Download Black Holes in Neutron Stars

Black Holes
in
N e u t ro n S ta r s
Submitted for
Master Thesis in Physics
M a rt i n Au t z e n
November 1st 2013
S u p e rv i s o r : C h r i s K o u va r i s
CP3 Origins and Institute for Physics,
C h e m i s t r y a n d Fa r m a c y
University of Southern Denmark
i
Abstract
This master thesis deals with the formation and evolution of
black holes formed by dark matter inside a neutron star. Neutron
stars are some of the densest objects in existence. Because of their
relatively small size and their high mass, they exert an enormous
gravitational attraction on the surrounding matter. This makes them
excellent traps for dark matter particles. Should enough dark matter
be accumulated inside the center of the star, there is a possibility
that these particles can become self-gravitating and facilitate their
own collapse into a black hole. The thesis addresses several issues
regarding the entrapment of dark matter particles inside neutron
stars. In addition it addresses the modifications on the Hawking
radiation of mini black holes formed in neutron stars by dark matter
particles due to the degeneracy of nuclear matter at the core of the
star.
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Resume
Dette speciale vil beskæftige sig med dannelsen og udviklingen af sorte
huller dannet af mørkt stof inde i neutron stjerner. Neutron stjerner er
blandt de objekter i Universet med den højeste massefylde. På grund af
deres relativt lille størrelse og høje masse udøver de en enorm gravitational
tiltrækning på det omkringværende stof. Dette gør dem til ideelle til indfangelse af mørkt stof. Hvis nok mørkt stof akkumuleres i kernen af stjernen
er der en mulighed for at disse partikler selv-graviterende og kan facilitere
deres eget kollaps til et sort hul. Dette speciale addreserer flere scenarier
angående indfangelsen af mørkt stof i neutron stjerner. Udover dette vil
der også undersøges modifikationen af Hawking strålingen fra mini sorte
huller dannet af mørkt stof inde i neutron stjernen grundet udartetheden
af nukleart stof i stjernens kerne.
iii
Conventions
Unless otherwise specified, we work in natural units of particle physics and
cosmology with h̄ = c = kB = 1. Newton’s constant is not set to one,
but denoted by G. This allows mass, time, distance and temperature to be
rewritten in terms of eV.
Thus, the following conversions will be used throughout
Unit
1 eV ≠1
1 eV
1 eV ≠1
1 eV
Metric Value
1.97 · 10≠7
1.78 · 10≠36 kg
6.58 · 10≠16 s
1.16 · 104 K
Derivation
= (1eV ≠1 ) h̄c
= (1eV ) /c2
= (1eV ≠1 ) h̄
= 1eV /kB
Any given spacetime metric is of the form ds2 = gµ‹ dxµ dx‹ , where the
signature of gµ‹ is (- + + +).
Acknowledgements
During the work on my thesis I have had the pleasure of being a part of
CP3 Origins. I would like to thank my supervisor Chris Kouvaris for taking
me as his student and fueling my interest for astrophysics. Thank you for
the numerous meetings and your patience in answering my questions when
I was stuck. It has been a pleasure to work with you.
Second I would like to thank all the wonderful people at CP3 . All the
professors and students for answering my questions and talks of chess and
other non-physics subjects. I special thanks goes to Martin Hansen for
writing the original code for solving the Lane-Emden equation.
Finally I would like to thank all my friends and family. Thank you for
all the support and good times during the past year of writing this work
and the four years of university life before.
Martin Autzen
Odense
v
Contents
Acknowledgements
v
Contents
vi
List of Figures
viii
List of Tables
x
Introduction
1
I Neutron Stars, Dark Matter and Black Holes
7
1 Neutron Stars
1.1 Stellar Evolution: The Road to a Neutron Star . . . . . . . .
1.2 Formation of a Neutron Star . . . . . . . . . . . . . . . . . .
1.3 Properties of Neutron Stars . . . . . . . . . . . . . . . . . .
9
9
12
13
2 Introduction to Black Holes
17
2.1 Schwarzschild Black Holes . . . . . . . . . . . . . . . . . . . 17
2.2 Kerr Black Holes . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Dark Matter
31
3.1 Inferring Dark Matter . . . . . . . . . . . . . . . . . . . . . 31
3.2 Accretion onto a Neutron Star . . . . . . . . . . . . . . . . . 36
3.3 Formation of a Black Hole from Dark Matter . . . . . . . . . 37
II Thermodynamics and Hawking Radiation
41
4 Black Hole Thermodynamics
43
vi
CONTENTS
4.1
4.2
4.3
vii
Black Holes and Thermodynamics . . . . . . . . . . . . . . .
The Holographic Principle . . . . . . . . . . . . . . . . . . .
A Microscopic Description of Black Holes . . . . . . . . . . .
43
45
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5 Hawking Radiation
49
5.1 Hawking Radiation . . . . . . . . . . . . . . . . . . . . . . . 49
5.2 The Information Paradox . . . . . . . . . . . . . . . . . . . . 51
6 Greybody Factors
55
6.1 Changing Hawking Radiation . . . . . . . . . . . . . . . . . 55
6.2 Black Hole Scattering Theory . . . . . . . . . . . . . . . . . 57
6.3 Low versus High Frequency . . . . . . . . . . . . . . . . . . 60
III Accretion and Radiation
63
7 Modeling Radiation
65
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.2 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.3 Adding Particles . . . . . . . . . . . . . . . . . . . . . . . . 70
8 Accretion and Radiation
8.1 Masses and Temperatures
8.2 Gluon Emission . . . . . .
8.3 Bondi Accretion . . . . . .
8.4 Geometric Accretion . . .
8.5 Comparison with limits . .
9 Effects of Rotation
9.1 Transport of Angular
9.2 Kerr Black Holes . .
9.3 a = 0.5 . . . . . . . .
9.4 a = 0.99999 . . . . .
9.5 Summing Up . . . .
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Momentum by Viscosity
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10 Comparison with IceCube
99
10.1 A Brief Summary . . . . . . . . . . . . . . . . . . . . . . . . 99
10.2 Black Hole Explanation: An Attempt . . . . . . . . . . . . . 100
11 Dark Matter Accretion
103
11.1 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
11.2 Uniform Density . . . . . . . . . . . . . . . . . . . . . . . . 105
11.3 r2 Density Profile . . . . . . . . . . . . . . . . . . . . . . . . 106
11.4 Lane-Emden Profile . . . . . . . . . . . . . . . . . . . . . . . 107
12 Conclusions
109
IVAppendices
113
A Geometrical Accretion
115
A.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
B Markovic derivations
117
B.1 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Bibliography
119
List of Figures
1.1
1.2
1.3
2.1
2.2
2.3
3.1
This figure shows the Proton-Proton 1 chain. . . . . . .
This figure shows the triple alpha process which fuses
into carbon. . . . . . . . . . . . . . . . . . . . . . . . .
Illustration of the possible structure of a neutron star.
. . . . .
helium
. . . . .
. . . . .
10
11
14
The Kruskal diagram showing the Schwarzschild solution in Kruskal
coordinates, where all light cones are at ±45¶ [1]. . . . . . . . . . 22
Regions of the Kruskal diagram. . . . . . . . . . . . . . . . . . . 23
Figure showing the cross section of a Kerr black hole. Here the
ergosphere is clearly visible between the outer and inner event
horizons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
This figure shows how the observed rotational curve compared
with the expected curve if only the disc is considered. . . . . . .
viii
33
List of Figures
3.2
3.3
In both figures, the red hues shows the X-ray emitted from
the two clusters hot intergalactic gas, captured by the Chandra X-ray observatory. The blue hues show the dark matter
inferred through gravitational lensing captured by the Hubble
Space Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparing Figure 3.3a and Figure 3.3b it can be seen that the
peaks of the Ÿ reconstruction of the weak lensing is located outside of the hot intergalactic gas clearly seen in Figure 3.3b . . .
4.1
The difference between the classic and quantum mechanical descriptions of a black hole. . . . . . . . . . . . . . . . . . . . . .
6.1
This figure shows how the thermal radiation from the horizon
(red) gets modified by the non-trivial geometry surrounding the
black hole (orange). . . . . . . . . . . . . . . . . . . . . . . . . .
When the Hawking radiation has been emitted it will have to
propagate through the non-trivial geometry created by the potential V(r). The potential will then filter the radiation; some of
the radiation will be transmitted by tunneling through the potential while the rest will be reflected back into the black hole.
The part that tunnels through will be modified and travels freely
to infinity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2
7.1
7.2
8.1
8.2
ix
35
36
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56
57
This figure shows the three emission curves for neutrinos, photons and gravitons. The curves pictured here are for a black hole
of M = 5 · 1014 g. . . . . . . . . . . . . . . . . . . . . . . . . . . 67
This figures shows the rescaling of both y and x-axis to the GeV
range. Note that the peak of the neutrino spectrum is located at
approximately 0.1 GeV which is consistent with its location in
Figure 7.1. These curves are for a black hole of mass M = 5·1014 g. 68
Ferynman diagram of a gluon creating an q q̄ pair . . . . . . . . .
These figures illustrates how gluons might be blocked by a fermi
momentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
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11.1 Figure taken from [2]. a) shows the scenario where – is small,
here we
see that the particle will only be deflected, since a neg1 22
du
ative d„ does not make physical sense. In b) – has been
increased, resulting in a temporarily spiraling solution. Finally
in c), – has reached the critical value and the particle is captured. We see that the solution
here can continue since it passes
1 22
du
through a minima where d„ = 0 and grows past this point. . 104
11.2 This shows the scenario considered for the uniform case. We see
that u0 = u2 = 1/R where R is the radius of the star. . . . . . . 105
11.3 Lane-Emden (orange) has a flat density profil close to the core
as well as in the outer layers. This is contrasted with the steep
function of r12 which goes to infinity as the radius goes to 0. . . 108
A.1 Illustrations for how geometrical accretion is envisioned. . . . . 116
List of Tables
7.1
7.2
7.3
7.4
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
9.1
Emission rates and powers for the dominant angular modes . . .
Total Power Output from Table 7.1 . . . . . . . . . . . . . . . .
Comparison between power outputs in ergs per sec for a black
hole of mass M = 5·1014 g from interpolation functions and those
calculated in [3] shown in Table 7.1 . . . . . . . . . . . . . . . .
Masses, number of species and polarizations for particles emitted
Black Hole Masses and Temperatures . . . . . . . . . . . . . . .
WIMP masses with the corresponding power output from up,
down, and strange quark along with the electron. The power
output here are in ergs sec≠1 , matching the units seen on Figures 7.1 and 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . .
Blocking of Quark Species and Electrons . . . . . . . . . . . . .
Comparison between the emission of photons based on the SBSH
power law and the data fitted from Page[3]. . . . . . . . . . . .
Critical Masses considering Bondi accretion . . . . . . . . . . .
Blocked percentage of power output at the critical mass for
Fermi blocking and boosted Fermi blocking with Bondi accretion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Critical masses for Non-Bondi accretion . . . . . . . . . . . . .
Blocked Percentage at the critical masses for Non-Bondi accretion
ks (a) for each of the three spin values considered. These change
with the critical angular momentum. . . . . . . . . . . . . . . .
x
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91
List of Tables
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
Critical masses for a black hole with a = 0.5 which accretes
matter by Bondi accretion. . . . . . . . . . . . . . . . . . . . . .
Blocked Percentages for the critical masses for a black hole using
Bondi accretion at a = 0.5. . . . . . . . . . . . . . . . . . . . . .
Critical masses for geometrical accretion in the case of a black
hole with angular momentum equal to half the critical value. . .
Blocked Percentages for each species of particles for geometrical
accretion at the critical masses with a = 0.5. . . . . . . . . . . .
Critical masses for a close the maximally rotating black hole
using Bondi accretion. . . . . . . . . . . . . . . . . . . . . . . .
Blocked Percentage for each particle species at the critical masses
for a close to maximally rotating black hole using Bondi accretion.
Critical masses for geometrical accretion onto a close to maximally rotating black hole. . . . . . . . . . . . . . . . . . . . . .
Blocked Percentages at the critical masses for geometrical accretion onto a close to maximally rotating black hole. . . . . . . .
xi
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Introduction
Neutron stars are some of the densest objects in the Universe, their surface
gravity and high densities means that they are possible candidates for the
entrapment of dark matter particles. Dark matter was first postulated in
1933 and has since become a major field of research. Recent measurements
of the Cosmic Microwave Background shows that dark matter may be ≥
85% of all matter in the Universe. Given that dark matter does not interact
with the electromagnetic force, hence the name dark, the detection of dark
matter particles is complicated. One of the methods of detection comes
by gravitational lensing of objects where dark matter bends light from the
source before it reaches the Earth. Other experiments like DAMA and
CoGent works on direct detection of dark matter. It is possible that dark
matter may open up avenues for physics beyond the Standard Model since
its makeup is as of yet still unknown. This thesis will consider dark matter
to be consisting of Weakly Interacting Massive Particles, WIMPS. Should
these become entrapped within a neutron star they may possibly collapse
into a black hole.
Black hole are amongst the most fascinating objects in physics. Predicted by Einsteins theory of general relativity, their existence is now supported by observational evidence. In his theory of general relativity, Einstein showed that mass bends spacetime; the higher the mass the larger
the curvature of local spacetime. Black holes are areas of spacetime where
the laws of physics, as we know them, might no longer be valid. This is
due to the fact that they might have a high mass compressed into a singular point, the singularity. This results in an extreme curvature of their
surrounding spacetime. The "edge" of the black hole, known as the event
horizon, ensures that the singularity remains hidden from the rest of the
Universe. Once beyond this limit even light is no longer able to escape
the extreme pull of the black hole. This is due to spacelike and timelike
directions switching roles forcing the object to move in a direction of decreasing radius before eventually hitting the singularity. The fact that even
light can not escape a black hole is what the term "black" derives from.
1
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List of Tables
What might not be obvious from this, is the fact that black holes are very
much thermodynamical systems. They have associated thermodynamical
quantities corresponding to entropy and temperature in the form of their
surface area and surface gravity respectively. While even light is not able to
escape there is a phenomena in which the black hole emits energy. This is
known as Hawking radiation, named after Stephen W. Hawking. By studying quantum field theory in the black hole background, he found that they
emit energy as would be expected from a thermal system. The radiation is
emitted with a characteristic blackbody spectrum, thus when considering
quantum mechanics black holes are no longer "black" but obey the laws of
thermodynamics, albeit versions which have been modified to them. While
this radiation is consistent with that of a blackbody exactly at the event
horizon, the spacetime geometry around the black hole will modify this by
the so-called greybody factors. This results in observers at infinity measuring not a perfect black body spectrum but a modification of this since
greybody factors are dependent upon both geometry and frequency. Radiation scales as the inverse of the black hole mass squared; the more massive
a black hole is, the less energy it will loose due to radiation. Further limits
on the radiation may occur if the black hole is not situated in the vacuum
of space, but at the heart of a star. In the case of a neutron star, the degenerate matter surrounding the black hole will impose a Fermi surface on
emitted fermions which will block any fermion with energy below a given
cutoff.
Not only will the black hole radiate energy but it may accrete energy
from its surroundings. Due to the gravitational pull of the black hole, matter beyond the event horizon may be pulled into paths which will take it
inside the event horizon eventually leading to an increase in the black hole
mass. Where the radiation goes as the inverse of the black hole mass, accretion scales as the mass of the black hole squared. This accretion of mass
may balance the radiation of energy leading to either the evaporation of the
black hole, a steady state where the black hole has reached a critical mass
or where accretion ultimately wins out over Hawking radiation. In the case
where accretion wins over Hawking radiation, the black hole will continue
to grow and might eventually devour the entire star if situated within one.
Otherwise, the black hole may simply evaporate before it has any significant
impact on its surroundings due to its low mass. Accretion can be limited
by several factors, such as the rotation of its surrounding matter which may
create an accretion disc or even a torus, but also by the spin of the black
hole itself. Don N. Page showed in 1977 that Hawking radiation is coupled
to the spin of the black hole. An increase in the angular momentum of the
black hole leads to an increase in Hawking radiation. This presents an inter-
List of Tables
3
esting scenario in which the accretion of angular momentum might change
not only the future accretion of matter but also the emission of energy.
Motivation
We see that the evolution of a black hole within a neutron star might
put limits on the mass of dark matter particles since the black hole may
eventually consume the star. As such, we are interested in the critical mass
of the black hole where accretion and radiation are perfectly balanced. This
will help in determining the WIMP mass needed to achieve a stable black
hole depending on the type of accretion used. Lighter WIMPs will be able
to create black holes heavy enough to continue accretion and growing in
mass. Since neutron star are rapidly spinning objects, their rotation may
significantly change the conditions for accretion and prevent small black
holes from even reaching this stage.
The goal of this thesis is to investigate how the critical mass of a black
hole changes when accretion and radiation is considered under different
circumstances. To do so, we will make use of the pioneering work of Page
to model the radiation from both Schwarzschild and Kerr black holes while
using the hydrodynamical, spherical accretion described by Bondi and a
more conservative geometrical accretion. Furthermore, we will attempt
to solve the accretion of dark matter particles onto the neutron star by
considering elastic Schwarzschild scattering in order to see if this might
alter the minimal cross section of particles which can be captured by a
neutron star.
Outline
This thesis has been written assuming that the target audience consists
of master students of theoretical physics. Therefore, anyone with a basic
knowledge of general relativity will be able to read it. The outline of the
thesis is as follows:
Part I
This part contains the basic background material for the ideas introduced
in the later parts of the thesis. Its inclusion is done in the hopes that this
thesis will be as self-contained as possible and, that it may provide a quick
introduction for people who are just starting to learn about the subjects
contained within. Chapter 1 contains the basics of stellar evolution leading
4
List of Tables
up to the formation of a neutron star. It further contains information about
their formation and their properties. In Chapter 2 we review the basics
of black holes in general relativity and introduce the static Schwarzschild
black hole as well as the rotating Kerr black hole. Finally, we provide an
introduction how dark matter was originally inferred in Chapter 3 as well
as discussing how dark matter might be accreted unto a neutron star and
eventually form a black hole.
Part II
In this part we cover the thermodynamics of black holes before moving to
Hawking radiation and how it is modified by greybody factors. Chapter 4
introduces the four laws of black hole mechanics and how these are analog to the laws of thermodynamics we know from classical theory. It also
provides a short introduction to the Holographic principle proposed by ’t
Hooft and how the entropy of a black hole differs between the description
from classical gravity and the description introduced by Bekenstein and
Hawking. In Chapter 5 we introduce the concept of Hawking radiation
by following Hawkings original argument before briefly covering how this
radiation might lead to the loss of information. Finally, in Chapter 6 we
introduce the greybody factors and how these arise.
Part II
This part contains the calculations of critical masses for different types
accretion as well for different values of critical angular momentum. First,
we show how the radiation was modeled using the work of Page in Chapter 7.
This will introduce the spectrum of Hawking radiation for three different
values of spin along with a comparison between our interpolation function
and the theoretical values before discussing how new particles might be
added to the emission. Then in Chapter 8 we discuss how the critical mass
of a black hole will change with different types of accretion. We will also
compare the WIMP masses found here with limits by DAMA and CoGent as
well as other similar works. Third, we consider how the rotation, not only of
the black hole but of the star, might change both accretion and radiation in
Chapter 9. This will be done by considering how viscosity might transport
angular momentum and ensure a spherical accretion, while also taking into
account the coupling between the critical angular momentum parameter of
the black hole and the emission of particles. In Chapter 10 we will compare
the 1 PeV neutrino peak observed by IceCube with Hawking radiation from
a black hole where the neutrino peak corresponds to this value. Finally,
List of Tables
5
in Chapter 11 we will present an attempt of using elastic Schwarzschild
scattering as a means of accreting dark matter onto a neutron star.
We conclude by analyzing and discussing our results. Finally, the appendices contain material that may help in clarifying parts of the thesis.
Part I
Neutron Stars, Dark Matter
and Black Holes
7
Chapter 1
Neutron Stars
This chapter will serve as a review of the basic concepts regarding stellar
evolution and neutron stars. This is included in the hopes that this thesis
will be as self-contained as possible since neutron stars play a significant
role in the further chapters due to some of their unique properties for the
potential study of dark matter.
1.1
Stellar Evolution: The Road to a
Neutron Star
Stars, such as our sun, are born from the gravitational collapse of dense regions within molecular clouds in interstellar space. Such regions are sometimes referred to as stellar nurseries. Interstellar clouds of a gas remain in
hydrostatic equilibrium for as long as the kinetic energy derived from the
gas pressure is balanced by the potential energy of the internal gravitational
force, this is expressed through the viral theorem. The viral theorem states
that the kinetic energy is equal to twice the potential energy
2ÈT Í = ≠
N
ÿ
n=1
ÈFn · rk Í,
(1.1)
where Fn is the force acting on the nÕ th particle located at position rn .
The viral theorem will become important again at later times during this
chapter. Thus for the cloud to remain in equilibrium, the gravitational
potential energy must equal twice the internal thermal energy. Should a
cloud be massive enough the pressure from the gas is insufficient to support
it, gravitational collapse will occur.
When the density of the in falling matter from the surroundings have
dropped below about 10≠8 g, the surroundings are sufficiently transparent
9
10
CHAPTER 1. NEUTRON STARS
to allow the radiation of energy away from the forming protostar. As the
forming protostar continues to collapse it will eventually get hot enough
for the internal gas pressure to support against further gravitational collapse forming a protostar [4]. Through further accretion of the surrounding
cloud, the protostar will continue to increase its mass and thus increase its
temperature by the viral theorem. Once the core of the protostar reaches
a temperature of 10 million kelvins, fusion of hydrogen starts to take place
and the protostar enters the main sequence as a star not unlike our Sun.
This first stage is known as the hydrogen burning stage and is determined
only by the amount of hydrogen available to fuse into helium. The PP1
chain for the fusion of hydrogen to helium is shown in Figure 1.11 .
Figure 1.1: This figure shows the Proton-Proton 1 chain.
Thus it might be expected that more massive stars will have far longer
lifetimes than lower mass stars, as they have more hydrogen available to
fuse. However the opposite is in fact true, due to the fact that the more massive stars also radiate away much more energy thus lowering their lifetimes
1
Source: http://en.wikipedia.org/wiki/File:FusionintheSun.svg
1.1. STELLAR EVOLUTION: THE ROAD TO A NEUTRON STAR 11
and are only expected to remain on the main-sequence for a few million
years contrasted with low mass stars which may have lifetimes of a trillion
years.
Once the star has used up most of its hydrogen supply, the internal gas
pressure will no longer be sufficient to balance the gravitational force and
will facilitate a new collapse of the star. As with the protostar, the collapse
will cause the temperature of the star to rise as described by the viral theorem. Should the mass of the star be sufficient that the core will reach the
new ignition temperature before the core becomes degenerate, the fusion of
helium will start to take place in the core. The core is then surrounded by
a shell of hydrogen burning. However for masses below approximately half
a solar mass, the star will never become hot enough to fuse helium in its
core [4].
The radiation pressure will begin to balance the gravitational forces
compressing the star and the star will start expanding to a larger radius
than before. This is called the Red Giant stage, here the star will continue
to fuse helium in its core until such a time that this supply is also mostly
used up. From here the process of collapse starts again, potentially reaching
temperature high enough to start the next stage of fusion in the core. At
this stage the star will fuse helium into oxygen and carbon through the
triple-alpha process shown in Figure 1.22 . For main-sequence stars with a
Figure 1.2: This figure shows the triple alpha process which fuses helium
into carbon.
mass between ≥ 0.5 ≠ 8M§ , further collapse will not create temperatures
sufficient for the fusion of carbon into neon. The remnant of this is known as
a white dwarf. Further fusion can occur until the the star starts producing
2
Source: http://en.wikipedia.org/wiki/File:Triple-Alpha_Process.png
12
CHAPTER 1. NEUTRON STARS
elements of the iron group: iron, nickel and cobalt. Beyond this group,
fusion no longer yields energy but requires energy. Thus at this stage fusion
in the core stops and the collapse begins anew.
1.2
Formation of a Neutron Star
Formation of a neutron star begins with a star of proper mass above 9M§ .
When the star reaches the red giant stage it has exhausted its sources of
energy as its core is formed by iron, nickel and cobalt. At this stage no
further fusion takes place since such processes requires energy and thus no
radiation pressure is available to counter the gravitational forces within the
star. By the viral theorem, as the star shrinks due to the lack of radiational
pressure, the core temperature rises but can not ignite further burning of
elements as these will not result in an energy yield, causing the star to
continue contracting. The temperature of the core will eventually become
high enough to produce photons of sufficient energies to initiate the breaking
of the star’s constituent elements into neutrons, protons and electrons. Free
neutrons can decay into protons and electrons by the following process
n æ p + e≠ + ‹¯e .
(1.2)
Protons and electrons can also recombine into neutrons by
p + e≠ æ n + ‹e .
(1.3)
For these reactions to be in thermal equilibrium, the chemical potentials
must be balanced such that
µn = µe + µp .
(1.4)
Using statistical mechanics we can write the density of states as
dN = gV p(k)
d3 k
4fik 2 dk 2 gV
gV
=
p(k)
= p(k) 2 k 2 dk,
3
3
8fi
2fi
(2fi)
(1.5)
where g is the degeneracy of the particles and p(k) is the distribution of the
particles[5]. Since the star at this point only consists of protons, neutrons
and electrons, it can be described by Fermi statistics
p(E) =
Exp
Ë
1
E≠µ
T
È
+1
.
(1.6)
1.3. PROPERTIES OF NEUTRON STARS
13
At this stage the temperature of the star is of the order 1010 K ¥ 1M eV
and the chemical potential µ ¥ 0.5GeV . This means that
T
π 1,
µ
(1.7)
which allows us the rewrite the Fermi-Dirac distribution in the relativistic
case where kF = EF , so the distribution now gives a step function for which
p(k) =
Y
]0,
[1,
k > kF
.
k Æ kF
(1.8)
where we have introduced the radius of the Fermi sphere in momentum
phase space, kF . Thus integrating above the Fermi momentum gives 0,
thus we can limit the integral up to the Fermi momentum. The number
density of states then becomes
n=
N
g k3
= 2 F.
V
2fi 3
(1.9)
We require that the star be electrically neutral, thus there must be an equal
amount of protons and electrons
ne = np
(1.10)
From this it is possible to derive that neutrons make up 90% of the mass
of the star, with protons making up the remaining 10%. As the pressure further increases due to the continued collapse of the star, it becomes
more energetically favorable for protons and electrons to recombine following Equation (1.3). However, the collapse continues further increasing the
pressure and making the neutrons degenerate. The gravitational pull of the
neutron core of the star becomes great enough that the outer layers hits
the core with a velocity so great that the star no longer can remain stable.
The potential energy of the core is released in a Type II supernova, blowing
away the outer layers of the star leaving only the neutron core. This was
first postulated by Zwicky and Baade[6] in 1934.
1.3
Properties of Neutron Stars
Due to their small size and relatively high mass, neutron stars are amongst
the densest objects in the universe. The outer regions of the star may have
densities as low as 109 kg/m3 , increasing towards the core where the density
14
CHAPTER 1. NEUTRON STARS
may reach 1017 kg/m3 . Another property of the neutron star, caused by the
mass, is that the surface gravity is several orders of magnitude larger than
that of the Earth. This makes them ideally suited for the purpose of this
thesis, as dark matter particle would have a good chance of getting trapped
in the neutron star because of the high gravity .
The equation of state for neutron stars, is not known as of yet. Some
predict that the surface of the star may be a solid crust or, for a young
neutron star with a surface temperature > 106 K, fluid and that the crust
solidifies as the star cools with age. The crust may be formed by heavier nuclei and as the density increases towards the center, the number of
neutrons in the nuclei increases, likely kept stable by the extreme pressure.
The nature of the core is still not known, but it could allow for exotic physicals states, such as pion condensate or degenerate quark matter[5, 7]. The
possible structure of a neutron star is shown in Figure 1.33
Since most stars rotate before the collapse into a neutron star, a neutron
Figure 1.3: Illustration of the possible structure of a neutron star.
star will also rotate. The radius of a neutron star is roughly 10km and due
to conservation of momentum the angular velocity of the neutron star is
much greater than that of its parent star. Observed neutron stars have periods ranging from several seconds down to a couple of milliseconds. While
neutron stars do slow down over time, this process is likely to be very slow.
3
Source:http://heasarc.gsfc.nasa.gov/docs/objects/binaries/neutron_star_structure.html
1.3. PROPERTIES OF NEUTRON STARS
15
Neutron stars posses some of the strongest magnetic fields in the universe
and these fields radiate away energy, slowing the rotation of the star over
time. These magnetic fields are also believed to accelerate particles at the
magnetic poles of the star, this radiation is primarily in the radio and x-ray
region of the electromagnetic spectrum. As the magnetic poles need not be
aligned with the axis of rotation, it is possible for this beam of energy to
"pulse" as it sweeps by the line of sight of an observer at the same rate as
the rotation of the neutron star, such neutron stars are referred to as pulsars.
While the neutron star slows down over time, so called giltches have
been observed where the star suddenly spins up instead of down, increasing
its angular velocity. Such glitches are thought of, in some theories, as being
the effect of star quakes. Rotation of such massive objects will have an
impact on the accretion rate of matter onto the object, thus rotation will
play an important role in this thesis as it might significantly alter the rate
of accretion, even for a black hole.
Chapter 2
Introduction to Black Holes
This chapter will review the basics of black holes in general relativity, including their general properties. As with the previous chapter, we include
this chapter in the hopes that the thesis be as self-contained as possible,
since these are the types of black holes which we will be working with
throughout this thesis. This section will not go into great depth with the
mathematical aspects. An excellent reference for a more detailed approach
see [8].
2.1
Schwarzschild Black Holes
With his theory of special relativity in 1905, Einstein showed that space
and time must be considered equally. Then in 1915, he published his theory of general relativity, completely changing the way in which we look at
the Universe and our understanding of how gravity works. By using Riemann geometry, he showed that gravity can be regarded as the curvature of
spacetime due to the presence of matter or, equivalently, energy. To derive
Einstein’s equation of general relativity in nonvacuum, we must consider
1
an action of the form S = 16fiG
SH + SM . Here
SH =
⁄
Ô
≠gRd4 x,
(2.1)
is the Hilbert action (sometimes known as the Einstein-Hilbert action)
where we R is the Ricci scalar and g is the determinant of the metric
tensor. The Ricci scaler is the simplest curvature invariant of a Riemann
manifold. The Hilbert action is the gravitational part of S and
SM =
⁄
Ô
≠gLM d4 x
17
(2.2)
18
CHAPTER 2. INTRODUCTION TO BLACK HOLES
is the matter-energy fields term. The problem at hand will determine the
Langrangian density LM used to define SM . Combining Equations (2.1)
and (2.2) gives the total action to be considered
⁄
Ô
1 ⁄ Ô
4
S=
≠gRd x +
≠gLM d4 x.
16fiG
(2.3)
By varying the action, Equation (2.3), with respect to the inverse g µ‹ we
obtain
3
4
1 ”S
1
1
1 ”SM
Ô
=
Rµ‹ ≠ Rgµ‹ + Ô
= 0.
µ‹
≠g ”g
16fiG
2
≠g ”g µ‹
(2.4)
We now define the energy-momentum tensor to be
≠2 ”SM
Tµ‹ = Ô
,
≠g ”g µ‹
(2.5)
which allows us to recover the complete Einstein equation
1
Rµ‹ ≠ Rgµ‹ = 8fiGTµ‹ ,
2
(2.6)
where Rµ‹ is the Ricci tensor and gµ‹ is the metric tensor of the spacetime.
Equation (2.6) tells us how the geometry of spacetime (the left-hand side)
will react to the presence of matter and energy (the right-hand side). It
is sometimes useful to rewrite Equation (2.6) by taking the trace of Equation (2.6), this yields, after rearranging some terms
3
Rµ‹ = 8fiG Tµ‹
4
1
≠ T gµ‹ .
2
(2.7)
Equation (2.7) is often useful when working in vacuum, where Tµ‹ = 0,
since this allows us to rewrite Einstein’s equation in the convenient form
Rµ‹ = 0
(2.8)
Physicists started working on finding solutions to the equation of general
relativity soon after it was published. The most obvious application is to a
spherically symmetric gravitational field. This particular scenario was the
solution Karl Schwarzschild found, the first analytical solution in vacuum
to the equation of general relativity. He considered a spherically symmetric,
stationary body of mass M and found that the generated metric, in spherical
coordinates (t, r, ◊, „) , is given by
3
ds2 = ≠ 1 ≠
4
3
2GM
2GM
dt2 + 1 ≠
r
r
4≠1
dr2 + r2 d
2
2,
(2.9)
2.1. SCHWARZSCHILD BLACK HOLES
19
where d 22 = d◊2 + sin2 ◊d„2 is the metric on a unit 2-sphere S 2 . Equation (2.9) is known as the Schwarzschild metric and it can be shown that
it is the unique vacuum solution with spherical symmetry and that it is
time-independent[8].
This is known as Birkhoff’s theorem, for a proof the reader is referred
to [8]. It can be seen from Equation (2.9) that far from the black hole, as
r æ Œ, we recover the Minkowski metric, thus the Schwarzschild metric is
asymptotically flat.
From Equation (2.9) it is immediately apparent that the metric is singular at r = 0 and at r = 2GM . The former is a true singularity of spacetime,
the latter, however, is rather an artifact due to the choice of coordinates, as
can be checked by calculating an invariant quantity (following [8] we choose
the curvature invariant scale)
Rµ‹fl‡ Rµ‹fl‡ =
48G2 M 2
r6
(2.10)
At r = 0 this scalar goes to infinity and we regard it as a true singularity.
At r = 2GM , however, this scalar retains a finite value, it can be proven
that this goes for all the curvature invariants. This suggests that r = 2GM
is not a true singularity but only appears to be so in Equation (2.9) due to
the choice of coordinate system. One way to understand the geometry of
spacetime is to explore the causal structure, as defined by the light cones.
A light cone is the path light, emanating from a single event, traveling in
all directions might take through spacetime. This even is sinuglar in space
and time. One cone represents all futuredirected paths, while the other
represents all pastdirected paths. Therefore, we consider radial null curves,
those for which ◊ and „ are constant and ds2 = 0:
3
4
3
2GM
2GM
ds = 0 = ≠ 1 ≠
dt2 + 1 ≠
r
r
2
4≠1
dr2 ,
(2.11)
from which it is seen that
3
dt
2GM
=± 1≠
dr
r
4≠1
.
(2.12)
Equation (2.12) measures the slope of the light cones on a spacetime diagram of t-r plane and it can be seen that at large r, the slope is ±1,
corresponding to a flat space, but as we approach r = 2GM we get that
dt/dr æ ±Œ and the light cones "close up". A light ray approaching
20
CHAPTER 2. INTRODUCTION TO BLACK HOLES
r = 2GM would thus never seem to get there, at least not in the coordinate system we have chosen so far, instead it would asymptote to this radius.
This can be solved by defining
a new
1
2 coordinate set, the so-called tortoise
r
ú
coordinate r = r + 2GM ln 2GM ≠ 1 , using these we can write the metric
to a form which is clearly well-behaved at r = 2GM
4
2
2GM 1
ds = 1 ≠
≠dt2 + drú2 + r2 d
r
2
3
2
,
(2.13)
where r is thought of as a function of rú . In this metric, the light cones no
longer close up, however the surface of interest at r = 2GM has just been
pushed to infinity. The next move is to define coordinates that are more
naturally adapted to the null geodesics. These could be
v = t + rú
u = t ≠ rú .
Then infalling radial null geodesics are characterized by v = constant, while
the outgoing ones satisfy u = constant. Going back to the original radial
coordinate r, but replacing the timelike coordinate t with the new coordinates v, this is known as the Eddington-Finkelstein coordinates1 and in
terms of these the metric is
3
ds2 = ≠ 1 ≠
4
2GM
dv 2 + (2dvdr) + r2 d
r
2
(2.14)
In the Eddington-Finkelstein coordinates the condition for null curves is
solved by
Y
(infalling)
dv ]0,
1
2≠1
=
.
(2.15)
2GM
dr [2 1 ≠ r
, (outgoing)
In this coordinate system the light cones remain well-behaved at r = 2GM
and this surface is at a finite coordinate value. Likewise there is no longer
a problem in tracing the paths of null or timeline particles past the surface,
but something important has happened in the coordinate change. While
the light cones do no close up, they do start to tilt, such that for r < 2GM
all future-directed paths are in the direction of decreasing r.
1
It should be noted that the tortoise coordinates are not the best coordinates to study
the full Schwarzschild metric. These are called Kruskal coordinates and the reader is
referred to [8]
2.1. SCHWARZSCHILD BLACK HOLES
21
Now, while being locally perfectly regular, the surface r = 2GM now
globally functions as the point of no return, once crossed it is no longer
possible to come back. This is called the event horizon and the radius
rS = 2GM is known as the Schwarzchild radius. Since nothing can escape
the event horizon, it is impossible for us to see inside it, hence the name
black hole. The event horizon serves as a spherical surface which remarkably divides spacetime in two regions r > rS and r < rS which are causally
disconnected.
In the case of the Schwarzschild solution, objects located outside the
event horizon will orbit the black hole much as it is the case for stars and
planets. It will not suck in everything around it anymore than the Sun
does. However, once an object crosses the event horizon, it will never be
able to again escape the gravitational force of the black hole. Inside the
event horizon timeline directions become spacelike and vice versa. Thus the
light cone of any object located below rS will become completely tilted and
the object will move towards the singularity at r = 0. Because of this, not
even light can escape from the black hole and the two spacetime regions
defined by the event horizon are causally disconnected. While it is possible
to pass the event horizon from infinity, the region beyond the event horizon
is causally disconnected from infinity since it is impossible to escape once
the event horizon is crossed.
For ordinary objects, rS is much smaller than the objects physical radius or size. In these cases we need not worry about the event horizon
because the Schwarzschild metric only applies in empty space, which we
are no longer in. However, if an object undergoes gravitational collapse,
as in the case for stars (mentioned in Chapter 1), the physical radius may
become smaller than the event horizon, forming a black hole. The empty
space surrounding the black hole, both inside and outside the event horizon,
is correctly described by the Schwarzschild metric. Thus Equation (2.9) can
be used to described the empty space outside of a star, a black hole or even
a planet, as well as the interior of a black hole.
We will now look at a Kruskal diagrams of the Schwarzschild solution
in Kruskal coordinates. These coordinates have not been introduced as
our interest is purely the diagram seen in Figure 2.1. This diagram represents the maximally extended Schwarzschild solution. While the original
Schwarzschild coordinates were useful for r > 2GM , this is only part of the
manifold covered by the Kruskal diagram. Here T is the timelike coordinate
and R is the spacelike coordinate. We see that surfaces with r = constant
22
CHAPTER 2. INTRODUCTION TO BLACK HOLES
Figure 2.1: The Kruskal diagram showing the Schwarzschild solution in
Kruskal coordinates, where all light cones are at ±45¶ [1].
are hyperbolae and that the event horizon is no longer infinitely far away.
It is convenient for us to divide the diagram into four regions. This is shown
in Figure 2.22 .
Region I corresponds to r > 2GM , the region in which our original
coordinates are well defined. If we follow future-directed null rays we end
up in region II and by following past-directed null rays we reach region
III. Had we instead explored spacelike geodesics we would have reached
region IV. The maximally extended Schwarzschild geometry reveal some
interesting spacetime. Region II is what we think of as the black hole.
Once something travels from region I into region II, it can never return.
Every future-directed path in region II ends up hitting the singularity at
r = 0. Thus once you have entered the event horizon there is no escape,
not only can you not escape back to region I but there is nothing to stop
you from moving in the direction of decreasing r as this is the timelike
direction. Regions III and IV are interesting and somewhat unexpected.
Region III is the time-reverse of region II, meaning that it is a part of
spacetime from which things can escape to us, but we can never get there.
This can be thought of as a white hole. The boundary of region III is
2
Source: http://commons.wikimedia.org/wiki/File:KruskalUniverse.png
2.2. KERR BLACK HOLES
23
Figure 2.2: Regions of the Kruskal diagram.
the past event horizon constrasted with the future event horizon as the
boundary of region II. Region IV can not be reached from region I, neither
forwards nor backwards in time. It is seperate asymptotically flat region of
spacetime mirroring our own.
2.2
Kerr Black Holes
Kerr black holes are different from Schwarzschild black holes in that they are
stationary but not static, they rotate. Finding exact solutions for the metric
is much more difficult than in the Schwarschild case, since the solution will
no longer have spherical symmetry. Instead we should look for solutions
with axial symmetry around the axis of rotation that also happen to be
stationary (timelike Killing vector). Although the Schwarzschild solution
was found soon after the publication of general relativity, the solution for a
rotating black hole was not found until 1963 by Kerr. The resulting metric,
the Kerr metric, is given by:
A
B
2GM r
2GM ar sin2 ◊
2
ds = ≠ 1 ≠
dt
≠
(dtd„ + d„dt)
fl2
fl2
5
6
2
fl2 2
sin2 ◊ 1 2
2
2
2 2
2
2
+ dr + fl d◊ +
r +a
≠ a sin ◊ d„2 ,
fl2
2
where
(r) = r2 ≠ 2GM r + a2
(2.16)
(2.17)
24
and
CHAPTER 2. INTRODUCTION TO BLACK HOLES
fl2 (r, ◊) = r2 + a2 cos2 ◊.
(2.18)
The constants M and a parametrize the possible solutions. a is the angular
momentum per unit mass,
a = J/M,
(2.19)
which runs from 0 (Schwarzschild black hole) to 1 (Maximally rotating
Kerr black hole). It is straightforward to see that as a æ 0 Equation (2.16)
reduces to the Schwarzschild metric. Keeping a fixed and letting M æ 0,
however, recovers flat spacetime but not in ordinary polar coordinates. The
metric now becomes
2
22
1
(r2 + a2 cos2 ◊) 2 1 2
2
2
2
2
2
dr
+
r
+
a
cos
◊
d◊
+
r
+
a
sin2 „d„2 ,
(r2 + a2 )
(2.20)
where the spatial part is flat space in ellipsoidal coordinates. They are
related to Cartesian coordinates in Euclidean 3-space by
ds2 = ≠dt2 +
1
x = r 2 + a2
1
y = r 2 + a2
z = r cos ◊
21/2
21/2
sin ◊ cos „
sin ◊ sin „
(2.21)
It can be shown using Killing vectors, that the metric is stationary but not
static. A static metric is defined on surfaces where t is constant whereas
all the components of a stationary metric are independent of t. This makes
sense since the black hole is spinning so it can not be static, but it is spinning in exactly the same way at all times so it is stationary. An alternative
argument is that the metric can not be static because it is not time-reversal
invariant, since time-reversal will reverse the angular momentum of the
black hole.
The choice of coordinates for the Kerr metric are such that the event
horizons occur at those fixed values of r for which g rr = 0. Since g rr = /fl2 ,
and fl2 Ø 0, this occurs when
(r) = r2 ≠ 2GM r + a2 = 0.
(2.22)
There are three solutions to Equation (2.22): GM > a, GM = a and
GM < a. These will be covered in the next three sections.
2.2. KERR BLACK HOLES
25
Solution 1: GM < a
In this case Equation (2.22) does not hold because will always be positive.
This has the effect of removing the event horizon thus leaving the metric
completely regular all the way to r = 0, which is still a true singularity.
The absence of an event horizon implies that the singularity located at
r = 0 is not hidden from us. This is called a naked singularity. Roger
Penrose, in 1969, made a conjecture called the cosmic censorship stating
that naked singularities, apart from the one at the Big Bang, do not exist.
His conjecture is based on the fact that if such a naked singularity existed,
things happening at the singularity itself would affect our universe.
The laws of physics, as we know them, break down at the singularity,
this would lead to us loosing predictive power. While Penrose’s conjecture
has yet to be proven, several studies suggest that naked singularities do not
form in the gravitational collapse of an object. This situation,GM < a, can
also be looked at in another way by noticing that the contribution from
angular momentum is greater than the total energy of the black hole which
is considered to be unphysical.
Solution 2: GM = a
This solution is also known as a extremal Kerr black hole because it has
the maximal angular momentum allowed given its mass. It should be noted
that such solutions are highly unstable, since the addition of even a tiny
amount of mass will bring it to Solution 3, detailed in the next section.
From Equation (2.22) it is possible to find that there will only be one event
horizon, located at r = GM .
Solution 3: GM > a
In this case there are two radii at which Equation (2.22) hold, given by
Ô
r± = GM ± G2 M 2 ≠ aa .
(2.23)
Both of these radii are null surfaces which will turn out to be event horizons.
Carroll [8] gives solutions for two surfaces given by
(r ≠ GM )2 = G2 M 2 ≠ a2 cos2 ◊,
(2.24)
for the stationary limit surface while the outer event horizon is given by
(r+ ≠ GM )2 = G2 M 2 ≠ a2 .
(2.25)
26
CHAPTER 2. INTRODUCTION TO BLACK HOLES
Thus there is a region between these two surfaces, which is known as the ergosphere, inside which it is only possible to move in the direction of rotation
of the black hole (the „ direction) while it is still possible to move toward
or away from the event horizon and entirely possible to exit the ergosphere.
It should be noted that the outer stationary limit is not an event horizon,
rather there is another event horizon, the inner event horizon, beyond the
outer event horizon. It is important to note that the singularity does not
occur at r = 0 in this spacetime but rather at fl = 0. Since fl is given by
Equation (2.18) which is the sum of two nonnegative quantities, this can
only be zero when both quantities are zero
r = 0,
◊=
fi
.
2
(2.26)
This singularity is not a point in space but rather a disk for which the set of
points r = 0, ◊ = fi/2 is the edge of the disk. Going inside the ring reveals
that this would cause a test particle to exit to another asymptotically flat
spacetime which is not an identical copy of the one it came from. This
new space would be described by the Kerr metric with r < 0, leading to
never vanishing and no horizons. Not only this but the region near the
ring singularity possess closed timelike curves [8], since such trajectories are
closed this would allow someone to meet themselves in the past.
Figure 2.3: Figure showing the cross section of a Kerr black hole. Here the
ergosphere is clearly visible between the outer and inner event horizons.
We now return to the ergosphere since this is the region of another
interesting physical property of a Kerr black hole, even before reaching the
2.3. ACCRETION
27
event horizon. Considering a photon emitted in the „ direction at a radius
in the equatorial plane (◊ = fi/2). There are two possible solutions for a null
trajectory, either the photon will move in the direction in which the black
hole is rotating but more interestingly if the photon moves in the opposite
direction. In this case it will not move at all in such a coordinate system,
in fact for an outside observer it will look stationary. This phenomena is
known as frame dragging. More massive particles must be dragged along
with the rotation of the black hole once these are inside of the stationary
limit surface.
2.3
Accretion
Accretion onto a black hole will be considered in several ways in this thesis,
including Bondi accretion. This section will primarily focus on Bondi accretion, following [7]. This is based on the assumption that the effective mean
free path of a gas particle collision is small enough that the flow can be
considered hydrodynamic. This corresponds to the typical dynamical conditions found in the interstellar medium and as such accretion onto compact
object will be hydrodynamic. Considering the steady, spherical accretion
of the surrounding gas onto a stationary, non-rotating black hole of mass
M, assuming that the gas flow is adiabatic to first approximation, the gas
can the be characterized as a polytropic fluid obeying
P = Kfl ,
(2.27)
where fl is the rest-mass density, P the pressure, K is a constant and is
the adiabatic index. The speed of sound can then be defined everywhere
to be a © (dP/dfl)1/2 = ( P/fl)1/2 . It is assumed that the gas is at rest at
infinity where the density is flΠ, the pressure is PΠand the sound speed is
aŒ . At very larges distances, r ∫ GM , the basic characteristics of the flow
can be described reasonably well by Newtonian gravity. What distinguishes
accretion onto a black hole from that of an uncollapsed star with a hard
surface, is that the black hole imposes some unique conditions at radii near
the Schwarzchild radius rS = 2GM [7].
Besides Equation (2.27), the flow is completely governed by the continuity equation
1 d 1 2 2
Ò · flu = 2
r flu = 0,
(2.28)
r dr
and the Euler equation
u
du
1 dP
GM
=≠
≠ 2 .
dr
fl dr
r
(2.29)
28
CHAPTER 2. INTRODUCTION TO BLACK HOLES
These equations holds for steady-state, spherical flow where the inward
radial velocity is denoted by u > 0. Integrating Equation (2.28) directly
gives an equation for Ṁ
Ṁ = 4fir2 flu = constant,
(2.30)
which is independent of r. Likewise Equation (2.29) can be integrated
to yield the Bernoulli equation, using Equation (2.27) and the boundary
conditions at infinity
1 2
u +
2
1
GM
a2 ≠
=
≠1
r
1
a2 = constant.
≠1 Œ
(2.31)
Once Ṁ and the distributions of P (r) and u(r) are known, the flow can be
determined. Different values of Ṁ leads to physically distinct solutions with
the same boundary conditions at infinity, this was shown by Bondi[9]. The
solution of interest in this case is where the velocity u rises monotonically
from 0 at r = Œ to free-fall velocity at small radii, u æ (2GM/r)1/2 as
r æ 0. To calculate the required accretion rate Ṁ , Equation (2.28) is
rewritten to the form
flÕ u Õ 2
+ + = 0,
(2.32)
fl
u
r
where ’ denotes d/dr, and Equation (2.29) in the form
uuÕ + a2
flÕ GM
+ 2 = 0.
fl
r
(2.33)
It is possible to solve Equations (2.32) and (2.33)) for uÕ and flÕ to get
uÕ =
D1
u2 ≠a2
ufl
where
,
D2
flÕ = ≠ u2 ≠a2 ,
(2.34)
ufl
2a2 /r ≠ GM/r2
,
fl
(2.35)
2u2 /r ≠ GM/r2
D2 =
.
u
(2.36)
D1 =
Equation (2.34) shows that there must be a "critical point" where
D1 = D2 =
u 2 ≠ a2
= 0,
ufl
(2.37)
2.3. ACCRETION
29
in order to ensure a smooth, monotonic increase in velocity with decreasing radius while simultaneously avoiding singularities in the flow. Using
Equation (2.35) -Equation (2.37), it can be found that at the critical radius
u2B = a2B =
1 GM
,
2 rB
(2.38)
so that this critical radius corresponds to the transonic radius at which
the flow speed equals the sound speed. Combining Equation (2.38) with
Equation (2.31), it is possible to relate aB , uB and rB to the known sound
speed at infinity
3
2
a2B = u2B =
5≠3
4
a2Π,
rB =
A
5≠3
4
B
GM
.
a2Œ
(2.39)
From Equation (2.30), the accretion can now be calculated using
fl = flŒ
3
a
aŒ
42/( ≠1)
.
(2.40)
This gives the accretion rate Ṁ as
Ṁ =
2
4fiflΠuB rB
3
aB
aŒ
42/( ≠1)
(GM )2
= 4fi⁄B flŒ
,
a3Œ
where ⁄B is a dimensionless parameter of order unity where for
⁄B = 0.707.
(2.41)
= 4/3,
Chapter 3
Dark Matter
While stars, planets and clusters are all massive objects when considered in
an everyday frame, they only make up around 15% [10] of the total matter
in the universe and only around 5% of the mass-energy. The remaining mass
is the so-called dark matter, first proposed in 1933, the structure of which
is still not known but for which several models exist within the current
constraints.
The reasons behind this obsession with determining the dark matter
content in the universe are many, of course the most obvious would be
curiosity regarding its structure, but it is also important when determining
the ultimate fate of the universe through the matter density and for that
the dark matter mass is needed.This chapter will deal with dark matter
relevant to the aim of this thesis.
3.1
Inferring Dark Matter
There is a large portion of baryonic matter in galaxies and clusters which
is left unaccounted for by simply counting the stars. However the vast majority of matter in the universe is not even baryonic, it is non-baryonic and
does not interact with the electromagnetic force. It can however be detected. One way of detecting this matter is by looking for its gravitational
effect on its surroundings. A classical way of detecting dark matter is by
looking at the orbital velocities of stars in spiral galaxies such as our own
or M31, spiral galaxies contain flattened discs of stars which are on nearly
circular orbital paths within the disc.
From classical physics it is known that an object on a circular path feels
31
32
CHAPTER 3. DARK MATTER
an acceleration
v2
,
(3.1)
R
directed towards the center of orbit, where R is the radius of orbit and v
is the orbital speed. If this acceleration is due to gravitational attraction,
such as that a star would feel towards the center of the galaxy, then that
acceleration is expressed by
a=
a=
GM (R)
,
R2
(3.2)
where M (R) is the mass contained within the sphere of radius R centered
on, in this case, the galactic center. This of course assumes that the distribution of mass is spherically symmetric, while this is not inherently true,
given that the disc has a flattened distribution, but this provides only a
small correction. Equating Equations (3.1) and (3.2) gives the relation
between v and M
Û
GM (R)
v=
.
(3.3)
R
A few scale lengths from the center of the spiral galaxy, the summed masses
of stars within R essentially becomes constant. If stars contributed most if
not all of the mass in a galaxy, it would
Ô be expected from Equation (3.3)
that the velocity would go as v à 1/ R at large radii. This relation is
known as Keplerian rotation, named after Johannes Kepler.
Orbital speeds of stars within a spiral galaxy is determined from observations, the first to detect the rotation of M31 was Vesto Slipher in 1914.
It was not until 1933 that a compelling case for the existence of a large
amount of matter which could not be seen, was made. This was done in
the groundbreaking work of Fritz Zwicky[11], after having been postulated
by Jan Oort the previous year. Zwicky, in studying the Coma cluster noted
that the dispersion in radial velocity of the cluster’s galaxies was very large,
around 1000km s≠1. The stars and gas visible within the galaxies of the
cluster did not provide enough gravitational pull to hold the cluster together. Zwicky concluded that, in order for the the Coma cluster to be held
together and not have its constituting galaxies be flung into the surrounding void, there must be a large amount of dunkle mature contained within
the galaxy.
While Slipher had detected the rotation of M31 in 1914 and Zwicky had
made a compelling case of the existence of dark matter, it was not until 1970
3.1. INFERRING DARK MATTER
33
that this was put together and gave observational proof for the existence of
dark matter, by Vera Rubin and Kent Ford. They looked at the emission
lines from regions of hot ionized gas in M31 and were able to find the orbital
speeds out to a radius of 24kpc or 4Rs . What they found was that there
was no sign of a Keplerian decrease in the orbital speed. Beyond this limit
the visible light from the galaxy was to faint for Rubin and Ford to detect,
this was later done by M. Roberts and R. Whiteburst who measured the
orbital speed out to ≥ 30kpc ¥ 5Rs and further supported the previous
findings.
Figure 3.1: This figure shows how the observed rotational curve compared
with the expected curve if only the disc is considered.
Figure 3.11 shows the comparison between the expected rotation curve
for just the visible matter, the curve labeled disc, with observed data points.
Here weÔ can see that observations do not fit the expected tendency of
v à 1/ R at large radii. Instead the observational data is almost constant up to large radii. This is consistent with a disc embedded in a dark
matter halo. From this it can be deduced that the visible stellar disc is
embedded within a dark matter halo, providing the gravitational anchoring
of high-speed stars and gas which prevents them from being flung into the
intergalactic space. This is not just the case for M31, most spiral galaxies,
1
Source: http://www.astro.rug.nl/ weygaert/app9/vanalbada.apj.jpg
34
CHAPTER 3. DARK MATTER
if not all of them appear to have dark matter halos. Even today, with the
discoveries and measurements of thousands of spiral galaxies, there is still
evidence that the orbital velocity remains constant at R > Rs . We have
so far seen that we can detect dark matter around spiral galaxies. This
is because it affects the motion of the stars and interstellar gas contained
within, as seen by Rubin and Ford. We have also seen that dark matter
can be detected in clusters, as Zwicky did, because it affects the motions
of galaxies and intercluster gas. But dark matter not only affects matter,
based on Einsteins theory of general relativity, dark matter should also be
able to bend light by altering the trajectory of photons.
This bending and focusing of light by dark matter is known as gravitational lensing and is used in the search for dark matter within the halo of
our own galaxy. Some of the dark matter in our galaxy may consist of massive compact object such as brown/white dwarfs, neutron stars and black
holes. These are collectively referred to as MACHOs. Should a photon
pass by one of these compact objects with an impact parameter b, it will
be deflected by an angle
4GM
–= 2
(3.4)
cb
due to the local curvature of space-time as described by general relativity,
where M is the mass of the compact object[12]. Since such a compact
object can deflect light it can act in much the same way as a lens. Suppose
that a MACHO within the halo of our galaxy passes between us and a
distant object. As the MACHO deflects the light from the distant object
it produces an image of the object which is both distorted and amplified
much in the same was a regular lens does. Should the MACHO be exactly
along the line of sight between the observer and the lensed object, the image
produced will be a perfect ring, with an angular radius of
3
4GM 1 ≠ x
◊E =
c2 d
x
41/2
(3.5)
where M is gain the mass of the lensing MACHO, d is the distance from
the observer to the lensed object and xd (where0 < x < 1) is the distance
from the observer to the lensing MACHO. The angle ◊E is known as the
Einstein radius. Should the MACHO not lie perfectly along the line of sight
to the object, then the image of the object becomes distorted into two or
more arcs instead of the single unbroken ring.
One case where gravitational lensing may very well prove the existence of dark matter is 1E 0657-558, commonly referred to as the Bullet
3.1. INFERRING DARK MATTER
(a) The picture shows a composite image of the Bullet Cluster
35
(b) The picture shows a composite image of MACS J0025.4-1222
Figure 3.2: In both figures, the red hues shows the X-ray emitted from the
two clusters hot intergalactic gas, captured by the Chandra X-ray observatory. The blue hues show the dark matter inferred through gravitational
lensing captured by the Hubble Space Telescope
cluster[13, 14]. It consists of two colliding clusters, where the major components are stars, gas and supposedly dark matter. The stars, which are
observable in visible light, were not affected greatly by the collision, mostly
passing right through only being slowed gravitationally. The hot gas contained in the two colliding clusters represents most of the baryonic matter
in the cluster pair, since these interact through the electromagnetic force
these were slowed considerably more than the stars. The gas is detectable
in the x-ray region of the electromagnetic spectrum.
The dark matter component was inferred from the weak lensing of background objects and it turns out that the lensing is strongest in two separate
regions near the visible galaxies, this provides support for the theory that
most of the mass in this cluster pair is in the form of dark matter which
does not self-interact as it does not appear to have collided. MACS J0025.41222 is another example where two clusters have collided and shows a clear
separation between the center of intergalactic gas and the colliding clusters.
Both Figure 3.2a2 and Figure 3.2b3 show a separation between the center
of visible matter in the clusters and the location of strongest lensing in the
clusters. This is even more evident in Figures 3.3a and 3.3b.
Figure 3.3a shows a color image from the Magellan images of the Bullet
cluster, the white scale bar indicates 200kpc, while Figure 3.3b shows 500ks
2
3
Source: http://apod.nasa.gov/apod/image/0608/bulletcluster_comp_f2048.jpg
Source: http://imgsrc.hubblesite.org/hu/db/images/hs-2008-32-a-full_jpg.jpg
36
CHAPTER 3. DARK MATTER
image from the Chandra X-ray. The green contours, shown in both panels,
are reconstructions of the Ÿ weak lensing. The white contours show the
errors on the Ÿ peaks positions and corresponds to confidence levels of ‡,
2‡ and 3‡.
(a) Shown here is a color image from
the Magellan images of the Bulletcluster
(b) This figure shows a 500ks Chandra image
Figure 3.3: Comparing Figure 3.3a and Figure 3.3b it can be seen that the
peaks of the Ÿ reconstruction of the weak lensing is located outside of the
hot intergalactic gas clearly seen in Figure 3.3b
3.2
Accretion onto a Neutron Star
This thesis will focus on WIMPs as possible candidates for dark matter
although such a particle with the required characteristics is not included
in the Standard Model meaning that such a WIMP likely is related to
physics beyond the Standard Model. Several candidates for dark matter
have been proposed ranging from supersymmetry to technicolor and KaluzaKlein eigenstates. Direct search experiments, such as CDMS and Xenon,
have put tight constrictions on the WIMP to nuclei cross section at the
level of ‡N Æ 10≠43 cm2 [15]. This can be found by requiring that the mean
free path of the WIMP is at most equal to the radius of the star:
⁄=
1
.
n‡
(3.6)
Using typical values for the number of scatterers, n, based on the average
density of the neutron star n = 1038 cm≠3 and a radius of R = 10km, one
obtains the cross section used above. This sets a minimum on the cross
section which can be probed by a neutron star.
3.3. FORMATION OF A BLACK HOLE FROM DARK MATTER
37
For dark matter to form a black hole inside of neutron star, which is the
focus of this thesis, the dark matter particles first have to be accreted onto
the neutron star. The reason for neutron stars, and other compact objects,
to be of particular interest in this case is due to their high baryonic densities
which will increase the probability of a WIMP scattering inside the star and
the eventually gravitational entrapment of the WIMP. For efficient WIMP
capture, the particle must scatter at least once per star crossing. Taking
relativistic effects into account the accretion rate of dark matter WIMPs
onto a neutrons star has been estimated by [15]:
8 flDM
F = fi2
3
m
3
3
2fi‹ 2
43/2
2
GM R 2 1
≠3E0 /‹ 2
‹
1
≠
e
f,
1 ≠ 2GM
R
(3.7)
where flDM is the local dark matter density, m is the WIMP mass, M and
R are the mass and radius of the neutron star respectively, v is the average
WIMP velocity far from the star, and E0 is the typical WIMP energy loss
in a single collision in the interior of the star. f describes the inefficiency
of DM capture in the case that the probability of collision in a single star
crossing is less than one. For cross sections larger than ≥ 10≠45 cm2 , f = 1
and f = ‡N /(10≠45 cm2 ) for ‡N < 10≠45 cm2 .
3.3
Formation of a Black Hole from Dark
Matter
After the WIMPs have been captured by the star, thermalization starts
through successive collisions between WIMPs and the nuclei inside of the
star. After a sufficient time this can be described by a Maxwell-Boltzmann
distribution in the velocity and distance form the center of the star. From
[16] it can be seen that the majority of the WIMPs are then concentrated
within the radius:
A
B1/2
9T
rth =
,
(3.8)
8fiGflc m
where T is the temperature of the star, flc is the core density of the star
and m is the mass of the WIMP. Once inside the neutron star the WIMPs
begin to thermalize once more towards the much smaller thermal radius of
the neutron star, typically a few centimeters for a TeV WIMP[16]. Given
a sufficiently large number of WIMPs having been accreted, these may
start to self-gravitate thus overcoming the gravitational pull of the star
and collapse gravitationally, given the absence of a repulsive force between
them. The onset of this self-gravitation occurs when the total mass of the
38
CHAPTER 3. DARK MATTER
WIMPs within the thermal radius exceeds that of the ordinary matter in
that very same region. This leads to [16]
NØ
A
T3
G 3 m 5 flc
B1/2
(3.9)
.
Considering Bosonic Dark Matter, it is possible to shiw that initially a
black hole formed in such a way is not heavy enough to survive Hawking
evaporation. The reason for this is that the WIMP-sphere can not collapse
instantaneously into a black hole due to the lack of loss of momentum and
energy of the WIMPs, even if self-gravitating. Before the WIMP-sphere
reaches its Scharwzschild radius, the density becomes high enough to form a
Bose-Einstein Condensate. This condensate will eventually, through further
concentration of WIMPs, become self-gravitating and collapse into a black
hole given
M > M0 =
3
4≠1
2Mpl2
m
= 9.5 · 1033 GeV
.
fimDM
10T eV
(3.10)
The initial density of the WIMP sphere is comparable to that of the nucleonic matter, flc = 5.6 · 1038 GeV/cm3 , however in order to form a black hole
the density needs to reach at least ≥ 1078 GeV/cm3 . However, the critical
density for the formation of a BEC is reached at
38
flcrit ≥ 4.7 · 10 GeV cm
≠3
3
m
10TeV
45/2
,
(3.11)
which means that the critical density for BEC forming is reached long before
the density initial density of the WIMP sphere. Once this density is reached,
the BEC will start to grow and any further compactification of the WIMP
sphere will only results in a increase in the BEC mass, not its density. In
the case of fermionic dark matter, the WIMP-sphere may collapse to a black
hole once the gravitational pull of the WIMPs exceeds the Fermi pressure
of the WIMPs. This onset of the gravitational collapse occurs when the
potential energy of a WIMP exceeds the Fermi momentum and therefore
the collapse can no longer be prevented by the Pauli exclusion principle.
This happens when
GN m2
> kF =
r
A
3fi 2 N
V
B
3
9fi
=
4
41/3
N 1/3
r
(3.12)
In this limit the WIMPs have been considered to be (semi)-relativistic,
which is justified since the WIMPs get closer and closer, building up a
3.3. FORMATION OF A BLACK HOLE FROM DARK MATTER
39
Fermi momentum that eventually corresponds to relativistic velocities once
the WIMPs have become self-gravitating. From Equation (3.12) the number
of WIMPs needed for the collapse can be deduced
N=
3
9fi
4
41/2 3
mpl
m
43
ƒ 5 · 1046
3
m
T eV
4≠3
(3.13)
in order to simplify the formation of the black hole, this thesis has used
M=
2 Mpl2
fi m
(3.14)
as the critical mass of the black hole, ignoring relativistic contributions,
beyond which the uncertainty principle can no longer prevent the collapse.
Part II
Thermodynamics and Hawking
Radiation
41
Chapter 4
Black Hole Thermodynamics
4.1
Black Holes and Thermodynamics
Using the second law of thermodynamics as a springboard, the fact that
classically nothing can come out of a black hole poses a problem. Consider
the following scenario: we take a system with a given entropy into a black
hole and throw it into a black hole. Imagine we are also somehow capable
of measuring the entropy of the entire Universe. As the system approaches
the event horizon it will vanish from our view and will eventually hit the
singularity. Were we to measure the entropy of the Universe again, we
would find it to be less than what we measure before sacrificing the system.
This would violate the second law of thermodynamics , one of the most
time-honored laws in physics. To solve this, Bekenstein [17] in 1973 proposed that black holes have an intrinsic entropy. We now return to the
scenario described above. If the black hole had an associated entropy Sbh
and the mass and energy of the remaining Universe had an entropy Sc , then
the total entropy would be,
d (Sbh + Sc )
Ø 0,
dt
(4.1)
and as stated would be non-decreasing thus avoiding the violation of the
second law of thermodynamics. Bekensteins proposal was in part motivated
by Hawking [18], who had stated that the total area of all black holes in the
Universe can not decrease in any physically allowed process. This sounds
noticesably close to the second law of thermodynamics , in fact this led
Bekenstein to propose that the entropy of a black hole is proportional to
its area.
Following these arguments suggests that there is a close analogy between
the laws of thermodynamics and the laws which govern the physics of black
43
44
CHAPTER 4. BLACK HOLE THERMODYNAMICS
holes. The four laws of black hole mechanics were proposed in 1973 [19],
which bear a striking similarity to the laws of thermodynamics given that
the surface gravity Ÿ1 and the area A of a black hole are like the temperature
and entropy respectively. The four laws are:
• Zeroth law: The surface gravity Ÿ is constant over the horizon of a
stationary black hole.
• First law: For a stationary black hole the change in mass is related
to the change in area, angular momentum and electric charge by
dM =
Ÿ
dA + dJ + dQ,
8fiG
(4.2)
where is the angular velocity, J is the angular momentum and
the electrostatic potential.
is
• Second law: The horizon area of a black hole must be nondecreasing
in any physically allowed process.
dA Ø 0.
• Third law: It is impossible to achieve Ÿ = 0 via a physical process.
While this analogy looks sound, it was noted that it was not consistent.
If a black hole had a temperature, it would be expected to radiate with a
characteristic Planck spectrum. By definition black holes do not radiate
since nothing can come out of them. This posed an interesting problem for
physicists and will be dealt with in detail in the next chapter. Staying in
the framework of thermodynamics, Hawking showed that the temperature
of a black hole is related to its surface gravity by:
TH ¥
Ÿ
.
2fi
(4.3)
This allows us to complete the analogy by inserting Equation (4.3) into
Equation (4.2). Recalling that the first law of thermodynamics is given by
dU = T dS + dW , from which it is possible to find the entropy. This is
called the Bekenstein-Hawking entropy and is given by:
Sbh =
1
A
,
4G
(4.4)
The surface gravity is defined as the acceleration need to keep a particle stationary
at the event horizon
4.2. THE HOLOGRAPHIC PRINCIPLE
45
which confirms Bekenstein’s conjecture that the entropy of a black hole is
related to its area. This should give a good indication that black holes are
indeed thermal systems which obey the laws of thermodynamics. However
this also leads to two profound puzzles: the microscopic description of black
holes and the information loss paradox. The information paradox will be
discussed in the next chapter, while the following sections will deal with the
microscopic description of a black hole while not going into great detail.
4.2
The Holographic Principle
As stated by Bekenstein, the entropy of a black hole scales with its area.
As such, the reader may have noticed that black holes behave in a very
different manner when compared to a normal physical system where the
entropy scales with the systems volume instead of the area. As a short
reminder, consider a system consisting of V cubes of unit volume each of
which can be in two states (for example spin up or down). The total volume
of the system would then be V and the system would have a number of
possible configuration given by:
= 2V ,
giving the total entropy of the system as
S = ln
= V ln 2.
This difference was investigated by ’t Hooft in 1993 [20] and he came to a
extraordinary conclusion; at Planckian scales, the world is two-, not threedimensional. For a simplified version of his arguments, consider a system,
much like the one described above, with an energy E contained within a
sphere of volume V with a radius R, where once more each unit volume has
two states available. The energy of the system is such that the Schwarzschild
radius is smaller than the physical radius of the system, this restricts the
density inside the sphere from being large enough to facilitate a collapse
into a black hole. ’t Hooft then showed that the entropy of the system has
an upper bound given by
fiR2
A
SÆ
=
,
G
4G
which means that the maximum entropy the system can have is that of
a black hole which fills the entire volume V . ’t Hooft build his argument
on two assumptions: that at Planckian scales the degrees of freedom are
46
CHAPTER 4. BLACK HOLE THERMODYNAMICS
discrete and that the evolution of the system is time reversible. ’t Hooft
then argues that most of the states of a regularized quantum field theory
have such high energies that they collapse into a black hole before they
can affect the time evolution of the system. This can explain why, when
calculating the number of accessible states of the system, it appears that we
overcount. It seems that when considering gravitational physics, the degrees
of freedom of the system is reduced. As such the number of states available
to the system scales exponentially with the area and not the volume. This
is known as the holographic principle, for further details on this area the
reader is referred to[21, 22].
4.3
A Microscopic Description of Black
Holes
Considering thermodynamical entropy as defined in statistical mechanics,
it is the number of microscopic configuration which give rise to the macroscopic properties such as temperature, pressure and volume. As one might
have expected, to calculate this entropy we need to know what the microscopic degrees of freedom are for the system. The so-called no-hair theorem
from classical gravity states that a black hole solution to the EinsteinMaxwell equations can be uniquely described by its three conserved charges;
mass, angular momentum and charge.
Figure 4.1: The difference between the classic and quantum mechanical
descriptions of a black hole.
All other information, the hair, about the matter which formed the
black hole, or is falling into it, disappears behind the event horizon and
4.3. A MICROSCOPIC DESCRIPTION OF BLACK HOLES
47
thus becomes inaccessible to any external observer. In this context it would
appear that the black hole only has one state and we would expect it to
have an entropy of
S = ln 1 = 0.
But as described in the previous section, black holes have an intrinsic entropy and this gives rise to the question of what microscopic freedoms give
rise to the Bekenstein-Hawking entropy
S = ln .
As an example, a one solar mass black hole would have an entropy S ≥
1018 [8], which would mean that the microscopic configurations goes as ≥
18
e10 . Thus we see that the classical and quantum pictures of black holes
are extremely different from one another as illustrated in Figure 4.1.
Chapter 5
Hawking Radiation
In this chapter, we will introduce the concept of Hawking radiation. Furthermore we will briefly discuss the information loss paradox which comes
about as a direct results of Hawking radiation.
5.1
Hawking Radiation
As we saw in the previous chapter, black holes are thermal objects. But
while the four laws of black hole mechanics provide a good analogy between
black holes and thermodynamics, the picture is incomplete. As mentioned,
this analogy only becomes complete if black holes emit black-body radiation, but this goes against the classical notion that nothing can come out of
a black hole. However Hawking [23] proved that black holes do in fact emit
radiation that is consistent with the thermodynamical analog. We shall
follow a simplified version of Hawkings arguments. Imagine a massless Hermitian scaler field, „, which obeys „;ab ÷ ab = 0, then „ can be expressed
as
„=
ÿÓ
Ô
fi ai + f̄i ai† ,
i
(5.1)
where fi is the complete orthonormal set of complex solutions to the wave
equation fi;ab ÷ ab = 0 containing only positive frequencies with respect to
the usual Minkowski time coordinate. Here the subscript ; denotes the
covariant derivative. The annihilation and creation operators for particles
in the ith state are given by ai and ai† respectively. We now define the
vacuum state |0Í as the state from which no particles can be annihilated,
i.e.
ai |0Í = 0 for all i.
49
(5.2)
50
CHAPTER 5. HAWKING RADIATION
In standard spacetime, the procedure is to decompose the field into positive and negative frequency components. However in curved spacetime one
may still consider a Hermitian scalar field operator „ which obeys the covariant wave equation „;ab g ab = 0 but it is no longer possible to decompose
it into positive and negative frequency parts. This is due to the fact that
these parts have no invariant meaning in curved spacetime.
If one considers a region of spacetime consisting of an initial flat region
(1) followed by a region of curvature (2) and finally another flat region (3).
Then the basis {f1i } containing only positive frequencies on region (1) will
not be the same as the basis {f3i } which contains only positive frequencies
on region(3). Thus the initial vacuum state |01 Í, satisfying a1i |01 Í = 0, will
not be the same as the final vacuum state |03 Í, meaning that a3i |01 Í =
” 0.
This can be interpreted as the creation of a certain number of particles of
the scalar field by the gravitational field or time dependent metric.
From Hawking’s argument it would seem as if the gravitational field of
the black hole creates particles with emission rates expected if the black
hole is considered as an ordinary object with a temperature given by Equation (4.3). The surface gravity Ÿ is inversely proportional to the mass.
While initially counter intuitive this must be the case since an increase
in mass will also cause an increase in the Schwarzschild radius leading to
expansion of the surface area.
Hawking showed that the temperature of a solar mass black hole (≥
2.8·1030 kg) is much lower than the surround cosmic microwave background.
Thus a black hole of such a size would be absorbing radiation from its surroundings faster than it is emitting radiation, leading to an increase in the
mass. This will only serve to further lower the emission rates. While such
black holes would be formed in stellar collapses there may be other options.
Smaller black holes may have been formed by density fluctuations in the
early universe, so-called primordial black holes or as is considered in this
thesis, by gravitational collapse of dark matter particles. Black holes of
small masses, such as these, would have a higher temperature causing them
to emit more than they absorb from their surroundings, neglecting for the
moment the accretion of matter. This emission process will lead to a decrease in their mass which only serves to further increase the emission rates
by elevating the temperature.
While we have only considered massless particles so far, as the temperature of the black hole rises it will eventually reach a point where emission of
particles such as electrons and muons takes place. Emission rates for neu-
5.2. THE INFORMATION PARADOX
51
trinos, photons and gravitons were calculated by Don. N. Page in [3, 24, 25]
and will be used in later chapters of this thesis. But the decrease in mass
of the black hole by emission seems to pose some problems; validity of the
second law of black hole mechanics and the information loss paradox.
The Second Law Revisited
The second law of black hole mechanics states that the surface of area of
black hole can never decrease by any physically allowed process. Then how
can Hawking radiation be a physical process since the decrease in mass
must lead to a decrease in the surface area. Hawking explained this by
stating that the flux of positive energy that is radiated to infinity by the
black hole must be balanced by a flux negative across the event horizon,
causing the violation. This flux of negative can be imagined in the following
way; a pair of virtual particles is created just outside of the event horizon,
one with positive energy and one with negative energy. The positive particle will be able to escape to infinity and become a part of the thermal
emission described previously. The negative particle is of more interest in
the scenario. It exists in a region which is classically forbidden, however it
can tunnel through the event horizon to a region where the timelike Killing
vector is spacelike. In this region the particle can exist as a real particle,
even though its energy is negative relative to infinity.
5.2
The Information Paradox
Basic quantum mechanics can give us a sensible approach to the problem
known as the information loss paradox. If we let H be the infinite Hillbert space spanned by the orthonormal pure quantum states |Âa Í, then the
density matrix for a given pure quantum state |Âa Í is given by
flpure = |Âa Í ÈÂa | .
This allows us to introduce mixed quantum states as being statistical ensembles of pure quantum states, in which case the density matrix will be
given by
ÿ
flmixed =
pa |Âa Í ÈÂa | ,
a
where pa is the probability for each of the pure states |Âa Í in the ensemble. When working with thermal systems the mixed state density matrix
becomes very valuable. Given a system at a finite temperature T , the probabilities pa are proportional to the Boltzmann factors e≠—Ea . This allows us
52
CHAPTER 5. HAWKING RADIATION
to rewrite the mixed state density matrix using statistical mechanics such
that
ÿ e≠—Ea
flmixed =
q ≠—E |Âa Í ÈÂa | .
a
ae
a
If we now consider an operator U acting on elements of H so that |Âb Í =
U |Âa Í and ÈÂb | = ÈÂa | U † . Given that ÈÂi |Âj Í = ”ij , we obtain that
1 = ÈÂi |Âj Í = ÈÂb | U † U |Âa Í ∆ U † U = 1
Because of this, should we demand that the elements of H are orthonormal,
then the conclusion is that U has to be a unitary operator. Physically this
means that for any process that is consistent with quantum mechanics,
we have to require that the evolution be unitary. In other words, a pure
quantum state can not in theory evolve into a mixed state using unitary
operators, which can be written in terms of the density matrices as
flmixed ”= U flpure U † .
Alternatively we now that for a pure ensemble the square of the density
matrix is equal the density matrix,
fl2pure = flpure .
We can write a mixed state as
fl(t) = U fl(t0 )U † ,
where fl(t0 ) is the density matrix at some initial time. Clearly if we take
the square of this matrix we obtain
fl2 (t) = U fl(t0 )U † U fl(t0 )U † = U fl2 (t0 )U † ,
if we have used unitary operators so U † U = 1. Thus a pure state can not
evolve into a mixed state through the use of unitary operators. Now, to
the problem at hand. If we take a pure quantum system and throw it into
a black hole, the system will disappear across the event horizon but the
black hole will still emit Hawking radiation. If the black hole then radiates
away all of its mass, it will disappear leaving only the thermal radiation
which is described as a mixed quantum state. In this regard, the black
hole has transformed a pure quantum state into a mixed quantum state,
but as described above this is not allowed by quantum mechanics because
it violates unitary evolution. This is what is known as the information loss
paradox. Once the black hole has evaporated, there is no longer an event
5.2. THE INFORMATION PARADOX
53
horizon hiding the states of the black hole which we may appeal to, only
the Hawking radiation produced by the black hole. Because this radiation
is exactly thermal there is no way of retrieving the vast information of the
system we have thrown into the hole unless there is a hidden correlation
between the outgoing particles.
We may throw to vastly different systems into two identical black hole
with the same spin, angular momentum and charge and the result will be
thermal radiation which will be indistinguishable from one another. Whatever information was contained in the system before appears to have been
erased by the black hole. Ever since Hawking first introduced the concept
of black holes emitting thermal radiation the question has raged whether
or not information is lost in the process. The AdS/CTF correspondence in
string theory for example provides a possible solution to this problem, but
will not be dealt with in this thesis.
Chapter 6
Greybody Factors
This chapter introduces the concept of greybody factors as well as covering
some qualitative ideas about black hole scattering theory. It will however
not include an explicit calculation of greybody factors as this is not the aim
of the thesis.
6.1
Changing Hawking Radiation
From the previous chapters it should be clear the black holes are indeed
thermal systems. They have an associated temperature and entropy which
means that they radiate away energy. Named after Stephen W. Hawking
who first postulated the existence of radiation from black holes in [23] where
he also showed that exactly at the event horizon, the emission rate in a mode
with frequency Ê from a black hole is exactly given by
(Ê) =
1
d3 k
,
e—Ê ± 1 (2fi)3
(6.1)
where — is the inverse of the Hawking temperature (since kB = 1) and the
minus (plus) sign should be used when considering bosons (fermions). This
formula is valid when considering both massless and massive particles. This
means that at the event horizon the spectrum of the Hawking radiation is
that of a black body and perfectly thermal. It does however leave out an
important fact; the geometry of spacetime surrounding the black hole is nontrivial. If we take this into account we might expect that once the radiation
has been emitted it gets modified by the non-trivial geometry so that at
infinity an observer will no longer see a black body spectrum. The geometry
outside the event horizon acts as a potential barrier. This potential filters
the Hawking radiation so that some of the radiation is transmitted and the
55
56
CHAPTER 6. GREYBODY FACTORS
Figure 6.1: This figure shows how the thermal radiation from the horizon
(red) gets modified by the non-trivial geometry surrounding the black hole
(orange).
rest is reflected back into the black hole. Thus the emission spectrum of
the black hole, as observed at infinity will be modified by a factor
‡(Ê) d3 k
(Ê) = —Ê
e ± 1 (2fi)3
(6.2)
where ‡(Ê) is the greybody factor for a given particle which depends on the
frequency. The name derives from the fact that they modify the spectrum of
Hawking radiation emitted by the black hole from a black body to that of a
grey body. This modification is not just relevant for Hawking radiation, but
in fact any field propagating in the black hole background will be modified
by the potential caused by the geometry surrounding the black hole. This
is to be expected as the potential V (r) only depends on the geometry of
spacetime. While part of the radiation will tunnel through the potential
barrier and reach infinity, part of it will be reflected back into the hole, this
is illustrated in Figure 6.2. The greybody factors can be defined in terms
of the reflection and transmission coefficients.
6.2. BLACK HOLE SCATTERING THEORY
57
Figure 6.2: When the Hawking radiation has been emitted it will have
to propagate through the non-trivial geometry created by the potential
V(r). The potential will then filter the radiation; some of the radiation will
be transmitted by tunneling through the potential while the rest will be
reflected back into the black hole. The part that tunnels through will be
modified and travels freely to infinity.
6.2
Black Hole Scattering Theory
In this section we will consider the scattering on a Schwarzschild black hole
(static and spherically symmetric) given by the metric Equation (2.9)
ds2 = ≠f (r)dt2 + f (r)≠1 dr2 + r2 d
2
,
(6.3)
where we have substituted f (r) = 1 ≠ rrS . The goal is then to study the
propagating field’s wave equation with frequency Ê, ÂÊ in the region outside
the event horizon given by rS < r < +Œ, this can be done using rudimentary quantum mechanics. Following the method described earlier, we now
switch to tortoise coordinates defined by
x©
⁄
dr
,
f (r)
so that the exterior of the black hole is now ≠Œ < x < +Œ. Now incoming waves will originate at x = +Œ and outgoing waves will originate
58
CHAPTER 6. GREYBODY FACTORS
at x = ≠Œ. We would expect that the potential V (r) æ 0 as x æ ±Œ
since it is caused by the geometry of spacetime surrounding the black hole.
Thus solutions to the wave equations would be expected to behave as plane
waves ÂÊ ≥ e±iÊx in these limits. The sign convention for the direction of
propagation of a plane wave is
e±i(Êx≠Êt) : Wave traveling in the positive x direction,
e±i(Êx+Êt) : Wave traveling in the negative x direction.
(6.4)
Here it should be noted that the wave number k will, with the units used
in this thesis, be k = Ê. Consider now a solution ÂÊ to the wave equation
which describes the scattering of an incoming wave, originating at x = +Œ,
that satisfies the boundary conditions
Y
]eiÊx
+ Re≠iÊx , x æ +Œ,
ÂÊ ≥ [ iÊx
Te ,
x æ ≠Œ,
(6.5)
where R(Ê) and T (Ê) are the reflection and transmission coefficients respectively. For the full picture we also have to consider the wave equation
on the other side of the potential barrier, Â≠Ê , with boundary conditions
Y
]e≠iÊx
 iÊx , x æ +Œ,
+ Re
Â≠Ê ≥ [  ≠iÊx
Te
,
x æ ≠Œ,
(6.6)
Â
where R(≠Ê)
and T (≠Ê) are some other reflection and transmission coefficients. From Equation (6.5) and Equation (6.6), we can construct the
conserved flux
A
B
1
dÂÊ
dÂ≠Ê
F=
Â≠Ê
≠ ÂÊ
(6.7)
2i
dx
dx
which does not depend on x, and evaluate it at x æ ±Œ. The flux should
be equal at both limits, which yields
 + T T = 1.
RR
(6.8)
When only considering Ê œ R this reduces to the familiar expression |R|2 +|T |2 =
1. Consider now the scattering of an outgoing wave originating at the black
hole outer horizon x = ≠Œ, described by the solution of the wave equation
ÂÊÕ , which satisfies the boundary conditions
ÂÊÕ ≥
Y
]T Õ e≠iÊx ,
[e≠iÊx
+Re
Õ iÊx
x æ +Œ,
.
, x æ ≠Œ
(6.9)
6.2. BLACK HOLE SCATTERING THEORY
59
Õ
Like before, we will also have to consider Â≠Ê
which is also a solution to
the wave equation, but again with different boundary conditions
ÂÊÕ ≥
Y
]TÂ Õ eiÊx ,
[e
iÊx
+Re
Â Õ ≠iÊx
x æ +Œ,
.
, x æ ≠Œ
(6.10)
The solutions Equations (6.9) and (6.10) for an outgoing wave can be expressed as linear combinations of the solutions for an incoming wave Equations (6.5) and (6.6)
ÂÊÕ = AÂÊ + BÂ≠Ê ,
(6.11)
Õ
Â≠Ê
= CÂÊ + DÂ≠Ê .
Using Equations (6.5), (6.6), (6.9) and (6.10) in Equation (6.11) it possible
to find that the coefficients, A, B, C and D are given by
Â
R
1
ÂÊÕ = ≠  ÂÊ +  Â≠Ê
T
T
Õ
and Â≠Ê
=
1
R
ÂÊ ≠ Â≠Ê
T
T
(6.12)
Moreover, we find that is possible to express the outgoing reflection and
transmission coefficients in terms of the incoming coefficients
T Â ÂÕ
TÂ
RÕ = ≠ Â R,
R = ≠ R,
T
T
Õ
Õ
Â
Â
T = T,
T = T.
(6.13)
 = RÕ R
Â Õ , therefore the greybody
From these we see that T TÂ = T Õ TÂ Õ and RR
factors can be defined in terms of the incoming and outgoing transmission
coefficients. Choosing to define them in terms of the former means that we
can write the greybody factor of the black hole given by Equation (6.3) for
a frequency Ê œ C as
‡(Ê) = T (Ê)T (≠Ê).
(6.14)
For real frequencies, this formula generalizes to ‡(Ê) = |T (Ê)|2 .
Greybody Factors as Absorption Cross Section
Equation (6.14) is an important equation, in that it shows that the greybody
factor for the incoming and outgoing wave scattering on the black hole is
the same. This equality implies that in the calculation of greybody factors,
it is only necessary to consider either option. Thus an incoming wave from
infinity that gets scattered on the black hole or the scattering of an outgoing
wave originating at the black hole horizon can be equally chosen. Perhaps
60
CHAPTER 6. GREYBODY FACTORS
somewhat counterintuitvely, in the case of the former the greybody factor
becomes the absorption cross section ‡abs of the black hole. This measures
the probability of the incoming being transmitted through the potential
barrier and being absorbed by the black hole. Should we choose to consider
greybody factors as absorption cross sections, the natural way to define
them would be to calculate the flux Equation (6.7) at the black hole horizon
and compare to the original flux at infinity. This would define the greybody
factor, or absorption cross section as
‡(Ê) = ‡abs (Ê) =
Fhorizon
.
Finf inity
(6.15)
It should be noted that the results derived above are valid for asymptotically
flat and asymptotically dS space times.
6.3
Low versus High Frequency
So far there have been no mention of the frequency Ê of the wave under
consideration. There are two frequency regimes that are of interest; low
frequency and high frequency. Naturally high and low frequency are relative
terms, so these will need to be defined in the context of the black hole. The
natural thing is to consider the scales set by the black hole, therefore we
can define the two regions of interest by
Low frequency:
Ê π TH ,
High frequency: Ê ∫ TH ,
M Ê π 1,
M Ê ∫ 1,
(6.16)
where we have chosen to follow the notation of Page[3], here TH is the
Hawking temperature of the black hole and M is the black hole mass.
These conditions specify the energy of the wave with respect to the scale
set by the black hole quantities, here the Hawking temperature and the
mass. It would also have been possible to use the horizon radius r0 instead
of the mass.
While the greybody factor equation, Equation (6.14), does not depend
on the frequency since it was derived only by considering the behavior of
the wave solutions, the methods used to compute the greybody factors will
be different in both frequency regimes. This particular aspect will not be
covered in this thesis, but it is worth noting the reasons for this difference.
In the low frequency regimes, the wavelength of the wave is much larger
than the length scale set by the black hole. In the high frequency regimes
6.3. LOW VERSUS HIGH FREQUENCY
61
the wavelength becomes comparable to the length scale. Therefore the wave
will see the curvature of spacetime surrounding the black hole. Greybody
factors for massless particles with spin s averaged over all orientations of
the black hole with rotation parameter a were calculated by Page[3] and
these will be used in this thesis, they are given by
Y
_
A,
_
_
_
_
]2fiM 2 ,
s = 0,
s = 12 ,
≥
‡(Ê)
4
2
Êæ0 _
A (3M
≠ a2 ) Ê 2 ,
s = 1,
_
_
9
1
2
_
_
[ 16
5
2
2 2
4
4
A 5M + 2 M a + a Ê , s = 2.
225
(6.17)
At high frequencies, the angle-averaged cross section for each different kind
of particle must approach the geometrical optics limit of 27fiM 2 for a static
black hole and roughly the same for a rotating black hole. Hence, the
cross section at lower frequencies must be smaller and we would expect
the curves for different species to diverge in this regime. As the frequency
goes to zero, the cross sections retain finite value for neutrinos and spin 0
massless particles but goes to zero as frequency squared for photons and as
the frequency to the fourth power for gravitons.
Part III
Accretion and Radiation
63
Chapter 7
Modeling Radiation
This section will serve as an introduction of how the Hawking radiation was
modeled for use in this thesis. The starting point was the data published
in the seminal work of Page[3] and we will look at how this paves the way
for the coming chapters.
7.1
Introduction
As the theory has made clear, black hole are thermodynamical systems and
they radiate away energy. In the mid-1970’s, Don N. Page published a
series of articles regarding the emission of particles from black holes. Here
we will discuss the first article in the series which deals with the emission
of massless uncharged particles from a stationary non-rotating black hole.
In this article, Page included a graph illustrating Hawking radiation for
three species of particles; neutrinos, photons and gravitons. Also included
is a table containing the emission rates and powers for the different particle
species as well as the dominant modes for each. This table is reproduced
below and will serve an important role throughout this chapter.
Table 7.1 allows us to calculate the power output for a given particle species
so long as we know its spin s and the mass M of the black hole. As
a standard [3] uses M = 5 · 1014 g. As can be seen from Table 7.1 the
neutrinos carry the majority of the power while the higher spin particles
carry increasingly less and less power. It should be noted that at the time
of publication, the tau neutrino ‹· had not been discovered and as such
is not included in the calculations in the article. However this can easily
be corrected for and will in fact be done throughout this thesis. The first
step will be to calculate the total power carried away by each of the three
65
66
CHAPTER 7. MODELING RADIATION
Table 7.1: Emission rates and powers for the dominant angular modes
2s
a
2l
b
For each mode
ratec
powerd
g
For each (s,l)
rate
power
e
1
1
1
1
3
5
1.191 · 10≠4
1.12 · 10≠6
9.5 · 10≠9
1.969 · 10≠5
3.75 · 10≠7
4.9 · 10≠9
8
16
24
9.531 · 10≠4
0.180 · 10≠4
0.002 · 10≠4
1.575 · 10≠4
0.060 · 10≠4
0.001 · 10≠4
2
2
2
2
4
6
2.55 · 10≠5
1.63 · 10≠7
1.1 · 10≠9
5.49 · 10≠6
6.67 · 10≠8
6.5 · 10≠9
6 1.463 · 10≠4
10 0.016 · 10≠4
14 0.0001 · 10≠4
0.330 · 10≠4
0.007 · 10≠4
0.0001 · 10≠4
4
4
4
6
1.10 · 10≠6
4.7 · 10≠9
3.81 · 10≠7
2.6 · 10≠9
10 0.110 · 10≠4
14 0.0007 · 10≠4
0.038 · 10≠4
0.0004 · 10≠4
a
s is the spin of the field, here doubled to give an integer
l is the total angular momentum of the mode
≠1
c
Rate in units of c3 G≠1 M ≠1 = 4.038 · 1038 (M/g) sec≠1
≠2
d
Power in units of h̄c6 G≠2 M ≠2 = 1.719 · 1050 (M/g) erg sec≠1
e
g is the number of modes for a given l and s, (2l+1)·(number of particle species
with the given s)·(number of polarizations or helicities for each species).
b
species of particles so that this may later be compared with calculations for
the fitted model. Calculating the total power output is done by multiplying
the power per mode by the number of modes, here it should be noted that
we will not yet include the tau neutrino. This calculation leaves us with
the following power outputs in the same units as in Table 7.1
Table 7.2: Total Power Output from Table 7.1
Species
Power
Neutrinos 0.000164
Photons
0.00005
Gravitons 0.000004
Table 7.2 clearly shows what was inferred from Table 7.1, namely that most
of the power output is in the neutrinos with the other species being one and
two orders of magnitude lower respectively. The values in Table 7.2 will
be used in the coming step to verify the interpolation while Table 7.1 will
be used when adding new particle species to the emission process. To get
an idea of the emission of particles, Figure 7.1, reproduced from [3], was
7.1. INTRODUCTION
67
used to model the emission across an energy range and to allow for the
calculation of emitted power. This graph was reproduced by copying the
Figure 7.1: This figure shows the three emission curves for neutrinos,
photons and gravitons. The curves pictured here are for a black hole of
M = 5 · 1014 g.
original from the article and then using PlotDigitizer1 to set the axis and
capture points along the individual curves. After this the resulting data sets
were plugged into Mathematica so that they could be used for calculations.
We can write the peak position in each of three spectra in terms of the
Hawking temperature.
Ê1/2 ≥ 0.18M ≠1 G≠1 ≥ 4.52389TH ,
Ê1 ≥ 0.23M ≠1 G≠1 ≥ 5.78053TH ,
Ê2 ≥ 0.35M ≠1 G≠1 ≥ 8.79646TH .
(7.1)
Scaling and Rescaling
We will briefly explain the scaling of the1graph 2to a given mass, using the
x-axis as example. The x-axis scales as 5·10M14 g h̄Ê, so if the mass of the
1
http://plotdigitizer.sourceforge.net
68
CHAPTER 7. MODELING RADIATION
Figure 7.2: This figures shows the rescaling of both y and x-axis to the
GeV range. Note that the peak of the neutrino spectrum is located at
approximately 0.1 GeV which is consistent with its location in Figure 7.1.
These curves are for a black hole of mass M = 5 · 1014 g.
black hole is 5 · 1014 g this will give exactly the graph shown in Figure 7.1 as
explained earlier. The peak of the neutrino spectrum at this mass is roughly
located at 100MeV, however, if we double the mass, the scaling becomes
2h̄Ê, this means that the previous position of the neutrino spectrum peak
now corresponds to 2h̄Ê.
Thus the new position of the peak will be at 50MeV, this is congruent
with the fact that the temperature, and thus the energy of emitted particles,
goes as 1/M. If we halved the mass instead, the peak would correspond to
0.5h̄Ê and the new position of the peak would be at 200MeV. This is also
the case for the y-axis.
For the our purposes it is more convenient to rescale from MeV to GeV
since this will be the scale at which the calculations will be done, this
rescaling is shown in Figure 7.2. The rescaling is done by changing the
standard mass of the black hole from M = 5 · 1014 g to M = 5 · 1011 g and
M = 5 · 1017 g for the x and y-axis respectively. This does in fact conserve
the total power output by scaling up the y-axis while scaling down the
x-axis by the same factor. Note that we have forced all the graphs to go
7.2. COMPARISON
69
through (0,0). This point is added during the processing of the data points
rather than being a point from the raw data but is physically sound. This
additional point causes some elongation of the photon and graviton curves
in both Figures 7.1 and 7.2 but is of little concern.
7.2
Comparison
Before using these curves for calculations it is important to check that the
method used to translate them from paper to digital points have in fact
been accurate. This was done by using the interpolation function in Mathematica to create a function which could be integrated over the energy range
and give the power output for each species.
The interpolation function is linear between each point, this means that
there will be a linear progression from (0,0) to the first point in the raw
data. While this is not entirely consistent with the profile of a planckian
spectrum, it will turn out that this approximation gives powers very close
to the actual power output for each species. The power outputs from the
interpolation functions are displayed in Table 7.3 with comparison to those
found in Table 7.1, note that these will be converted into ergs per second
to match those of the interpolation functions.
Table 7.3: Comparison between power outputs in ergs per sec for a black
hole of mass M = 5·1014 g from interpolation functions and those calculated
in [3] shown in Table 7.1
Interpolation function
Neutrino
Photons
Gravitons
Total Power
1.1158 · 1017
2.32553 · 1016
2.78157 · 1015
1.37617 · 1017
Theoretical from Table 7.1 Percentage
1.12517 · 1017
2.31144 · 1016
2.64478 · 1015
1.38276 · 1017
99.1672%
100.61%
105.172&
99.5232%
By looking at Table 7.3 it is clear that the power outputs found using the
interpolation functions are indeed very close to those found theoretically
by Page. Even though the interpolation function has a linear step between
the origin and the first data point from the raw data, the power outputs
for each species is within reasonable bounds.
The neutrino and photon powers are incredibly close to those calculated
by Page while the gravitons differ by a little more. Still 5% is not a major
70
CHAPTER 7. MODELING RADIATION
difference, especially considering that the total power carried by gravitons
or other spin 2 particles will only contribute approximately 1% to the total
power output. As can been seen in Table 7.3 the total power output from
the interpolation functions is 0.5% below the mark. These factors gives us
confidence that the method used to model future radiation will indeed be
very reasonable.
7.3
Adding Particles
So far we have only considered the particles outlined in [3]; ‹e , ‹¯e , ‹µ , ‹¯µ , “,
and G. But as already discussed it will be necessary to add the ‹· and ‹¯·
since these had not yet been discovered at the time of publication. Page
also considers neutrinos to be massless, but today we know that they may
have some mass. This means that they will have two helicites instead of
only one as considered by Page. Furthermore, the emission of particles
will lower the mass of the black and further increase the emission rate as
outlined previously. Thus as the mass of the black hole decreases the temperature will increase and we can no longer ignore the emission of massive
particles such as electrons, positrons, and muons as well as up, down, and
strange quarks along with gluons.
The addition of these particles will require a calculation of the number of modes for each species for comparison with those already found in
the datasets. Their masses become relevant when modeling the emission of
these particles with respect to the temperature of a given black hole. These
values are shown in Table 7.4, which will allow us to calculate the number
of modes for a given l and s as shown in Table 7.1
Here ni denotes the number of species and hi denotes the number of helicities or polarizations for the given particle. With Table 7.4 we are now
ready to start calculations with these new particle species. For this we will
need the formula for the number of modes, this was given as a footnote in
Table 7.1, but for the sake of transparency will be given below
#modes = (2l + 1) · (number of particle species with the given s)·
(number of polarizations or helicities for each species).
(7.2)
In order to create a formula for the total power output of the black hole
we will need to sum over fermions as well as spin 1 and 2 bosons. A full
formula for the power output can be written as
P = k1/2 (a)Pf ermions + k1 (a)PBosons + k0 (a)PScalar + k2 (a)PSpin 2 particles ,
(7.3)
7.3. ADDING PARTICLES
71
Table 7.4: Masses, number of species and polarizations for particles emitted
Mass [GeV] ni
hi
≥0
≥0
0
0
0.51 · 10≠3
0.51 · 10≠3
0.105
0.105
1.7 · 10≠3
1.7 · 10≠3
4.1 · 10≠3
4.1 · 10≠3
0.120
0.120
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Neutrino
Anti Neutrino
Photon
Graviton
Electron
Positron
Muon
Anti-Muon
Up Quark
Anti Up Quark
Down Quark
Anti Down Quark
Strange Quark
Anti Strange Quark
Gluons
3
3
1
1
1
1
1
1
3
3
3
3
3
3
8
where Pi describes the power output for that particular family of particles
and ks (a) is a coefficient which depends on the rotation parameter of the
black hole. For a Schwarzschild black hole these equal 1. Each of these can
be decomposed in the following way;
71
PF ermions = 9.11535 · 10
PBosons = 3.74514 · 1071
3
3
3
M
GeV
M
GeV
4≠2
4≠2
4
GeV2
GeV2
ÿ
ÿ
ni hi ,
ni hi ,
(7.4)
(7.5)
ÿ
M ≠2
PSpin 2 = 4.28524 · 10
GeV2
ni hi ,
(7.6)
GeV
where we have summed the power output in each angular mode with the
factor of 2l + 1 using Equation (7.2). The number of helicities or polarizations as well as the number of species can be found in Table 7.4. From
Equation (7.3) it is possible to calculate the total power output from a
black using Equations (7.4) to (7.6) in tandem with Table 7.4. Here it
is possible to add and subtract specific particles, for instance it would be
entirely possible to add a new spin 2 boson with two helicities and check
how that might change the power output. We have not listed any formula
for the emission of scalar particles, while both the W± and Z bosons could
70
72
CHAPTER 7. MODELING RADIATION
be emitted as wells as the Higgs, the masses for these are at least 90 GeV
which is far above the temperature of any black hole considered here.
However, it should be of note that the emission of quarks, muons, electrons, and positrons is not entirely similar to those modeled in [3], since
these are not uncharged particles. Emission rates for charged leptons was
calculated by Page in [25], but the effects of these will not be considered
at the moment, we will assume that these behave similar to non-charged
particles of the same s and l.
The emission of particles by Hawking radiation we have presented so
far, based on the work by Page, is a little complex. If we consider the
black hole as a perfect black body, then we should be able to estimate
the thermal radiation by known formulae. This approach results in the
Stefan-Boltzmann-Schwarschild-Hawking (SBSH) power law. We can derive
this simply by combining the Schwarschild radius rS with the Hawking
temperature of the black hole
TH =
h̄c3
.
8fiGM kB
We then insert the Schwarzschild surface area AS = 4firS2 and the Hawking
temperature into the Stefan-Boltzmann power law
P = AS j ú = AS ‘‡T 4 ,
where ‡ is the Stefan-Boltzmann constant, AS is the Schwarzschild sphere
surface area and ‘ = 1 assuming that a black hole is a perfect black-body.
This yield the SBSH power law
h̄c6
P =
.
15360fiG2 M 2
(7.7)
A solar mass black hole would thus have a power output of P = 9 · 10≠22
erg sec≠1 which would have very little impact on the mass. It would of
course be favorable if the SBSH power law described the thermal radiation
in the same way as the work by Page. If the total power output of these two
approaches are consistent with one another, even for just a single species,
it will make calculations easier. For a black hole of mass M = 5 · 1014 g, as
has been used so far, the SBSH, Equation (7.7), gives a power output of;
P = 1.43069 · 1016 ergs sec≠1
(7.8)
This is consistent with 12.7% of the four species neutrino power from Table 7.3, 61.9% of the photon power, and 540.9% of the graviton power. Since
7.3. ADDING PARTICLES
73
Equation (7.7) is based on the thermal radiation from a black hole assumed
to be a perfect black-body, the resulting power should be compared to that
of the photons. As such the power output calculated by Page in only photons is quite in excess of the power law approximation, but this could be
due to the simpler nature of the derivation of Equation (7.7).
Chapter 8
Accretion and Radiation
This section will deal with results from the Mathematica code which was
briefly outlined in the previous chapter. The main goal is to investigate
the black hole critical mass where the accretion rate is perfectly balanced
by the emission rate. This will allow us to determine whether a black hole
formed by dark matter in the core of a neutron star will be able to devour
the entire star or if it will evaporate before this occurs.
8.1
Masses and Temperatures
Before looking at the critical masses, let us first turn to the masses and
temperatures of black holes formed by dark matter. These black holes are
created through the processes outlined in the theory. They will have masses
given by Equation (3.14) and temperatures given by
1
(8.1)
8fiGM
where we use the gravitational constant in units of Planck mass. The masses
and temperatures for black holes formed by WIMPS in the mass range of
1 ≠ 10 GeV is displayed in Table 8.1. From Equations (3.14) and (8.1) we
would expect the temperature of the black hole to increase linearly with
WIMP mass, since the temperature scales as the inverse of the black hole
mass which in turn scales as the inverse of the WIMP mass.
TH =
These masses cover a range from 8.48 · 10≠20 M§ down to 8.48 · 10≠21 M§ ,
thus these are far below the solar mass black holes traditionally formed
by stellar collapse. As such black holes formed in this way could pose
interesting new avenues of research, as they also have significant emission
of particles compared to that of solar mass black hole.
75
76
CHAPTER 8. ACCRETION AND RADIATION
Table 8.1: Black Hole Masses and Temperatures
mDM [GeV]
M [GeV]
TH [GeV]
1
2
3
4
5
6
7
8
9
10
9.48 · 1037
4.11 · 1037
3.16 · 1037
2.37 · 1037
1.89 · 1037
1.58 · 1037
1.35 · 1037
1.18 · 1037
1.05 · 1037
9.48 · 1036
0.0625
0.125
0.1875
0.25
0.315
0.375
0.4375
0.5
0.5625
0.625
Comparing Table 8.1 with Table 7.4 it is clear that for a WIMP mass as
low as 1 GeV it is not possible to ignore the emission of massive particles.
Here particles such as up and down quarks, with the strange quark and
muons being emitted for WIMP masses as low as 2 GeV. As such we would
expect these particles to significantly increase the emitted power as they
will be based on the neutrino spectrum which contributed the majority of
the total power output.
Since the temperature increases as the WIMP gets heavier (the black
hole getting lighter), the contribution from massive particles will only increase due to the scaling seen on Figures 7.1 and 7.2. In fact we would
expect an increase in power output by two orders of magnitude between a
WIMP mass of 1 GeV and 10 GeV.
In fact, this is clearly shown in Table 8.2, where the power outputs
for the three quark species and the electron is shown. We also see that a
doubling of the WIMP mass results in an increase of the power output by
a factor of four, which is what we would expect from the scaling.
The difference in number of modes between the quarks and the electron
shows up in the difference in their power outputs. These outputs can be
compared with that of a solar mass black hole, as seen earlier, which was
of the order ≥ 10≠21 ergs sec≠1 . The power output in even just one of these
particle species from one of these smaller black hole is ≥ 39 ≠ 41 orders of
magnitude larger than that of a solar mass black hole. As such Hawking
radiation will have a major influence on their evolution as they will radiate
away their mass rapidly. This is only considering a single particle species,
the total emitted power containing all the particle species will significantly
8.1. MASSES AND TEMPERATURES
77
Table 8.2: WIMP masses with the corresponding power output from up,
down, and strange quark along with the electron. The power output here
are in ergs sec≠1 , matching the units seen on Figures 7.1 and 7.2
WIMP Mass [GeV]
1
2
3
4
5
6
7
8
9
10
Power output [ergs sec≠1 ]
Up
Down
Strange
Electrons
1.46655 · 1018
5.86622 · 1018
1.3199 · 1019
2.34649 · 1019
3.66639 · 1019
5.2796 · 1019
7.18612 · 1019
9.38595 · 1019
1.18791 · 1020
1.46655 · 1020
1.46655 · 1018
5.86622 · 1018
1.3199 · 1019
2.34649 · 1019
3.66639 · 1019
5.2796 · 1019
7.18612 · 1019
9.38595 · 1019
1.18791 · 1020
1.46655 · 1020
Not Emitted
5.86622 · 1018
1.3199 · 1019
2.34649 · 1019
3.66639 · 1019
5.2796 · 1019
7.18612 · 1019
9.38595 · 1019
1.18791 · 1020
1.46655 · 1020
4.88851 · 1017
1.95541 · 1018
4.39966 · 1018
7.82162 · 1018
1.22213 · 1019
1.75987 · 1019
2.39537 · 1019
3.12865 · 1019
3.9597 · 1019
4.88851 · 1019
increase, thus further speeding up the process. Even the heaviest black
hole formed by the WIMPs considered here, will have considerable output
in the quarks and the electron even with the strange quark blocked by the
temperature.
Limiting the emission is the fact that the black hole exists inside of a
neutron star with a Fermi surface. The Fermi surface will limit the emission
process by blocking states up to the Fermi energy, thus only particles with
energy above the Fermi energy will be emitted. The up, down and strange
quarks will have a Fermi momentum given by Equation (8.2) which are
taken from [26].
puF = µ ≠
m2s
6µ,
pdF = µ +
m2s
,
12µ
psF = µ ≠
5m2s
,
12µ
(8.2)
where µ is the chemical potential of the neutron star, typically ≥ 500 MeV
and ms is the mass of the strange quark which can be found in Table 8.1.
The chemical potential of the electron is given by
µe =
m2s
.
4µ
(8.3)
78
CHAPTER 8. ACCRETION AND RADIATION
Any up, down or strange quark with energy below those given by Equation (8.2) will not be emitted thus limiting the emitted power. Electrons will
be blocked up to Equation (8.3). There is no blocking of muons, photons,
gravitons or any anti-particles. Table 8.3 shows the percentage blocked for
each of these four particle species for WIMP masses in the range 1-10 GeV.
Table 8.3: Blocking of Quark Species and Electrons
WIMP mass [GeV]
0.1
1
2
3
4
5
6
7
8
9
10
Blocked Percentage
Up
Down
Strange
Electrons
100
100
Not Emitted
0.327464
93.8925
94.3408 Not Emitted 0.000582476
30.4125
31.7824
29.0617
0.000145619
7.1875
7.58735
6.80221
0.0000647195
2.40409
2.54177
2.27292
0.0000364047
1.04305
1.10008
0.988457
0.000023299
0.536446 0.564859
0.509052
0.0000161799
0.307107 0.323677
0.291311
0.0000118873
0.191371 0.201205
0.181883
9.10118 · 10≠6
0.127017 0.133525
0.12074
7.19106 · 10≠6
0.0881836 0.0927234
0.0837967
5.82476 · 10≠6
With Fermi blocking, the power output for a black hole formed by 1 GeV
WIMPs decreases drastically. This is due to both the up and down quarks
being almost completely blocked with barely 6% of the power output from
Table 8.2 being emitted in these two species. All quarks are completely
blocked by Fermi blocking and temperature if we decrease the WIMP mass
to 100MeV, but there is still very little blocking of electrons. As the WIMP
gets heavier, the Fermi blocking of quarks decreases and gets less significant.
However for a WIMP mass of 2 GeV where the temperature is high
enough to emit strange quarks, almost a third of the quark power output
per species is blocked by the Fermi surface. As such, the Fermi blocking can
become crucial when trying to balance accretion and radiation since this
reduction in emission will also reduce the required accretion rate and thus
the black hole mass. However, for WIMP masses above 3 GeV the reduction
in the power output from up, down, and strange quarks is at most 2.5%
here the blocking becomes negligible. Even though the blocking becomes
8.2. GLUON EMISSION
79
negligible we will still be taking it into account to obtain more accurate
results.
8.2
Gluon Emission
While we have discussed the emission of quarks and electrons, gluon emission will be different. Although gluons are bosons, meaning they are not
subjugated to the Fermi blocking discussed above, the pair creation of gluons can be problematic. When the gluon pair is created, one tunnels inside
the black hole while the other escapes and this creates the problem. A gluon
Figure 8.1: Ferynman diagram of a gluon creating an q q̄ pair .
can always be imagine as carrying q q̄ quantum number as seen in Figure 8.1,
where the q quark and q̄ quark each carry different colors. This is the case
both inside and outside the black hole. The two gluons will form a color flux
tube extending from the q̄q pair inside the black hole to the escaping q q̄ pair.
This color flux tube can break once it become energetically favorable to
create an intermediate gluon pair which can then neutralize the escaping
gluon as well as the gluon inside of the black hole. Since this gluon pair
can also be written as a q q̄, this process can potentially create two escaping
pions through hadronization, as well as two pions inside of the black hole.
This will be a possible scenario for a black hole emitted gluons in vacuum,
however when we consider the black hole to exist inside of a neutron star,
this slightly complicates matters due to the presence of a Fermi surface
created by the degeneracy of matter.
For the intermediate gluon pair needed for hadronization to be created,
an energy of twice the Fermi energy of the degenerated quarks is needed.
We estimate the Fermi surface to be the value for the up quark, which is
0.4925 GeV. While the strange quark has a slightly lower Fermi surface,
0.488 GeV, the strange quark is severely limited by the temperature of the
80
CHAPTER 8. ACCRETION AND RADIATION
(a) The color flux tubes extending from quark to anti-quark.
(b) The color flux tube breaks when it becomes energetically
favorable to create q q̄ pairs instead.
Figure 8.2: These figures illustrates how gluons might be blocked by a fermi
momentum.
black hole when compared to the up quark. Once one of the intermediate
gluons have tunneled inside the black hole, it can become a q q̄ pair. This
process is limited by the degeneracy of the quark matter, thus it will require
two times the Fermi energy to create this q q̄ pair. Since the creation of this
pair requires two times the Fermi energy of the degenerate quarks and the
fact that in the creation of the gluon pair, both gluons most have the same
energy, we find that the creation of this intermediate gluon pair requires
energy equal to four times the Fermi surface.
Thus, we will consider gluons to be blocked up to 1.97 GeV, much like
the Fermi blocking of quarks and electrons. This effectively means that for
8.3. BONDI ACCRETION
81
WIMP masses below ≥ 2.3GeV, gluons will be completely blocked according
to the arguments given above.
8.3
Bondi Accretion
So far we have only considered the radiation of particles from a black hole
without the black hole accreting any matter. Given the nature of black
holes, it would be likely to assume that they will accrete some matter, even
if this might be a very small amount. Accretion onto a black hole was briefly
discussed in Chapter 2, the first case discussed dealt with a black hole with a
temperature lower than the surrounding CMB. This would of course absorb
radiation from its surrounding. The second case was the hydrodynamical,
spherically symmetric accretion of matter onto a Schwarzschild black hole,
which was first derived by Bondi[9]. Here we will consider the accretion of
matter onto the black hole as being described by the expression derived by
Bondi, Equation (2.41).
In Equation (2.41), we see that the accretion rate scales as the mass
squared, where we previously saw that Hawking radiation goes as M ≠2 .
This means that not only will small black holes emit more energy than more
massive ones, they will also accrete less matter from their surroundings. If
a black hole is to survive it must accrete matter from its surroundings that
will, at least, balance the emitted energy. Thus the change in mass with
respect to time can be written as
dM
P (a)
= CM 2 ≠
,
dt
M2
(8.4)
where C is the expression for a given type of accretion and P is the total
power emitted. For the black hole to survive, these two contributions must
balance, so that Equation (8.4) is zero. This would mean that there is no
change in the mass over time, assuming that the black hole can keep accreting matter at a constant rate. We will be interested in the critical mass of
the black hole, by this we mean the mass at which the accretion is exactly
balanced by the Hawking radiation.
Let us start by considering a simple case; Hawking radiation as described
by the SBSH power law, Equation (7.7). We previously saw that this could
not account for the total photon power as calculated by Page. However,
given that we have the full analytical expression for this radiation model,
82
CHAPTER 8. ACCRETION AND RADIATION
we can easily obtain an expression for the critical mass in this case
M=
A
3
–Œ
61440fi 2 G4 ⁄flc
B1/4
= 5.7 · 1036 GeV,
(8.5)
where we have used a core density of flc = 5·1038 GeV/cm3 . This black hole
mass corresponds to a WIMP mass of 16.64 GeV, however it is interesting
to compare this with the photon only emission of Page. The change from a
Table 8.4: Comparison between the emission of photons based on the SBSH
power law and the data fitted from Page[3].
mDM [GeV]
M [GeV]
SBSH
Page
16.64
5.7 · 1036
14.9992
6.32967 · 1036
description of photon emission by SBSH to one consistent with Page results
in a 9.86% decrease in the WIMP mass. This decrease leads to an increase
in the black hole mass by the same percentage. The fact that the WIMP
mass decreases is consistent with our expectations, since the emitted power
from Page is significantly higher than what was obtained using the SBSH
power law.
We will consider a number of different scenarios of Hawking radiation
and see how that impacts the critical mass, but first we should note a few
things regarding this scenario. The core density of the neutron star given
as 5 · 1038 GeV cm≠3 is merely an estimate for the actual energy density.
Calculations of the energy density for the up, down and strange quark with
a Fermi momentum of 0.5 GeV gives an energy density of 6.45271 · 1038 GeV
cm≠3 which we will use for Bondi accretion henceforth. Furthermore, due
to the accretion of matter onto the black hole, the density of states around
the black hole will increase. This is shown in Appendix G of [7]. Here we
choose to work with = 4/3 which means that the increase in density of
states is given by
3 4
nh
⁄ 1 3
=
= 35.976,
(8.6)
nŒ
4 as
where nh is the density of states at the Bondi radius and nΠis the density of
states far from the horizon. Since the each number density is proportional to
kF3 , this increase in the density of states will increase the Fermi momentum
by a factor of 35.9761/3 = 3.30119. Thus we should consider a boost in
8.3. BONDI ACCRETION
83
the Fermi momentum due to the accretion of matter onto the black hole.
It should be noted that while accretion increases the Fermi momentum,
every state is boosted by the same factor meaning that there may be some
of the lower states which will not be blocked since they are suddenly free.
However, while these may be freely emitted at the horizon, further from the
black hole these states will once more be blocked and the emitted particles
be blocked at this state.
We will consider three cases when calculating the critical mass for Bondi
accretion; the completely free emission of all particles, Fermi blocking of
relevant particles (up, down and strange quark as well as electrons and
gluons) and the boosted Fermi blocking of relevant particles. Table 8.5
Table 8.5: Critical Masses considering Bondi accretion
Free Emission
mDM [GeV]
M [GeV]
Fermi Blocking Boosted Fermi Blocking
4.87814
1.9453 · 1037
4.956595
1.91451 · 1037
5.26383
1.80276 · 1037
shows the critical masses for each of the three cases we have considered. We
see an increase in the WIMP mass, corresponding to a decrease in black
hole mass, as we go from free emission to Fermi blocking and on to the
boosted Fermi blocking. The regular Fermi blocking increases the WIMP
mass by 1.08% compared to the case where everything is freely emitted,
while the boosted Fermi blocking results in an increase in WIMP mass of
7.9%. The boosted Fermi blocking leads to an increase in WIMP mass
of 6.2% compared to the regular Fermi blocking. Inspection of Table 8.6
Table 8.6: Blocked percentage of power output at the critical mass for Fermi
blocking and boosted Fermi blocking with Bondi accretion.
Blocked Percentage
Fermi Blocking Boosted Fermi Blocking
Up Quark
Down Quark
Strange Quark
Electron
Gluon
1.0771%
1.13622%
1.02056%
2.37 · 10≠5 %
56.0523%
55.6826%
57.4108%
53.9247%
2.29 · 10≠4 %
100%
shows that the boosted Fermi blocking leads to a significant increase in the
blocking of the quarks. Essentially this changes the blocking of quarks from
84
CHAPTER 8. ACCRETION AND RADIATION
negligible to over half of their individual power outputs. This increase in
blocking of quarks is likely the contributor to the increase in WIMP mass
between regular Fermi blocking and the boosted Fermi blocking. Due to
their low chemical potential, the electrons are almost entirely unblocked
even with the boosted in Fermi blocking. While it may appear strange
that such a large increase in blocking only leads to a 6% increase in WIMP
mass, consider that the total power output of Fermions still include the
three anti-quarks which remain unblocked, as well as neutrinos, electrons
and muons with their respective anti-particles. Furthermore, a change in
power output has to be quite large as the change in mass goes as the fourth
root of the power output.
8.4
Geometric Accretion
In the previous section we considered the accretion of matter unto a black
hole through the hydrodynamical, spherically symmetric Bondi accretion.
However here we run into a problem, at the critical masses for Bondi accretion, the Schwarzschild radius of the black hole becomes much smaller
than the radius of the surrounding neutrons. Thus we change the accretion
to a simpler case where the black hole only accretes matter which passes
through the Schwarzschild radius;
Ṁ =
G2 M 2 4
kF ,
fi
(8.7)
where kF is the Fermi momentum which we will set equal to 0.5 GeV. This
accretion rate is significantly lower than Bondi accretion, as such the critical
black hole mass must increase in order to accommodate this decrease in accretion. As such we would expect Fermi blocking to play a more significant
role, as the emission spectrums shifts to lower energies meaning that more
of the power is blocked by the Fermi momentum. Furthermore, we would
expect the difference in critical mass between Fermi blocking and boosted
Fermi blocking to be less than for Bondi accretion, as the Fermi blocking
likely already blocks a substantial part of the power. Comparison of TaTable 8.7: Critical masses for Non-Bondi accretion
Free Emission
mDM [GeV]
M [GeV]
1.1397225
8.32609 · 1037
Fermi Blocking Boosted Fermi Blocking
1.2667715
7.49104 · 1037
1.290796
7.35161 · 1037
8.5. COMPARISON WITH LIMITS
85
bles 8.5 and 8.7 reveals that the critical mass has decreased significantly
from Bondi accretion. In fact, for the completely free emission of particles, the change in accretion has lead to a 76.6% decrease in WIMP mass.
With the geometrical accretion Fermi blocking increases the WIMP mass
by 11.14% confirming our expectations that Fermi blocking should be more
significant with this accretion than with Bondi accretion. The boosted
Fermi blocking increases the WIMP mass by 13.25% compared with the
completely free emission of particles but only by 1.89% when compared
with the regular Fermi blocking. Once more this confirms what we would
expect, that at these low masses, regular Fermi blocking already blocks a
significant amount of the power output, thus the increased blocking only
removes a little of the power output, as always with the exception of the
electrons. As Table 8.8 shows, the regular Fermi blocking blocks around
Table 8.8: Blocked Percentage at the critical masses for Non-Bondi accretion
Blocked Percentage
Fermi Blocking Boosted Fermi Blocking
Up Quark
Down Quark
Strange Quark
Electron
Gluon
80.579%
81.7865%
Not Emitted
3.62 · 10≠4 %
100%
100%
100%
Not Emitted
3.85 · 10≠3 %
100%
80% of the power from up and down quarks and all of the power from the
gluons. This means that the boosted Fermi blocking will result in the complete blocking of up and down quarks as well as the gluons, however the
total impact of this is less than what was already in effect from the regular
Fermi blocking.
8.5
Comparison with limits
In this section we have seen how the critical masses change with the emission
of particles as well as the temperature in the case of geometrical accretion.
The emission of particles, as we have seen, is dependent upon the mass of
the black hole. Thus a change in the critical mass can change the emission
significantly, especially when considering Fermi blocking. When considering
Bondi accretion we found the critical WIMP mass to be ≥ 4.9 ≠ 5.2GeV.
With Bondi accretion we saw that regular Fermi blocking has little impact
86
CHAPTER 8. ACCRETION AND RADIATION
at these high masses, but that boosted Fermi blocking can actually have a
significant impact even compared with regular Fermi blocking. [27] found a
lower bound on the WIMP mass of mDM > 3.3GeV. The results presented
in this section for Bondi accretion is consistent with these results. However, once the change is made to geometrical accretion the critical WIMP
mass is less than half of this limit. Here we also saw that Fermi blocking
becomes more significant at low masses, while the boosted Fermi blocking
is closer to the regular Fermi blocking. The change to geometrical accretion
was made because the radius of the black hole became smaller than that of
the surrounding neutrons at the critical masses found with Bondi accretion.
While the masses for Bondi accretion is within the limit presented in
[27] it is however below the preferred mass of asymmetric dark matter at
≥ flDM mn /flbaryon ≥ 5.7GeV[28]. Furthermore, almost all critical masses
presented here is below the limits of 5-15 GeV favored by DAMA[29] and
CoGent[30], although the masses for Bondi accretion is only just slightly
below the lower bound from these experiments. Finally for Bondi accretion the code gives 99.83% of the theoretical critical mass for the case of
completely free emission. The geometrical accretion gives 99.85% of the
theoretical critical mass in the same scenario. However, for the geometrical accretion it is important to that one keeps in mind the temperature of
the resulting black hole, as neither the strange quark nor the muon can be
emitted due to the low temperature of the black hole.
Chapter 9
Effects of Rotation
In this section we will consider the effects of rotation on the critical mass
of the black hole. We will consider rotation of the black hole itself but also
how the rotation of the surrounding star might affect the accretion rate.
9.1
Transport of Angular Momentum by
Viscosity
We will now consider the accretion onto a black hole found inside of a
rotating star. The accretion of matter onto the black hole will be determined
by the angular momentum of the accreting matter. For a piece of matter
to fall onto the black hole its specific angular momentum must below the
angular of the innermost stable orbit. This is given by
Ô GM
liso = 2 3Â
,
c
(9.1)
where  = 1 for a Schwarzschild hole and  = 1/3 for an extreme Kerr
hole [31]. If the angular momentum of the matter piece is above this value,
it will be expelled from the system. It is however possible that the angular
momentum of the infalling matter is great enough to stall the accretion.
The specific angular momentum of a piece of matter located at radius r0
from the star’s center rotating with an angular velocity Ê0 , is given by
l = Ê0 r02 . For the matter at radius r0 to fall into the black hole, all the
star’s matter up to this radius should already have been consumed by the
black hole. This corresponds to a mass M = 4/3fiflc r03 . Thus, the condition
that l < liso gives a lower limit of the mass of the black hole, above which
87
88
CHAPTER 9. EFFECTS OF ROTATION
rotation can be safely ignored which can be written as
Mrot
1
= 3/2
12
A
3
4fiflc
B2 3
Ê0
G
43
1
.
Â3
(9.2)
Our interest will be how nearby old neutron stars, with periods ≥ 5 miliseconds, constrain this mass. Using a typical core density flc = 5·1038 Gev/cm3 ,
gives Mrot = 1.6·1053 GeV. Once a black hole has reached a mass above this
limit, infalling rotating matter will no longer be able to stall the accretion.
However, until the black hole reaches Mrot , the rotation of infalling matter
can and will stall the acrretion unless there is a mechanism for getting rid
of the extra angular momentum.
As stated by Markovic[31], there can be a viscous transport of angular
momentum so that, under certain conditions, the viscosity can keep the
accretion practically spherical. This is based on braking by molecular viscosity, here the accretion of angular momentum will depend on the viscosity
of the neutron star. This in turn depends on the density of the neutron star
and the temperature. Markovic further considered the transport of angular momentum by magnetic braking and by convection induced turbulent
viscosity. We will only consider the transport of angular momentum by
viscosity. [31] points out that a black hole accreting matter through Bondi
accretion, viscosity can enforce a rigid rotation of matter at radii larger
than
G2 M 2
r‹ = 3 ,
(9.3)
–Œ ‹
where ‹ is the kinematic viscosity of the star
cm2
‹ ¥ 20
s
3
T
1.5 · 107 K
45/2 A
150 g cm≠3
fl
B
(9.4)
and M is the mass of the black hole. As long as the radius is larger than
Equation (9.3), the infalling matter will have an angular velocity Ê0 . For
radii smaller than Equation (9.3), the viscosity will not be able to effectively
brake the rotation which induces Ê ¥ Ê0 r‹2 /r2 , where we have used the expression approximated by Markovic. We will consider the two case to be
when r‹ < rB , where rB is the Bondi radius given in Equation (2.39), and
rB < r‹ . The transition between these two cases happens at M = 2 · 1051
GeV, which puts the black holes we have considered so far firmly in the
case of r‹ < rB . Since this is the case, in order for the specific angular
momentum at rB to not stall the Bondi acccretion, the specific angular momentum l(rB ) should be much smaller than liso . We recall that once matter
reaches the Bondi radius it goes supersonic and we can assume conservation
9.2. KERR BLACK HOLES
89
of angular momentum.
In this case, r‹ < rB , matter reaches rB with an angular velocity Ê0 and
2
thus specific angular momentum l = Ê0 rB
. For the condition lB π liso to
57
hold, we must have that M π 8 · 10 GeV, for 5 millisecond pulsar. Up
to this mass the black hole will grow by Bondi accretion, however as we
saw before, at masses larger than M = 2 · 1051 GeV, the black hole transitions to rB < r‹ . Thus the specific angular momentum at rB becomes,
by Markovic’s approximation, lB = Ê0 r‹2 . Once more, the condition for
Bondi accretion to hold is that lB < liso . This condition hold so long as
M < 3.26 · 1053 GeV, above this mass viscosity can no longer effectively
subtract angular momentum. As we mentioned before, once M > Mrot ,
the specific angular momentum of the infalling matter can no longer stall
Bondi accretion. Since Mrot is only half of 3.26 · 1053 GeV, by the time viscosity can no longer reduce the angular momentum effectively, the angular
momentum no longer plays any role in accretion.
So far we have only considered the approximate expression for Ê when
rB < r‹ . The full expression for the angular velocity in this case is given by
Ê = Ê0
A
B
r‹ 1 r‹2
1+
+ 2 .
r
2r
(9.5)
If we use this to calculate the mass as before we find that for lB π liso , the
mass has to satisfy the condition M π 4.09 · 1053 GeV. This is an increase
of 24% compared to the approximate expression for the induced Ê in the
case rB < r‹ . Still, Mcrit is much smaller than this value, which means that
viscosity will be able to effectively remove angular momentum past Mcrit .
This demonstrates that the rotation of the neutron star can not be ignored
although it can be eventually due to the effects of viscosity. Here we have
considered the effect of viscosity as a means of maintaining Bondi accretion,
others have considered Bondi accretion under rotation[32, 33].
9.2
Kerr Black Holes
Up until now we have considered a Schwarzschild black hole. However, as
the black hole accretes matter it will also accrete some angular momentum.
When the black hole begins to rotate, the accretion rate may decrease as the
rotation increases. A maximally rotating black hole will have no accretion
and will thus only radiate energy away. We will consider accretion under
90
CHAPTER 9. EFFECTS OF ROTATION
rotation as following Bondi and geometrical accretion, as detailed in the
previous chapter, but let us for the moment consider the arguments from
the previous section in the context of a rotating black hole.
In the case of rotating black hole, the parameter  will no longer be 1,
instead for an extremal Kerr black hole  = 1/3. We will now consider an
extremal Kerr black hole in the context of the arguments made previously
for a Schwarzschild black hole using [31]. The first step will be to calculate
Mrot,Kerr similar to that for a Schwarzschild black hole. Equation (9.2) is
still valid for an extremal Kerr black hole, but will yield a different result
since  has changed. It is easy to see that Mrot,Kerr = 9Mrot , due to the
factor of (1/3)≠3 introduced because of the rotation. This means that for
masses above Mrot,Kerr = 1.44 · 1054 GeV, the rotation of infalling matter
will no longer play a part in the accretion. Just as in the previous section
we will consider here the two scenarios where r‹ < rB and where rB < r‹ .
The solution to both cases follows the same steps as for a Schwarzschild
black hole, but the results yield some interesting new scenarios. When
r‹ < rB , the condition lB π liso requires that the mass now satisfies
M π 2.781 · 1057 GeV. This is still well above the transition to the case
where rB < r‹ . In this case for the condition lB π liso to hold the mass
must satisfy the condition M π 2.834 · 1053 GeV. Above this mass, the
rotation will not be able to effectively reduces the angular momentum of
the infalling matter, just as in the Schwarzschild case, but there is a difference. In the Schwarzschild case Mrot was well below this limit which meant
that the rotation of the infalling matter would not play a role in accretion.
However in the case of an extremal Kerr black hole, 2.834 · 1053 < Mrot,Kerr ,
meaning that viscosity will not be able to reduce the angular momentum
long enough for this to no longer play a role. Thus an extremal Kerr black
hole will not be able to accrete matter through Bondi accretion since the
rotation of the neutron star will completely stall accretion. In fact, there is
no accretion onto an extremal Kerr black hole even when its surroundings
are static.
However, it is not only the accretion rate which is influenced by the rotation of the black hole. Page showed that Hawking radiation also changes
when the black hole is rotating. The rotations increases the emitted power,
though particles are affected differently depending on their spin. Table 9.1
shows the relative factors ks (a), which were introduced in Equation (7.3).
We see that fermions, although carrying the majority of the total power,
9.2. KERR BLACK HOLES
91
Table 9.1: ks (a) for each of the three spin values considered. These change
with the critical angular momentum.
a
0
0.01
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.96
0.99
0.999
0.99999
1
k1/2 (a)
k1 (a)
k2 (a)
1
1
1
1.00038133 1.00130592 0.999636023
1.0196923 1.06496589 1.217762063
1.07921407 1.2687373 2.010191348
1.18175774 1.6435577 3.884151414
1.33098994 2.25189716 8.101081531
1.53754394 3.21274627 17.73606489
1.81743071 4.74177552 40.92138103
2.20487223 7.28817442 101.6274958
3.47108489 11.940707 287.0216306
3.9049705 22.3702333 1067.751664
5.58184671 39.05866653 3392.782862
8.19860472 63.9871966 9302.204659
11.3091368 90.9385681 18851.39351
13.1265675 105.752898 25439.37188
13.358788 107.567505 26310.31614
are less significantly affected as the black hole spins up when compared with
spin 1 and 2 bosons. This poses an interesting scenario, for a high spin black
hole, spin 2 particles will significantly dominate the power output due to
their link to the black hole rotation. Spin 1 particles are affected more by
rotation than fermions, the power output for these are almost one order of
magnitude higher, but it is still significantly lower than for spin 2 particles.
Inspection of Table 9.1 shows that fermions increase less than 10% for a
rotation parameter 0.2. After this, fermions start to increase more between
each step, although a steep rise in k1/2 (a) is not present until the black hole
reaches 90% of the critical angular momentum.
Now, if we consider the spin of the black hole to only affect the Hawking
radiation, we would expect the spin to affect the critical mass of the black
hole even with optimal accretion rates. As the black hole spins up we would
consider this to only further decrease the WIMP mass, as the radiation
part of Equation (8.4) will grow with the critical parameter a as seen from
inspection of Table 9.1 and equation (7.3). We will consider three values
of the critical parameter a; 0, 0.5 and 0.99999. We have already discussed
the results for a Schwarzschild black hole in the previous section. We have
92
CHAPTER 9. EFFECTS OF ROTATION
chosen these three values to get an idea of the critical masses close to both
extremes (Schwarzschild and Kerr) as well as the middle of the spectrum.
We will not consider the effects the rotation of the neutron star will have
on the accretion in these sections.
9.3
a = 0.5
Inspection of Table 9.1 reveals that at a = 0.5, the fermions will have
a 53.7% increase in their power outputs, the spin 1 bosons will have their
power output tripled while the spin 2 bosons will be the most affect, reaching
almost 18 times their power output at a = 0.
Bondi Accretion
If at first we consider Bondi accretion onto such a black hole, we see from
Table 9.2 that there is a decrease in the WIMP mass. Comparison between
Table 9.2: Critical masses for a black hole with a = 0.5 which accretes
matter by Bondi accretion.
Free Emission
mDM [GeV]
M [GeV]
4.234885
2.24078 · 1037
Regular Fermi Blocking Boosted Fermi Blocking
4.404355
2.15456 · 1037
4.734359
2.00438 · 1037
Tables 8.5 and 9.2 shows that on average the WIMP mass has decreased
by 11.46% as the black hole went from not spinning at all to spinning at
half its critical value. Since the mass has decreased we would expect the
significance of Fermi blocking to be higher than for the Schwarzschild case.
While the difference between regular and boosted Fermi blocking would
be expected to have decreased since the mass decreased, the increase in
power output due to the spin will have increased what power is emitted.
Thus the difference between these regular and boosted Fermi blocking
should increase. Regular Fermi blocking increases the WIMP mass by 4%
compared with the completely free emission, while boosted Fermi blocking
increases the WIMP mass by 11.79%. Boosted Fermi blocking furthermore
leads to an increase of 7.49%. This difference is slightly larger than for
the Schwarzschild case. We see that as expected the blocking for each
species has increased between the non-rotating black hole and the case
where the black hole has an angular momentum equal to half the critical
9.3. a = 0.5
93
Table 9.3: Blocked Percentages for the critical masses for a black hole using
Bondi accretion at a = 0.5.
Blocked Percentage
Regular Fermi Blocking Boosted Fermi Blocking
Up Quark
Down Quark
Strange Quark
Electron
Gluons
1.67464%
1.76641%
1.58515%
3 · 10≠5 %
73.7443%
68.1709%
69.7716%
66.507%
2.8 · 10≠4 %
100%
value. Here we see that the gluons are almost 75% blocked without the boost
in Fermi momentum, while the quarks have increased by close to 60% from
the Schwarzschild black hole when considering regular Fermi blocking. Still,
regular Fermi blocking is much less severe on the quarks than the boosted
Fermi blocking which blocks almost 70% of the power output for the three
species of quarks.
Geometrical Accretion
We have already seen that geometrical accretion results in significantly lower
critical masses than Bondi accretion. When the black hole starts spinning
up these critical masses should be pushed even further down due to the
extra power output from each species. As with Bondi we would expect
Fermi blocking to have an even more significant role now that the black
hole is spinning. However the only species which can be significantly more
affected by Fermi blocking is the up and down quark which were almost
completely blocked in the case of a non-rotating black hole. This means
that the boosted Fermi blocking can likely only affect the electrons which
are only blocked very little. Here we see that there is actually a difference
Table 9.4: Critical masses for geometrical accretion in the case of a black
hole with angular momentum equal to half the critical value.
Free Emission
mDM [GeV]
M [GeV]
0.97917305
9.69127 · 1037
Regular Fermi Blocking Boosted Fermi Blocking
1.1285245
8.40871 · 1037
1.1403503
8.32151 · 1037
between regular and boosted Fermi blocking, more than would be expected
94
CHAPTER 9. EFFECTS OF ROTATION
Table 9.5: Blocked Percentages for each species of particles for geometrical
accretion at the critical masses with a = 0.5.
Blocked Percentage
Fermi Blocking Boosted Fermi Blocking
Up Quark
Down Quark
Strange Quark
Electron
Gluons
88.65%
89.42%
Not Emitted
4.57 · 10≠4 %
100%
100%
100%
Not Emitted
5.21 · 10≠3 %
100%
from electrons alone. Comparison between Tables 8.8 and 9.5 shows that
blocking increases only slightly due to the effects of rotation. The reason for
this small increase in blocking, when comparing with Bondi accretion is due
to the fact that the Fermi momentum is already located at the tail of the
spectrum which contains only a small part of the power output. We still see
a significant decrease in WIMP mass when the black hole starts spinning.
In fact, the WIMP mass decreases by an average of 12.5% due to rotation.
Inspection of Table 9.4 shows that Fermi blocking leads to a 15.25% increase
in WIMP mass while the boosted Fermi blocking increases the WIMP mass
by 16.46%. However, we also see that the difference between regular and
boosted Fermi blocking is quite small, in fact boosted Fermi blocking only
increases the WIMP mass by 1.04% over the regular Fermi blocking.
9.4
a = 0.99999
Here we consider a close to maximally rotating black hole. Inspection of
Table 9.1 shows that for a such black hole, the fermion power output will
increase by a factor of 13.12, while spin 1 bosons increase by 105.75 and
spin 2 bosons by 25439.37. These show a massive increase in Hawking
radiation, even compared to the case of a = 0.5. Especially the gravitions
receive an extreme boost in their power output. Thus we would expect them
to dominate the total power output for such a black hole. Furthermore, we
would expect that Fermi blocking has an even more significant impact on
the critical mass of the black hole while the difference between regular and
Fermi blocking should be the smallest of the three rotation cases we have
considered.
9.4. a = 0.99999
95
Bondi Accretion
Once again we will start by considering Bondi accretion of matter. Table 9.6
Table 9.6: Critical masses for a close the maximally rotating black hole
using Bondi accretion.
Free Emission
mDM [GeV]
M [GeV]
Regular Fermi Blocking Boosted Fermi Blocking
1.742375
5.44626 · 1037
1.84032
5.1564 · 1037
1.854505
5.11696 · 1037
shows that the WIMP mass has decreased by an average of 59.3% from
a = 0.5 and 63.97% from a non-rotating black hole. As seen before, Fermi
blocking increases the critical WIMP mass significantly at low masses, however in the case for a close to maximally rotating black hole there is one
thing to consider. In this case, the fermions no longer dominate the total
power output, instead this is dominated by spin 1 and 2 bosons. Although
we do consider the blocking of gluons, their power pale in comparison to that
of the gravitons. Here regular Fermi blocking increases the WIMP mass by
5.6% while the boosted Fermi blocking increases the WIMP mass by 6.4%.
Here we see the effect of the spin 1 and 2 bosons. Even though the boosted
Table 9.7: Blocked Percentage for each particle species at the critical masses
for a close to maximally rotating black hole using Bondi accretion.
Blocked Percentage
Fermi Blocking Boosted Fermi Blocking
Up Quark
Down Quark
Strange Quark
Electron
Gluons
38.87%
40.48%
Not Emitted
1.7 · 10≠4 %
100%
99.97%
99.98%
Not Emitted
4.5 · 10≠4 %
100%
Fermi blocking almost completely blocks the up and down quark, while the
regular Fermi blocking only blocks around 40%, the difference between the
two critical masses is 0.77%. This is due to the fact that the photons and
gravitons dominate the power output for such a rapidly rotating black hole.
96
CHAPTER 9. EFFECTS OF ROTATION
Geometrical Accretion
We now turn to the case of geometrical accretion. Much like with Bondi
accretion for such a rapidly rotating black hole, we would here expect that
Fermi blocking will a rather diminished effect compared with the previous
cases. However there should also be no difference between the regular and
boosted Fermi blocking expect for a very small difference resulting from
the blocking of electrons. Comparison with Tables 8.7 and 9.4 shows that
Table 9.8: Critical masses for geometrical accretion onto a close to maximally rotating black hole.
Free Emission
mDM [GeV]
M [GeV]
Regular Fermi Blocking Boosted Fermi Blocking
0.379922084
2.49773 · 1038
0.4047642
2.34444 · 1038
0.4047655
2.34443 · 1038
the critical WIMP masses for geometrical accretion has decreased by an
average 63.25% compared with a = 0.5 and by 67.8% from a non-rotating
black hole. Compared with Bondi accretion, we see here that the critical
masses for geometrical accretion are more affected by the rotation of the
black hole in this particular case. For a close to maximally rotating black
hole we can see that regular Fermi blocking leads to 6.53874% increase in
the WIMP mass while boosted Fermi blocking increase the WIMP mass
by 6.53908%. Much as expected these two values are extremely close to
one another, but boosted Fermi blocking does increase the WIMP mass by
3.2 · 10≠4 %, which is negligible. Even though the blocking of electrons goes
Table 9.9: Blocked Percentages at the critical masses for geometrical accretion onto a close to maximally rotating black hole.
Blocked Percentage
Fermi Blocking Boosted Fermi Blocking
Up Quark
Down Quark
Strange Quark
Electron
Gluons
100%
100%
Not Emitted
3.5 · 10≠3 %
100%
100%
100%
Not Emitted
0.16%
100%
up by a factor of 50, they are still not significantly blocked nor do they
have a massive impact on the critical masses. Once more this is due to
9.5. SUMMING UP
97
the contribution from the gravitons which dominates for black holes with
values of a close to 1. We do see that at very low masses, the emission of
electrons are the only difference between the two types of Fermi blocking
which we have considered.
9.5
Summing Up
This section showed that rotation of either the neutron star or the black
hole may significantly change the scenario considered in Chapter 8. We
saw in the second section that the viscosity of the neutron star will not
be sufficient to insure Bondi accretion unto an extremal Kerr black hole
when the star is rotating. However more importantly we used the work of
Page[24] to show that the rotation of the black hole itself can impact the
critical masses. While this would be clear by just looking at the original
article, we have considered here also the effect of Fermi blocking. In fact
we see that even at low masses the importance of Fermi blocking may be
overturned by the boost in radiation of unblocked particles, especially the
graviton.
Chapter 10
Comparison with IceCube
In this section we will use our model to see if it might explain the PeV
neutrinos[34] observed at IceCube.
10.1
A Brief Summary
The IceCube neutrino observatory (IceCube) is a neutrino telescope located
the Amundsen-Scott South Pole Station in Antarctica. It houses thousands
of sensors distributed over a cubic kilometre of volume under the antarctic ice. While neutrinos do not interact strongly with matter, when they
do react with the water molecules in the ice they may create charged leptons. Should these charged particles be energetic enough they may emit
Cherenkov radiation. Cherenkov radiation is emitted when a charged particle passes through a dielectric medium with a velocity greater than of light
in that medium[35].
While Einsteins theory of special relativity states the speed of light in
vacuum is a universal constant, the propagation of light through matter can
be lower than this value. This makes it possible for particles to move faster
than the speed of light in that medium while still being below c. When
these charged particles move, they disturb the local electromagnetic field
in the medium. This causes the electrons in the mediums atoms to become
displaced making the atoms polarized as the EM field of the charged particles passes. As this disruption passes, the electrons reach equilibrium and
fall into their ground state, emitted photons.
Under normal circumstances, these photons will interfere destructively
and no radiation is detected, however, when the disruption moves faster
99
100
CHAPTER 10. COMPARISON WITH ICECUBE
than the light can be propagated through the medium, the photons interfere constructively and amplify the radiation. An analogy can be made
to the sonic boom of super-sonic aircrafts, in which the aircraft travels
faster than the speed of sound in air, this means that the sound waves
can not propagate forward from the aircraft and instead form a shockfront.
While Cherenkov radiation requires particles with high energies, these can
be found in nuclear reactors, Cherenkov radiation is the source of nuclear
reactors characteristic blue glow, but also from astrophysical sources.
IceCube has several experimental goals such as finding a point source of
high-energy neutrinos, gamma ray bursts, indirect detection of WIMP dark
matter, neutrino oscillations, galactic supernovae and string theory. Our
interest is the discovery of neutrinos with energies of 1 PeV in 2013[34].
10.2
Black Hole Explanation: An Attempt
We will now try to explain the occurrence of the PeV signals observed by
IceCube. Previously we saw that the peak in the neutrino spectrum goes
as ≥ 4.5TH . This means that we can calculate the mass of the black hole
needed for a peak at 1 PeV.
Ê‹ =
4.52389
mDM .
16
(10.1)
From this we get that the required WIMP mass for this peak is 3.53678 · 106
GeV resulting in a black hole mass of 2.68307·1031 GeV. Using Table 7.1, we
can find the total rate of neutrino emitted across all modes, now including
the ‹ and ‹¯, to be
dN
= 0.001456 sec≠1 .
dt
If we assume that the black hole evaporates at a constant rate, given that
no accretion rate we have studied can stabilize this mass, the time it will
take to evaporate the black hole is given by
5120fiG2 M 3
,
h̄c4
This gives us a lifetime of 0.009126 seconds. Meaning that the black hole
will emit 1.1235 · 1026 neutrinos before it evaporates. If we assume that this
black hole is located inside the closest neutron star, 100 parsecs away, then
the average number of neutrinos hitting the earth per square meter is
tevp =
N
neutrinos
= 9.42458 · 10≠13
A
m2
(10.2)
10.2. BLACK HOLE EXPLANATION: AN ATTEMPT
101
The number of events will the depend on the surface area of the detector.
IceCube is contains 86 strings, we will assume that these run from a depth
of 50m down to 2450m, furthermore, based on [36] we estimate the radius
of each string to be 20m. This means that the combined surface area of the
detector is
A = 86 · 2 · fi · 2400m · 20m = 2.5937 · 107 m2
(10.3)
Combining Equations (10.2) and (10.3), we find the number of events we
would expect from this model to be 2.4 · 10≠5 . Thus, our model may not be
the most likely for the PeV event.
Chapter 11
Dark Matter Accretion
In this section we will consider another type of dark matter accretion than
the one described earlier. We will test to see if this method may allow for a
lower cross section while still maintaining accretion onto the neutron star.
11.1
Scattering
We will base this chapter on the work outlined in [2]. We have already seen
that the Schwarzschild metric is given by Equation (2.9), but if we take the
motion of the particle to be in the ◊ = fi/2 plane, then we may write the
timelike geodesic equation as
A
du
d„
B2
= 2mu3 ≠ u2 + 2–2 mu + –2 (— 2 ≠ 1) © 2m(u ≠ u1 )(u ≠ u2 )(u ≠ u3 ),
(11.1)
where we have defined u = 1/r and m = GM . The constants of motion
– and — are expressed in terms of the mass of the scattered particle µ, its
energy E and its angular momentum l at asymptotic radial distances[2]
–=
µ
,
l
—=
E
.
µ
(11.2)
We will be interested in the scenarios where u1 Æ u2 Æ u3 are all real, a
complex radius has no physical meaning, The test particle will initially be
located at infinity and we may treat the neutron star as a point mass in
this scenario, however as the particles reaches the radius of the star we will
no longer be able to treat this as such. This is where there will be some
deviation from [?]. As the particles moves further into the star, the mass
enclosed within its orbit will decrease according to its density profile. Due
103
104
CHAPTER 11. DARK MATTER ACCRETION
Figure 11.1: Figure taken from [2]. a) shows the scenario where – is small,
1 22
du
here we see that the particle will only be deflected, since a negative d„
does not make physical sense. In b) – has been increased, resulting in a
temporarily spiraling solution. Finally in c), – has reached the critical value
and the particle is captured. We see that
the solution here can continue
1 22
du
since it passes through a minima where d„ = 0 and grows past this point.
to this we will be changing the mass as u changes. There can in this case
be two possibilities; the particle is scattered when u1 Æ 0 < u2 < u3 and
capture of the particle when u1 Æ 0 < u2 = u3 .
Considering this, Equation (11.1) leads to three scenarios. When – is
very small, ie. l is large, the particles trajectory will be largely unaltered
from that of a straight line, the trajectory may become slightly curved but
there will be great deviation. If – is increased, this deviation also increases
until the particle returns to its original trajectory, this can lead to several
spirals around the center of the star as – is increased, but eventually the
particle will be expelled. When – reaches the critical value of –c the particle
will no longer be able to escape but will spiral towards the center. The
11.2. UNIFORM DENSITY
105
Figure 11.2: This shows the scenario considered for the uniform case. We
see that u0 = u2 = 1/R where R is the radius of the star.
critical value of alpha is approximated as
–c =
A
—(9— 2 ≠ 8)3/2 ≠ 27— 4 + 36— 2 ≠ 8
32m2
B1/2
.
(11.3)
While — is constant through the motion of the particle, m can change so
long as we do not regard the neutron star as a point particle. As the particle
goes through the neutron star, –c will increase as the encompassed mass
decreases. So long as – > –c the particle will continue the spiral, however
when this is no longer valid the spiral will end and the particle will be
expelled to infinity.
11.2
Uniform Density
The simplest case will be to consider the neutron star as being isotropic
throughout. This means that we can define the density as fl = M/V . Thus
we can write Equation (11.1) as
A
du
d„
B2
= k1 ≠ u 2 +
k2
,
u2
(11.4)
where we have defined k1 and k2 as
8
k1 = fiGfl + –2 (— 2 ≠ 1),
3
8
k2 = fi–2 Gfl.
3
(11.5)
Using this we can integrate Equation (11.4) to get
A
B
1
k1 ≠ 2u2
±„ + C = ≠ tan≠1 Ô
.
2
2 k2 + k1 u 2 ≠ u4
(11.6)
We assume that the particle enters at an angle „0 at u0 . The particle
then has its closest approach to the center at u1 , note that this u1 is not
the same as the one used in Equation (11.1), where we required that at
k2 + k1 u2 ≠ u4 = 0. From here the particle moves out of the star, exiting at
u2 , which is also different from the one in Equation (11.1). For simplicity
we require that „(u0 ) = „(u2 ) = 0. We will the plus (minus) sign of
106
CHAPTER 11. DARK MATTER ACCRETION
Equation (11.6) for the angle u1 ≠ u0 (u2 ≠ u1 ). We can the see that the
total angle deflection can be written as
„ = („2 ≠„1 )+(„1 ≠„0 ) = ≠„1 +„1 = tan
≠1
A
B
k ≠ 2u2
Ô 1
, (11.7)
2 k2 + k 1 u2 ≠ u 4
where we have made use of the sign convention explained previously. Given
the nature of tan≠1 we see that the angle can at most become fi meaning
that the particle will at most be deflected by half a rotation. However, we
have more density profiles to consider.
11.3
r2 Density Profile
We now assume a steeper density profile, as opposed to the flat, uniform
profile considered in the previous section. We now assume that the density
goes as fl = flc /r2 , which means that the reduced mass m is given by
m = 4fiG
⁄
flr2 dr = 4fiGflc
where we have used that
tion (11.1) gives us
A
B2
dr
du
⁄
≠
1
1
du = 4fiGflc ,
2
u
u
(11.8)
= ≠ u12 . Inserting Equation (11.8) into EquaA
A
BB
—2 + A ≠ 1
= Au ≠ u + A– + – (— ≠ 1) = (A ≠ 1) u + –
,
A≠1
(11.9)
where we have defined A = 8fiGflc . We can now integrate Equation (11.9)
which gives us
1Ô
2
1
±„(u) = Ô
ln
B 2 + u2 + u ,
(11.10)
A≠1
du
d„
2
Ò
2
2
2
2
2
2
2
+A≠1
where B = – — A≠1
. However, inspection of A reveals that A π 1, meanÔ
ing that A ≠ 1 will be complex. However, we can rewrite Equation (11.9)
so that
A
B2
A
A
BB
2
du
2
2 — +A≠1
= (1 ≠ A) ≠u + –
,
(11.11)
d„
1≠A
which can be integrated to give
A
where C = –
be fi.
Ò
B
1
u
±„(u) = Ô
tan≠1 Ô 2
,
1≠A
C ≠ u2
A+— 2 ≠1
.
1≠A
(11.12)
Once again we have an a deflection that can at most
11.4. LANE-EMDEN PROFILE
11.4
107
Lane-Emden Profile
A more accurate description of the density profile of a neutron star would
be that of a polytropic fluid, Equation (2.27) with polytropic index
n=
1
“≠1
or
“ =1+
1
.
n
(11.13)
For a neutron star we set n = 1.5. We furthermore introduce the dimensionless parameter ◊ as a measure of fl
fl = flc ◊ n
(11.14)
and the measure › as the distance to the centre given by
r = –›,
where
–2 =
(n + 1)Kfl1/n≠1
c
.
4fiG
(11.15)
Having introduced these we can introduce the Lane-Emden equation given
by
A
B
1 d
2 d◊
›
= ≠◊n ,
(11.16)
› 2 d›
d›
where the solution ◊ = ◊n (›) is called the Lane-Emden function. The LaneEmden equation is a rewritten form of Equation (2.27), where we have
introduced ◊, › as well as the definitions dm/dr = 4fir2 and
dP
Gmfl
=≠ 2 .
dr
r
(11.17)
For this section we have used a code to solve the Lane-Emden equation
numerically for n = 1.5 and where we have bounded the mass and radius
according to
K = Nn GM (n≠1)/n R(3≠n)/n ,
(11.18)
where Nn = 0.42422 for n = 1.5. The resulting density profile is shown
below along with a scaled r2 model. So long as the r12 profile is steeper than
the Lane-Emden the particle should continue its spiral. From inspection
of Figure 11.3 we see that this is the case from 10,000m down to roughly
2,000m. At best this gives us fi/2 from Equation (11.12).
108
CHAPTER 11. DARK MATTER ACCRETION
Figure 11.3: Lane-Emden (orange) has a flat density profil close to the core
as well as in the outer layers. This is contrasted with the steep function of
1
which goes to infinity as the radius goes to 0.
r2
Chapter 12
Conclusions
We have in this thesis shown how the critical mass of a black hole will
depend on several factors including its spin and its accretion rate. The goal
was to observe how Hawking radiation might impact the formation of such
a dark matter black hole inside of a neutron star. We used the work of
Page[3, 24] to model the radiation and how it changes with the accretion of
angular momentum onto the black hole. Furthermore, we looked at how the
Hawking radiation might be limited by the surrounding degenerate matter
by imposing a Fermi surface. While the Fermi blocking of quarks and
electrons had been done in [26], we have presented here a possible way in
which the emission of gluons from the black hole might be limited by the
Fermi surface although these are not fermions themselves.
We will conclude by discussing some of subtleties of our calculations as
well as ways in which to improve these calculations in the future.
Interpolation Function
We have seen some small fluctuations in the results calculated by the interpolation functions made from [3]. The reason for these are likely that
the software used does not autotrace the functions, something which could
difficult due to the crossing of some of the curves in the original figure.
Instead each point of data used for the interpolation function had to be selected manually along with the range of the axes. As such it is possible that
some points were slightly above their actual value while others may have
been below. Given that the interpolation function gives critical masses very
close to those found by Equation (7.3), the overall use of the interpolation
function appears justified.
109
110
CHAPTER 12. CONCLUSIONS
Accretion Rates
The results presented in this thesis show that the critical black hole mass
will depend on the type of accretion. We have considered only Bondi accretion and an estimated accretion rate based on geometrical arguments
when the radius of the black hole is below the radius of the surrounding
neutrons. The derivation of Bondi accretion[9, 7] assumes that K and are
constant, however [27] points out that the equation of state might change
if n reaches a critical value. It would be of interest to use data from [37]
to calculate the Bondi accretion for the ultra-dense material at the core of
the neutron star. Furthermore, it would be interesting to use the solution
of the Lane-Emden equation to calculate how the change in density might
affect Bondi accretion. The geometrical accretion assumes that the will be
a constant flux of neutrons through the cross section of the black hole. Due
to the extreme density of the core this may not entirely be the case, however, the geometrical accretion was introduced as a conservative estimate
for accretion at such low radii. While we have only considered these two
types of accretion, it is possible that in the regime just above the radius
of the neutron that Bondi accretion may not be the appropriate choice,
as such, future work could include multiple types of accretion for a wide
spectrum of masses when it comes to comparison.
Rotation of the Black Hole
While we have considered how the rotation of the black hole affects the
Hawking radiation, we assumed that Bondi and geometrical accretion still
holds under these circumstances. We already know that accretion onto an
extremal Kerr black hole, although considered here, is not physical actuality.
However, for a rotating black hole with a < 1, accretion will still take place
but it may not follow Bondi or geometrical accretion. The black holes we
have worked with have been assumed to be spherically symmetric, but as we
have seen this is not the case for the rotating black hole. Here the spherical
symmetry is no longer valid and the new accretion will have to take this into
account. Works by Pariev [38] and Pal et. al[39] have considered accretion
onto a rotating black hole. Pal et. al assumes a black hole moving through
a surrounding gas medium with a constant velocity. When considering flow
aligned with the rotation axis of the black hole, Pal et. al shows that the
rotation of the black hole lowers the accretion of the surrounding matter.
This happens since the effective cross section drops, where the effective cross
section is defined as a circle in the plane perpendicular to the rotation axis
at infinity. It would be interesting to use these accretion rates when the
111
black hole is rotating to see how these modified accretion rates will change
the critical black hole mass.
Rotation of the Neutron Star
We showed that the effects of the neutron stars rotation may eventually
be disregarded in the presence of viscosity. For the case of a Schwarzschild
black hole we saw that viscosity may effectively carry off the excess angular momentum to ensure Bondi accretion onto the black hole, however, the
mass for which this is the case is far above the critical mass of black hole
using Bondi accretion. For an extremal Kerr black hole we saw that viscosity is not sufficient to ensure Bondi accretion when the star is rotating. We
only considered viscosity as a way of transporting angular momentum and
ensuring that Bondi accretion takes place, however Markovic[31] only considers the effects of magnetic torques and convection as ways of transporting
angular momentum. Considering that neutron stars have very strong magnetic fields it is possible that magnetic braking may be able to effectively
transport away angular momentum. Even if this is the case, it would be
useful to consider how the formation of a disc or torus may change the
accretion rate.
Evolution of the Black Hole
This thesis has not considered what happens beyond the critical masses
outlined, the sole interest has been to find these limits. Future works could
include how a black hole which is formed with a mass different from the
critical mass evolves. Newly formed black holes with masses below the
critical mass may evaporate before they reach critical mass unless another
type of accretion takes place which is sufficient to stall the evaporation.
Once above the critical mass the black hole may continue to grow until it
has consumed the star. This means that old neutron stars may constrain
the mass of dark matter WIMPs. Both of these cases would be interesting
to consider dynamically so as to show how a black hole of a given mass
may evolve with time including the accelerated radiation (accretion) when
the mass decreases(increases) from the critical mass as well as how the
dynamical accretion of angular momentum from its surrounding can impact
its evolution.
Part IV
Appendices
113
Appendix A
Geometrical Accretion
In this chapter we will cover the derivation for the geometrical accretion.
A.1
Derivation
We will consider the scenario where the black hole only accretes particles
which crosses the event horizon located at rS . We will consider the energy flux across a surface area dS in the time interval ”t. If we consider a
box of volume V where one side has area dS, then the energy flux across
this area will depend on the density of the particles within as well as their
distribution. The particles in velocity phase space may be given be a distribution such that not all particles cross the surface area. As illustrated in
Figure A.1b we are only interested in particles which have a velocity that
takes them across the event horizon of the black hole. Thus the energy
transported across the event horizon in the time ”t can be written as
d3 k
dE = Ágvp(v) cos ◊dS”t
,
(2fi)3
(A.1)
where we have introduced the momentum phase space described by the k
vector and p(v) is the distribution of the particles in velocity phase space.
Furthermore we have used the degeneracy g and the energy per particle Á.
We can then decompose the volume integral in momentum phase space and
rewrite Equation (A.1)
dE
1
= 3 Ágv cos ◊p(v)k 2 dkd„d cos ◊.
dS”t
8fi
To find the flux we need to integrate both sides of Equation (A.2)
Flux =
⁄
⁄
1 ⁄
2
Ágvp(v)k
dk
cos
◊d
cos
◊
d„.
8fi 3
115
(A.2)
(A.3)
116
APPENDIX A. GEOMETRICAL ACCRETION
(a) Picture showing the surface
area dS and how the volume is
calculated.
(b) Picture showing the flux
through the surface area dS and
through the surface area of the
black hole.
Figure A.1: Illustrations for how geometrical accretion is envisioned.
In order for us to be able to carry out this integral we need not only the
angular limits but also a way of relating k to the velocity v. Since we are
dealing with relativistic particles we know that
v=
k
.
Á
(A.4)
Inserting this into Equation (A.3) allows us to rewrite it to only depend on
k to
⁄
⁄
g ⁄
3
Flux = 3 p(v)k dk cos ◊d cos ◊ d„.
(A.5)
8fi
Given that this is in the core of a neutron star we can set the degeneracy
factor g = 2 but it also allows us to use the Fermi-Dirac distribution for
the particles. Thus we can integrate from 0 to the Fermi momentum kF
as outlined by Equation (1.8). The angular limits will be 0 Æ „ Æ 2fi and
0 Æ cos ◊1, which simplifies Equation (A.5) to
1 ⁄ kF
1 4
Flux = 2
p(k)k 3 dk =
k .
4fi 0
16fi 2 F
(A.6)
The accretion of energy onto the black hole will then be the flux across the
total surface area spanned by the Schwarzschild radius
Accretion = 4firS2 · Flux =
G2 M 2 4
kF
fi
(A.7)
Appendix B
Markovic derivations
We will here outline the derivations of the formulae used in Markovic’s
paper leading to the full expression for the angular velocity Equation (9.5).
B.1
Derivations
We will write Bondi accretion Ṁ = CM 2 , which allows us to write the
radial velocity as
CM 2
vr = ≠
.
(B.1)
4fiflr2
We will consider the effects of rotation to be most pronounced near the
equator of the star, thus we limit the motion to a thin slice of star between
≠a/2 < ◊ < a/2, where ◊ is the angular latitude with ◊ = 0 at the equator.
The total angular momentum enclosed within the ring-shaped part of the
slice from r to r + r can be found be approximating the angle a as small
and then calculating the mass enclosed within the ring. Thus the total
angular momentum is given by
L = 2afi( r)flr2 l,
(B.2)
where l is the specific angular momentum. Then we can write the change
in L is given by
d
L =
dt
A
B
ˆ
ˆ
+ vr
L = T (r) ≠ T (r +
ˆt
ˆt
ˆ
¥ ≠ r T (r),
ˆr
where we have defined
ˆ
T (r) = ≠2afifl‹r4
ˆr
117
r)
(B.3)
(B.4)
(B.5)
118
APPENDIX B. MARKOVIC DERIVATIONS
as the viscous torque exerted on the side of the ring at radius r,‹ is
kinematic viscosity, = l/r2 is the angular velocity, and fl‹rˆ /ˆr is
viscous stress. Combining Equation (B.1)-Equation (B.5) along with
conservation of mass, d(flr2 r)/dt = 0, we get that the equation for
transport of angular momentum can be written as
C
ˆ
CM 2 ˆ
1 ˆ
ˆ
l≠
l= 2
fl‹r2
2
ˆt
4fiflr ˆr
flr ˆr
ˆr
3
1
l
r2
4D
.
the
the
the
the
(B.6)
Assuming that M does not change significantly over the course of the accretion, we neglect the time derivative. This allows us to integrate Equation (B.6) and write it as
4fifl‹ 4 ˆ
l+
r
CM 2 ˆr
3
4
1
l =K
r2
(B.7)
where K is the constant of integration. We can rewrite this equation to a
more convenient form using = l/r2
d
dr
=≠
where we have defined
D©
D
r2
+
KD
,
r4
CM 2
.
4fifl‹
(B.8)
(B.9)
If we assume that D is everywhere constant we get the stationary solution
of the viscous angular momentum equation
=
0
C
3
D 1 D
1+ +
r
2 r
42 D
(B.10)
where we have substituted 0 © C/D2 . This is the same as Equation (9.5)
although that used small Ê as the angular velocity.
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