University of Amsterdam

Mass constraints on MACHOdark matter

Author:Harm van Leijen

Supervisor:dr. Christoph Weniger

Second Supervisor:dr. Shin’ichiro Ando

August 19, 2017

Abstract

There is still a lot of debate concerning the nature of dark matter. Oneof the explanations is that dark matter is made of MACHOs: Massivecompact halo objects. Here, we derive constraints on the possible massesof these MACHO dark matter components. Microlensing measurementshave excluded MACHO masses between 0.15 and 20 M� to form all ofthe dark matter at an 95 % CL. Simulations to the effects of dark matterMACHOs on the evolution of a galaxy have excluded masses from 1 up toa hundreds of solar masses to be the only component of dark matter andMACHOs with masses of a few tens of solar masses are even excluded toform more than 4 % of the total dark matter in the universe. We also showthat the constraints from dynamical heating strengthen when a range ofMACHO masses is present.

Contents

1 Introduction 2

2 Dark Matter 22.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Dark matter components . . . . . . . . . . . . . . . . . . . . . . . 32.3 MACHOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3.1 Primordial black holes . . . . . . . . . . . . . . . . . . . . 4

3 Constraints on MACHO dark matter 53.1 Constraints by microlensing measurements . . . . . . . . . . . . . 53.2 Constraints from galactic dynamics . . . . . . . . . . . . . . . . . 8

3.2.1 Dynamical heating . . . . . . . . . . . . . . . . . . . . . . 83.2.2 Dynamical heating for a range of MACHO masses . . . . 113.2.3 Mass segregation . . . . . . . . . . . . . . . . . . . . . . . 14

4 Conclusion and outlook 18

5 Acknowledgement 18

6 References 18

7 Appendix 217.1 The mean value of The log-normal distribution. . . . . . . . . . . 217.2 Populair dutch summary . . . . . . . . . . . . . . . . . . . . . . . 22

1

1 Introduction

One of the greatest mysteries of modern day astrophysics is the phenomenoncalled dark matter; a type of matter that doesn’t interact with light but doeshave a gravitational impact on its surrounding. Although most scientist agreethat there is something called dark matter, there is still a lot of debate con-cerning the nature of dark matter [1]. The explanation for the fundamentalcomponents of dark matter even range from ultralight axions [2] to super mas-sive black holes [3]. For this paper we assume that dark matter is made ofMACHOs (Massive Compact Halo Objects). The goal of this thesis will be tofind constraints on the mass range of these dark matter components. A lotof research has already been done to these constraints : Brandt (2016) andKoushiappas (2017) placed constraints by looking at the dynamical impact ofthese MACHOs on a galaxy [4] [5] and alcock (2001) placed constraints by look-ing at the microlensing effects of these MACHOs [6]. Here, I will reproduce theresults of these studies and strengthen some of the constraints by looking at thecase were the MACHOs are made off a range of masses.

2 Dark Matter

2.1 History

One of the first scientist who mentioned the existence of dark matter was theSwiss astronomer Fritz Zwicky, he measured the velocity dispersion of galaxiesin the Coma cloud cluster. He found that the velocity dispersion had a valuearound the 1000 km/s. If the total mass of this cluster only existed off starsthan the total gravitational pull of this system wouldn’t be sufficient to keepthe stars with such velocities bound. He concluded that a part a the galaxiesmass may be in the form of ’dark bodies’ [7].One of the major breakthroughs came in 1980 when Vera Rubin and KentFord published their paper : ”Rotational Properties of 21 Sc Galaxies witha Large Range of Luminosities and Radii from NGC 4605 (R=4kpc) to UGC2885 (R=122kpc)” [8]. In this paper they measured the rotational curve of thegalaxy and found that the rotational velocity remained relatively constant after acertain radius(figure 1). This doesn’t match with the expectations from Kepler’slaw. This difference can be explained by assuming that there are big quantitiesof dark matter in the outer layers of a galaxy. since then a lot of studies foundevidence for the existence of dark matter: measurements to gravitational lensing[9], X-ray measurements [10] and evidence coming from the Cosmic microwavebackground [11].At this points most of the scientist agree that there is a lot of mass in thegalaxies that we can’t see. The questions about how and when it was formedand what it is made of however still remain open.

2

Figure 1: The measured rotational velocity of a galaxy(B) vs the expectedrotational velocity (A). The difference can be explained by assuming that theouter layers of a Galaxy contain big quantities of dark matter.

2.2 Dark matter components

There are a lot of possible candidates for the fundamental components thatform dark matter. These candidates even cover a mass range of 90 orders ofmagnitude [12]. All those candidates can be roughly divided into two groups;Baryonic and non-baryonic matter components. Baryonic matter is matter thatis made from Baryons, a type of particle that is made of 3 quarks. Most of thematter we know, such as protons and neutrons, are baryonic matter. Underbaryonic matter we also include astronomical bodies such as white and browndwarfs, neutron stars and black holes. These objects are collectively known asMassive compact halo compact objects (MACHOs). Non Baryonic dark mattercandidates are more theoretical particles like axions and WIMPS (Weakly in-teracting massive particles). Wimps have been subject to a lot of research andalthough the R-parity-conserving supersymmetry predicts a Wimp-like particle[13], they have yet to be found. If these Wimp particles indeed exist, than theyhave been formed in huge quantities in the early universe. These Wimp particleswould than annihilate with each other to form two normal particles. As the uni-verse expanded the chance that two Wimp particles would collide decreased andthese annihilation’s would become more and more rare. Simulations to theseevents show that we would end up with around 5 times more mass in Wimpsthan in normal matter [14], this closely resembles the amount of dark mattermass predicted by measurements to the cosmic microwave background [15] [16].This among other things makes wimps for a lot of scientist the most favorableexplanation to the dark matter issue.

Another possible explanation for the missing matter is the axion. The ax-ion was first mentioned as a solution to the strong CP problem in quantumchromodynamics (QCD) [17]. Violation of the CP symmetry would result in aneletric dipole moment of the neutron of around 10−18e∗m, experiments howeversuggest a fraction of this value [18]. In contrary to Wimps, axions are relativelylight particles, (Berenji 2016) even placed the upper limit of the axion mass at7.9 ×10−2 eV [19].

It is possible that the total dark matter in our universe consists of a combi-nation of some of the suggested fundamental components.

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2.3 MACHOs

During this project we look into the possibility that MACHOs are the funda-mental components that form dark matter. The fact that they are both likelyto exist and objects that emit little to no light makes them among the firstmentioned explanations for the dark matter issue. However, since most of the’natural’ MACHOs (neutron stars and black holes) are the result of supernovaeexplosions, which are very rare, it is unlikely that there would be enough to formthe total dark matter in the universe. One thing that could solve this problemis the existence of primordial black holes, black holes that formed at the earlieststages of the universe [20].

2.3.1 Primordial black holes

If we look at the Cosmic microwave background (CMB) we get an image whichshows us the temperature fluctuations in the 379,000 years old universe [21]

Figure 2: An image from scholarpedia [21] of the temperature fluctuations whenthe universe was 379.000 years old. The image has been obtained by measuringthe Cosmic microwave background radiation.

This image (figure 2) tells us that even at the earliest stages of the universe thematter was not equally distributed. These density fluctuations have started theformations of stars and galaxies. Some researchers suggest that if these densityfluctuations where high enough, they could have resulted in the formation ofprimordial black holes (PBH) [22]. Since these black holes don’t need massivestars for there formation, they can span a broad range of masses. In 1971Hawking even suggested that these PBH could have masses which ranges from10−8 kg to thousands of solar masses [23]. The relatively light ones (with a massbelow 1011 kg) however would have been evaporated by now due to hawkingradiation.

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3 Constraints on MACHO dark matter

3.1 Constraints by microlensing measurements

Since Einstein stated his theory of relativity, we know that mass curves thespace around it and that even a massless light particle is effected by gravity[24]. Einstein also predicted a phenomenon called gravitational lensing; whenmatter passes in front of a light-source it can bend the light as it travels fromthe source to the observer. This can be very useful when we are looking forexoplanets. As an exoplanet moves in front of the star, it will temporary bendthe light towards us which we can measure as in increase in Luminosity comingfrom the star (see figure 2).

Figure 3: An image from Encyclopedia Britannica Online [25] showing thatmatter can act as a lens when it bends the light to an observer. The observerfrom earth will temporary measure an increase in luminosity coming from thestar.

Dark matter MACHOs can be found using the same technique. But since theduration of the luminosity increase depends on multiple variables; lens velocity,lens mass and lens distance, it isn’t possible to determine the lens mass for aunique event. However, K Griest made an estimate for the average duration ofa microlensing event for a given halo model. He found that the mass of a lenscould be given by [26]:

m

M�'(

t

130d

)2

, (1)

with m/M� being the mass of the lens in solar masses and t being the time ittakes for this lens to cross the Einstein ring of the source star in days.

5

These microlensing effects can be measured from earth. The efficiency of detect-ing these effects have been calculated by Alcock 2001 [6], for there calculationsthey considered fluctuations caused by seeing, the weather, telescopic failuresand effects caused by blending. They got the following efficiency for detectinglong-duration microlensing effects:

Figure 4: graph from Alcock 2001 [27] showing the efficiency of detecting long-duration microlensing effect.

If we follow Alcock 2001 [27] and we assume that the MACHO masses aredistributed according to a δ- function, we can get a figure which shows theexpected amount of measurements set out to the mass of those MACHOs:

Figure 5: Number of expected dark matter MACHO measurements if darkmatter is only made of MACHOs [6].

During there 5.7 year during survey to 11.9 million stars, Alcock et al found zero

6

long duration (> 150 days) microlensing effects caused by dark matter MACHOs[6]. To use this to put constraints on the mass of dark matter MACHOs, wecalculate the probability of detecting 0 events. Since the expected measurementsare distributed as a Poisson distribution, these probabilities can be given by:

P (zero events) = e−α. (2)

With P(zero events) being the chance that zero events are measured, e beingthe natural logarithm and α being the amount of measurements expected. Thismeans that all masses who have more than 3 expected measurements can beruled out at a 95 percent certainty: P(zero events) = e−3 = 0.05. They howevercan still make up a fraction of the total dark matter. If we combine equation 2with the data from figure 5, we get the following constraints on MACHO darkmatter:

Figure 6: Figure from alcock (2001) showing the constraints on MACHO darkmatter masses by microlensing measurements. The left y-axis represent the totalfraction of the dark matter in the form of MACHOs and the x-axix gives themass of those MACHOs. The right y-axis gives the totak possible dark mattermass in the form of these MACHOs. The section above the line is excluded atan 95 percent certainty. MACHOs with masses between 0.3 and 20 solar massesare ruled out to be the only components of dark matter.

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3.2 Constraints from galactic dynamics

3.2.1 Dynamical heating

If the dark matter in galaxies is indeed made of MACHOs, than these MACHOswill have a dynamical impact on there surrounding. Gravitational interactionswill lead to exchange of energy. These interactions will dynamically heat thesystem, causing it to expand. To calculate the growth of a star cluster we assumethat the system can be treated as a diffusion problem, with weak scatteringchanging the velocities of the stars. If we assume that the velocities of the starsare distributed according to the Maxwell–Boltzmann distribution, we get thatthe diffusion coefficient can be given by [28]:

D[(∆v)2] =4√

2πG2fdmρma ln(Λ)

vdisp

[erf(X)

X

]. (3)

Where fdm is the fraction of the dark matter that is in the form of MACHOs, ρis the total dark matter density, vdisp is the velocity dispersion of the MACHOsand ma is the mass of the MACHOs. X is the ratio between the velocities ofthe stars to the velocity of the MACHOs (v∗/

√2vdisp), with erf(X) being the

error function of X. This error function can be given by:

erf(X) =2√π

∫ X

0

e−t2

dt. (4)

We will assume that the relative temperature of the MACHOs is much higherthan that of the stars, so vdisp > v∗ and X << 1. If we take the Tayler expansionof the integral close to 0 we get:

erf(X) ' x− x3

3+x5

10+O(x6). (5)

So when X is close to zero, erf(X)/X becomes∼ 1. ln(Λ) is known as the coulomblogarithm, a factor which describes the amount that small angle collisions aremore effective than large angle collisions :

ln(Λ) = ln( rhvdispG(m+ma)

). (6)

This diffusion coefficient describes the mean change of the velocities per unittime. The potential energy per unit mass of the star cluster can be given by [4]:

U

M= C − αGM∗

rh+ βGρr2h , (7)

in which C is a constant, M∗ is the total stellar mass of the cluster, ρ is thedark matter density, α and β are constants that depend on the mass distributionof the system and rh is the half light radius : the radius in which half of theclusters light is emitted.If we now take the derivative of Equation 5 we get:

8

1

M

d

dt(U) = (α

GM∗r2h

+ 2βGρrh)drhdt

. (8)

We now have a formula which gives the change in the total kinetic energy overtime (equation (3)) and a formula which gives the change in the total potentialenergy over time (equation (8)). We can combine these two with the use of thevirial theorem, a theorem which gives a relation between the potential and thekinetic energy. In our case this virial theorem is : Etot = − 1

2U [4]. If we usethis, we get the following expression for the change in the half light radius:

drhdt

=4√

2πGfdmma

vdispln(Λ)

(αM∗ρr2h

+ 2βrh

)−1. (9)

To Simulate this effect we used the known data from a star cluster near the centreof Eridanus II. Eridanus II is a Ultra-faint dwarf galaxy which are perfect for thestudy to dark matter because they are relatively high dark matter dominated[29]. We numerically solved this equation (using python), this gave us thefollowing change in half-light radius due to dynamical heating:

Figure 7: Change in the half light radius of a 6000 M� star cluster due todynamical heating. We used that all of the dark matter mass was in the form offMACHOs with mass 30 M� and that the velocity dispersion of the MACHOswas 5 kms−1. This figure gives the same results as the simulations done by(brandt 2016) [4]

9

If we now combine equation 7 with the observed data from Eridanus II and werequire that the timescale for dynamical heating takes longer than the clusterscurrent age, we can put constraints on the MACHO dark matter masses [4].The system has an observed half-light radius of 13 parsec and we assume thatat t = 0 its half-light radius was 2 pc. If we use this data, python gives usthe following constraints for two different velocity dispersion’s and dark matterdensities:

Figure 8: MACHO dark matter mass constraints for a 3 Gyr old star clusterwith a total mass of 2000 M�. the units of the velocity dispersion and thedensity are given in m/s and in M�/pc

3.

10

3.2.2 Dynamical heating for a range of MACHO masses

In the previous section we only looked at the case where all the MACHOs havethe same mass. It is however more likely that those MACHOs will span a rangeof masses. In this section we look at the equations for dynamical heating whenthe dark matter is made of MACHOs with a range of masses and look at theeffects that this will have on the mass constraints.

We already have the formula which describes the diffusion coefficient if all theMACHOs have the same mass ma:

D[(∆v)2] =4√

2πG2fdmρma ln(Λ)

vdisp

[erf(X)

X

]. (10)

The only parameters in this equation that depend on ma are ρ, Λ and ma. Soto simplify equation (1) we write:

D[(∆v)2] = Aρmaln(Λ) , (11)

with

A =4√

2πG2fdmvdisp

, (12)

now we split the density of the dark matter ρ into:

ρ = nma , (13)

with n the number density of the MACHOs and ma the mass of such an object.Equation (9) than becomes:

D[(∆v)2] = Anm2aln(Λ). (14)

If we assume that the MACHOs span a range of masses, we need to integrateover all the possible masses:

D[(∆v)2] = Antot

∫ mmax

mmin

P (ma)m2aln(Λ)dma. (15)

Were P (ma) stands for the probability density function of the MACHO massesand ntot for the total number density of the MACHOs. For this project weassume that the MACHO masses are lognormaly distributed, in this case P (ma)can be given by:

P (ma) =1

σ√

2πma

e−(ln(ma)−µ)2

2σ2 . (16)

With σ being the standard deviation of the logarithmic mass and µ being themean of the logarithmic mass. If we put this into equation (13) and integrateform 0 to infinity we get:

11

D[(∆v)2] =Antot

σ√

2π

∫ ∞0

e−(ln(ma)−µ)2

2σ2 maln(Λ)dma. (17)

This ntot is not a known parameter so we substitute ntot with :

ntot =ρtot

< ma >. (18)

With ρtot being the total dark matter density and < ma > being the averageMACHO mass of the system. In appendix 1 I prove that we can write < ma >as:

< ma >= eµ+σ2

2 . (19)

Equation (15) than becomes :

D[(∆v)2] =Aρtot

σ√

2πeµ+σ2

2

∫ ∞0

e−(ln(ma)−µ)2

2σ2 maln(Λ)dma. (20)

If we now write out A and Λ we get:

D[(∆v)2] =4√πG2fdmρtot

σeµ+σ2

2 vdisp

∫ ∞0

e−(ln(ma)−µ)2

2σ2 ln

(rhvdisp

G(m+ma)

)madma , (21)

which describes the diffusion coefficient when a range of MACHO masses ispresent. The equation for the potential energy per unit mass is :

U

M= C − αGM∗

rh+ βGρr2h. (22)

So if we combine equation (19) with (20) and use the virial theorem (Etot =− 1

2U) we get the following formula for the change in half-light radius over time:

drhdt

=4√πGfdm

σeµ+σ2

2 vdispI

(αM∗r2hρ

+ 2βrh

)−1, (23)

with I1:

I =

∫ ∞0

e−(ln(ma)−µ)2

2σ2 ln

(rhvdisp

G(m+ma)

)madma. (24)

12

If we simulate this equation for multiple values of sigma we get the followingfigures:

Figure 9: The change in the half light radius due to dynamical heating when arange of dark matter MACHO masses is present. The average MACHO mass is30 M� and the total Stellar mass is 6000 M�.

It becomes visible that the change in half-light radius increases as the standarddeviation gets higher. If we use this in the same way as in the previous sectionto put constraints on the MACHO masses we get the following results:

Figure 10: mass constraints from galactical dynamics. I used that the averageMACHO mass was 30 M�, the velocity dispersion was 10 km/s, the dark matterdensity was 1 M�/pc3 and the the total system had a stellar mass of 2000 M�

The figure shows that the mass constraints on the MACHOs becomes strongerif a range of masses is present.

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3.2.3 Mass segregation

In the section about MACHOs, we assumed that most of the dark matter MA-CHO mass is in the form of primordial black holes, these PBHs have been aroundsince the formation of the galaxies and star clusters. If this is the case than theexchange of kinetic energy will have let to mass segregation: A process wherethe kinetic energies of two objects tend to equalize during an encounter. Thisequalization will cause the relatively heavy object(MACHOs) to lose velocityand the relatively light ones (The stars) to gain in velocity. Kepler’s second lawtells us that this will cause the MACHOs to move to the center of the galaxywhile the stars will move away from the center.

In (Koushiappas 2017) it is calculated that this will lead to a mean changeof kinetic energy of the stars: [5]

dEsdt

=

√96πG2msρBH ln Λ

[〈v2s〉+ 〈v2BH〉]32

[mBH〈v2BH〉 −ms〈v2s〉

]. (25)

With 〈vs〉 and 〈vBH〉 being the average velocities of the stars and Black holes.

If mBH〈v2BH〉 = ms〈v2s〉, the kinetic energies have equalized and the system hasreached an equilibrium.

For the simulations to these mass segregation we use the known data fromthe Segue 1 dwarf galaxy. Segue 1 has a half-light radius of 29 pc. Within itshalf-light radius, a total mass of 2.6 ∗ 105M� is located. To calculate how theradial shells of Segue 1 would have evolved, we use equation (7):

dr

dt=

4√

2πGfdmma

vdispln(Λ)

(αM∗ρr2

+ 2βr

)−1. (26)

If we simulate this (in python) we get that the radial shells have evolved asfollows:

Figure 11: Change of the radius of radial shells due to mass segregation. Leftshows the effects when 1 percent of the total dark matter is in the form of 30M� MACHOs and the right figure shows the effect when 10 percent of the darkmatter is in the form of 30 M� MACHOs

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The Figure shows that all the shells increase in radius with the change decreasingas the radius increases and that in the center of the galaxy a lack of stars becomesvisible. To calculate the stellar number density as a function of radius, weassume that the stars were initially distributed according to Plummers model:

ρ(r) =

(3M

4πrs

)(1 +

r2

r2s

)− 52

. (27)

Were M stands for the total stellar mass in the star cluster and rs for the scaleradius, which depends on the size of the core. For Segue 1, this scale radius is16 parsec [5]. If we use this distribution and use python to simulate how thisdistribution changes when mass segregation takes place, we get the followingfigure for the stellar distribution:

Figure 12: Number density of stars as a function of radius. The left figure showsthe distribution when 10 percent of the dark matter is in 30 M� MACHOs andthe right figure shows the distribution when 1 percent of the dark matter is in30 M� MACHOs. The blue line shows the initial distribution.

From earth it isn’t possible to measure this distribution of stars, it is howeverpossible the measure the projected stellar surface density from the measuredlight coming from Segue 1. So if we apply a line of sight integral over the datafrom figure 12 and do this for multiple fractions of MACHO dark matter we getthe following figure:

15

Figure 13: figure from (Koushiappas 2017) showing the projected stellar surfacedensities. The black line represents the distribution where there is no darkmatter in the form of MACHOs and the vertical black lines represent the datameasured from observation [30].

If we make a 3d plot of these line of sight distribution we will get the followingfigure:

Figure 14: These figures from (Koushiappas 2017) show the simulated effects ofMACHOs dark matter in a galaxy: The left figure shows the stellar distributionwhen there is no dark matter in the form of MACHOs, the one in the middleshows the effects when 1 percent of the dark matter is in 10M� MACHOs andthe right figure shows the effects when 10 percent of the total dark matter isin the form of 30 M� MACHOs. It becomes visible that, when a fraction ofthe dark matter was in the form of MACHOs, mass segregation resulted in adepletion of stars in the centre.

To put constraints on the MACHO masses and the fraction that it can form ofthe total dark matter (fdm), we compare the simulated mass segregation effectswith the observed data (shown in figure 13). We do this by applying the χ2 test:

χ2 =∑ (o− e)2

o, (28)

16

where o stands for the observed values and e for the expected values. For eachvalue of fdm and mBH we calculate how the line of sight integral looks andcompare the 3 measured data points from figure 13 (o in equation 28) with thesimulated points at the same radii (e in equation 28). To get the probabilitythat this null hypothesis (a specific fdm and mBH) is true, we use the followinggraph:

Figure 15: Line which shows the relation between χ2 value and the p-value:The probability of obtaining the data, or if the null hypothesis is true

If we use this graph, equation 28 and the observed and simulated data we getthe following constraints on MACHO dark matter:

Figure 16: figure from (Koushiappas 2017) showing the constraints on MACHOdark matter masses from measuring the star distribution. The black lines rep-resent the p-values written above them. These measurements have excludedMACHO masses above 1 solar mass as the only component of dark matter.

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4 Conclusion and outlook

During this project I tried to find mass constraints on dark matter MACHOs. Bylooking at the microlensing effects and at the dynamical impact on the evolutionof a galaxy, we got the figures (6),(8),(10) and (16) which give constraints onthe possible MACHO masses.These figure show that all MACHOs with masses from 0.15 M� up to very highmass are excluded to form all of the dark matter in our universe and MACHOswith a few tens of solar masses are even excluded to form more than 4 percent ofthe total dark matter. Figure (10) shows that the constraints strengthen whenwe assume that the MACHOs have a whole range of masses instead of one mass.

There is certainly a lot of room for improvements on these constraints. For thesimulations, a lot of assumptions had to be made, if more precise data wereto be available, the constraints could strengthen. Another way to strengthenthe constraints is by looking at other effects that the MACHOs would have onthe universe. I for example excluded the researches on the effects of MACHOson the cosmic microwave background, which sets constraints on the low end ofthe mass spectrum [31]. So maybe in the future, when more data is available,MACHOs can be completely excluded as the main components of dark matter.

5 Acknowledgement

I would like to thank my supervisor Christoph Weniger for guiding me throughthis project. Despite his busy schedule, he made time to answer all of myquestions. This project gave me an insight in the work he is doing and in thechallenges he is facing.

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[22] Tomohiro Harada, Chul-Moon Yoo, and Kazunori Kohri. “Threshold ofprimordial black hole formation”. In: Physical Review D 88.8 (2013),p. 084051.

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[23] Stephen Hawking. “Gravitationally collapsed objects of very low mass”.In: Monthly Notices of the Royal Astronomical Society 152.1 (1971), pp. 75–78.

[24] Max Born. Einstein’s theory of relativity. Courier Corporation, 1965.

[25] Encyclopædia Britannica Online. C. Possibilities for Dark Matter. url:https://imagine.gsfc.nasa.gov/educators/galaxies/imagine/

dark_matter.html.

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[29] D Crnojevic et al. “DEEP IMAGING OF ERIDANUS II AND ITS LONESTAR CLUSTERThis paper includes data gathered with the 6.5 m Mag-ellan Telescopes at Las Campanas Observatory, Chile.” In: The Astro-physical Journal Letters 824.1 (2016), p. L14.

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7 Appendix

7.1 The mean value of The log-normal distribution.

We want to calculate the mean value of a random log-normally distributed func-tion: Y = exp(x) where x in a normally distributed variable.

For a normally distributed variable the probability density function is :

PDF (x) =1

σ√

2πe

−(x−µ)2

2σ2 . (29)

So to compute the mean value of Y we need to compute the following integral:

< ex >=

∫ ∞−∞

ex1

σ√

2πe

−(x−µ)2

2σ2 dx. (30)

In order to do this we need to carry out variable substitution, we take y = x−µwith dy = dx. The integral now still has the same limits so it becomes:

< ex >=

∫ ∞−∞

ey+µ1

σ√

2πe

−(y)2

2σ2 dy , (31)

µ doesn’t depend on y, so it can be taken outside of the integral:

< ex >= eµ∫ ∞−∞

1

σ√

2πe−

y2+y2σ2

2σ2 dy. (32)

Now we can rewrite the exponent to get:

< ex >= eµ∫ ∞−∞

1

σ√

2πeσ2

2 e−(y−σ2)2

2σ2 dy , (33)

< ex >= eµ+σ2

2

∫ ∞−∞

1

σ√

2πe−

(y−σ2)2

2σ2 dy. (34)

The σ in this function doesn’t depend on y and it can have any value so if wereplace σ2 with µ we end up with the probability density function inside theintegral:

< ex >= eµ+σ2

2

∫ ∞−∞

PDF (y)dy. (35)

And since this PDF is normalized, the integral from −∞ to ∞ will be 1 and weend up with:

< ex >= eµ+σ2

2 . (36)

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7.2 Populair dutch summary

Ongeveer 75 jaar geleden begon de Zwitserse astronoom Fritz Zwicky zijn on-derzoek naar de spreiding van de snelheden van sterrenstelsels in een cluster vansterrenstelsels. Hij vond dat deze snelheden veel hoger waren als verwacht. Zewaren zelfs zo hoog dat als de massa van de cluster alleen uit sterren bestond datde zwaartekracht nooit sterk genoeg zou kunnen zijn om de sterrenstelsels bijelkaar te houden. De verklaring hiervoor vond hij in donkere materie: materiedie je niet ziet maar die wel zwaartekracht uitoefent op zijn omgeving. Sinds-dien is hier heel veel onderzoek naar gedaan en zijn de meeste onderzoekers heter wel over eens dat er iets is met de eigenschappen van donkere materie. Eenvraag die echter nog onbeantwoord is, is de vraag waar deze donkere materie uitbestaat. De verklaringen hiervoor lopen uiteen van superlichte neutrino deelt-jes tot superzware zwarte gaten. Een mogelijke verklaring voor de ontbrekendedonkere materie bestandsdelen zijn de MACHOs : Massive Compact Halo Ob-jects. MACHOs zijn een verzameling van astronomische objecten die geen lichtuitzenden, je kunt hierbij denken aan witte- en bruine dwergen, neutronenster-ren en zwarte gaten. Tijdens dit project nam ik aan dat donkere materie uitMACHOs bestaat en heb ik geprobeerd hier massa restricties voor te vinden.Een van de manieren waarop dat gedaan is, is door naar de microlensing ef-fecten van MACHOs te kijken. Sinds Einsteins relativiteits theorie weten wenamelijk dat ook massaloze lichtdeeltjes beınvloed worden door zwaartekracht.Dit zorgt ervoor dat, wanneer een MACHO voor een ster langs beweegt, hetlicht van de ster naar de waarnemers op aarde wordt toegebogen. Dit kunnenwe op aarde meten als een toenamen van de lichtkracht van de ster. Er zijnechter nul van deze microlensing effecten gemeten, met behulp van een statis-tische test kunnen hiermee MACHOs tussen 0.15 en 30 zonsmassa uitgeslotenworden als het enige bestandsdeel van donkere materie. Een andere manier ommassa restricties te vinden is door naar de impact van MACHOs op de evolutievan sterrenstelsels te kijken. Door voor meerdere MACHO massa’s te simulerenwat de impact zou zijn en dit te vergelijken met de waargenomen data van ster-renstelsels, kunnen MACHOs met massa’s tussen 1 en honderden zonmassa’suitgesloten worden als de enige bestandsdeel van donkere materie. MACHOsvan enkele tientallen zonsmassa’s kunnen zelfs uitgesloten worden om meer dan4 % van de totale donkere materie in ons heelal te maken. Met meer onderzoekzouden deze restricties in de toekomst nog versterkt kunnen worden waardoorMACHOs in de toekomst misschien helemaal uitgesloten kunnen worden als hetenige bestandsdeel van donkere materie.

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