can express the trajectory of the dark matter particles as follows [39]:(du
dφ
)2
= u
(2Mu2 − u+
2M
J2
)+
2E
J2(4.9)
where u = 1r , 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 factthat at the periastron u is independant of φ, one has the relation between the kinetic energy and theangular momentum of the dark matter particles needed to respect this condition:
E =J2
2R2
(1− 2M
R
)− M
R(4.10)
In the classical case, when E was supposed small, J2 = 2MR. If one takes also E very small in Eq.
(4.10), one has J2 = 2MR
(1− 2M
R
)−1
. Thus there is a corrective factor which needs to be taken
into account when integrating Eq. (4.8). Also, in practise, E0 is bigger than v2/3 for a neutron star[38]. Finally one has to take into account the fact that even if the particle goes trough the neutronstar, 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 factorf = σ/σcrit, with σcrit some critical value (of the order of 10−45cm2). If the cross-section is bigger thanthis critical value, f = 1 and every particle will be captured on the first passage on average. With allthis information, one finally gets the number of accreted particles over some time t:
Nacc = Facc
(1− 2M
R
)f t = 8π2n0
MR
1− 2M
R
( 3
2πv2
) 32 v2
3f t
=
√6π
v
ρDMm
RSR
1− RSR
σt
σcrit
(4.11)
(4.12)
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 valuesv = 220 km s−1, ρDM = 0.3 GeV cm−3, R = 104 m and RS = 5 103 m are fixed. mtot = mNacc is thetotal 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 thebaryons in it and all the particles will go inside some radius called the thermal radius. After enoughparticles 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 ofparticles needed. The first problem is now to determine the value of Ncoll, assuming self-interaction ofthe dark matter particles.
35
It is assumed that the attractive potential between two particles is a Yukawa potential VY (r) givenas follows [36]:
VY (r) =a e−br
r(4.13)
where a is a parameter to be determined and b the mass of the mediator particle of the self-attractioninteraction. The value of Ncoll can then be determined using the virial theorem of a homogeneousspherical dark matter cloud of radius Rc:
2〈K〉 = −〈V 〉 = −〈Vgrav + Vext + VY,tot〉 (4.14)
where 〈K〉 and 〈V 〉 are respectively the mean kinetic and potential energy, Vgrav the gravitationalpotential energy of the dark matter cloud, Vext the potential energy due to the rest of the neutron starand VY,tot the total potential energy coming from the Yukawa interactions between all the dark matterparticles.
One can show that VY,tot for the dark matter cloud can be expressed as follows [40]:
VY,tot = − 3N2
4b5R6c
(3− 3b2R2
c + 2b3R3c − 3e−bRc(1 + bRc)
2)
(4.15)
where N is the number of dark matter particles. Developing each potential energy term in Eq. (4.14),one has
2〈K〉 = N2 3m2
5Rc+N
8πρcmR2c
5+
3N2
4b5R6c
(3− 3b2R2
c + 2b3R3c − 3e−bRc(1 + bRc)
2)
(4.16)
where ρc is the core density of the neutron star. Moreover, Eq. (4.16) allows to get an approximateexpression 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):
〈K〉 ' 8πρcmR2c
10=
3T
2⇔ rth =
(15T
8πρcm
) 12
(4.17)
Some values of the thermal radius for different masses are given in Table 4.2.
m (TeV) rth (m)10−6 8810−3 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. Herethe typical values T = 105 K and ρC = 1018 kg m−3 are fixed.
Finally to determine the value of Ncoll, one needs to take into account all the terms of Eq. (4.16). Aslong as the parameters a and b are small, one can see an unstable solution for r → 0 for N sufficientlysmall. 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 particleslarger than Ncrit, such that the degeneracy pressure won’t be enough to stop the collapse (the situationis very similar to the Chandrasekhar limit for the white dwarfs or the Oppenheimer-Volkoff one forthe neutron stars). It is possible to get a simplified expression for Ncrit which depends only on theparameters 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 thesum of its kinetic energy K, its gravitational potential energy Vgrav and the self-interaction potentialenergy VY . The latter two have already been expressed in Eq. (4.16). The last term needed is thus thekinetic energy. To get its expression, we follow the same reasoning as for the pressure of a relativisticdegenerate gas of fermions in Chapter 1. The only difference is the fact that here instead of the pressurewe compute the kinetic energy by the following relation:
K = V ρ =V
4π3
∫ pF
04πp2Etot(p)dp =
4πm4R3c
3J(xF ) (4.18)
where ρ is the kinetic energy density of the dark matter cloud, V the volume of the cloud, xF = pFm
and J(x) is given by
J(x) =1
8π2
[x(1 + x2
) 12(1 + 2x2
)− ln
[x+
(1 + x2
) 12
]](4.19)
where x = pm . Applying the same reasoning to obtain the number of particles N , one has:
N = V n =V
4π3
∫ pF
04πp2dp =
m3x3
3π2(4.20)
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 simplifiesthe expression of J(xF ) (4.19) since it means that xF is small by definition. In this approximation onehas for the kinetic energy:
K =4πm4R3
3J(xF ) ' mN +
2
15π
(9π
4
) 53 N
53
mR2c
(4.21)
The short range approximation bRc >> 1 simplifies the expression of Eq. (4.15) as follows:
VY ' −3aN2
2b21
R3c
(4.22)
Combining Eqs. (4.21), (4.22) and the expression of Vgrav from Eq. (4.16), one gets an expressionfor Etot:
Etot = mN +2
15π
(9π
4
) 53 N
53
mR2c
−N2
(3a
2b21
R3c
+3m2
5Rc
)(4.23)
Let us find the extrema of Eq. (4.23) by determining the roots of its derivative with respect to Rc:
dEtotdRc
= 0⇔ Rc =2
9π
(9π
4
) 53 N−
13
m3±
√4
81π2
(9π
4
) 103 N−
23
m6− 15a
2b2m2(4.24)
There are two different solutions for Rc. However one can determine the value of N for which thesolutions are equal (i.e. the discriminant is equal to zero), corresponding to the unstable situation ofthe collapse, which gives the value of Ncrit:
4
81π2
(9π
4
) 103 N
− 23
critm6
− 15a
2b2m2= 0⇔ Ncrit = π2
(9
4
)5( 8
1215
) 32(b
a12
)3 1
m6' 0.3
(b
a12
)3 1
m6
(4.25)
Depending on the values of a and b, one can then compute Mcrit = mNcrit. A graphical illustrationis 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 supposean asymmetric non-annihilating dark mater which will accrete in a neutron star until the numberof particles is sufficiently high to start a collapse and the formation of a black hole. The key pointhere 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 theblack hole, in this section mainly the pressure due to the zero point energy and the possible formationof 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 arefermions or bosons since the differences between them are not involved in the computations. Howeverfor this section we will follow this time a slighty different approach [41] and compare it to the approachused for the fermionic case, neglecting this time the self-attraction of the dark matter and consideringstill at first that the dark matter particles follow a Maxwell-Boltzmann velocities distributions. It ispossible to show that in that case that the accretion rate is given by the following relation [42]:
Facc = 4π
∫ R
0
dFaccdV
r2dr (4.26)
The differential term is given by
dFaccdV
=
(6
π
) 12
σn0(r)nB(r)v2esc(r)
v
[1− 1− e−B2
B2
]min
(∆p
pF, 1
)(4.27)
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:
B2 = 6
(vesc(r)
v
)2 mmB
(m−mB)2 (4.28)
where mB is the mean mass of the ordinary baryonic particles of the neutron star. The minimumfactor in Eq. (4.27) comes from the possible case where the momentum transfer when a dark matterparticle is scattered by the baryons is inferior to the Fermi momentum. If we consider n0, nB and vescindependent of the radial coordinates, Eq. (4.26) becomes
Facc '(
6
π
) 12 σNBρDMv
2esc
mv
[1− 1− e−B2
B2
]min
(mvescpF
, 1
)(4.29)
One can look for the value of m for which the momentum transfer will be lower than the Fermimomentum. The latter is given by Eq. (1.10) where the electronic density is replaced by the baryonicdensity :
pF =(3π2nB
) 13 =
(3π2ρBmB
) 13
' 0.58 GeV (4.30)
where ρB is the baryonic mass per unit volume and has been taken equal to 1.4 1015 g cm−3. As such,one can determine the mass for which the minimum factor equal to 1:
mvescpF
> 1⇔ m >pFvesc' 1 GeV (4.31)
where vesc = 1.8 105 km s−1. Finally one has the number of particles accreted over time Nacc = Facct:
• If m is over 1 GeVNacc = 2.3 1030 ρDM
mf t (4.32)
• If m less than 1 GeVNacc = 3.4 1032ρDM f t (4.33)
where the typical values v = 220 km s−1 and NB = 1.7 1057 are fixed, and ρDM is expressed inGeV cm−3, t in years and m in TeV. The order of magnitude of Nacc for the same parameters that inTable 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.Here the typical values v = 220 km s−1 and ρDM = 0.3 GeV cm−3 are fixed. mtot = mNacc is the totalmass 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 thesame order of magnitude (by a factor ∼ 2) than that for the fermionic case, which is in agreementwith the fact that for the computation of Nacc no hypothesis has been made on the exact nature ofthe dark matter particles. The slight difference comes from several factors: even if in both cases anasymmetric dark matter interacting with baryons is assumed, for the bosonic dark matter the possibleself-interaction between the particles is neglected. Moreover, from a computational point of view, somefactors are taken into account in one case and not in the other and vice versa (for example, the numericdensity of baryons and the escape velocity are directly considered for the bosonic dark matter and notfor the fermionic dark matter, and on the opposite the radius of the dark matter cloud is not involvedin 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 mattercloud. As a first approximation, it is still possible to estimate the number of particles over whichthe cloud becomes self-gravitating [41]. Indeed, if one considers that the collapse will begin when thedensity of dark matter particles is higher than the baryonic density within the thermal radius, one has:
ρDM > ρB ⇔Nm
4πr3th
3
> ρB ⇔3mN
4π
(4πρBm
15T
) 32
> ρB ⇔ N & 1042
(100GeVm
) 52(
T
105K
) 32
(4.34)
Finally, one can also estimate the value of Ncrit. However, contrarily to the accretion of fermionicdark matter, there is no Fermi pressure to oppose the gravitational collapse. The pressure that will playthis role comes from the zero-point energy. Since there is no exclusion principle, the average distancebetween 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 orderof the inverse of the radius of the cloud. Then the average energy 〈E〉 of a boson inside the cloud is
〈E〉 = V +K ' −Nm2
Rc+
1
Rc(4.35)
Deriving Eq. (4.35) with respect to Rc, one finds an approximation of the critical value over whichthe gravitational term is larger than the kinetic one :
d〈E〉dRc
= 0⇔ Ncrit '1
m2(4.36)
Some values of Ncrit in function of m are indicated in Table 4.4.
m (TeV) Ncrit10−2 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 casewhere the particles are bosonic.
Comparing Tables 4.2 and 4.4, one sees that due to the more constraining constraint imposed bythe Pauli principle for the fermions, a lot fewer particles are needed to create a black hole, for the darkmatter 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 ofbosons falls below some critical temperature Tc, which depends mainly of the mass of the bosons andtheir density, and given by the following relation (supposing that the interactions between the darkmatter particles are negligible and that all the particles are within the thermal radius):
Tc =2π
ζ23 (1.5)
(n0
m32
) 23
(4.37)
where ζ is the Riemann zeta function. When experimenting on Earth, we have to get to extremely lowtemperatures 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 acritical temperature above the actual temperature in the center of the neutron star (which is also thetemperature of the dark matter particles within the thermal radius by definition). When such stateis attained, the Maxwell-Boltzmann distribution is no longer a valid approximation, and one has touse instead the Bose-Einstein distribution. Inverting Eq. (4.37), one obtains the critical number ofparticles NBE over which a Bose-Einstein condensate forms:
Tc ' 3.3
(3N
4πr3thm
32
) 23
⇔ NBE =4πr3
thm32
3
(Tc3.3
) 32
' 0.32
(Tc√ρB
)3
(4.38)
In an analogous way to the thermal radius, there is a radius rBE under which theses particles inthe condensate state are located given by [41]
rBE =
(3
8πρBm2
) 14
(4.39)
The number of particles over which the ones in the Bose-Einstein condensate begin to self-gravitatecan be determined by using Eq. (4.34), replacing the thermal radius rth by rBE :
N & 1023
(100GeVm
) 52
(4.40)
Comparing Eqs. (4.34) and (4.40), one can see that for the same mass fewer particles are needed tostart 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 wouldeventually accumulate enough dark matter particles that it will collapse into a black hole. However itis pretty obvious that the value of the interaction dark matter-baryon cross-section will play a big rolein the actual time needed. Since we observe neutron stars, it means these still didn’t collapse. Thusthe population of neutron stars could be used to help estimate the fraction of neutron stars that havecollapsed 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 dependsin particular on three main factors: the dark matter density ρDM , the root-mean-squared velocity vand the interaction cross-section σ. The first two are not equally distributed in a same galaxy (theMilky 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]. Oneneeds to take into account all these elements to get the impact of the cross-section. The idea tocompute 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 thecollapse of the neutron star. It can be achieved by looking at when the number of accreted particlesNacc is larger than the critical number of particles Ncrit, for example in the fermionic case:
Nacc > Ncrit ⇐⇒ρDMv
>0.3 σcrit√
6π
1− RSR
RSR
( b√a
)3 1
m5
1
σt(4.41)
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 theparameters involved in the Yukawa interaction, the time period studied and of course the interactioncross-section. This equation also shows explicitly that if we consider a smaller period or a smallercross-section, less neutron stars will be converted into black holes, since the radio ρDM/v needs to behigher to do so and thus less regions satisfy this condition. The same logic can also be applied tobosonic 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 shownbelow, where the dark matter distribution follows a Burkert profile described as follows:
ρDM (r) = ρs
[(1 +
r
rs
)(1 +
(r
rs
)2)]−1
(4.42)
where r is the radial distance, ρs = 3.15 Gev cm−3 and rs = 5 kpc.
Figure 4.2: Fraction of neutron stars which should have collapsed in t = 5 109 years with respect tothe interaction dark matter-baryons cross-section in the fermionic case and in the Milky Way. Thedifferent curves correspond to different combinations of values of a, b (of the Yukawa interaction) andm, so that the curves have the same shape. The velocity distribution is supposed to increase linearlyup to 220 km s−1 at 0.5 kpc from the center of the galaxy, then the velocity has constantly this valuebeyond 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 theirconstraints
One other class of low-mass black holes is constituted by the primordial black holes, which could havebeen 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 thefact that it is possible to put constraints on these primordial black holes, partly due to observationaldata. 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 earlyin the Universe history in the classical models of accretion.
• Similarly, some very bright galaxies and dusty environments have also been discovered at highredshift 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 masseven with more than ten billion additional years.
• The primordial black holes can also be candidates for solar mass and intermediate mass blackholes (∼ 103 − 104M).
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 confirmedsince 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 severalsatellites. However, despite its success, several elements are not explained by the classical Big Bangmodel. For example, the CMB has a quasi-uniform temperature of about 2.7 K and is a quasi-perfectblack body spectrum. However all the regions seen on the CMB images were not initially causallyconnected, begging the question of how the temperature is so uniform. This problem, known as thehorizon problem, is one among other issues that the classical Big Bang model doesn’t address. One canalso 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. Thegeneral idea is that in the very early history of the Universe, it experienced a short phase of exponentialexpansion, resolving some of the problems not addressed by the hot Big Bang model. To obtain suchan 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:
ds2 = dt2 − a2(t)
[dr2
1− kr2+ r2
(dθ2 + sin2(θ)dφ2
)](5.1)
where k is the curvature and can be equal to -1, 0 or 1 and a(t) the scale-factor with a(t0) = 1, wheret0 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 (takingΛ = 0) [43, 44]:
a
a= −4π (ρ+ 3p)
3(5.2)
H2(t) =
(a
a
)2
=8πρ
3− k
a2(5.3)
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 thespatial coordinates, the potential V much greater than the kinetic energy term φ2 and V » ∂V/∂φ:
H2(t) ' V
3⇐⇒ a(t) = a(t0)et
√V/3 (5.4)
As long as the condition V » φ2 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 ofthis 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 highdensity compared to their neighbourhood. If the characteristic lengths of these regions are smallerthan their corresponding gravitational radius, they undergo a gravitational collapse and create whatis called a primordial black hole. It is important to keep in mind however that a priori there is notnecesseraly a reason for which a primordial black hole should be in a particular range of masses. Itcould be a very light black hole or a massive one. But if the black hole was too light when it has beenproduced, 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]:
Mcrit ' 1012( α
4 10−4
) 13
(t
13.8 109
) 13
(5.5)
where α is a coefficient depending on the species emitted [47] and t is expressed in years. If one takest ' t0, α ' 4 10−4 and subsequently Mcrit ' 1012 kg.
5.2 The constraints derived from the observations
Considering the primordial black holes sufficiently massive not to have evaporated by now, the obser-vations (electromagnetic waves and in the future the gravitational waves) can help put constraints ondifferent ranges of primordial black hole masses, but also on wheter or not primordial black holes couldbe candidates to be a sizeable fraction of the missing dark matter. In the following section, we willfocus 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 theseobjects 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 deflectionand the characteristics of the images won’t be the same and thus it allows to some extent to putconstraints on the primordial black holes.
One of the main way of studying gravitational lensing is by using the microlensing, i.e. the angulardistance 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 onthe different surveys of microlensing, let us remind the reader of some notions of gravitational lensingwhich 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 thedifferent 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 S′Q in Fig.5.1:
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 thesource, and between the plane of the observer and the plane of the source respectively. The deflectionangle α is given by
α =4ML
θdL(5.7)
where ML is the mass of the lens (here the mass of the primordial black hole) and dL the angulardiameter distance between the plane of the observer and the plane of the lens. Fig. 5.1 illustratesthese distances.
Figure 5.1: Schematic representation of the different planes and distances used commonly in gravita-tional lensing. O corresponds to the observer, M to the lens (or deflector), S to the source and S’ tothe 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:
RE = 2
√MLdLdLS
dS(5.8)
It corresponds to the radius of the Einstein ring (full ring of deflected light) which appears whenthe 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 lightfrom the source passes inside the circle defined by the Einstein radius on the lens plane (when itis much smaller than unity)" [48] and it can be expressed as follows:
τ =
∫ dS
0nPBH(dL)σd(dL) = 4π
∫ dS
0ρPBH(dL)
dLdLSdS
d(dL) (5.9)
where nPBH is the number density of primordial black holes, σ the lensing cross-section (equalhere to πR2
E) 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 theEinstein radius. To show this, one first multiplies Eq. (5.6) by θd2
L/dS , using Eq. (5.7) and notingr = θdL and r0 = βdL:
r2 − r0r −R2E = 0⇔ r± =
r0 ±√r2
0 + 4R2E
2(5.10)
In the thin-lens approximation, there are two images located at the positions given by Eq. (5.10). Fora small source and supposing a circular symmetry of the lens, the magnification A of an image is givenby [49]
A =θ
β
dθ
dβ=
r
r0
dr
dr0(5.11)
Noting u the ratio of r0 to the Einstein radius, one has using Eq. (5.11):
|A±| =∣∣∣∣r±r0
r±dr0
∣∣∣∣ =
∣∣∣∣ 1
4u
(u±
√u2 + 4
)(1± u√
u2 + 4
)∣∣∣∣ =
∣∣∣∣∣±u√u2 + 4 + u2 + 2
2u√u2 + 4
∣∣∣∣∣ (5.12)
The ratio of the two magnifications in Eq. (5.12) thus gives
R =
∣∣∣∣A+
A−
∣∣∣∣ =
∣∣∣∣∣u2 + 2 + u√u2 + 4
u2 + 2− u√u2 + 4
∣∣∣∣∣ (5.13)
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 atwhich 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 tolink the optical depth and the fractional density fPBH of dark matter which is due to compact objectsin our galaxy dark halo. For example, B. Paczynski demonstrated that there is a relation between theoptical depth and fPBH in the simple case of an isothermal model [50] with sources in the Magellanicclouds: τ ∼ 10−6fPBH.
46
Following this, many observations have been performed to try and determine if primordial blackholes could explain at least partially the "missing" mass needed to explain notably the velocity curveof 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 LensingExperiment). Some of them are grouped in Table 5.1. It is also important to remember that theresults are for the fraction of compact objects, not necessarily primordial black holes. As such theresults in the table should be taken as upper limit for the proportion of primordial black holes in allcases.
As one can see when looking at Table 5.1, the duration of the surveys needs several years to attaina low number of microlensing events. Because of this, the incertitude on the optical depth can be verylarge if the number of events is very low (EROS for the Small Magellanic Cloud). It is also interestingto highlight the fact that even if no events have been detected (which is the case of EROS for the LargeMagellanic Cloud)), it is still a piece of information that helps to put an upper limit on the fraction ofdark matter constituted of compact objects.
τ (10−7) fPBH (%) Number of events Period (years) Mass range (M) Cloud Study1.2+0.4−0.3 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 theLarge Magellanic Cloud (LMC) and the Small Magellanic Cloud (SMC). The corresponding fractionof compact objects (value taken from the articles themselves), the number of microlensing eventsconsidered, the period of observation and the mass range of the compact objects concerned are alsoindicated.
Finally, in general the primordial black hole moves at some (tangential) speed vt with respect to thesource-observer axis. Because of that the value of the brightness amplification due to the microlensingwill change over time. The charateristic time-scale (in years) between the amplification maximum andits near-minimum is the ratio of the Einstein radius and the tangential speed of the lens:
tscale =REvt' 2
√dLdLSd2S
(ML(M)
10
)(dS(kpc)
100
)(200
vt(km s−1)
)(5.14)
If one considers as lenses primordial black holes in the Milky Way dark matter halo and as sourcesstars 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 afew 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 somecelestial objects
Constraints on primordial black holes can also be obtained by looking at the gravitational perturbationsthey produce on the celestial bodies or regions they encounter. In the case of a celestial body, let ustake as an example a white dwarf [55]. If some circumstances make a primordial black hole encountera white dwarf, provided that the black hole has a sufficiently low mass (if it is too heavy, the blackhole will directly destroy the white dwarf), it can go through the white dwarf, transferring some of itskinetic 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 startthe carbon fusion which in turn causes a thermonuclear runaway (since by definition a white dwarf iscomposed of a state of degenerated electrons, which cannot really control the temperature to keep thewhite dwarf stable) if the thermal diffusion is slow enough and eventually a supernova explosion. Themain difference with a classical type Ia supernova would be the fact that the white dwarf wouldn’tneed 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 slowenough compared to the fusion rate F to lead to a thermonuclear runaway and a supernova. Indeed thetime-scale tdiff of the decrease in temperature of a hotter region of temperature T (which in our casewould 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-dimensionalheat equation:
∂T
∂t= α
∂2T
∂x2=
κ
ρcp
∂2T
∂x2(5.15)
where α is the thermal diffusivity, κ the thermal conductivity, ρ the density of the medium consideredand cp the specific heat of the medium.
Reformulating Eq. (5.15) in term of dimensional analysis one has
T
tdiff=
κ
ρcp
T
L2⇐⇒ tdiff =
(ρcpκ
)L2 (5.16)
Thus when L is bigger than some minimal value Lmin, tdiff is larger than F−1. In that case thefusion reactions will have enough time to develop before the excess thermal energy is diffused. As themedium 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 functionof the temperature:
κ ' 4aT 3
ρκop(5.17)
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. photonsand electrons. If ρ > 108 g cm−3, then the electrons are dominant and κop ÷ ρ−2, while when ρ < 108
g cm−3, photons dominate and κop÷ρ [55]. Combining these information, one finally has the followingcondition to have the thermonuclear runaway (considering F ÷ ρ):
F tdiff = 1⇔ L2 =κ
R ρcp÷ T 3
ρ3κop
⇔ L2min ÷ ρ−0.5 (for electrons)
⇔ L2min ÷ ρ−2 (for photons)
(5.18)
(5.19)
(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
R ' 104GMPBH
√log
(R
GMPBH
)> Lmin (5.22)
A primordial black hole respecting this condition will heat the medium enough to lead to a supernovaevent. Since all black hole masses cannot make a white dwarf explode, and since some ranges of massesare already constrained by other means, one can also use the population of existing white dwarfs andthe frequency of type Ia supernovae to put constraints on the importance of the primordial black holesin the dark matter content.
48
Finally one can also find the mass of a white dwarf that a given black hole can destroy, using anequation of state to have the relation between the density of the medium and the mass of the whitedwarf and Eq. (5.22) as illustrated in Fig. 5.2.
Several other objects perturbed by their gravitational interaction with potential primordial blackholes 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 itselfby 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 goingthrough it. The speed of the primordial black hole is supposed to be roughly equal to the escapevelocity 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 onthe 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 ormore recently should emit, from their accretion disks, in X-rays and radio waves. Thus one cansearch 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 ofprimordial black holes is presented in Fig. 5.3. However some of the constraints in Fig. 5.3 have beenupdated recently. On the one hand, the OGLE survey data have been updated. On the other hand,the constraints on the very low-mass primordial black holes (below ∼ 10−13 M), notably from thefemtolensing with gamma-ray bursts is argued not to be reliable by Katz et al. [57], if the non-pointlike nature of the source and the lens is taken into account. The concerned mass ranges are shown inFig. 5.4.
49
Figure 5.3: Graphics regrouping the different methods of constraints on the fraction of primordialblack holes in the dark matter depending on the mass of the primordial black holes (expressed insolar 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 thegeneral microlensing, and to black lines for the millilensing and femtolensing (the name comes fromthe size of the angular separation in arcsec). The red curves are the constraints derived from accretionconsiderations 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 toEri-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 extendthese constraints. Another example linked to gravitational lensing is the possible use in the future ofthe fast radio bursts [60], phenomena lasting a few milliseconds. Using a primordial black hole againas a lens (or to be more general any MACHO object), one could detect, if the mass of the black holeis 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 asfollows [60]:
∆t = 4ML(1 + zL)
[u√u2 + 4
2+ log
(√u2 + 4 + u√u2 + 4− u
)](5.23)
where zL is the redshift of the primordial black hole. Two conditions need to be fulfilled to have asufficiently strong lensing:
• The observed time delay must be greater than some reference time delay (greater than thecapability of the instruments at least). This condition ∆tobs > ∆tref imposes u > umin whereumin(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 bemonitored. This condition Robs < Rref imposes u < umax where umax is some upper boundcorresponding to Rref.
Figure 5.4: Top: Updated constraints on the fraction of primordial black holes in the dark matterfrom the OGLE 5-year survey. The red area is the region of 95 % confidence level in the case of nomicrolensing events by primordial black holes detected by the OGLE survey [59]. Bottom: Projectionsof the future constraints (in purple) depending on the number of correctly identified gamma-ray burstsand taking into account the extended size of the source. as is the actual transverse size of the emissionregion in the case of an extended source and corresponds to an upper limit under which constraintscan be set. The cut-off for the HSC constraints is due to the limit of the validity of the geometricalapproximation used in the HSC case. EGγ corresponds to the constraint from the extragalactic photonsfrom PHH evaporation [57].
51
Like previously, one important thing left is the determination of the optical depth τ . Since ucannot take all values because of the above constraints, the cross-section won’t be a disk anymorebut an annulus which inferior and superior radii are given by umin and umax respectively. Let us alsointroduce the comoving distance χ defined as follows [44] and which can be linked to the angulardiameter distance via the redshift (last equality):
χ = a(t0)Θk(r) = a(t0)
∫ r
0
dr′
1− kr′2= dP (1 + zP ) (5.24)
where r corresponds here to the radial coordinate of Eq. (5.1) and zP is the redshift corresponding todP . With these elements, one can formulate in another way Eq. (5.9) by integrating on the comovingdistance:
τ =
∫ dS
0nPBHπR
2E
(u2max − u2
min)d(dL) =
∫ dS
0NPBH (1 + zL)3 πR2
E
(u2max − u2
min)d(dL)
=
∫ zS
0NPBH (1 + zL)3 πR2
E
(u2max − u2
min) dχ
(1 + zL)
=
∫ zS
0(1 + zL)2NPBHσanndχ
(5.25)
(5.26)
(5.27)
where NPBH is the number density per unit comoving volume (justifying the (1 + zL)3 term) and σannis the annulus cross-section. Then, from Eq. (5.27), one integrates on the redshift and express theresult as fPBH:
τ =
∫ zS
0(1 + zL)2NPBHσanndχ =
∫ zS
0
ΩM,0
ΩM,0(1 + zL)2NPBHσann
dzLH(zl)
= ΩM,0
∫ zS
0
(3H2(t0)
8πρDM,0
)(1 + zL)2NPBHσann
dzLH(zL)
=3ΩM,0
8π
∫ zS
0
H2(t0)
H(zL)(1 + zL)2 NPBH
ρDM,04πMPBH
dLdLSdS
(u2max − u2
min)dzL
=3ΩM,0
2
∫ zS
0
H2(t0)
H(zL)(1 + zL)2
(NPBHMPBH
ρDM,0
)dLdLSdS
(u2max − u2
min)dzL
= fPBH3ΩM,0
2
∫ zS
0
H2(t0)
H(zL)(1 + zL)2 dLdLS
dS
(u2max − u2
min)dzL
(5.28)
(5.29)
(5.30)
(5.31)
(5.32)
where H(zL) is the Hubble parameter at the time corresponding to the redshift zL, ΩM,0 the currentmatter density parameter and ρDM,0 the current matter density. Finally, taking into account that thereis a distribution NFRB(z) of fast radio bursts, the integrated optical depth τint is given by
τint =
∫τNFRB(z)dz (5.33)
Having a relation between fPBH andML (through ymin), one can then put constraints from it, dependingon 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 objectsin 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 issupposed there have been 10 000 fast radio bursts and none of them have been lensed. The constraintsfrom the MACHO and EROS projects are also presented, along the constraint from the study of widebinary 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, beingone of the most extreme known object in the universe. Following the development of the knowledge onrelativity along the 20th century, different metrics have been discovered in an attempt to describe thespace-time around the black holes, which depend only on the mass, angular momentum and electriccharge of the black hole. The Schwarzschild metric, when the black hole doesn’t spin and is electricallyneutral, is a simpler metric than the other ones. However, in reality the black holes spin, and it ismore appropriate in general to use at least the Kerr metric in that case. By taking into accounta non-zero angular momentum, different properties are different from a Schwarzschild black hole, oreven appear solely because of it. The curvature singularity has a toroidal geometry (in the appropriatecoordinates), but also the coordinate singularity (the horizon) is no longer unique and doesn’t coincidewith the curvature singularity. Because of this a region appears, called the ergosphere, in which therecannot 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 certainlimit, 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 coalescenceof black holes, and even neutron stars. The third run is already ongoing this year. On the other handthe first close-up image of a (supermassive) black hole, M87∗, has been revealed in April 2019, showingthe light emitted by the accretion disk bent by the gravitational field of M87∗ and the shadow of theblack hole, within which resides the event horizon.
These recent feats revived the interest in the subject of black holes. Among other things, the searchfor the low-mass black holes is an important topic. If one looks at the different measured neutron starand 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 starshave a maximum mass, the Oppenheimer-Volkoff limit, due to the fact than the neutron degeneracypressure is not enough to counterbalance the large mass (which is analogous to the Chandrasekharlimit for the white dwarfs). Beyond that, it would collapse into a black hole. Even if this limit isnot 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 wayof 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 isalready very massive. As such, if solar-mass (or lighter) black holes are discovered one day, one shouldconsider other mechanisms to explain their origin. Two interesting possibilities are the creation ofsolar-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 differentinvestigations have been and are currently done to attempt to resolve this problem. One option isto consider the dark matter as particles with unusual properties compared to the ordinary baryonicmatter. This consideration has lead to the investigation of the possibility of creating black holes fromdark matter. One of the ways to achieve this is by using a neutron star as a seed to accumulate darkmatter until their gravitational collapse into a black hole, eating what is left of the neutron star. Theidea is that the dark matter particles accumulate into the neutron star, thermalizing with the mediumand going within a cloud of some thermal radius.
55
Two thresholds have to be crossed: the number Ncoll of particles over which the gravitational col-lapse begins and the number Ncrit over which the black hole is created (similarly to the Chandrasekharand the Oppenheimer-Volkoff limits). The values of these numbers will depend on whether the darkmatter is composed of fermions or bosons. In the first case, what will act against the gravitationalcollapse is the Fermi pressure, while in the second case the constraint of the Fermi pressure doesn’texist, but a Bose-Einstein condensate can form. Globally the apparition of a Bose-Einstein condensateof 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 numberof particles required, for a same mass of the particles, will also be lower compared to a fermionic darkmatter. Combining the estimates of the value of the interaction cross-section between the dark matterand the baryonic matter, and the population of neutron stars, one can then set constraints on thiscross-section.
The other way is to consider that these low-mass black holes don’t come from the stellar evolutionbut originate from the very early Universe. To resolve some issues not addressed by the hot Big Bangmodel, the inflation models have been developed. These models state there has been an exponentialexpansion 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 theywere 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 afraction of the dark matter mass. Many different methods have been used to put constraints on thefraction of primordial black holes in dark matter (to be more precise the fraction of massive compactobjects): 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 widerange 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 ofexisting instruments (which increases the expected number of detected events for LIGO and Virgo forexample) and the creation of new ones. The gravitational waves direct detection can lead to a new eraof observation to explore domains which were not accessible before. Combining this multi-messengerastronomy with the research on the dark matter, it could potentially lead to very interesting discoveriesconcerning its nature, wheter it is composed of particles or of a set of compact objects like primordialblack holes.
56
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