Stockholm University
Black Holes as standard candles through their Hawking radiations
Tabatha Dufour
Physics department Bachelor’s Degree - 15 credits
Black holes as standard candles through their Hawking’s radiations Spring term 2021
Supervisor: Edvard M¨ortsell
CONTENTS CONTENTS
Contents
1 Background 3
1.1 Energy peak . . . . 3
1.2 Temperature . . . . 4
1.3 Flux . . . . 5
1.4 Lifetime . . . . 5
2 Theory 6 2.1 Parallax . . . . 6
2.2 Standard candles . . . . 7
2.3 Types of Black Holes . . . . 8
3 Observations 9 3.1 The Fermi-LAT telescope . . . . 9
3.2 Energy and lifetime range . . . . 10
3.3 Highest sensitivity . . . . 11
Appendices 18
A Code Python f(x)=x 18
B Code Python Planck Integral 19
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. Our goal is to determine at which distance it is possible to measure those radiations with current telescopes. Being capable of measuring those radiations would mean that we can use them as standard candles i.e. we could use it as distance indicator. To do so, we are first going to compute the different characteristic of a black hole such as its energy peak, temperature, flux and lifetime. Knowing that, we will be able to describe how, theoretically, we could use black holes as standard candles. Finally, we will take the Fermi-LAT telescope as an example of what we 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 be released by black holes (BH), due to quantum effects near the black hole event horizon. The event horizon is the most important feature of a black hole. It is an imaginary surface which acts as a boundary for the black hole as we can see on figure 1. The singularity in the center of a BH is a gravitational singularity. It is a one-dimensional point which contains a huge mass in an infinitely small space, where density and gravity become infinite and space-time curves infinitely, the laws of physics as we know them cease to operate. RSch refers to the Schwarzschild radius (sometimes historically referred to as the gravitational radius). It is the radius defining the event horizon.The Schwarzschild radius is given as rs = 2GMc2 . We can see that the radius is directly proportional to the mass. The bigger the black hole is, the heavier it gets.
Figure 1: Structure of BH diagram. (10)
1 BACKGROUND
Hawking radiation reduces the mass and rotational energy of black holes and is therefore also known as black hole evaporation. Because of this, black holes that do not gain mass through other means are expected to shrink and ultimately vanish. The smaller the black hole is, the faster it will evaporate and consequently, the more Hawking radiation it will emit. Thus, we are interested in small black holes. 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 of energy, the temperature, the flux and the lifetime of black holes by assuming that the Hawking radiation is a perfect black body. In the second section, we will discuss the different types of BHs and how show we can theoretically use BH as standard candles. In the third and last section, we will apply the theory to practice by analysing at which distance BHs can be observed by current telescopes, 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 law that describes B(f, T ) the spectral radiance for a blackbody where f is the frequency and T is the temperature (11) :
B(f, t) =2hf3 c2
1
ekThf−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 c2
1
ekThf−1 −2hf3 c2
hekThf kT
ehfkT − 1)2 (2)
⇐⇒ 2hf3 c2
1 ekThf−1 ·
3
f − hekThf kT
ehfkT − 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 − hekThf kT
ekThf − 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 or with Python (code in appendix A). We find:
x ∼ 2.8 ⇐⇒ hf
kT ∼ 2.8. (6)
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 = 2M c2, (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 wavelength of the size λ of a black hole and energy = hcλ. The total energy is then:
U = N = N hc
λ = N hc3
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 a system 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 we start with a lot of particles or a few, the final entropy must be of the order of the maximum N. We can find :
S = 8π2kGM2
hc . (13)
We can rewrite this for a BH as:
S = 8π2k(M c2)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 convert it to eV knowing the relation E = kT using k ∼ 8.617 × 10−5 eV/K.
T (eV) = hc3
16π2GM (16)
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 unit surface area of a blackbody is directly proportional to the fourth power of the blackbody. We can find the same result by integrating Planck’s law from 0 to ∞.
Φ = σT4 (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
c4 of a black body with the flux.
P = σAT4
= 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 given by the differential equation:
dU
dt = −σAT4 (21)
⇐⇒ dM
dt = − H
M2 where H = 2hc4
(30720)π2G2. (22) We can integrate to obtain the lifetime τ (in second) of a blackbody:
Z 0 Mi
M2dM = −H Z τ
0
dt (23)
⇐⇒ −Mi3
3 = −Hτ (24)
⇐⇒ τ = Mi3
3H. (25)
By using this result and plugging in the numerical values of the constants c = 3 × 108 m/s, G = 6.67 × 10−11m3.kg−1.s−2, h = 6.63 × 10−34J.s = 4.14 × 10−15eV.s,P = 3.86×1026W 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)
2 THEORY
This gives us a global idea of how the BHs are behaving. If the BH is smaller (so lighter), the lifetime will be shorter, but the temperature and the power will increase. This is why the smaller the black hole is, the more Hawking radiation we have. We are now going to discuss the theory behind 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, the fact 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 is called parallax. The principle of parallax is very simple, to understand it, we can put our finger in front of us and alternately close one eye. The distance between the two eyes produces an effect of perspective, which is called parallax. This effect is all the more marked the closer the observed object is, that is to say, its distance is smaller in front of the gap between the eyes. It can be measured by the angle the light rays make on the drawing. In the triangle formed by the eyes and the finger, we know the distance between the eyes, and we measure the angle and we deduce the distance. The idea of astronomers was to increase the gap between the eyes. To simulate this, they took two sky photos six months apart. In these photos, there are very distant stars, which play the role of the wall, and nearby stars which play the role of the finger. This method is based the apparent displacement of an object because of a change in the observer’s point of view. As we can see in figure 2, by measuring the amount of displacement – the parallax angle – and knowing the distance between the Sun and the Earth, astronomers can determine the star’s distance using simple trigonometry.
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 the triangle; it is the diameter of the Earth’s orbit. We measure the angle α (parallax on figure 2).
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 angle at which we see the radius of the Earth’s orbit is called the parallax. We use the radius of the Earth’s orbit, because it is the astronomical unit. This method gave a new unit of distance: the parsec is the distance corresponding to a parallax of one second angle.
The Hipparcos telescope (High Precision PARallax COllecting Satellite) of the European Space Agency (ESA), launched in 1989, observing outside the atmosphere, has increased the precision of parallax measurements. The European Space Agency has built a successor to Hipparcos, named Gaia. 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 50 times more accurate than Hipparcos. Thus, parallax method makes it possible to measure stellar distances 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 used anymore. In that case, standard candles can be used. A standard candle is a source that has a known luminosity (8). Light sources which are further away appear fainter because the light is spread out over a greater area. If we know how luminous a source really is, then we can estimate its distance from how bright it appears from Earth. The larger the distance is, the smaller the power that we measure is.
There are several types of standard candle objects for which we can predict the luminosity from some other 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 like BHs merge, they produce gravitational waves, which can be picked up by observatories such as the Laser Interferometer Gravitational-Wave Observatory (LIGO). The emitted gravitational waves have 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 Hertzsprung and Henry Norris Russell, it plots the temperature of stars against their luminosity as shown in figure 3. As we can see it applies for object with a temperature between 2000 K and 30000 K. For higher temperatures, we can use that the luminosity is equal to the total power output, measured in watts (W) or solar luminosity (L ) where 1 L = 3.84 × 1026W).
The relation between the distance r and the luminosity is given by:
Φ = L
4πr2 ⇐⇒ r =
r L
4πΦ. (30)
Flux is measured. Luminosity is not directly measurable, but is instead derived from its relation to some observable as shown in (28).
For BHs, we will see that the temperature is a lot greater than 30000 K. Therefore, we will not use 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 still need to discuss what do we want to measure.
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 greater than 106 M (for reminder, M is a solar mass and 1 M = 2 × 1030 kg). It is theorized to exist in the center of almost all massive galaxies. The largest SMBH in the Milky Way’s vicinity appears to be that of Messier 87 (i.e. M87), at a mass of 6.3 × 109M . This is the first 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 few IMBH have been discovered. An example is M74, discovered by Chandra X-Ray Observatory with 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 evolutionary endpoints of high mass stars. Indeed, when a star with a large enough mass has no hydrogen left to burn, it will expand in a red supergiant, and will then explode in a supernovae before becoming, 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 a mass of about 4 M
Primordials Black Hole (PBH): it is a hypothetical type of black hole that formed soon after the Big Bang. In the early universe, high densities and heterogeneous conditions could have led sufficiently dense regions to undergo gravitational collapse, forming black holes. PBH has a mass that is less than M . We know that the lighter the black hole is, the faster it will evaporate. In that sense, PBHs are the best candidate for studying Hawking radiations. Using equations (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.
3 OBSERVATIONS
We can see that the values for larger PBHs are very low. But lighter and lighter black holes have increasingly large Hawking temperatures so we will focus on Asteroid-Mass Primordial Black 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 been obtained in (1) by looking for gamma-ray emission for PBHs with a mass of M =∼ 106 kg and T ∼ 10 TeV and lifetime τ ∼ 0.4 s. A search for such PBHs has been carried out by searching for high-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 gamma rays 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 sensitive to 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 intended for the study of high-energy gamma rays emitted by celestial objects. This satellite was launched on June 11, 2008 and placed in a 565 km circular low-earth orbit. Its characteristics are given in figure 4. Since the main LAT instrument observes gamma rays from 20 MeV to 300 GeV, we are going to compute the power of black holes with corresponding energy to obtain roughly the range of 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 is given for energy greater than 100 MeV. Using equations (8), (27), (26), (28) (note that we don’t need
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 of W.m−2 using:
Flim= N · Epeak. (31)
We find:
Flowlim= 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 × 1015K, τhigh= 8.8 × 107s, Mhigh= 8.9 × 107kg,
Fhighlim = 2.9 × 10−12W.m−2 (33)
The sensitivity of the Fermi-LAT telescope is given for 1 year sky survey. Note that 1 year = 3.15 × 107s. We can see the lifetime of the black holes we can observe are greater that 1 year but we should take this lifetime into account anyway if we want to compute the flux. We also need to take into 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 temperature and then over the lifetime.
ΦR(Ti) = Z τ
0
Z E2/h E1/h
2hf3 c2
1 ekTihf−1
df dτ0, (34)
where E1 = 100 MeV, E2 = 300 GeV, T1 = 4.4 × 1011 K and T2 = 1.3 × 1015 K. We change our variables with x = kThf
i =kTE
i and we get:
ΦR(Ti) = Z τ
0
2k4T4 h3c2 dτ0
Z x2 x1
x3
ex− 1dx. (35)
We can solve the second integral numerically using Python (code in appendix B (13)) or with any online integral calculator. The code is made of two main parts: the first part consist in calculating the flux at a given temperature (and so, energy) in the Fermi-LAT energy range (which is between 100 MeV and 300 GeV). This is the planck integral function. In order to do so we first compute the x integral, and call ”a” the solution (”res” in the code). Once this is done, we need to replace the temperature using (27) to obtain the τ dependency (we have to pay attention to the temperature which needs to be convert from K to TeV to obtain the lifetime with 1 TeV=11604 × 1012 K). We get:
ΦR(Ti) = Z τ
0
2k4a h3c2
50 × 11604 × 1012× τ0−1/34 dτ0
=6k4a
h3c2 50 × 11604 × 10124
τ−1/3. (36)
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 the code.
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 like we did in (20). We have:
Alow= 1.24 × 10−30 m2 Ahigh= 2.18 × 10−37 m2
=⇒ Plow= 6.3 × 1020W Phigh= 1.5 × 1017W
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 × 1017m, rhigh= 6.43 × 1013m.
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 exactly astronomical 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 × 1016m.
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. By plugging-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 distance we just computed, we can see that the BH has to be extremely close to be detected only through its Hawking’s radiation by the Fermi-LAT telescope. If it is not obvious, we can use Proxima Centuri as a scale. Indeed, it is the closest star from the sun and it is located at a distance of 4.2 ly . The closest BH candidate detected yet is 1500 ly away. It was named ”the Unicorn” and is a binary companion to V723 Mon, a red giant star. It was discovered recently by analysis of the light curves of 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. It is much greater than the distance at which we can measure the PBHs. It seems better to use the parallax 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 more sensitive to a population of PBHs with ”low” temperature (less than 10 GeV) or to a population with ”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 temperatures specified in the labels, with the corresponding lifetimes shown in parentheses. PBHs with lifetimes
3.3 Highest sensitivity 3 OBSERVATIONS
longer than four years (Fermi observation time) are shown as solid lines, while PBHs with lifetimes shorter than four years are shown as dashed lines. By analysing, it is found that for the differential point source (PS) sensitivity in 4 years of observation, the Fermi LAT is most sensitive to PBHs with 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 PS differential sensitivity (2).
In this part, we are going to check if our previous result agree with this distance. We use the same reasoning as before but since we already have the distance, we will check if the power we obtain using eq. (30) is the same as the power we obtain using eq. (34). For 16 GeV PBH, the sensitivity of the telescope will be Flimeg = 1.5 × 10−13 W/m2. We therefore find, plugging in the distance of 0.03 pc in (30), that the power should be 1.61 × 1018 W. We can calculate the temperature and lifetime 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 × 1053W/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 × 1018W/m2.
We obtain a power that is close from the one in (2). The distance at which we can detect 16 GeV PBH is 0.03 pc, which is equivalent to 0.1 light years which is extremely close. It is then not the smartest choice to use those PBHs as standard candles.
To put in perspective and compare the differences between the PBHs we studied, we can gather the characteristics of in one table using charact function from appendix B as shown in Table 1.
3.3 Highest sensitivity 3 OBSERVATIONS
Energy 100 MeV 16 GeV 300 GeV
Temperature 4.4 × 1011K 6.6 × 1013K 1.3 × 1015 K Lifetime 1.2 × 1018s 6.8 × 1011s 8.8 × 107 s
Masse 2.1 × 1011 kg 1.75 × 109 kg 8.9 × 107kg 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 × 1050W/m2 1.1 × 1053 W/m2 7.1 × 1053 W/m2 Power detected 6.3 × 1020 W 8.36 × 1018W 1.5 × 1017W
Distance 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). Which also makes sense with what we already discussed: the hotter the PBHs is, the more radiations it will emit, the faster it will evaporate. On the contrary, the mass is increasing with the lifetime as predicted by eq. (26). We can also note that PBHs with higher temperature are lighter since temperature is inversely proportional to mass as shown by eq. (15). Which also makes sense since they are smaller.
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 it is too faint to be measured for big BH such as SMBH, IMBH and stellar BH. However, another type of smaller (and so, hotter) BH might exist: primordial black holes. The potential existence of PBHs that emit detectable Hawking radiation is one of the most intriguing features of some theories of cosmological evolution. In addition to providing evidence for these theories, the possibility for direct observation of Hawking radiation was the main motivation for the search of PBHs with the Fermi-LAT. Our goal was to determine at which distance they could currently be detected to know if they could then be used as standard candles.
To achieve that, we started in the first section by computing the important characteristics of a BH such as its energy peak, temperature, mass, surface area, flux, lifetime and power by assuming BHs are a perfect blackbodies. By knowing only one parameter, we can then find out all of those characteristics, which is extremely convenient. That is what we did using Python. In our code, the only 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 standard candles, we can then calculate distances.
The question was then to determine at which distance should be those standard candles (PBHs in our case) to be observed by current telescopes. By computing the frequency of PBHs, we have checked 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 the same with 16 GeV PBHs since it happens to be the energy for which the Fermi-LAT is the most sensitive.
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 standard candles at this distance but it would be more accurate and way easier to simply use parallax to measure 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-like velocity dispersion, PBHs are not fixed and one of the difficulties in observing a moving point source is that the flux becomes spread into an track following the trajectory of the source. Estimation of the Fermi-LAT sensitivity to a moving source has been obtained by integrating the flux during the time when the moving source appears as a point-like source for 16 GeV PBHs detection with the Fermi-LAT as we can see in figure 6. The solid line corresponds to the detectability distance for a PBH as a function of initial temperature T from the Fermi-LAT point source sensitivity. The dashed–dotted line is the detectability distance taking into account relative motion of PBHs. The break at 16 GeV results from lower-temperature PBHs having longer lifetimes than the observation time (4 years in this case). We see on this figure that the detectability distance with proper motion is even smaller than the one we compute.
3.3 Highest sensitivity 3 OBSERVATIONS
Figure 6: Comparison of the detectability distance of 16 GeV PBHs for 4 years observation with the Fermi-LAT with and without proper motion (2).
However, further studies could include an analysis of the capacities of other telescopes than the Fermi-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 and GRAMS (3).
3.3 Highest sensitivity 3 OBSERVATIONS
Acknowledgments
I express gratitude to my supervisor Edvard M¨ortsell who gave me the opportunity to do this inter- esting project on this topic. I am thankful for his patience, support and knowledge. His guidance helped 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.
REFERENCES REFERENCES
References
[1] Carr, B., Kohri, K., Sendouda, Y., and Yokoyama, J. Constraints on primordial black holes.
[2] Collaboration, L., et al. Search for gamma-ray emission from local primordial black holes with the fermi large area telescope.
[3] Coogan, A., Morrison, L., and Profumo, S. Direct detection of hawking radiation from asteroid-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. The Astrophysical Journal 434 (1994), 557–559.
[6] Gispert, J. Mesure des distances. Universit´e de la M´editerran´ee, Association Androm`ede (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 in monoceros: the 3 solar mass dark companion to the bright, nearby red giant v723 mon is a non-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). Oxford University Press, 2021.
[12] Teˇsi´c, G., Collaboration, Veritas, et al. Searching for primordial black holes with the veritas gamma-ray experiment. In Journal of Physics: Conference Series (2012), vol. 375, IOP Publishing, p. 052024.
[13] Widger, W. K., and Woodall, M. P. Integration of the planck blackbody radiation function. Bulletin of the American Meteorological Society 57, 10 (1976), 1217–1219.
[14] Wikipedia. Hertzsprung–russell diagram, Feb 2021.
A CODE PYTHON F(X)=X
Appendices
A Code Python f(x)=x
# -*- coding: utf-8 -*-
from numpy import exp, linspace
from pylab import plot, figure, grid, show, xlabel, ylabel, title def g(x):
y = 3*(1-exp(-x))-x return (y)
def g1(x): # derivative of g y = exp(-x)-1
return (y)
def NewtonsMethod( x, epsilon=0.0001):
n = 0
dif = 2 * epsilon while dif > epsilon :
x1 = x - g(x) / g1(x) dif = abs(x1 - x) x = x1
n = 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()
B CODE PYTHON PLANCK INTEGRAL
B Code Python Planck Integral
# -*- coding: utf-8 -*-
from numpy import exp,pi,sqrt from scipy.integrate import quad
#constants
Planck = 6.6260693e-34 Planck third= Planck**3 Boltzmann = 1.380658e-23
Boltzmann fourth = Boltzmann**4 Speed of light = 299792458.0
Speed of light sq = Speed of light * Speed of light c2 = Boltzmann fourth/(Planck third*Speed of light sq)
#define function that will be integrated def 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 lifetime
temperatureTeV=temperature/(11604e12) lifetime=(50/temperatureTeV)**3
#compute E (in J) and x, the dimensionless spectral coordinate E1=E1*1.6e-19
E2=E2*1.6e-19
x1 = E1 / (Boltzmann*temperature) x2 = E2 / (Boltzmann*temperature)
#integrate function
res, err = quad(function, x1, x2)
#return flux, in units of W/m2
B=((11604e12)**4)*c2 * res * (lifetime**(-1/3))*6 return(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
#Temperature
T=EnergyJ/(2.8*Boltzmann) print(”Temperature = ”,T,”K”)
B CODE PYTHON PLANCK INTEGRAL
#Lifetime
TeV=T/11604
tau=(TeV/50e12)**(-3) print(”Lifetime = ”,tau,”s”)
#Mass
M=2e5*(tau**(1/3)) print(”Mass = ”,M,”kg”)
#Surface area
A=16*pi*(G**2)*(M**2)/(Speed of light**4) print(”Surface area = ”,A,”m2”)
#Flux
Phi=planck integral(100e6,300e9,T) print(”Flux = ”,Phi,”W/m2”)
#Power P=A*Phi
print(”Power = ”,P,”W”)
#Fermi-sensitivity F=6e-5*EnergyJ
print(”Flim = ”,F,”W/m2”)
#Distance
rm=sqrt(P/(4*pi*F)) rpc=rm/3.086e16 print(”r = ”,rpc,”pc”) return(” ”)
#print the characteristics for the wanted energies print(”Characteristics 100MeV BH:”)
print(charact (100e6))
print(”Characteristics 16GeV BH:”) print(charact (16e9))
print(”Characteristics 300GeV BH:”) print(charact (300e9))
