a phenomenological study of black holes bouncing into white …1052415/... · 2016. 12. 6. ·...

Master of Science Thesis

A Phenomenological Study of Black Holes

Bouncing into White Holes in Loop Quantum

Cosmology

Celine Weimer

Department of Theoretical Physics,School of Engineering Sciences,

KTH Royal Institute of Technology, 106 91 Stockholm, Sweden

Stockholm, Sweden 2015

TRITA�FYS 2015:68ISSN 0280-316XISRN KTH/FYS/�15:68�SE

c© Celine Weimer, September 2015

Abstract

Loop quantum gravity enables a new way to consider black holes. The black hole is hereconsidered a stage in a bounce that will quantum tunnel into a white hole. This bounceseems frozen in time due to the huge gravitational time dilation. The bounce time, τ , andthe mass, M , of the black hole are related by a constant k of unknown value. Using k as afree parameter the detection characteristics of a bouncing black hole are investigated. Twodi�erent energy signals are studied, one based on dimension and the other on the contentsof the early Universe. From these the maximum detection distance to a bouncing blackhole is calculated and an integrated spectrum from several bouncing black holes is plotted.Both these suggest signals that detection of bouncing black holes, if they exist, is possible.

Sammanfattning

Denna rapport diskuterar hur kvantgravitation, speci�kt loopkvantgravitation, möjliggörett nytt sätt att se svarta hål och dess egenskaper. Svarta hål betraktas här som en fasunder ett bounce, som genom kvanttunnling kommer övergå till ett vitt hål. Denna övergångobserveras som fryst i tiden pga av den enorma gravitionella tidsdilatationen. Livstiden, τ ,och massan, M , för svarta hål relateras till varandra genom konstanten k med okänt värde.Med k som en fri parameter undersöks detektionsegenskaperna för två olika energisignaler.Den första baserad på dimensioner och den andra på egenskaperna hos det tidiga universum.Från dessa signaler är det maximala detektionsavståndet till ett vitt hål beräknat och ettintegrerat spektrum från �era vita hål konstruerat. Båda dessa antyder att detektion avdessa vita hål, förutsatt att de existerar, är möjligt.

iii

Preface

The work of this master thesis was conducted in Grenoble, France, at the Laboratoire dePhysique Subatomique & Cosmologie LPSC, IN2P3 (CNRS), Université Grenoble Alpesunder the supervision of Prof. Aurélien Barrau. For the Department of Theoretical Physicsat KTH Royal Institute of Technology in Stockholm, Sweden with Prof. Tommy Ohlssonas supervisor.

The results in this thesis are also presented in the article "Phenomenology of bouncingblack holes in quantum gravity: a closer look" which can be found on arXiv:1507.05424.

v

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiSammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Preface v

Contents vii

Introduction ix

1 Background 1

1.1 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Hawking Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Bouncing Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Primordial Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Investigation 5

2.1 Interval of the parameter k . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Low Energy Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 High Energy Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Particles Emitted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1 Pythia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5.1 Number of Required Photons . . . . . . . . . . . . . . . . . . . . . . 132.5.2 Detector Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5.3 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6 Integrated Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6.1 Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6.2 Integrated Low Energy Signal . . . . . . . . . . . . . . . . . . . . . . 162.6.3 Integrated High Energy Signal . . . . . . . . . . . . . . . . . . . . . 172.6.4 Mass Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Result 21

3.1 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Integrated Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Discussion 29

5 Conclusions and Future Prospects 33

vii

viii Contents

Bibliography 35

Introduction

In 1687 Sir Isaac Newton laid the foundation for the theory of gravitation [1]. Within hiswork on Newton's law of universal gravitation he describes that all objects are attracted toeach other with a certain force, gravity. This force is proportional to the product of themass of the objects and the distance between them. More elegantly written as

F = Gm1m2r

,

where G is the gravitational constant, m1 and m2 the masses of the objects and r thedistance between the objects' center of mass.

Fast forward to 1905 where another great scientist, Albert Einstein, �nds the relation-ship between time and space, the space-time. Creating special theory of relativity, with twoimportant rule:

• The principle of relativity, which states that in all inertial frames the laws of physicsare the same.

• Einstein's law of light propagation, saying that the speed of light in vacuum, c, is thesame in all inertial frames, directions and time.

The work on special relativity also lead to the world famous equation E = mc2 where themass, m, is related to an energy, E, by multiplying it with the square of the speed of lightin vacuum, c. Special relativity is used to describe fast objects, with velocities close to thespeed of light, but it neglects gravitation and is therefore not applicable for heavy objects.

The general theory of relativity includes the force of gravity into special relativity. It was de-veloped by Einstein in 1915. General relativity uni�es Newton's law of universal gravitationand the special theory of relativity, describing fast and heavy objects.

In modern physics general relativity is used to describe gravitation, while classicalphysics still uses Newton's models. In general relativity gravity is considered a geomet-ric property of space-time. General relativity suggests the existence of black holes and istherefore important for the work of this thesis.

Using Einstein's �eld equations energy and momentum are related to the curvature ofspace-time. The �rst solution to the non-linear Einstein's �elds equations was quite quicklyfound by the physicist Karl Schwarzschild, with the Schwarzschild metric. It is the mostsimple solution but can only be used to describe black holes without electrical charge andangular momentum, Schwarzschild black holes.

General relativity has been con�rmed in all experiments and observations up to date.

Another large paradigm shift came with the development of quantum theory. Quantum

ix

x Introduction

theory is said to have been founded by Max Planck in the early 20th century, but manyother great scientists worked on it as well i.e. Erwin Schrödinger. Quantum theory pro-vides a fundamental explanation of nature at small scales. When studying extremely smallscales, in the order of the Planck constant, objects cease to behave as they do in everydaylife. Particles, like electrons and photons, do not only behave as particles but also as waves.Finding the wave nature of particles opened a new way of studying the quantum realm insize scales of an atom or even smaller objects.

In quantum theory particles behave probabilistically. A famous aspect of quantum theorycomes with the uncertainty principle where one can determine either the particles posi-tion or its momentum with great certainty, but not both simultaneously. With increasingcertainty of the position the certainty of the momentum decreases, and vice versa. Theuncertainty principle is described by the following equation

σxσp ≥~2,

where the σx and σp are the standard deviations for position, x, and momentum, p, respec-tively and ~ is the reduced Planck constant.

Quantum theory also solved the problem with the stability of the atom. According toclassical mechanics and electromagnetism the electrons circulating the nucleus would grad-ually loose energy and eventually collide with the nucleus. In quantum theory the electronis no longer considered as a �rm particle with absolute position and energy, therefore it willnot loose energy and collide with the nucleus.

One of the most memorable example of quantum theory is the thought experiment withSchrödinger's cat, where a cat is trapped in a box with an arrangement of a radioactivecompound and a �ask of poison which has a probability to break and kill the cat. Witha closed box it is not possible to know whether the cat is alive or dead, it is thereforeboth simultaneously. This thought experiment is to help describe how quantum systemsare based on probability. With a simple measurement (opening the box) one may �ndout what state the cat truly is in. Although, this is not applicable to objects a�ected byquantum e�ects, since measurements changes the state of the particles. This odd behaviourof particles a�ected by quantum e�ects was shown in the Stern-Gerlach experiment in 1922.

In this thesis quantum theory will be used along side with general relativity in order tostudy black holes further.

The discovery of black holes has had a major impact in science and even though theirexistence have been known for a long time a lot remains to be discovered. It is believed thatall which falls into a black hole is lost and therefore all its physical information. This callsfor the black hole information paradox, where an issue arise with the fact that informationmust always be preserved. There are several theories to resolve this paradox, whereas oneis presented here.

Black holes are believed to be the end stage for massive stars, with masses > 5M�. Butwhat if they can reach a further stage in their development? It is just not observed dueto the great gravitational time dilation caused by their massiveness. In order to answerthis question quantum gravity is required. Quantum gravity is the theory which combinesquantum theory, explaining the e�ects on the in�nitely small, and general relativity, han-dling the immensely massive objects. A black hole, being something so dense it is both

xi

immensely massive and incredible small, needs a combination of both the theories to beexplained further.

The search for a uni�cation theory, to unify both theories has been a long quest andis not yet completed. There are several approaches to explain quantum gravity. The mostpopular ones being loop quantum gravity and string theory. In this work loop quantumgravity, LQG, is used with its e�ect on cosmology, called loop quantum cosmology, LQC.

With the help of quantum gravity new ways of studying black holes are opened, provid-ing an alternative view of the classical black hole, enabling further stages in the black hole'slife. If the quantum gravity e�ects can counteract the weight collapse of the black hole itwill be able to bounce out into a white hole, emitting a lot of energy and all the contentswhich formed the black hole. Within the contents, all its physical information still intact,solving the information paradox.

This thesis will study the theoretical bounce of a black hole into a white hole and cer-tain aspects for detection of the emitted photons. It presents the detection distance forblack holes bouncing in our time epoch and also how an integrated signal from severalwhite holes would appear if measured at Earth.

Chapter 1

Background

The properties of black holes are explained in the �rst section with details on Hawkingradiation. It is followed by a di�erent view on black holes, using quantum gravity, allowingthem to bounce into white holes. Thereafter primordial black holes are presented. Thelast section of the background explains previous work on this speci�c theory and how tocontinue from it.

1.1 Black Holes

Black holes are mathematically de�ned as spacetime regions where, due to strong gravita-tional pull, no particles are known to escape, except the theorized Hawking radiation (moredetails on Hawking radiation are presented in section 1.1.1). Black holes are extremely denseobjects and only described with three parameters: their mass, electrical charge and angu-lar momentum (spin). The most primitive black holes are only characterized by their massand are called Schwarzschild black holes, that is non-rotating black holes with no charge [2].

The size of a black hole is de�ned as its Schwarzschild radius which for non-rotating blackhole also acts as the event horizon. Anything that passes through the event horizon, whichis a boundary in spacetime, will not be able to escape the gravitational �eld. In laymen'sterms it is known as "the point of no return". The Schwarzschild radius, rs is de�ned inSI-units as [2]

rs =2GM

c2, (1.1)

whereM is the mass of the black hole, G the gravitational constant and c the speed of lightin vacuum.

1.1.1 Hawking Radiation

In 1974 Stephen Hawking �rst published his theory that particles can escape the huge grav-itational pull from black holes, so-called Hawking radiation [3].

One intuitively way to describe Hawking radiation is through the spontaneous creationof a pair of particles, with the regular particle and its antiparticle. One of the createdparticles falls through the event horizon and is pulled into the black hole whilst the othermanages to escape, thanks to tidal forces. The particle which manage to escape has positive

1

2 Chapter 1. Background

energy and is the Hawking radiation. The particle which tunnels through the event horizonmust have negative energy in order to preserve the energy. The black hole will then loosemass due to the negative energy from the infalling particle [3].

If the black hole looses more matter than it accretes, it will eventually vanish. The blackhole will evaporate mass over time at di�erent rates and eventually vanish at evaporationtime, ending with a big explosion.

For non-rotating, uncharged black holes the evaporation time, tev, is [4]

tev =5120πG2M3

~c4, (1.2)

where ~ is the reduced Planck constant.

There is so far no experimental detection of Hawking radiation but it is a well recognizedtheory. With black holes loosing mass over time the lifetime of black holes is consideredlimited. This opens a new way of studying black holes as they are not static, eternal cosmicobjects [4].

In this thesis Hawking radiation will only be considered a small correction to the phe-nomenon which causes the black hole to explode.

1.2 Bouncing Black Holes

Similar to the theory of the Big Bounce black holes may only be one of the stages in abounce. The Big Bang is a singularity, thus better models are strived for in order to haveit removed. Remaking the Big Bang into a collapsing universe tunnelling into an explodingone would solve this singularity. Instead of just a bang there is instead a Big Bounce. Forthis to work quantum gravity is required. There are several approaches to quantum gravityand here loop quantum gravity, LQG, is favoured. LQG is non-perturbative and backgroundindependent, and its application to cosmology, loop quantum cosmology uses mostly sym-metry reduced models [5].

A black hole can be formed from the gravitational collapse of a dying massive star, itmay then reach a further stage, forming a white hole. If the quantum-gravitational pressureis strong enough it may impede the weight collapse and the black hole may bounce out withan explosion. This bounce consists of the collapse (black hole) and then ejection of mass(white hole). Due to the great gravitational time dilation of the black hole this bounce willseem very long for an observer from a large distance, that is how observations on black holesare seen today. It is a bounce frozen in time which can be misinterpret as a static object. Inits own time-frame the bounce will just collapse and immediately bounce out again, into auniverse which will have developed and changed remarkably during the bounce. The bouncecould be considered a transport to the future, where the content has remained the same [4].

An important aspect which was shown is that quantum gravity e�ects occur when the

energy density, ρE , is of the order of Planckian, ρEP =c7

~G2 , and not only when the sizereached Planck length [4]. This enables quantum gravity e�ects for considerably larger ob-

jects than the Planck length and also enables measurements. The Planck length lP =√

~Gc3

is theorised to be the shortest possible measurable length, according to the generalized un-certainty principle [6]. If there were only quantum gravity e�ects for objects in the size on

1.4. Previous Work 3

lP they would be long gone and there would be no chance of detection [4].

The e�ective metric for the collapsing part and the expanding part are glued togetherin Ref. [7] where the collapse quantum tunnels into the expanding, exploding part, and ismathematically shown. Quantum e�ects are shown to pile up, therefore after su�cient timequantum e�ects will arise even if it is not expected for shorter times.

With this solution the black hole will not have any singularities since the collapsing matteris stopped with the bounce. It also solves the black hole information paradox. The blackhole information paradox is that all the physical information which falls into a black hole islost forever, violating the preservation of information. This is no longer the case since withLQC the size of the black hole can be larger than the Planck length and it is large enoughfor the information to be stored [4].

1.3 Primordial Black Holes

In order for a black hole to be exploding today its mass would have to be signi�cantly smallerthan in the magnitude of solar masses. Black holes formed from gravitationally collapsedstars are thus too massive, with masses of > 5M�. For smaller masses to be possible onemust consider the so-called primordial black holes (henceforth abbreviated to PBH ). Thereis so far no evidence of the existence of PBH but there are several theories on how theywere formed. PBH were not formed as regular black holes through gravitational collapses,instead they were formed due to the immense compression from the Big Bang and largedensity perturbations. PBH have a large mass range and would provide the small massrequired to have black holes with a lifetime in the same order as the age of the Universe.Meaning that the emitted energy from the bounce could be measured today [8].

PBH are one of several dark matter candidates [8], which will here be used as an upperlimit for the number density of PBH.

1.4 Previous Work

Previous work on the bouncing black to white holes shows that gluing together the metrics,for the collapsing and expanding part, one �nds the relation between the bounce time andthe mass. The bounce time, τ , is the life time of the exploding black hole. The relation, inPlanck units, is given by [9]

τ = 4kM2 , (1.3)

where k is a unitless constant of unknown value, and M is the mass of the black hole. Theminimum value of k was found in Ref. [9] to be kmin = 0.05. For values below kmin quantume�ects will not allow a bounce. The energy from the ejected particles from a bouncing blackhole was also calculated using kmin to get the lowest possible energy. In this thesis k willnot have a �xed value but varied over a large range and treated as a variable. In Ref. [10]the maximum detection distance to a bouncing black hole was calculated as well as anintegrated spectrum over several bouncing black holes. This report will continue on thiswork with an improved model for the energies and with k as a free parameter.

Chapter 2

Investigation

The investigation consist of two main parts, the �rst one is �nding how far away an explodingblack hole can be detected. The second part is studying an integrated spectrum of combiningseveral exploding black holes and how it would appear measured at Earth.

To reach these main goals several smaller investigations are required. These are: theinterval of the parameter k in section 2.1, the low-energy signal in section 2.2, the high-energy signal in section 2.3 and the particles emitted in section 2.4. The distance is theninvestigated in section 2.5 and the integrated signal in section 2.6.

In this work all black holes are assumed to be non-rotating and uncharged, so the onlyknown component is the mass. This assumption is reasonable since charge and angularmomentum are lost during quantum emission faster than the mass [8].

Only detection of photons will be considered in this work, since they have no charge anddo not decay. Charged particles are de�ected by the magnetic �elds and neutrinos are hardto detect because of their small interaction cross section [10].

There are two theories for the energy signal, they will be refereed to as the Low Energysignal and the High Energy signal.

The �rst one is based on dimension, the only known component from the white holeis its size, calculated from its mass, before bouncing out. The low energy signal is thenassumed to have its wavelength in the same order of magnitude as the size of the whitehole.

The second signal, the high energy one, is derived from the contents of the Universewhen the black hole was formed. It had to be created from something. Considering that inits own time frame not a long time has passed since it was created, it is justi�ed to assumethat the ejecting particles are the same as the collapsed ones. It is then assumed that theenergy will be of the temperature as the Universe was at the formation time [9].

Starting with eq 1.3 giving the relation between bounce time and mass which in SI-unitsbecomes

τ = 4kM2√G3

c7~. (2.1)

What is interesting is a bouncing black hole which can be observed today, therefore the

5

6 Chapter 2. Investigation

bounce time is required to be in the same magnitude as the age of the Universe. That isτ = tH where tH is the Hubble time ≈ 14 billion years such that

tH = 4kM2

√G3

c7~. (2.2)

For larger distances it is important to take into account the time it takes for the signal totravel this distance, so the time then becomes τ = tH − R/c, where R is the distance andR/c the time it has taken the photons to travel this distance. This will be used for theintegrated signal.

Rewriting eq 2.2 gives the mass of the bouncing black hole as a function of k

M(k) =

(t2Hc

7~16G3

) 14 1√

k. (2.3)

2.1 Interval of the parameter k

The parameter k has a minimum value acquired from previous work, kmin = 0.05, and to�nd how far it varies the maximum value is required. The maximum value, kmax, ariseswhen the Hawking radiation no longer can be considered a small correction but insteadthe main contribution. Hence, when the bounce time, eq 2.1, would equal the Hawkingevaporation time, tev from eq 1.2

tev = τ ↔5120πG2M3

~c4= 4kmaxM

2

√G3

c7~↔ kmax = 1280π

√G

~cM .

Using eq 2.2 this gives the value of kmax

kmax = (640π)23

(t2Hc

5

~G

) 16

' 3 · 1022 .

These values of k give the mass range of the bouncing black hole using eq 2.3 and is shownin �g 2.1. It ranges from M(kmin) ≈ 1.37 ·1023 kg to M(kmax) ≈ 1.72 ·1011 kg with bouncetime ≈ 14 billion years. Considerably smaller than a solar mass M� ≈ 1030 kg, showingthe necessity of PBH in order to have bounces in the current time epoch.

Figure 2.1. Mass depending on k. Double logarithmic plot over how the mass, M , dependon k from kmin to kmax. With the mass in kg on the y-axis and k on the x-axis.

2.3. High Energy Signal 7

2.2 Low Energy Signal

The size of a white hole is the only known size and there is no reason to believe that thewavelength of the emitted photons would di�er immensely from this magnitude. Seeing asit is a strong explosion in a small region and this tends to emit particles with the same sizeas the given region [9].

The low energy signal is calculated from the mass using eq 1.1, since white holes havethe same characteristics as black holes i.e. size determined by the Schwarzschild radius.Assuming the size of the wavelength is in the same order as the size of the hole, the wave-length λlow is then

λlow =2GM

c2. (2.4)

From this equation the energy, Elow, can be calculated using the Planck-Einstein relation,E = hν, where h is the Planck constant and ν is the frequency from ν = cλ . The energyElow is then given by

Elow = hνlow =hc

λlow=

hc3

2GM.

Inserting the value of the mass from eq 2.3 the relation between Elow and k is derived to

Elow(k) =

(2πh3c5

Gt2H

) 14 √

k , (2.5)

expressed in J. For k between kmin = 0.05 to kmax = 1022 the low energy signal varies as

Elow = 0.006 eV− 4.8 · 109 eV as can be seen in �g 2.2. The energy increases with k whichis due to that higher k gives a smaller radius, thus a smaller wavelength and higher energy.

2.3 High Energy Signal

The other possibility is that the emitted particles from the exploding black hole would havethe same characteristics as the particles which formed the black hole. In this aspect thematter which formed the black hole has just travelled in and then bounced out in the timeit takes for light to travel across the black hole's radius [4]. The content has not had signif-icant amount of time to change radically [9].

To estimate the characteristics of the energy one must examine how the Universe appearedwhen the PBH were formed. The high energy signal is expected to behave as a black bodywith energy corresponding to the temperature of the Universe at formation time.

The relation between the time of formation and the black hole's mass is found by com-paring the density for black holes and the cosmological density at a certain time after theBig Bang, tform. It shows that the mass, M , is in the order of the particle horizon mass,MH , which gives the following relation [8]

MH =c3

Gtform . (2.6)


Using MH = M and eqs 2.3 and 2.6 the formation time depending on k is derived:

tform =

(t2H~G16c5

) 14 1√

k. (2.7)

The temperature, T , of the Universe at a given time, t, after the Big Bang is provided fromthe following approximation [11]

t = 0.30mpl√g∗T 2

∼(

1 MeV

Ehigh

)2seconds . (2.8)

Here mpl is the Planck mass and g∗ the number of degrees of freedom for the particles.

With t = tform from eqs 2.7 and 2.8 the high energy signal Ehigh becomes:

Ehigh =√

2 · 106(

c5

~Gt2H

) 18k

14 (√seconds · eV) . (2.9)

This gives the range Ehigh = 1.7 · 1012 eV − 1.5 · 1018 eV with k between kmin = 0.05 tokmax = 10

22 plotted in �g 2.2. A higher k means that the PBH were formed earlier andtherefore during a hotter more energetic Universe resulting in higher energy for higher k.

Figure 2.2. Energy depending on k. Double logarithmic plot over how Ehigh (black, plainline) and Elow (blue, dashed line) depend on k from kmin to kmax. With the energy in eV onthe y-axis and k and the x-axis.

2.4 Particles Emitted

The total number of emitted particles are found by taking the total energy equivalent ofthe black hole's mass Etot = Mc

2 divided by the mean energy of the particle, either Elow

2.4. Particles Emitted 9

or Ehigh.

This gives for the low energy signal

Nburstlow =EtotElow

=Mc2(

2πh3c5

Gt2H

) 14 √

k

=

(t2Hc

7~16G3

) 14 1√

k· c2(

2πh3c5

Gt2H

) 14 √

k

,

reduced to

Nburstlow =

√t2Hc

5

32πGh

1

k. (2.10)

For the high energy signal, note that Etot is in J while Ehigh in eV so a conversion factor isapplied

Nbursthigh =Etot

Ehigh · 1.602177 · 10−19 J/eV=

=Mc2

√2 · 106

(c5

~Gt2H

) 18k

14 (√seconds · eV) · 1.602177 · 10−19 J/eV

,

simpli�ed to

Nbursthigh =1013

9

(t6H~3c25

G5

) 18

k−34 (√s · J)−1 . (2.11)

For both the low and high energy parts the number of particles will decrease with the in-crease of k since the mean energy increases while total energy decreases, resulting in lessparticles.

Gravity does not favour any particles as long as the energy is high enough for its cre-ation. Assuming there are no preferences on which fundamental particles to be ejectedfrom the explosion. The amount of photons are given by the internal degrees of freedomfor photons, divided by the total amount of internal degrees of freedom for all possiblefundamental particles, at given energy. For higher mean energy the fraction of photons willdecrease since fundamental particles with larger mass are able to form [10].

The high energy signal, Ehigh ≥ 1.7 TeV, always has high enough energy to create allthe di�erent kinds of fundamental particles so the fraction of photons will be

γfrachigh =internal degrees of freedom for photons

internal degrees of freedom for all fundamental particles=

2

120,

with the fundamental particles and their degrees of freedom from table 2.1. For particleswith undermined mass the upper limit is used.

Even though gluons are massless they will not be considered for the low energies. Thisdue to the fact that gluons cannot exist on their own. They are taken into account whenquarks begin to form.

For the low energy signal the fraction of photons will decrease with the increase of k seeingas it spans from Elow(kmin) = 0.006 eV to Elow(kmax)4.8 ·109 eV, which makes the fraction


of photons range between γfraclow(kmin) =24 to γfraclow(kmax) =

298 .

Table 2.1. Fundamental particles with their mass and internal degrees of freedom, takenfrom Particle Data Group, PDG, [12].

Fundamental particle Mass [eV] degrees of freedom, g

u 2.3 · 106 12d 4.8 · 106 12c 1.275 · 109 12d 95 · 106 12t 173.21 · 109 12b 4.18 · 109 12e− 0.510 · 106 4µ− 105.65 · 106 4τ− 1776.82 · 106 4νe < 2 2νµ < 2 2ντ < 2 2g · 8 0 2γ < 10−18 2

W+ 80.38 · 109 3W− 80.38 · 109 3Z0 91.18 · 109 3H 125.7 · 109 1

graviton < 6 · 10−32 2

The number of photons, Nγ , directly ejected from the explosion will then simply be thefraction of photons, γfrac, multiplied by the total number of particles from the burst, Nburst,

Nγ = γfrac ·Nburst . (2.12)

The majority of photons will not come from directly emitted photons but instead from de-cayed unstable hadrons. This is taken into account for energies higher than the con�nement-decon�nement transition at Tc = 160 MeV, the limit where hadrons begin to form anddecay [12]. In order to obtain a proper approximation of the decayed photon energies andamount a simulation program, Pythia, was used. Pythia is a tool, written in C++ developedby T. Sjöstrand, using Lund Monte Carlo methods to generate high-energy collisions andenabling the study of the decay. Pythia is limited for low energies where the lowest possibleenergy beam is at 10 GeV [13].

2.4.1 Pythia

In Pythia the decay of u quark jets of di�erent energies is simulated and histograms overthe decayed photons are obtained. A curve is �tted to the histograms see �g 2.3 and theparameters are extrapolated for lower and higher energies.

2.4. Particles Emitted 11

Figure 2.3. Histograms (gray) acquired using Pythia, with �ts (black). With initial jetenergies 10 GeV (top left), 100 GeV (top right), 1000 GeV (bottom left) and 10000 GeV(bottom right). With the decayed photon energy, E, in GeV on the x-axis and the number ofphotons, N , on the y-axis from one quark jet.

Trying several di�erent �t functions, an exponential �t with 4 parameters, A, B, C and D,is found most favourable:

Hfit(E , E) = A · EB exp (C · E) +D , (2.13)

where A, B, C and D are parameters depending on the initial energy, E. Fits over thesevalues for di�erent E provided the following:

A(E) = 7.60523 · E0.025442 ,B(E) = 3.07294 · E−0.0152561 − 2 ,C(E) = −10.7232 · E−0.0614127 ,D(E) = 0.0375857 · E0.287030 ,

where E is the jet energy (Elow or Ehigh) and E is the measured energy, both in GeV. Thesehistograms and �ts are made for large values of E therefore the �ts may not apply well forenergies remarkably lower than 10 GeV.

Histograms over d and s quarks were also made and compared with the histogram of theu quarks. Only three di�erent quarks were tested due to the limited time. With a similarshape and a variation in number of photons being a maximum of 5% the �t for the u quarkis used as an approximation for all kinds of quarks. Only the decay of quarks is taken intoaccount due to the fact that the majority of decayed photons will come from hadron decay,thus the rest are discarded.


Pythia also provided the number of decayed photons per quark jet and their mean en-ergy. A �t over the relation between number of decayed photons and the jet energy isplotted in �g 2.4, with the �t

Nγ/jet = 4.21477 · E0.37334 ,

where E is the jet energy measured in GeV.

Figure 2.4. Number of photons from Pythia. Values given by Pythia on how many photons,Nγ , (y-axis) decaying from one u-quark at energy E (x-axis) in GeV.

A �t over the mean energy of the decayed photons, Edecmean , resulted in

Edecmean(E) = 0.285121 · E0.0361813 , (2.14)

also measured in GeV.

Thus, the total amount of decayed photons is

Nγdec = qfrac ·Nburst ·Nγ/jet , (2.15)

where qfrac is the fraction of quarks, i.e. the internal degrees of freedom for the quarksdivided by the degrees of freedom for all the available fundamental particles, at a certainenergy.

For the high energy signal this fraction is, for all k, 72120 since the energy is high enough forall fundamental particles to form. For the low energy signal this fraction will vary and forvalues of k corresponding to energy lower than Tc it will be 0.

2.5 Distance

The �rst main part is to calculate the maximum distance of detection, Rdet, to an explodingblack hole from Earth.

In order to calculate this distance one needs to know how many detected photons are

2.5. Distance 13

required for the detection to be valid, how large surface the detector has and how the pho-tons are absorbed during their travel through space [10].

The distance can be calculated by considering a sphere spanned by the distance, Rdet,as illustrated in �g 2.5. With the detector's surface, S, the number of photons at the ex-plosion, Nburst, and measured at the detector, Nmes. Assuming isotropic distribution overthe explosions the ratio of photons per area becomes

NburstSurface of sphere

=NmesS

,

with the surface of the sphere 4πR2det being

4πR2detNburst

=S

Nmes.

Giving in the distance:

Rdet =

√S ·Nburst4πNmes

. (2.16)

Determination of Nmes and S are found in the sections 2.5.1 and 2.5.2 below, and Nburstin section 2.4 above.

Figure 2.5. The distance Rdet between the exploding black hole (in the middle) and Earthspans a sphere with isotropic distributed photons in the shell.

2.5.1 Number of Required Photons

Depending on the energy band used for the measure, di�erent photon-counts are needed tobe certain that the measurement has statistical signi�cance. The detection needs to be sep-arable from the background �uctuation and therefore be several standard deviations higher.Energies with low background does not need as high count as the optical band which has amore di�use background [10].

Reasonable values for the required amount of photons are Nmes(1 TeV) = 10 for E = 1 TeVand Nmes(1 eV) = 50 for E = 1 eV [10]. Using, as a �rst approximation, a linear relationNmes = aE + b and adding these values the approximated value of Nmes becomes


Nmes = −4 · 10−11E + 50 , (2.17)

with E measured in eV.

2.5.2 Detector Surface

Detectors available today have di�erent size depending on in what wavelength they aredesigned to measure. This is due to several reasons. For IR, UV, X-ray and soft gamma-rays the detectors need to be placed on satellites due to the absorption in the atmosphere.Detectors placed on satellites have restricted size because of the expenses of sending larger,the size is of magnitude 1 m in diameter [10, 14].

Groundbased detectors can be larger and are available for optical light. Groundbaseddetectors are of size 10 m [14].

For high energy photons (hard gamma-rays), above 100 GeV, satellites are not e�ectiveand the original photon will not reach the ground, fortunately air showers will occur. Thephoton creates extensive air showers which emit Cherenkov radiation. These showers willcover large ground so a detector placed in the area of the shower will have an e�ective areaof the Cherenkov light pool which is 100 m [14].

2.5.3 Absorption

An important aspect for photons travelling in space is the absorption. If the photons areabsorbed before reaching our detectors they can evidently not be detected.

For low-energy photons the Universe can be considered transparent and the photons willtravel freely, this is the case for the low-energy signal [14].

For high-energy photons, of energy 1 TeV and above, this is not the case. Due to thehigh amount of infrared photons in the Universe the high-energy photons will be inter-cepted for long distances. The high-energy photons will couple with the IR photons andcreate a positron-electron pair, as γTeV + γIR → e+ + e−. This e�ect will have a signi�cantimpact for the high-energy photons [14].

The photon's blocking gamma-ray horizon depending on energy can be observed in �g2.6, the photon horizon. With data points from Ref. [14]. This is the furthest the radiationcan travel.

2.6. Integrated Signal 15

Figure 2.6. Double logarithmic plot over the photon horizon, the absorption of high-energyphotons in the Universe. With the energy in eV on the y-axis and the distances in m on thex-axis. Data have been obtained from Ref. [14]

2.6 Integrated Signal

The second main part of this thesis is to determine how the integrated energy spectrumfrom several exploding black holes would appear. Assuming they are uniformly distributedin the Universe when dealing with cosmological distances.

The number of measured photons, N , per unit energy, dE , gives the spectrum accordingly

dN

dE=

∫ RH0

φind(E(1 + z), R) · n(R) ·Acc ·Abs(E , R) ·1

4πR2· 4πR2dR , (2.18)

where φind(E(1 + z), R) is the individual �ux depending on distance, R, and measured en-ergy, E , which is a�ected by the redshift z due to large distances. The quantity n(R) isthe number of exploding black holes within distance, R, whereas Acc is the acceptance ofthe measurement and Abs(E , R) is the absorption. The factor 14πR2 is the solid angle and4πr2dR is the integrated element, which cancels each other out.

The mean energy Elow and Ehigh depend on the distance R since the bounce time willdepend on R, as τ = tH − Rc .

The individual �ux, φind, is calculated from the number of photons from a single bouncingblack hole multiplied by the probability of the energy.

The integrated curve will have an arbitrary magnitude since the number density of PBH isnot known. It is only known that it is limited by the dark matter density, ΩDM . The upperlimit of the integral is chosen as the Hubble length, RH =

cH0

, with H0 being the Hubbleparameter. The Hubble length is how far photons has been able to travel during the Hubbletime, tH =

1H0

. The upper limit can be picked arbitrarily as long as it is large enough tohave signi�cant content, higher values only give a larger magnitude of the integrated signal.


An isotropic distribution of white hole is assumed, meaning exploding black holes fromall directions in the Universe is evenly distributed. Therefore the acceptance, Acc will onlya�ect the magnitude and is thus discarded.

2.6.1 Redshift

Due to the large distances involved it is important to take into account the redshift. Signalsfrom further away will be more redshifted and thus have lower energy. The redshift, z,distance, R, relation is given by [15]

R(z) =c

H0

(z + 1)2 − 1(z + 1)2 + 1

. (2.19)

This relation is plotted in �g 2.7 over z = 0 to 7.

Figure 2.7. Plot over the relation between distance in m on the y-axis and redshift on thex-axis. Using the value of the Hubble parameter H0 = 70 km/s/Mpc.

2.6.2 Integrated Low Energy Signal

There will be two contributions to the individual �ux one from the directly emitted photonsand (for the higher k values) one from the decayed hadrons.

It is not physical that all photons would radiate with a Dirac peak at one energy, Elow.Therefore the low energy signal from the direct photons is approximated with a Gaussiandistribution, with Elow as the mean value.

The large integration distances a�ects the bounce time, τ , which is now set as τ = tH − Rc .


Using this bounce time in eq 2.10 for the number of photons and in eq 2.5 for Elow with aGaussian curve the �ux becomes

φindlowdir (E(1 + z), R) = Nburstlow ·Gaussian =

=

√(tH − Rc

)2c5

32πGh

1

k· 1σ

exp

(− (E(1 + z)− Elow)

2

2σ2

).

The width, σ, is not known so three di�erent widths are tested, σ = 0.1 · Elow, 0.2 · Elowand 0.3 · Elow, in order to see how much they a�ect the integrated signal.

The fraction of photons, γfrac, is taken into account later on after normalization of thecurve.

The decay part uses the �t and approximation acquired from Pythia eq 2.13, giving the �ux

φindlowdec (E(1 + z), R) = Nburstlow ·Hfit =

√(tH − Rc

)2c5

32πGh

1

k·Hfit ,

where Hfit depends on E and E, E does in turn depends on the distance, R.

The total individual �ux for the low energy signal is the sum of the both:

φindlow(E(1 + z), R) =φindlowdirnormdir

γfraclow +φindlowdecnormdec

qfraclowNγ/jet . (2.20)

The integration over the �ux is performed in two terms one for the direct and one for thedecay. They are thereafter normalized, each divided by a normalisation factor normdec andnormdir. The terms are multiplied with the fraction of photons, γfrac, or fraction of quarks,qfrac. The term with the decayed photons is also multiplied with the number of photonsper quark jet, Nγ/jet. The decayed term is non-zero only for k values where the energiesare high enough to produce decaying hadrons, i.e. k over the con�nement-decon�nementtransition level at Tc = 160 MeV. The energy Tc corresponds to k ≈ 1019, for the low energysignal.

2.6.3 Integrated High Energy Signal

For the high energy signal, one would expect it to behave like a black body spectrum ofcertain temperature, T , given by Ehigh. Since the PBH were formed during the early Uni-verse its contents are expected to have black body behaviour, like the particles in the earlyUniverse.

From Planck's law the spectral radiance, I, is

Iplanck(ν) =2hν3

c21

exp(hνkBT

)− 1

,

where kB is the Boltzmann constant.


With E = hν the individual �ux for the direct photons in the high energy signal is given by

φindhighdir (E(1 + z), R) = Nbursthigh · Iplanck =

=1013

9

((tH − Rc

)6 ~3c25G5

) 18

k−34 (√s · J)−1 · 2 (E (1 + z))

3

h2c21

exp(E(1+z)kBT

)− 1

.

The �ux from the decayed photons is derived in the same way as for the low energy signal

φindhighdec (E(1 + z), R) = Nbursthigh ·Hfit =

=1013

9

((tH − Rc

)6 ~3c25G5

) 18

k−34 (√s · J)−1 ·Hfit .

Adding the terms for the direct and decayed part, normalized and multiplied with theparticle fraction and number of photons per quark the total �ux for the high energy signalbecomes

φindhigh(E(1 + z), R) =φindhighdirnormdir

γfrachigh +φindhighdecnormdec

qfrachighNγ/jet . (2.21)

The integrated signals are tested with di�erent values of k in order to see how much ita�ects the spectrum, for both the low and high energy signals.

2.6.4 Mass Spectrum

The number of existing primordial black holes, n, is not known but it is limited by the darkmatter distribution. How the number density depends on its mass, M , is shown to be [16]

dn

dM∝M−α , (2.22)

where α = (1 + 3β)/(1 + β) + 1 from the equation of state in the early Universe [16], andβ is the relation of pressure to energy density at the formation epoch [3].

Integrating this relation for a small time period, dt, with the limits M(t) to M(t + dt),one can de�ne n as

n(R) =

∫ M(t+dt)M(t)

dn

dM(M)dM ≈ dn

dM(M(t))dM ,

using dt = 8k dM√

G3

c7~ derived from eq 2.2

n(R) ≈ dndM

(M(t))dt

8k

√c7~G3

,

with t = tH − Rc

n(R) ≈ dndM

(M

(tH −

R

c

))dt

8k

√c7~G3

.


Inserting eq 2.22 and eq 2.3, also discarding all unnecessary constants the number of PBHbecomes

n(R) ∝

√ tH − Rc4k

−α 18k

. (2.23)

Then α needs to be decided. Depending on the equation of state α will have di�erent values.A very soft equation of state gives β = 0 and thereby α = 2, a very sti� equation of stategives β = 1 and α = 3. The radiation equation state has β = 1/3 with α = 5/2. The mostprobable is the last one, from a non-interacting relativistic gas with β = 1/3 and α = 5/2 [3].

All these di�erent values for α plus using a constant mass spectrum are tested and plottedin �g 2.8. This is the low energy signal and with k = 100 and σ = 0.1 · Elow. As can beseen in the �gure the mass spectrum does not a�ect the shape of the curve radically andtherefore the most probable value α = 5/2 is used.

The constraints from the dark matter density distribution, ΩDM , would give an upperlimit to the number of PBH as ∫

dN

dMdM ≤ ΩDM .

The number of PBH is not calculated, as it highly depends in the bounds of the integralwhich are not known.

Figure 2.8. Double logarithmic plot over di�erent mass spectra. With M−25 (grey, plain

line), M−2 (red, dashed line), M−3 (green, dotted line) and constant (blue, dashed line).With energy in eV on the x-axis and dN

dE in arbitrary units on the y-axis.

Chapter 3

Result

3.1 Distance

Low energy signal distance

The distance for the low energy signal is the following

Rlow(k) =

√NγlowS

4πNmes, (3.1)

where Nγlow is given by the sum of eq 2.12 for the directly emitted photons and eq 2.15 forthe decayed photons. The decayed photons are all approximated with their mean energywhich is of the same magnitude as the initial jet energy, calculated with eq 2.14. The surfacearea, S, of the detector is either Soptical = π5

2 m2 (diameter of 10 m) or Slower = π0.52 m2

(diameter of 1 m), depending on the wavelength of the measured energy, see section 2.5.2.Nmes varies with initial energy and is given by eq 2.17. The relation between the distance,R, and k is plotted in �g 3.1.

21

22 Chapter 3. Result

Figure 3.1. Double logarithmic plot over the maximum distance for the low energy signal.The upper dotted line is the Hubble radius RH and the lower dashed line represents thegalactic scale. There are several steps which corresponds to either change of detector, anincreased diversity of fundamental particles or the start of hadron decay. With the distance,R, on the y-axis in m and k on the x-axis with corresponding energy values in eV.

The maximum distance for an exploding black hole for the low energy signal is plotted in�g 3.1. The step up at k ≈ 104 is where the energy is in the optical light and the detectorsare of larger size, Soptical. The small steps down are due to the energy being high enoughfor new fundamental particles to form and thereby γfrac decreases. The �rst step down,which is in the step up for the optical detector k ≈ 104, is when neutrinos begin to form,it is here assumed that the mass is the upper limit mν < 2 eV [12]. The step down shortlyafter k ≈ 1014 is when the energy reaches the electron mass decreasing γfrac further. Thenext step down at k ≈ 1016 is when the �rst quark, u, forms and thereby also gluons. Thestep up at k ≈ 1019 is when the energy is high enough to form decaying hadrons, at theQCD con�nement scale, and the decayed photons vastly increase the number of photons.The upper dotted line is the Hubble radius, RH , and the lower dashed is the galactic scale.The longest distances are for low values of k where the Hubble radius RH is the limit. Thedistance for kmax is Rlow(kmax) = 1.16 · 1019 m. Although the shortest distance is reachedjust before the QCD con�nement scale at k = 1019.

High energy signal distance

For the high energy signal the distances are considered separately for the direct and thedecayed photons as their energy are of di�erent magnitudes. For the direct part the distancedepending on k is

Rhighdir (k) =

√NγhighS

4πNmes, (3.2)

with Nγhigh given by eq 2.12, S will for these high energies be Scher = π502 m2 (diameter

of 100 m) and Nmes given by eq 2.17.


For the decayed photons the distance is

Rhighdec(k) =

√NγdecS

4πNmes, (3.3)

Nγdec is given by eq 2.15, S is Slower and Nmes given by eq 2.17. The distances are plottedin �g 3.2, with k varying between kmin = 0.05 and kmax = 3 · 1022.

Figure 3.2. Double logarithmic plot over the maximum distance for the high energy signal.The upper curve (black, plain line) is for the decayed photons and the lower one (blue, dashedline) is the directly emitted photons. The dashed black line at 1021 m represents the galacticscale. With the distance, R, in m on the y-axis and k on the x-axis with corresponding energyvalues in eV.

In �g 3.2 the maximum distances for the high energy signals are plotted. The upper line isfrom the decayed photons and varies between Rhighdec = 1.63 ·1024 m and 2.49 ·1016 m. Thelower one is for the directly emitted photons with a distance varying between Rhighdir =1.53 · 1023 m and 1.82 · 1014 m.

3.2 Integrated Signal

The integrated signal is calculated with

dN

dE=

∫ RH0

φind(E(1 + z), R) · n(R)dR .

Plots over integrated low energy signal

With eq 2.20, eq 2.23 and α = 5/2 the integrated signal, for di�erent k values varyingbetween kmin and kmax, results in the following spectra �gs 3.3 to 3.6.


Figure 3.3. Integrated low energy signal for kmin. With expected measured photon energy,E, in eV on the x-axis and dN

dE on the y-axis in arbitrary units. Plotted for di�erent values forσ. With the plain line being σ = 0.1 ·Elow, the dashed line σ = 0.2 ·Elow and the dotted lineσ = 0.3 · Elow.

Figure 3.4. Integrated low energy signal for k = 109. With expected measured photonenergy, E, in eV on the x-axis and dN

dE on the y-axis in arbitrary units. Plotted for di�erentvalues for σ. With the plain line being σ = 0.1 · Elow, the dashed line σ = 0.2 · Elow and thedotted line σ = 0.3 · Elow.


Figure 3.5. Integrated low energy signal for k = 1019 corresponding to the energy valuewhen unstable hadrons begin to form. With photon energy, E, in eV on the x-axis and dN

dEon the y-axis in arbitrary units. Plotted for di�erent values for σ, with the plain line beingσ = 0.1 · Elow, the dashed line σ = 0.2 · Elow and the dotted line σ = 0.3 · Elow. Bothdirect photons (peak on right) and decayed photons (bump on left) are taken into account,distinguishable for σ = 0.1 · Elow.

Figure 3.6. Integrated low energy signal for kmax. Double logarithmic plot with the decayedpeak on the left and a small direct peak on the right. Plotted for di�erent values for σ.With the plain line being σ = 0.1 · Elow, the dashed line σ = 0.2 · Elow and the dotted lineσ = 0.3 ·Elow. With photon energy, E, in eV on the x-axis and dNdE on the y-axis in arbitraryunits.


Plots over integrated high energy signal

The integrated spectrum for the high energy signal using eqs 2.21 and 2.23 for kmin isplotted in �g 3.9, with separate plots for only the directly emitted photons in 3.7 and onlythe decayed photons in 3.8. Figure 3.10 shows an enhanced plot over the direct peak.

Figure 3.7. Integrated signal for directphotons for kmin. Double logarithmicplot with measured photon energy, E, ineV on the x-axis and dN

dE on the y-axisin arbitrary units.

Figure 3.8. Integrated signal for de-cayed photons for kmin. Double loga-rithmic plot with measured photon en-ergy, E, in eV on the x-axis and dN

dE onthe y-axis in arbitrary units.

Figure 3.9. Integrated high energy signal for kmin. Double logarithmic plot over the fullspectrum, both the direct (�g 3.7) and decayed photons (�g 3.8). Decayed peak on the leftand the direct peak at E ≈ 1012, enhanced in �g 3.10. With measured photon energy, E, ineV on the x-axis and dN

dE on the y-axis in arbitrary units.


Figure 3.10. Enhanced on direct peak kmin. Double logarithmic plot zoomed in on thedirect peak at E ≈ 1012. With measured photon energy, E, in eV on the x-axis and dN

dE on they-axis in arbitrary units.

The integrated high energy signal is quite similar for all k values with the most di�erencebetween kmin and kmax. The spectrum for kmax can be seen in �gs 3.11 to 3.14.

Figure 3.11. Integrated signal for di-rect photons for kmax. Double logarith-mic plot with measured photon energy,E, in eV on the x-axis and dN


Figure 3.12. Integrated signal for de-cayed photons for kmax. Double loga-rithmic plot with measured photon en-ergy, E, in eV on the x-axis and dN

dE onthe y-axis in arbitrary units.


Figure 3.13. Integrated high energy signal for kmax. Double logarithmic plot over thefull integrated high energy signal, both the direct (�g 3.11) and decayed photons (�g 3.12).Decayed peak on the left and the direct peak not visible at E ≈ 1018 shown enhanced in �g3.14. With measured photon energy, E, in eV on the x-axis and dN

dE on the y-axis in arbitraryunits.

Figure 3.14. Enhanced on direct peak kmax. Double logarithmic plot zoomed in on thedirect peak at E ≈ 1018. With measured photon energy, E, in eV on the x-axis and dN


Chapter 4

Discussion

The lifetime of a black hole according to the discussed theories gives a dependence on themass as M2 from eq 1.3 instead of M3 given by the Hawking radiation eq 1.2. Giving asigni�cant lower lifetime and the possibility of black holes with higher mass to explode inour time epoch.

Distance

The maximum distance for the low energy signal plotted in �g 3.1 has several bumps. It isimportant to remember that this plot does not demonstrate a physical behaviour, it simplygives the maximum distance depending of what the real k value has. The parameter k is aconstant of unknown value, which is why it in this report has been treated as a variable.

A higher k would give a higher mean energy, Elow, and in turn a shorter distance, dueto the less amount of photons, Nburst =

EtotElow

, being emitted. With decreasing total energy,Etot, divided by increasing mean energy, Elow. Higher energy also enables the formationof more kinds of fundamental particles giving a lower fraction of photons. Although whenthe energy reaches higher values than the QCD con�nement scale at Tc = 160 MeV corre-sponding to k ≈ 1019 decayed photons from unstable hadrons will begin to emit as well.The decayed photons give a boost to the number of total photons, as they have energies inthe same magnitude as the directly emitted photons.

As for the maximum distance of the high energy signal, as can be observed in �g 3.2,there are no steps since the energies are for all k higher than the mass of the heaviest fun-damental particle, the t quark at mass mt ≈ 173 GeV. Therefore the fraction of particlesare distributed the same way for all k with regards to their internal degrees of freedom.

The distances for the decayed photons are longer than for the directly emitted ones. Thisis due to the vastly higher numbers of photons coming from decayed hadrons as oppose ofbeing emitted directly. Not only is the fraction of quarks higher than the fraction of photons,qfrac > γfrac, but also each quark decays into several photons, Nγ/jet, which can be seenin �g 2.4.

The distances' relation to k is not the same for the direct and decayed curves, they bothdecrease but the direct one has a steeper curve. This due to the fact that the decayed curvegets more photons when increasing k. Although this does not compensate for the totalnumber of particles decreasing with higher k, thus the distances decreases.

29

30 Chapter 4. Discussion

It can be seen when comparing the plot over the high energy signals maximum distance�g 3.2 with �g 2.6, over the absorption, that the gamma horizon is further away than thecalculated distances, meaning there are other factors which inhibit the maximum distance.Therefore no factor for absorption is included in the calculations.

Integrated Signal

On the integrated signal for the low energy, �g 3.3 and 3.4, there is a bump on the leftof the peak, more prominent for σ = 0.1 · Elow this is a result of the non-linear relationbetween the redshift and the distance eq 2.19, which makes the signal accumulate on otherlocations than just the peak.

For the integrated signal for the low energy signal with energies over Tc where decayinghadrons are formed, that is in �gs 3.5 and 3.6, one can see peaks from both the decayedand direct photons, for k = 1019 the peaks are only distinguishable for σ = 0.1 ·Elow. Withthe decayed on the left since they are clearly of lower energy, E , than the direct one. Forkmax the decayed peak is dominating and the direct peak is quite short in comparison, sothe di�erences in the curves for di�erent σ are barely visible as they only a�ect the directpeak.

The di�erent values of the width σ on the Gaussian curve give signi�cantly di�erent shapesin the integrated spectra as can be seen in e.g. �g 3.3. Therefore all di�erent values arekept since the proper distribution is not known.

For both the low and the high energy signals the integrated curves are very similar in shapeto the unintegrated one. The integrated low energy signal is still similar to a Gaussian curve,�gs 3.3 and 3.4, and the integrated high energy signal is of black body characteristics, �gs3.7 and 3.11. This is caused by compensation from the redshift. A black hole bouncing faraway would be more redshifted than a closer bounce giving it a lower energy, but it wouldalso be younger with a shorter time to bounce, giving a higher emitted energy. For thelow energy signal this is due to the fact that the mass is smaller and thereby the size andwavelength, resulting in higher energy. For the high energy signal this is because a shorterbounce time in turn means that the PBH were formed earlier during a hotter Universe andthus higher energy.

For both cases the redshift somewhat compensates for the higher energy and conse-quently suppresses some of the e�ects from the integration.

An integrated curve for a single bouncing black hole is expected to change more in shapeover integration.

Comparing �gs 3.3 and 3.4 (over kmin and k = 109 for the low energy signal) one can

observe the similarities between them, the shape is the same, all which di�ers is the higherenergy, E on the x-axis, due to the higher source energy. Meaning that k does not a�ectthe shape of the integrated curve for the directly emitted photons. In combination with thedecayed photons the shape does change with k as can be seen in �gs 3.5 (k = 1019) and3.6 (kmax) where higher k also gives a larger decayed peak, dominating the direct for thelargest values of k.

31

For the high energy signal this is also the case. The integrated curve for direct anddecayed part separately have the same shape for all k, with the integrated decayed partbecoming increasingly wider with higher k. Compare the �gures for kmin �gs 3.7 and 3.8with the ones for kmax �gs 3.11 and 3.12. Although, in the same way as for the low energysignal, the decayed peak will become larger with higher values of k and the direct peak inthe combined integrated curve swiftly goes from barely visible to invisible (unless zoomingin) when increasing k. As can be seen in the combined integrated curve in �g 3.9 for kminand �g 3.13 for kmax, with the peak visible when enhancing the plot in �gs 3.10 and 3.14.

Also noteworthy is the fact that the direct and decayed peaks moves further away fromeach other with the increase of k this due to that the directly emitted energy is increasingwhile the energy of the decayed photons stays of the same magnitude.

Chapter 5

Conclusions and Future Prospects

The relation between a black hole's lifetime, τ , and its mass, M , depends on a constant kof unknown value. Treating k as a free parameter several characteristics, from a bouncingblack hole into a white hole, have been calculated. This was done with emphasize on itsdetection possibility.

The maximum distance for how far away a black hole could be bouncing and detectedin the current time epoch is found for two di�erent signals, the low energy signal and thehigh energy signal. The distances are for the low energy signal varying between the Hubbleradius, RH , and down to a distance of Rlow(kmax) = 1.16 · 1019 m, with the exceptionof a smaller distance for energies just before the QCD con�nement scale. The distancesfor the high energy signal are divided in directly emitted photons and decayed photonsfrom unstable hadrons. The decayed part provides the longest distances and varies betweenRhighdec = 1.63 · 1024 m − 2.49 · 1016 m, decreasing with k. These distances are largeenough to make it possible for experimental detection, the low energy signal even has forlower values of k the size of the observable Universe as its limits. If there somewhere inthe observable Universe exists bouncing primordial black holes detection should be possible.

An integrated spectrum for several exploding PBH within the distance of the Hubble radiushas been calculated and plotted for di�erent values of k. The shape of the integrated curvefor the directly emitted photons does not change with k, only the range. These integratedspectra should be compared with experimental measurements of unknown sources sincethere is a possibility that one of these may be from bouncing black holes. If this provesunsuccessful there are frequently more measurements being conducted, with better equip-ment, and there is a possibility that the emission from bouncing black holes may be amongthem.

This work consists of only photons and their detectability. What would be interestingis to continue the work with other particles preferably antiparticles such as the antiprotonand e+. Antiparticles have much lower background and would be easier to distinguish.Although one needs to take into account that charged particles will be de�ected by themagnetic �elds.

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