Stockholm University

Black Holes as standard candles throughtheir Hawking radiations

Tabatha Dufour

Physics departmentBachelor’s Degree - 15 credits

Black holes as standard candles through their Hawking’s radiationsSpring term 2021

Supervisor: Edvard Mortsell

CONTENTS CONTENTS

Contents

1 Background 31.1 Energy peak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Theory 62.1 Parallax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Standard candles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Types of Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Observations 93.1 The Fermi-LAT telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Energy and lifetime range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Highest sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Appendices 18

A Code Python f(x)=x 18

B Code Python Planck Integral 19

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CONTENTS CONTENTS

Black holes as standard candles through their Hawking radiations.

Tabatha Dufour

Stockholm University

Abstract

As they are evaporating, black holes are emitting radiations called Hawking radiations. Ourgoal is to determine at which distance it is possible to measure those radiations with currenttelescopes. Being capable of measuring those radiations would mean that we can use them asstandard candles i.e. we could use it as distance indicator. To do so, we are first going tocompute the different characteristic of a black hole such as its energy peak, temperature, fluxand lifetime. Knowing that, we will be able to describe how, theoretically, we could use blackholes as standard candles. Finally, we will take the Fermi-LAT telescope as an example of whatwe can observe in practice and at which distance.

Key words

Hawking radiation, Primordial Black Hole, Black Hole, Standard Candle, Fermi-LAT Telescope,Blackbody, Distance, Luminosity, Planck law, Spectral Radiance, Energy, Temperature, Flux,Lifetime, Gamma-Ray.

Introduction

Hawking radiation is named after the physicist Stephen Hawking, who provided a theoretical ar-gument for its existence in 1974. Hawking radiation is blackbody radiation that is predicted to bereleased by black holes (BH), due to quantum effects near the black hole event horizon. The eventhorizon is the most important feature of a black hole. It is an imaginary surface which acts as aboundary for the black hole as we can see on figure 1. The singularity in the center of a BH is agravitational singularity. It is a one-dimensional point which contains a huge mass in an infinitelysmall space, where density and gravity become infinite and space-time curves infinitely, the lawsof physics as we know them cease to operate. RSch refers to the Schwarzschild radius (sometimeshistorically referred to as the gravitational radius). It is the radius defining the event horizon.TheSchwarzschild radius is given as rs = 2GM

c2 . We can see that the radius is directly proportional tothe mass. The bigger the black hole is, the heavier it gets.

Figure 1: Structure of BH diagram. (10)

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1 BACKGROUND

Hawking radiation reduces the mass and rotational energy of black holes and is therefore also knownas black hole evaporation. Because of this, black holes that do not gain mass through other means areexpected to shrink and ultimately vanish. The smaller the black hole is, the faster it will evaporateand consequently, the more Hawking radiation it will emit. Thus, we are interested in small blackholes. But what does ”small” mean for a black hole? In which range of mass are we interested in?In the first section, we are going compute the different characteristics of BH such as the peak ofenergy, the temperature, the flux and the lifetime of black holes by assuming that the Hawkingradiation is a perfect black body. In the second section, we will discuss the different types of BHsand how show we can theoretically use BH as standard candles. In the third and last section, wewill apply the theory to practice by analysing at which distance BHs can be observed by currenttelescopes, and especially the Fermi-LAT telescope.

1 Background

In this section, we will assume BHs to be a perfect blackbodies to compute their characteristics.

1.1 Energy peak

First of all, we want to know the energy peak of a BH. We start by writing down Planck’s lawthat describes B(f, T ) the spectral radiance for a blackbody where f is the frequency and T is thetemperature (11) :

B(f, t) =2hf3

c21

ehfkT −1

. (1)

c is the velocity of light in vacuum, h is the Planck constant and k is Boltzmann constant. The

frequency is maximum at fpeak when∂B(f, T )

∂f= 0:

⇐⇒ 6hf2

c21

ehfkT −1

− 2hf3

c2he

hfkT

kT(e

hfkT − 1)

)2 (2)

⇐⇒ 2hf3

c21

ehfkT −1

·

3

f− he

hfkT

kT(e

hfkT − 1)

) = 0. (3)

We recognize that the first factor is the Planck’s distribution B(f, t) and is then not equal to zero.Therefore:

⇐⇒

3

f− he

hfkT

kT(e

hfkT − 1)

) = 0. (4)

Let x =hf

kT. We then have:

xex

ex − 1= 3 ⇐⇒ x = 3 (1− ex) . (5)

Let f(x) = 3 (1− ex). We are looking for f(x) = x. We can compute the answer numerically orwith Python (code in appendix A). We find:

x ∼ 2.8 ⇐⇒ hf

kT∼ 2.8. (6)

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1.2 Temperature 1 BACKGROUND

Therefore:

fpeak =2.8kT

h, (7)

Epeak = 2.8kT. (8)

1.2 Temperature

Now that we have the energy, we want to know the temperature of a BH, depending on its mass.We know its total energy (11) is:

U = 2Mc2, (9)

where M is it total mass. The radius of the BH is the Schwarzschild radius:

rs = λ =2GM

c2, (10)

where G is the universal gravitational constant. Let’s consider N photons, each with the wavelengthof the size λ of a black hole and energy ε = hc

λ . The total energy is then:

U = Nε =Nhc

λ=Nhc3

2GM. (11)

Hence,

N =4GM2

hc. (12)

The entropy of a system is of the same order as the number of particles in the system. If we take asystem of N particles and compress it to form a BH, the second law requires that when we are done,the entropy of the BH is still at least of order N . But since the end result is the same whether westart with a lot of particles or a few, the final entropy must be of the order of the maximum N. Wecan find :

S =8π2kGM2

hc. (13)

We can rewrite this for a BH as:

S =8π2k(Mc2)2

hc5=

8π2kU2

hc5. (14)

To find temperature, we simply have to derive the entropy with respect to the total energy. We find:

1

T=dS

dU=

16π2kU

hc5=

16π2kGM

hc3

⇐⇒ T =hc3

16π2kGM(15)

This temperature is also known as ”Hawking’s Temperature”. It is given in K but we can convertit to eV knowing the relation E = kT using k ∼ 8.617× 10−5 eV/K.

T (eV) =hc3

16π2GM(16)

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1.3 Flux 1 BACKGROUND

1.3 Flux

We can now derive the flux using Stefan’s law that states that the total energy radiated per unitsurface area of a blackbody is directly proportional to the fourth power of the blackbody. We canfind the same result by integrating Planck’s law from 0 to ∞.

Φ = σT 4 (17)

with σ the Stefan-Boltzmann constant:

σ =2π5k4

15h3c2. (18)

Plugging in (15) and (18) into (17), the flux is then:

Φ =2π5k4

15h3c2

(hc3

16π2kGM

)4

=hc10

(491520)π3G4M4. (19)

1.4 Lifetime

We are now able to find the lifetime of a blackbody. We use the radiated power P that we calculate

by multiplying the surface area A =16πG2M2

c4of a black body with the flux.

P = σAT 4

=

(2π5k4

15h3c2

)(16πG2M2

c4

)(hc3

16π2kGM

)4

=hc6

(30720)π2G2M2. (20)

The power radiated is the same as the blackbody energy decreases, so the rate of decrease is givenby the differential equation:

dU

dt= −σAT 4 (21)

⇐⇒ dM

dt= − H

M2where H =

2hc4

(30720)π2G2. (22)

We can integrate to obtain the lifetime τ (in second) of a blackbody:∫ 0

Mi

M2dM = −H∫ τ

0

dt (23)

⇐⇒ −M3i

3= −Hτ (24)

⇐⇒ τ =M3i

3H. (25)

By using this result and plugging in the numerical values of the constants c = 3 × 108 m/s, G =6.67×10−11 m3.kg−1.s−2, h = 6.63×10−34 J.s = 4.14×10−15 eV.s,P� = 3.86×1026 W in equations(25), (15) and (20) we can note that:

M ∼ 2× 105τ1/3 kg, (26)

T ∼ 50× τ−1/3 TeV, (27)

P ∼ 5× 10−5 × τ−2/3P�. (28)

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2 THEORY

This gives us a global idea of how the BHs are behaving. If the BH is smaller (so lighter), thelifetime will be shorter, but the temperature and the power will increase. This is why the smallerthe black hole is, the more Hawking radiation we have. We are now going to discuss the theorybehind standard candles and the different types of black holes so we can choose the more convenient.

2 Theory

We have already shown how to compute different characteristic of the BH. But how in theory, thefact that the distance and the flux are known, can be employed to use the BH as a standard candle?

2.1 Parallax

In astronomy, there are different ways to measure distances. The most common is to use what iscalled parallax. The principle of parallax is very simple, to understand it, we can put our fingerin front of us and alternately close one eye. The distance between the two eyes produces an effectof perspective, which is called parallax. This effect is all the more marked the closer the observedobject is, that is to say, its distance is smaller in front of the gap between the eyes. It can bemeasured by the angle the light rays make on the drawing. In the triangle formed by the eyes andthe finger, we know the distance between the eyes, and we measure the angle and we deduce thedistance. The idea of astronomers was to increase the gap between the eyes. To simulate this, theytook two sky photos six months apart. In these photos, there are very distant stars, which playthe role of the wall, and nearby stars which play the role of the finger. This method is based theapparent displacement of an object because of a change in the observer’s point of view. As we cansee in figure 2, by measuring the amount of displacement – the parallax angle – and knowing thedistance between the Sun and the Earth, astronomers can determine the star’s distance using simpletrigonometry.

Figure 2: Parallax diagram. (4)

The distance between the two pictures is the dimension of the Earth’s orbit: 300 million kilometers.With this, we can hope to measure the distance of the nearest stars. We know the base of thetriangle; it is the diameter of the Earth’s orbit. We measure the angle α (parallax on figure 2).

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2.2 Standard candles 2 THEORY

Knowledge of the angle α is therefore equivalent to that of distance. The relation used is:

A

sin(a)=

B

sin(b)=

C

sin(c)(29)

where A, B and C are the sides, and a, b, c the angles respectively opposite to the sides. The angleat which we see the radius of the Earth’s orbit is called the parallax. We use the radius of theEarth’s orbit, because it is the astronomical unit. This method gave a new unit of distance: theparsec is the distance corresponding to a parallax of one second angle.The Hipparcos telescope (High Precision PARallax COllecting Satellite) of the European SpaceAgency (ESA), launched in 1989, observing outside the atmosphere, has increased the precision ofparallax measurements. The European Space Agency has built a successor to Hipparcos, namedGaia. Gaia is part of ESA’s Horizon 2000 science program, comprising Rosetta, Herschel, Planck,Lisa, BepiColombo and Gaia. It was launched in 2013 and arrived at his post in 2014. Gaia is 50times more accurate than Hipparcos. Thus, parallax method makes it possible to measure stellardistances from the ground up to a distance of 30 pc (6).

2.2 Standard candles

Unfortunately, for distances which are too large, the parallax becomes to small and cannot be usedanymore. In that case, standard candles can be used. A standard candle is a source that has a knownluminosity (8). Light sources which are further away appear fainter because the light is spread outover a greater area. If we know how luminous a source really is, then we can estimate its distancefrom how bright it appears from Earth. The larger the distance is, the smaller the power that wemeasure is.There are several types of standard candle objects for which we can predict the luminosity from someother measurement. Two of the most important are Cepheid variable stars and type Ia supernovae.Merging binary black holes (BBHs) can also serve as standard candles. When compact objects likeBHs merge, they produce gravitational waves, which can be picked up by observatories such asthe Laser Interferometer Gravitational-Wave Observatory (LIGO). The emitted gravitational waveshave a characteristic energy, meaning that these mergers could be used to measure distances as“standard sirens”.The luminosity L can be found by measuring the temperature and then refer to the Hertzsprung-Russell diagram (HR diagram). Developed independently in the early 1900’s by Ejnar Hertzsprungand Henry Norris Russell, it plots the temperature of stars against their luminosity as shown infigure 3. As we can see it applies for object with a temperature between 2000 K and 30000 K. Forhigher temperatures, we can use that the luminosity is equal to the total power output, measuredin watts (W) or solar luminosity (L�) where 1 L� = 3.84× 1026 W).The relation between the distance r and the luminosity is given by:

Φ =L

4πr2⇐⇒ r =

√L

4πΦ. (30)

Flux is measured. Luminosity is not directly measurable, but is instead derived from its relation tosome observable as shown in (28).

For BHs, we will see that the temperature is a lot greater than 30000 K. Therefore, we will notuse the HR relation but the luminosity instead. To resume: we measure the temperature by com-paring the flux from two different wavelengths, that gives the luminosity, which gives us the distance.It seems quite doable but what is actually measurable with current telescopes? To find that, we stillneed to discuss what do we want to measure.

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2.3 Types of Black Holes 2 THEORY

Figure 3: Hertzsprung-Russell diagram. (14)

2.3 Types of Black Holes

There are four categories of black holes depending on their mass.

Supermassive Black Hole (SMBH): it is is an extremely large black hole. It has a mass greaterthan 106 M� (for reminder, M� is a solar mass and 1 M� = 2 × 1030 kg). It is theorizedto exist in the center of almost all massive galaxies. The largest SMBH in the Milky Way’svicinity appears to be that of Messier 87 (i.e. M87), at a mass of 6.3 × 109 M�. This is thefirst black hole ever picture by the Event Horizon Telescope (EHT) in 2019.

Intermediate Black Hole (IMBH): it has a mass between 100 M� and 10000 M�. Only a fewIMBH have been discovered. An example is M74, discovered by Chandra X-Ray Observatorywith a mass of roughly 10000 M� and a diamater a bit larger than the planet Uranus.

Stellar Black Hole: it has a mass between 3 M� and 30 M�. It is one of the possible evolutionaryendpoints of high mass stars. Indeed, when a star with a large enough mass has no hydrogenleft to burn, it will expand in a red supergiant, and will then explode in a supernovae beforebecoming, depending on its mass, a neutron star, a supernovae remnant or a stellar black hole.The smallest black hole yet discovered is a stellar black hole called GRO J0422+32 with amass of about 4 M�

Primordials Black Hole (PBH): it is a hypothetical type of black hole that formed soon afterthe Big Bang. In the early universe, high densities and heterogeneous conditions could haveled sufficiently dense regions to undergo gravitational collapse, forming black holes. PBH hasa mass that is less than M�. We know that the lighter the black hole is, the faster it willevaporate. In that sense, PBHs are the best candidate for studying Hawking radiations. Usingequations (26) we can easily compute that the lifetime of PBHs will be less than τ ∼ 1075

years. Then using equations (27), (28), (8), using the appropriate units, we find that:

T > 5× 10−12 eV, =⇒ T > 5× 10−8 K,

P > 2× 10−28 W

Epeak > 10−11 eV.

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3 OBSERVATIONS

We can see that the values for larger PBHs are very low. But lighter and lighter black holeshave increasingly large Hawking temperatures so we will focus on Asteroid-Mass PrimordialBlack Holes (3).

3 Observations

We know what we want to observe so we can now wonder: what can be observed in practice?Using which instrument? And at which distance? Constraints on PBH evaporation rate have beenobtained in (1) by looking for gamma-ray emission for PBHs with a mass of M =∼ 106 kg andT ∼ 10 TeV and lifetime τ ∼ 0.4 s. A search for such PBHs has been carried out by searching forhigh-energy gamma rays (between 100 GeV and 50 TeV) with Cherenkov telescopes (12).The Compton Observatory EGRET has also the capability of detecting directly the bursts of gammarays peaked near 250 MeV predicted by Hawking to be emitted by PBHs at the end of their life (5).Fermi Large Area Telescope (LAT) is a pair conversion telescope (meaning that it detects high-energy gamma rays using standard particle-physics techniques to detect particles) which is sensitiveto gamma rays from ∼20 MeV to 300 GeV (2).

3.1 The Fermi-LAT telescope

In this paper we are going to focus on the Fermi-LAT telescope. The Fermi-LAT (formerly Gamma-ray Large Area Space Telescope, or GLAST) is a space telescope from the US Space Agency intendedfor the study of high-energy gamma rays emitted by celestial objects. This satellite was launchedon June 11, 2008 and placed in a 565 km circular low-earth orbit. Its characteristics are given infigure 4. Since the main LAT instrument observes gamma rays from 20 MeV to 300 GeV, we aregoing to compute the power of black holes with corresponding energy to obtain roughly the rangeof the distance at which we can measure them.

Figure 4: LAT characteristics.(9)

For the lowest energy, we will take 100 Mev, because the sensitivity of the telescope in figure 4 isgiven for energy greater than 100 MeV. Using equations (8), (27), (26), (28) (note that we don’t need

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3.2 Energy and lifetime range 3 OBSERVATIONS

the mass to calculate the power but it is interesting to see the behavior), we obtain respectively:

Tlow = 4.4× 1011 K,

τlow = 1.2× 1018 s,

Mlow = 2.1× 1011 kg,

From the table of figure 4 we have that the point source sensitivity is F = 6 × 10−9 ph.cm−2.s−1.In order to calculate the distance, we need to convert this photon flux in power density in unit ofW.m−2 using:

Flim = N · Epeak. (31)

We find:

F limlow = 9.6−16 W.m−2. (32)

We use the exact same reasoning for the highest energy, which is 300 GeV and we obtain:

Thigh = 1.3× 1015 K,

τhigh = 8.8× 107 s,

Mhigh = 8.9× 107 kg,

F limhigh = 2.9× 10−12 W.m−2 (33)

The sensitivity of the Fermi-LAT telescope is given for 1 year sky survey. Note that 1 year =3.15×107 s. We can see the lifetime of the black holes we can observe are greater that 1 year but weshould take this lifetime into account anyway if we want to compute the flux. We also need to takeinto account that not all photons radiated from the BH will be within the correct frequency range.

3.2 Energy and lifetime range

To do so, we need to integrate Planck’s law (1) over a finite energy range for a given temperatureand then over the lifetime.

ΦR(Ti) =

∫ τ

0

∫ E2/h

E1/h

2hf3

c21

ehfkTi−1dfdτ ′, (34)

where E1 = 100 MeV, E2 = 300 GeV, T1 = 4.4 × 1011 K and T2 = 1.3 × 1015 K. We change ourvariables with x = hf

kTi= E

kTiand we get:

ΦR(Ti) =

∫ τ

0

2k4T 4

h3c2dτ ′∫ x2

x1

x3

ex − 1dx. (35)

We can solve the second integral numerically using Python (code in appendix B (13)) or with anyonline integral calculator. The code is made of two main parts: the first part consist in calculatingthe flux at a given temperature (and so, energy) in the Fermi-LAT energy range (which is between100 MeV and 300 GeV). This is the planck integral function. In order to do so we first computethe x integral, and call ”a” the solution (”res” in the code). Once this is done, we need to replace thetemperature using (27) to obtain the τ dependency (we have to pay attention to the temperaturewhich needs to be convert from K to TeV to obtain the lifetime with 1 TeV=11604 × 1012 K). Weget:

ΦR(Ti) =

∫ τ

0

2k4a

h3c2

(50× 11604× 1012 × τ ′−1/3

)4dτ ′

=6k4a

h3c2(50× 11604× 1012

)4τ−1/3. (36)

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3.3 Highest sensitivity 3 OBSERVATIONS

The planck integral function is now gonna give us the flux that we print in the second part of thecode.

ΦR(T1) = 5.1× 1050 W.m−2,

ΦR(T2) = 7.1× 1053 W.m−2.

We can compute the power by multiplying the flux with the surface area of the black hole just likewe did in (20). We have:

Alow = 1.24× 10−30 m2

Ahigh = 2.18× 10−37 m2

=⇒ Plow = 6.3× 1020 W

Phigh = 1.5× 1017 W

Finally, we use (30) and the sensibility of the telescope at the given energy from eq. (32) and (33)to compute the distance and we obtain:

rlow = 2.28× 1017 m,

rhigh = 6.43× 1013 m.

In order to visualize better the distance, we should convert the meters in parsec. The parsec,with the symbol pc, is a unit of length used in astronomy. It is defined as being equal to exactlyastronomical units, or about 3.26 light years. The name parsec is the contraction of ”parallax-second”, an expression relating to its historical definition, now obsolete. 1 pc = 3, 086× 1016 m.

rlow = 6 pc = 19.6 ly,

rhigh = 0.003 pc = 0.01 ly.

In the second part of the code, we define a function charact depending only on the energy. Byplugging-in the energy (in eV), the function is going to give us all the characteristics of the corre-sponding black hole and the distance it must be to be measured using the calculation we just made.The characteristics for the studied black holes are summarized in Table 1. If we look at the distancewe just computed, we can see that the BH has to be extremely close to be detected only through itsHawking’s radiation by the Fermi-LAT telescope. If it is not obvious, we can use Proxima Centurias a scale. Indeed, it is the closest star from the sun and it is located at a distance of 4.2 ly . Theclosest BH candidate detected yet is 1500 ly away. It was named ”the Unicorn” and is a binarycompanion to V723 Mon, a red giant star. It was discovered recently by analysis of the light curvesof the Kilodegree Extremely Little Telescope (KELT), and Transiting Exoplanet Survey Satellite(TESS) (7).We also have seen in the previous section that parallax can be use up to a distance of 30 pc. Itis much greater than the distance at which we can measure the PBHs. It seems better to use theparallax method instead of standard candles as a tool of measure.

3.3 Highest sensitivity

Even if the Fermi-LAT has a large energy range, one of the main question is whether it is moresensitive to a population of PBHs with ”low” temperature (less than 10 GeV) or to a populationwith ”high” temperature (greater than 10 GeV).Figure 5 shows the comparison of spectra of PBHs with different initial temperatures and Fermi-LAT PS differential sensitivity. The lines correspond to PBHs with different initial temperaturesspecified in the labels, with the corresponding lifetimes shown in parentheses. PBHs with lifetimes

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3.3 Highest sensitivity 3 OBSERVATIONS

longer than four years (Fermi observation time) are shown as solid lines, while PBHs with lifetimesshorter than four years are shown as dashed lines. By analysing, it is found that for the differentialpoint source (PS) sensitivity in 4 years of observation, the Fermi LAT is most sensitive to PBHswith temperature T ∼ 16 GeV (2). Using Monte Carlo simulations, it is then shown that the corre-sponding distance to which a PBH can be detected is roughly 0.03 pc.

Figure 5: Comparison of spectra of PBHs with different initial temperatures and Fermi-LAT PSdifferential sensitivity (2).

In this part, we are going to check if our previous result agree with this distance. We use the samereasoning as before but since we already have the distance, we will check if the power we obtainusing eq. (30) is the same as the power we obtain using eq. (34). For 16 GeV PBH, the sensitivityof the telescope will be F eglim = 1.5 × 10−13 W/m2. We therefore find, plugging in the distance of0.03 pc in (30), that the power should be 1.61 × 1018 W. We can calculate the temperature andlifetime of such PBHs to be:

Teg = 6.6× 1013 K,

τeg = 6.8× 1011 s.

Using Python, we obtain the flux to be:

Φeg = 1.1× 1053 W/m2.

We calculate the surface area A to be A = 7.6× 10−35 m2. Therefore, we obtain the power to be:

Peg = 8.36× 1018 W/m2.

We obtain a power that is close from the one in (2). The distance at which we can detect 16 GeVPBH is 0.03 pc, which is equivalent to 0.1 light years which is extremely close. It is then not thesmartest choice to use those PBHs as standard candles.To put in perspective and compare the differences between the PBHs we studied, we can gather thecharacteristics of in one table using charact function from appendix B as shown in Table 1.

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3.3 Highest sensitivity 3 OBSERVATIONS

Energy 100 MeV 16 GeV 300 GeV

Temperature 4.4× 1011 K 6.6× 1013 K 1.3× 1015 KLifetime 1.2× 1018 s 6.8× 1011 s 8.8× 107 sMasse 2.1× 1011 kg 1.75× 109 kg 8.9× 107 kg

Surface area 1.2× 10−30 m2 7.6× 10−35 m2 2.2× 10−37 m2

Sensitivity 9.6× 10−16 W/m2 1.5× 10−13 W/m2 2.9× 10−12 W/m2

Flux 5.1× 1050 W/m2 1.1× 1053 W/m2 7.1× 1053 W/m2

Power detected 6.3× 1020 W 8.36× 1018 W 1.5× 1017 WDistance 6 pc 0.03 pc 0.003 pc

Table 1: Recap of studied PBHs characteristics.

Note: we can check that those results make sense with the derivations of the first section. Indeed,we can see that the temperature increase as the lifetime decrease as predicted by eq. (27). Whichalso makes sense with what we already discussed: the hotter the PBHs is, the more radiations itwill emit, the faster it will evaporate. On the contrary, the mass is increasing with the lifetimeas predicted by eq. (26). We can also note that PBHs with higher temperature are lighter sincetemperature is inversely proportional to mass as shown by eq. (15). Which also makes sense sincethey are smaller.

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3.3 Highest sensitivity 3 OBSERVATIONS

Discussion and conclusion

Hawking radiation is an interesting property of BH. It is released when the BH evaporate so itis too faint to be measured for big BH such as SMBH, IMBH and stellar BH. However, anothertype of smaller (and so, hotter) BH might exist: primordial black holes. The potential existence ofPBHs that emit detectable Hawking radiation is one of the most intriguing features of some theoriesof cosmological evolution. In addition to providing evidence for these theories, the possibility fordirect observation of Hawking radiation was the main motivation for the search of PBHs with theFermi-LAT. Our goal was to determine at which distance they could currently be detected to knowif they could then be used as standard candles.

To achieve that, we started in the first section by computing the important characteristics of aBH such as its energy peak, temperature, mass, surface area, flux, lifetime and power by assumingBHs are a perfect blackbodies. By knowing only one parameter, we can then find out all of thosecharacteristics, which is extremely convenient. That is what we did using Python. In our code, theonly input is the energy peak, however, we could have used any parameter.In the second section, we discussed the main methods that are used to measure distances in astron-omy. Among them, we have seen that parallax can be used with accuracy up to a distance of 30 pc.Then for further distances, standard candles are used. By knowing the luminosity of the standardcandles, we can then calculate distances.The question was then to determine at which distance should be those standard candles (PBHsin our case) to be observed by current telescopes. By computing the frequency of PBHs, we havechecked that they emitted gamma ray radiations. We decided to focus on the Fermi-LAT telescope.The energy range of this telescope is between ∼20 MeV and 300 GeV. Thus, we compute the char-acteristics of PBHs with corresponding energy in order to obtain a range of distance. We did thesame with 16 GeV PBHs since it happens to be the energy for which the Fermi-LAT is the mostsensitive.

We have found that the distance at which we could measure PBHs should be between 0.003 pc(for ”high” energy PBHs) and 6 pc (for ”low” energy PBHs).PBHs are still hypothetical and haven’t been detected yet, but according to our results, if they were,it would be at a relatively cloth distance (less than 6 pc). In theory, we could use them as standardcandles at this distance but it would be more accurate and way easier to simply use parallax tomeasure distances.

Furthermore, we haven’t take into account the rotation of the PBHs or the relative displacement be-tween the telescope and the PBH. Indeed, due to orbital motion in the Galaxy and dark-matter-likevelocity dispersion, PBHs are not fixed and one of the difficulties in observing a moving point sourceis that the flux becomes spread into an track following the trajectory of the source. Estimation ofthe Fermi-LAT sensitivity to a moving source has been obtained by integrating the flux during thetime when the moving source appears as a point-like source for 16 GeV PBHs detection with theFermi-LAT as we can see in figure 6. The solid line corresponds to the detectability distance fora PBH as a function of initial temperature T from the Fermi-LAT point source sensitivity. Thedashed–dotted line is the detectability distance taking into account relative motion of PBHs. Thebreak at 16 GeV results from lower-temperature PBHs having longer lifetimes than the observationtime (4 years in this case). We see on this figure that the detectability distance with proper motionis even smaller than the one we compute.

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3.3 Highest sensitivity 3 OBSERVATIONS

Figure 6: Comparison of the detectability distance of 16 GeV PBHs for 4 years observation with theFermi-LAT with and without proper motion (2).

However, further studies could include an analysis of the capacities of other telescopes than theFermi-LAT. For example, EGRET, COMPTEL or Cherenkov telescope are also high energy tele-scope in the looking of PBHs. Moreover, several other missions are in the proposal, planning,or construction phase such as AdEPT, AMEGO, eASTROGAM, GECCO, MAST, PANGU andGRAMS (3).

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3.3 Highest sensitivity 3 OBSERVATIONS

Acknowledgments

I express gratitude to my supervisor Edvard Mortsell who gave me the opportunity to do this inter-esting project on this topic. I am thankful for his patience, support and knowledge. His guidancehelped me in all the time of research and writing my Bachelor’s project.

I would also like to thank Markus Hennrich and Tim Linden for their devotion.

I offer my sincere appreciation for the learning opportunities made possible by the ERASMUS+program and my coordinators.

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REFERENCES REFERENCES

References

[1] Carr, B., Kohri, K., Sendouda, Y., and Yokoyama, J. Constraints on primordial blackholes.

[2] Collaboration, L., et al. Search for gamma-ray emission from local primordial black holeswith the fermi large area telescope.

[3] Coogan, A., Morrison, L., and Profumo, S. Direct detection of hawking radiation fromasteroid-mass primordial black holes.

[4] ESA/ATG. Parallax, 2013.

[5] Fichtel, C., Bertsch, D., Dingus, B., Esposito, J., Hartman, R., Hunter, S., Kan-bach, G., Kniffen, D., Lin, Y., Mattox, J., et al. Search of the energetic gamma-ray experiment telescope (egret) data for high-energy gamma-ray microsecond bursts. TheAstrophysical Journal 434 (1994), 557–559.

[6] Gispert, J. Mesure des distances. Universite de la Mediterranee, Association Andromede(Observatoire de Marseille) (Sep 2017).

[7] Jayasinghe, T., Stanek, K., Thompson, T. A., Kochanek, C., Rowan, D., Vallely,P., Strassmeier, K., Weber, M., Hinkle, J., Hambsch, F.-J., et al. A unicorn inmonoceros: the 3 solar mass dark companion to the bright, nearby red giant v723 mon is anon-interacting, mass-gap black hole candidate.

[8] Liddle, A. An introduction to modern cosmology. Wiley, 2015.

[9] NASA. Fermi data - overview of the lat, Nov 2018.

[10] Pogosian, D. Lecture 20: Black holes. University of Alberta (2003).

[11] Schroeder, D. An introduction to thermal physics (7.4 - Blackbody Radiation). OxfordUniversity Press, 2021.

[12] Tesic, G., Collaboration, Veritas, et al. Searching for primordial black holes with theveritas gamma-ray experiment. In Journal of Physics: Conference Series (2012), vol. 375, IOPPublishing, p. 052024.

[13] Widger, W. K., and Woodall, M. P. Integration of the planck blackbody radiationfunction. Bulletin of the American Meteorological Society 57, 10 (1976), 1217–1219.

[14] Wikipedia. Hertzsprung–russell diagram, Feb 2021.

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A CODE PYTHON F(X)=X

Appendices

A Code Python f(x)=x

# -*- coding: utf-8 -*-

from numpy import exp, linspacefrom pylab import plot, figure, grid, show, xlabel, ylabel, title

def g(x):y = 3*(1-exp(-x))-xreturn (y)

def g1(x): # derivative of gy = exp(-x)-1return (y)

def NewtonsMethod( x, epsilon=0.0001):n = 0dif = 2 * epsilonwhile dif > epsilon :

x1 = x - g(x) / g1(x)dif = abs(x1 - x)x = x1n = n+1

return (x, n)

x = float(input(”Start value? : ”)) #We can choose any value, it will be the same result.

(x,n) = NewtonsMethod(x)

print(’x=%f n=%d’ % (x, n) )tab x = linspace(-3,4,100)tab y = g(tab x)

figure()title(’g(x)=3*(1-exp(-x))-x’)xlabel(’x’)ylabel(’g(x)’)plot(tab x,tab y, color=’red’)grid()show()

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B CODE PYTHON PLANCK INTEGRAL

B Code Python Planck Integral

# -*- coding: utf-8 -*-

from numpy import exp,pi,sqrtfrom scipy.integrate import quad

#constantsPlanck = 6.6260693e-34Planck third= Planck**3Boltzmann = 1.380658e-23Boltzmann fourth = Boltzmann**4Speed of light = 299792458.0Speed of light sq = Speed of light * Speed of lightc2 = Boltzmann fourth/(Planck third*Speed of light sq)

#define function that will be integrateddef function(x):

return (x**3/(exp(x)-1))

#integral of spectral radiance from E1 (eV) to E2 (eV).def planck integral(E1,E2,temperature):

#compute the lifetimetemperatureTeV=temperature/(11604e12)lifetime=(50/temperatureTeV)**3

#compute E (in J) and x, the dimensionless spectral coordinateE1=E1*1.6e-19E2=E2*1.6e-19x1 = E1 / (Boltzmann*temperature)x2 = E2 / (Boltzmann*temperature)

#integrate functionres, err = quad(function, x1, x2)

#return flux, in units of W/m2B=((11604e12)**4)*c2 * res * (lifetime**(-1/3))*6return(B)

#define a function that will give all the wanted characteristics by plugging-in the energy (in eV)def charact (Energy):

EnergyJ=Energy*1.6e-19

#TemperatureT=EnergyJ/(2.8*Boltzmann)print(”Temperature = ”,T,”K”)

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B CODE PYTHON PLANCK INTEGRAL

#LifetimeTeV=T/11604tau=(TeV/50e12)**(-3)print(”Lifetime = ”,tau,”s”)

#MassM=2e5*(tau**(1/3))print(”Mass = ”,M,”kg”)

#Surface areaA=16*pi*(G**2)*(M**2)/(Speed of light**4)print(”Surface area = ”,A,”m2”)

#FluxPhi=planck integral(100e6,300e9,T)print(”Flux = ”,Phi,”W/m2”)

#PowerP=A*Phiprint(”Power = ”,P,”W”)

#Fermi-sensitivityF=6e-5*EnergyJprint(”Flim = ”,F,”W/m2”)

#Distancerm=sqrt(P/(4*pi*F))rpc=rm/3.086e16print(”r = ”,rpc,”pc”)return(” ”)

#print the characteristics for the wanted energiesprint(”Characteristics 100MeV BH:”)print(charact (100e6))

print(”Characteristics 16GeV BH:”)print(charact (16e9))

print(”Characteristics 300GeV BH:”)print(charact (300e9))

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