Cluster Dark Matter: Comparing Two Methods of ﬁtting Gravitational Lensing Data with an NFW Proﬁle

UNIVERSITYOF AMSTERDAM

Cluster dark matter: Comparing two methods of ﬁtting gravitational lensing data with an NFW proﬁle

Author: Supervisor: Cornelis J. JONGENS Dr. Theodorus M. NIEUWENHUIZEN Second supervisor: Dr. Christoph WENIGER

July 30, 2018 i

University of Amsterdam Abstract

Cluster dark matter: Comparing two methods of ﬁtting gravitational lensing data with an NFW proﬁle

by Cornelis J. JONGENS

One of the big unanswered questions in modern astrophysics is about the nature of dark matter. To describe dark matter in a cluster of galaxies, an NFW mass profile is often used. This profile is used to fit observational data and constrain the model for the cluster. These constraints can tell us something about the mass in the cluster and therefrom about the nature of dark matter. In this thesis we fit the NFW profile to observational lensing data of different clusters in two different ways and replicate the results of Nieuwenhuizen. The fit parameters were A, the characteristic mass density for the cluster and R, the characteristic radius for the cluster. Through A we can also find the concentration parameter c.The first method is using an ad- hoc constant and the second method is using an intra-bin fluctuation regularization to construct the best fit. The main difference lies in the way of regularizing small eigenvalues in the covariance matrix, the method using binning puts forward a way to regularize the eigenvalues with a diagonal term that stems from the data itself. In contrast to the method using an ad-hoc constant, which is not as well founded from the data. The first method performed on cluster A1689 had a χ2/ν of 1.8, an R of 400.0 kpc and a c of 7.38. The intra-bin fluctuation regularization method had 2 15 a χ /ν 13.3, an R of 485.9 ± 1.3 kpc, a c of 5.24 ± 3.8 and an M200 of 2.37 × 10 M . We also reproduced the results for cluster A1835: χ2/ν = 5.5, R = 159.0 ± 1.5 kpc, 14 c = 9.54 ± 0.067 and M200 = 6.32 × 10 M . The intra-bin fluctuation regularization method was also applied to cluster A1703 and that gave the following results: 2 14 χ /ν = 26.2, R = 245.4 ± 2.2 kpc, c = 6.60 ± 0.24 and M200 = 7.49 × 10 M . ii

Contents

Abstracti

1 Introduction1

2 Theory2 2.1 Dark Matter...... 2 2.1.1 Existence...... 2 2.1.2 Candidates...... 2 2.1.3 Gravitational lensing...... 3 2.2 Navarro, Frenk White mass distribution proﬁle...... 4 2.3 Lensing data of cluster Abell 1689...... 5

3 Results8 3.1 Fitting the data to the NFW proﬁle...... 8 3.1.1 Fitting the data of Abell 1689 with different regularization constants...... 8 3.1.2 Fitting the data of Abell 1835 using intra-bin ﬂuctuation regularization...... 9 3.2 Fitting the binned data for multiple clusters...... 12 3.3 M200 and r200 of the clusters...... 13

4 Conclusions and outlook 16

A Appendix 17 A.1 Dutch Summary...... 17

Bibliography 18 1

Chapter 1

Introduction

What is dark matter (DM)? This is a question that many astrophysicists are attempt- ing to answer. There is profuse evidence for the existence of DM and just as much discussion about the nature of this substance. The existing evidence is of purely gravitational origin [9] and exists on many scales. So it exists, but what is it? There are numerous DM candidates, ranging in mass from axions with m = 10−5eV = −72 4 9 ∗ 10 M , to black holes of mass m = 10 M [10]. And there is just as much research trying to find out which one of these candidates might be DM [3]. To describe DM there are a number of different mass density profiles which work quite well, but this report is just concentrating on the most popular one: the Navarro, Frenk and White (NFW) profile [12]. The aim of this report is to compare two methods of regularizing lensing datasets to fit the NFW profile to the gravitational lensing data. In doing so the results of Nieuwenhuizen [16] are taken as a guide line. 2

Chapter 2

Theory

2.1 Dark Matter

2.1.1 Existence The evidence for the existence of DM is based on gravitational effects and varies widely in scale. The ﬁrst questions surrounding DM arose when Oort studied stars near the sun and researched their velocities. He came to the conclusion that the stars he analyzed couldn’t amount to the gravitating matter implied by their velocities [17]. Shortly thereafter it was Frits Zwicky who found that galaxy clusters needed DM to account for the velocity dispersions [20]. Then in 1980 Rubin, Ford and Thonnart studied rotation curves of spiral galaxies and found out that the mass wasn’t centrally condensed, but that there is also substantial mass at a large radius. They concluded: "It is inescapable that nonluminous matter exists beyond the opti- cal galaxy." [19]. The so-called Oort discrepancy and the observations of the rotation curves of spiral galaxies are on a galactic scale and the evidence Zwicky found is on a scale of galaxy clusters. So the question whether DM exists or not isn’t relevant anymore, the focus now lies in ﬁnding out more about the nature of DM. There is no shortage of theories about this important question, each with their own candidates for DM [3]. It could even be that the DM on different scales consists of different matter.

2.1.2 Candidates There are many different candidates ranging from tiny axions to massive black holes. To create order in the chaos of candidates it is helpful to categorize them. There are categorization schemes, figure 2.1 is a basic scheme based on an article by Jungman [10]. The first distinction is between baryonic and nonbaryonic matter. The focus of this project is on describing DM with the CDM assumption, so for the baryonic DM and hot DM only the main candidates will be stated. Because although the evidence for non-baryonic DM is compelling, the baryonic DM can’t be ruled out altogether [3, 10]. The main CDM candidates on the other hand will be highlighted more thor- oughly. The main candidates for baryonic DM are massive compact halo objects (MACHOs), these include, among others, brown dwarfs, white dwarfs, and neutron stars. Then among the nonbaryonic candidates there is the distinction between cold dark matter (CDM), warm dark matter and hot dark matter. This is based on the speed at which the candidate moved when galaxies could just start to form. If it was moving at relativistic speeds it is classified as hot and if not it’s classified as cold [10]. Warm DM is cooled down hot DM [5]. The main candidate for hot DM is the light neutrino. An interesting candidate for warm DM is the sterile neutrino of keV mass, in particular the case of 7 keV, for which there is support from the detection Chapter 2. Theory 3 of a 3.5 keV gamma ray line [6,7]. Within the CDM model the main candidates are axions and weakly interacting massive particles (WIMPs). WIMPs are the largest class of CDM candidates, their masses are in the range from 10 GeV to a few TeV. These wimps are stable particles and an extension to the standard model. If such a stable particle would exist there would be a relevant abundance of it and the es- timated annihilation cross section is surprisingly close to the value needed for it to account for the DM. Theoretically a good candidate, but it hasn’t been found yet. Another theory which hasn’t been proven yet is the theoretical framework for supersymmetry, which produces multiple DM candidates like neutralinos, sneutrinos, gravitinos, and axinos. This supersymmetry is the symmetry between fermions and bosons, this would mean symmetry between matter and interactions [3]. Although it’s an appealing theory to many, supersymmetry hasn’t been found yet [8]. Then there is the axion, this particle was proposed as an attempt to solve the problem of CP violation in quantum chromodynamics. Research has constrained the axions to be very light (0.01 eV)[3], there are estimates that place the mass in an ever lower range (0.11 meV)[2]. The problem with the axion is again the fact that it hasn’t been found. So it seems there are many eligible candidates, but what the DM consists of remains a mystery as of yet. There are techniques that could tell us more about the DM from observing the galaxy, one of these techniques is gravitational lensing. In the next section it will be introduced.

FIGURE 2.1: Categorization scheme DM.

2.1.3 Gravitational lensing Gravitational lensing is the phenomenon where massive bodies act like a lens for light rays; the mass deﬂects the light along the line of sight from the source to the observer and can distort, enlarge or multiply the image. A cluster of galaxies can act like a gravitational lens, in Figure 2.2 the cluster Abell 1689 with arcs caused by this lensing effect can be seen. The light from a galaxy behind a cluster of galaxies can be seen as an Einstein ring, if the source, the massive body and the observer are aligned. Otherwise it can be seen as an arc. Because the bending of the light depends Chapter 2. Theory 4

FIGURE 2.2: Gravitational lensing by galaxy cluster Abell 1689 pho- tographed by the Hubble Space Telescope’s Advanced Camera for Surveys. The gravitational arcs are the distorted and enlarged images of galaxies behind the cluster. only on the mass distribution of the structures that form the lens, the studying of this lens offers possibilities to find out more about the mass profiles of these structures [11, 18]. The lensing data gives insights into the composition of a galaxy cluster and with that could give insights into the nature of DM. Lensing phenomena can be split up into 2 categories, strong and weak lensing. It’s called strong when there are easily discernible arcs or a complete Einstein ring and weak when the effects are much smaller. Weak lensing by galaxy clusters gives rise to weakly distorted images of faint background galaxies (arclets), these are used to reconstruct the cluster mass distribution. The idea is that the image distortions due to the tidal field, which is called gravitational shear, and the surface mass density are linear combinations of second-order derivatives of the lensing potential, so they are related through the potential [1]. Weak lensing analysis roughly consists of four parts [18]:

1. shear estimation from the measurement of the background galaxy ellipticities

2. inversion methods to derive a dark matter mass map from a shear map

3. ﬁltering of the dark matter mass map to reduce the noise level

4. statistical analysis of the weak lensing data to constrain the cosmological model.

In this project statistical analysis of the Navarro-Frenk-White model for mass distribution plays a central role, so it would fall under the fourth category. In the next section we will take a closer look at the NFW proﬁle.

2.2 Navarro, Frenk White mass distribution proﬁle

To work with lensing data of galaxy clusters the CDM assumption is often used because it works so well [15]. Also, a CDM universe predicts the range of masses of galaxies and these predictions match the observed data [4]. So from a practical Chapter 2. Theory 5 point of view these are good reasons for working with the CDM assumption. For describing the CDM density a mass profile proposed by Navarro, Frenk and White is used. In 1997 they used N-body simulations to study the density profiles of DM halos in hierarchically clustering galaxies. By doing so they showed that the NFW profile (2.1), described the CDM density quite well [12].

AR3 ρ (r) = (2.1) NFW r(r + R)2

r200 Where the R = c is a characteristic radius for the cluster, it is the radius where the density dependency changes from 1/r to 1/r3. A is the characteristic mass den- 3H2 sity for the cluster described in equation 2.2, ρc = 8πG in equation 2.2 is the critical density with H the value of Hubble’s constant. We are using: H = 70 km/s/Mpc. And ﬁnally c is the concentration parameter for the cluster.

3 3 200c ρc(1 + z) A = c (2.2) 3[log(1 + c) − (1+c) ] In the next section we will take a closer look a the lensing data

2.3 Lensing data of cluster Abell 1689

The galaxy cluster Abell 1689 (A1689) is one of the most extensive studied galaxy clusters with large lensing arcs. It is a relaxed cluster, which means it seemingly isn’t a colliding cluster and it is spherically symmetric [14]. On top of that there is also good data for strong lensing and weak lensing available [15]. All these properties make A1689 a good cluster to test the NFW mass proﬁle (2.1) on. Strong lensing analysis of the data from Limousin by Nieuwenhuizen yielded data for the mass density along the line-of-sight,

∞ p Σ(r) = dzρ( r2 + z2). (2.3) ˆ−∞ From this the 2d mass, the mass in a cylinder of radius r around the cluster centre, r was inferred; M2d(r) = 2π 0 du uΣ(u). The average of this 2d mass over the disk is 2 the Σ(r) = M2d(r)/πr , so´ we need to work out a formula for the 2d mass and then for the Σ(r). The integral we need consists of two parts. The part where we integrate over the cluster centre, shown in 2.4:

1 2 Σcenter(r) = 2 u sin(θ)ρ(u)dudθdφ. (2.4) πr ˚center And the part where we integrate over the two halves of the cylinder, shown in 2.5:

2 2 Σcylinder(r) = 2 u sin(θ)ρ(u)dudθdφ. (2.5) πr ˚cylinder Now we need to deﬁne the boundaries and perform the integrals, in 2.4 we integrate over a sphere with radius r. This integral is pretty straightforward and gives us the ﬁrst part of the total Σ(r):

r 4 2 Σcenter(r) = 2 du u ρ(u). (2.6) r ˆ0 Chapter 2. Theory 6

FIGURE 2.3: 2D Drawing of cluster centre and cylinder of radius r around it

The boundaries for the cylinder part of the integral follow from ﬁgure 2.3, we integrate u from r to inﬁnity, but since we have to stay inside the cylinder around the cluster center, the θ runs from 0 to arcsin(r/u). Where the upper boundary follows from: sin(θ) = r/u. This gives us the second part of the total Σ(r):

∞ 4 2 p 2 Σcylinder(r) = 2 u 1 − 1 − (r/u) ρ(u)du. (2.7) r ˆr We can now rewrite 2.7 and get the following integral:

∞ 4 p 2 2 Σcylinder(r) = 2 u u − u − r ρ(u)du. (2.8) r ˆr √ If we now multiply and divide 2.8 by u + u2 − r2 the second part of the total Σ(r) becomes: ∞ 4uρ(u) Σ (r) = du √ . (2.9) cylinder 2 2 ˆr u + u − r

Which brings us to the following formula Σcenter(r) + Σcylinder(r) for the average of the 2d mass over the disk: 4 r ∞ 4uρ(u) Σ(r) = du u2ρ(u) + du √ . (2.10) 2 2 2 r ˆ0 ˆr u + u − r Now the next problem is that we have an underdetermined problem, so an en- semble of 1001 compatible 2d mass maps was generated from observed strong lensing arclets. From these mass maps the M2d values and Σ values of the dataset were produced [16]. The Nieuwenhuizen dataset [16] consists of a set of radii rn with corresponding Σn and their covariance matrix Γmn. The set of radii rn with n = 1,..., 149 ranges from r1 = 3.15 kpc to r149 = 876 kpc. In the next section this dataset is fitted to the NFW profile, but before we begin the fitting, let’s take a closer look at the data. In the center of the cluster there is only data from a few arclets, so the radii with corresponding M2d(rn) data and covariance matrix datapoints that contain no new information on Σn have to be thrown away. First the M2d(rn) val- 2 ues are calculated from: Σn = M2d(rn)/πrn. Then the duplicate M2d datapoints are thrown away with the DeleteDuplicatesBy function of Mathematica and now the Chapter 2. Theory 7

M2d(rn) dataset has 117 datapoints. The duplicates in the M2d dataset correspond to datapoints in: the set of radii, the set of Σn and the set of Γmn. So all the corresponding datapoints in those sets are also thrown away. The new dataset has N = 117 datapoints and the index n = 1,...,N is relabelled. In the next chapter we will ﬁt the NFW proﬁle to the data and see how well it describes our dataset. 8

Chapter 3

Results

3.1 Fitting the data to the NFW proﬁle

To see if the NFW mass distribution is suited to the dataset, the theoretical data is fitted to the observational data using a χ2 test. This is a test to see how well the theoretical predictions describe the observational data. So the theoretical Σ(r) with ρ(u) = ρNFW from 2.10 is fitted to the observational Σn from the dataset. The reduced χ2 is a summary for the model fit and reads: χ2/ν with ν the number of datapoints minus the parameters. If the reduced χ2 is 1 the model is a perfect fit. Before the fitting can take place the measure for the goodness of fit, the χ2 function, needs to be defined. For a set of data di(i = 1,...,N) modeled by a theoretical fit ti = ti(q1, . . . , qp) involving p fixed parameters q1, q2, . . . , qp with deviations ∆i = 2 di − ti, the χ -test reads as follows:

N 2 X −1 χ = ∆iΓij ∆j (3.1) i,j=1 −1 With Γij as the inverse covariance matrix. We are fitting equation 2.10 where equation 2.1 has 2 fit parameters; A and R, so the theoretical fit used is: Σi = 2 Σi(A, R). Because the dataset runs from i = 1 to i = 117 the χ reads:

117 2 X h i −1h i χ = Σi − Σi(r) Γij Σj − Σj(r) (3.2) i,j=1 2 Where Σi is the data and Σi(r) are the theoretical values of Σ(r). Since χ de- −1 pends on Γij it is important to take a look at the eigenvalues of the covariance matrix, this is because they are a measure for the spread in the data. The covariance matrix of the data has very small eigenvalues ranging from roughly 0.7 to 6 × 10−19gr2/cm4. These tiny values aren’t physical and need to be regularized. In the next two sections, two different methods for ﬁtting the data and regularizing the small eigenvalues will be shown.

3.1.1 Fitting the data of Abell 1689 with different regularization constants The ﬁrst method for ﬁtting the data and regularizing the small eigenvalues is by implementing a regularization constant so that the new covariance matrix will be:

2 Cij = Γij + σ I. (3.3) Chapter 3. Results 9

2 TABLE 3.1: Best ﬁt parameters, with reduced χ for A1689 with a regularization constant σ = 0.0450 gr/cm2

2 3 χ /ν A(mN/cm ) R(kpc) c 1.8 0.118 400.0 7.38

Where σ is an ad-hoc constant, for the value of σ the typical value from the diago- P117 1/2 2 nal elements of the covariance matrix is taken σ = [ i=1 Cii/117] = 0.0450gr/cm [13]. And now the χ2 in terms of C is:

117 2 X h i −1h i χ = Σi − Σi(r) Cij Σj − Σj(r) . (3.4) i,j=1 With the χ2 defined, the actual fitting can be done. The fitting is done using Wolfram Mathematica 11.2.0.0, all the data is imported and the functions 2.1, 2.10 and 3.4 are defined. Now the best fit parameters are found by applying the built- in procedure FindMinimum to χ2. This procedure searches for a local minimum of a function, starting at a certain point. To get an indication in which range the parameters would be and to find out what a good starting point for the search might be, a graphical fit to the data is done by plotting the Σ(r) with different values for A and R and looking for which values it fits graphically. The graphical fit is shown 3 in figure 3.1. The fit parameters used for this graphical fit were: A = 0.078 mN/cm 3 and R = 550 kpc, so as a starting place for the search A = 0.058 mN/cm and R = 400 kpc are taken. Different starting points are tried, to make sure the evaluated χ2 is the actual minimum for the function and not just a local minimum. Also to get a better feeling for how the constant effects the fitting, different values for σ around the typical value from the diagonal elements are tried, ranging from 2 × 10−3 to 0.1 gr/cm2. A graph with all these values of σ and the corresponding χ2/ν with ν = 115, is shown in figure 3.2. χ2/ν < 1 is unphysical, because χ2/ν = 1 is the minimum value for Gaussian errors. This restraint determines the a maximum value for σ. As can be seen the χ2/ν strongly depends on σ. When the σ is chosen too small, the unphysical eigenvalues start playing a role again and when the σ is chosen too large the constant itself is fitted. So we must be careful with our choice of σ. For a given σ the best fit parameters found by Mathematica are shown together with the concentration parameter obtained from equation 2.2 in table 3.1 and the theoretical fit of the NFW profile to the data can be seen in figure 3.3. As can be seen the NFW profile describes the data fairly well, except for the tail of the function also from the reduced χ2 in table 3.1 we can tell it fits reasonably well. In [13] Nieuwenhuizen found: χ2/ν = 2.18 this was after using a binning procedure with σ = 0.0362gr/cm2. This difference is most likely due to the more precise observational data available now.

3.1.2 Fitting the data of Abell 1835 using intra-bin fluctuation regularization The second method of fitting the data and regularizing the small eigenvalues will be by using an intra-bin fluctuation regularization procedure of Nieuwenhuizen [16]. Where the first method was used on the A1689 cluster, the second method will first be applied to cluster Abell 1835 (A1835) and later on A1689 and Abell 1703 (A1703). Chapter 3. Results 10

FIGURE 3.1: The observational data in green and the graphical ﬁt for cluster A1689 of Σ in gr/cm2 as a function of the radius in kpc in orange.

2 FIGURE 3.2: χ /ν with ν = 115, for cluster A1689 as a function of the regularization constant σ, the red line is the line where χ2/ν = 1. Chapter 3. Results 11

FIGURE 3.3: The observational data in green and the best theoretical ﬁt using a regularization constant σ = 0.0450 gr/cm2 for cluster A1689 of Σ in gr/cm2 as a function of the radius in kpc in blue.

By using this intra-bin fluctuation regularization method on A1835, the results of Nieuwenhuizen [16] are being tested. The first step in the intra-bin fluctuation regularization procedure is binning the observational data. The datapoints are grouped into Nbin = 17 bins with each bin containing ni = 7 points, except for 1 bin with ni = 5 points. The smallest errors occur in bin 9, so to minimize bias bin n8 = 5 and n10 = 5 are tried. Now the index is relabeled n → {ik}, where i = 1,...,Nbin is the bin number and k = 1, . . . , ni is the position within the bin. So the set of radii rn, the set of Σn and the set of Γmn now read: rik, Σik and Γik;jl. The next step after the binning of the data is to define the bin centre ri, it is defined as the geometrical average. The geometrical average normalizes the radii being averaged, so no radius dominates the weighting

n ! 1 Xi ri = exp log rik . (3.5) ni k=1 Next, the theoretical value Σ(r) in the binning is divided out. So that the data is binned in the following way:

ni bin 1 X Σik Σi = Σ(ri) . (3.6) ni Σ(r ) k=1 ik The binned correlations are now

ni nj bin Σ(ri)Σ(rj) X X Γik;jl Γij = . (3.7) ninj Σ(r )Σ(r ) k=1 l=1 ik jl bin −14 2 4 The eigenvalues of Γij are between 0.07 and 7 × 10 gr /cm , so there is more Chapter 3. Results 12 regularization needed. As Nieuwenhuizen stated [16]: "the deﬁnition (3.6) puts forward a measure for the intra-bin ﬂuctuations",

2 ni bin ! bin ! Σ (ri) X Σik Σi Σil Σi γi = − − . (3.8) n2 Σ(r ) Σ(r ) Σ(r ) Σ(r ) i k,l=1 ik i il i This is a square; without absolute values it would vanish due to the deﬁnition of bin Σi . The γi is now used as a diagonal regulator to deﬁne the new binned covariance matrix C. This means that the regularization now has its roots in the data instead of being an ad-hoc constant. And the new binned covariance matrix C reads

bin Cij = Γij + δijγi. (3.9) −7 2 4 The smallest eigenvalues of Cij are now of the order 10 gr /cm , so further regularization with an ad-hoc constant is not needed. The last step is deﬁning the new χ2 for the goodness of ﬁt,

Nbin " # " # 2 X bin −1 bin χ (Σ) = Σi − Σ(ri) Cij Σj − Σ(rj) . (3.10) i,j=1 This new χ2 takes the binning with the fit function Σ(r) into account. All the functions (3.5, 3.6, 3.7, 3.8, 3.9 and 3.10) are defined in Mathematica so we can again use the FindMinimum routine to find a measure for the best fit. The most difficult thing in programming all these functions in Mathematica proved to be making sure that all the functions perform the way they should. To make sure that the functions iterate over the right values of r, Σ(r) and Γ and still have a fast program was cum- bersome. It took quite some time to get al the functions working and defined the right way, all the testing of the final χ2 also took up a lot of computation time. Now before we use the FindMinimum routine to find the best fit, we also need to estimate the errors in the fit parameters A and R. It is assumed the data involves Gaussian errors and the main errors of χ2 are defined as follows,

δχ2(Σ) = (δ∆ − ∆C−1δC)C−1(δ∆ − δCC−1∆). (3.11) bin Where ∆i = Σi − Σ(ri), the errors are δ and the errors in ∆i are, ! ! ∂∆i ∂∆i δ∆ = δA + δR. (3.12) i ∂A ∂R

2 The errors in Cij are defined in the same way. The covariances are now δχ (Σ) ≡ P −1 A,R(X )ARδAδR with hδAδRi = XAR and the errors in the parameters are δA = 1/2 1/2 (XAA) and δR = (XRR) . Now we can fit the binned data to the NFW profile (2.1, 2.2) and find the best fit. For binning with bin n8 = 5 and n10 = 5 the results are displayed in table 3.2. So the best fit is with bin n8 = 5, also the results from Nieuwenhuizen are reproduced [16].

3.2 Fitting the binned data for multiple clusters

In the previous sections the data of A1689 en A1835 was ﬁtted in two different ways, now the intra-bin ﬂuctuation regularization method is applied to both A1689 and the cluster A1703. For these clusters the smallest errors occur in different bins. So Chapter 3. Results 13

TABLE 3.2: Best ﬁt parameters for A1835 using the intra-bin ﬂuctua- tion regularization with different bin sizes and corresponding χ2/ν

2 3 χ /ν A(mN/cm ) R(kpc) c

n8 = 5 5.5 0.43 ± 0.1 159.0 ± 1.5 9.54 ± 0.067 n10 = 5 5.6 0.43 ± 0.1 160.7 ± 1.4 9.48 ± 0.068

TABLE 3.3: Best ﬁt parameters for A1689 using the intra-bin ﬂuctua- tion regularization with different bin sizes and corresponding χ2/ν

2 3 χ /ν A(mN/cm ) R(kpc) c

n8 = 5 13.3 0.088 ± 1.7 485.4 ± 1.3 5.24 ± 3.8 n10 = 5 13.7 0.089 ± 1.7 488.9 ± 1.4 5.26 ± 3.8

TABLE 3.4: Best ﬁt parameters for A1703 using the intra-bin ﬂuctua- tion regularization with different bin sizes and corresponding χ2/ν

2 3 χ /ν A(mN/cm ) R(kpc) c

n8 = 5 27.4 0.18 ± 0.71 242.1 ± 2.3 6.60 ± 0.24 n16 = 5 26.2 0.18 ± 0.71 245.4 ± 2.2 6.53 ± 0.19

TABLE 3.5: Calculated r200 in kpc and M200 in M for the clusters A1689, A1703 and A1835 using the intra-ﬂuctuation regularization method

Cluster r200(kpc) M200(M ) A1689 1784 2.37 × 1015 A1703 1172 7.49 × 1014 A1835 1062 6.32 × 1014

for A1689 binning with bin n8 = 5 and n10 = 5 is tried. For A1703 binning with bin n8 = 5 and n16 = 5 is tried. The results for A1689 are found in table 3.3 and for A1703 in table3.4. The errors in the A parameter are large, so it will not give a very accurate description for the cluster. The best fits are shown in figures 3.4, 3.5 and 3.6. It can already be seen that the fit for cluster A1703 is not very good, because it has more structure in its mass distribution.

3.3 M200 and r200 of the clusters

Now that the parameters for all the clusters have been found, we can calculate the M200 and the r200 of the different clusters. The r200 is the radius of a cluster where the density is 200 times the critical density and it is an indication for the size of the cluster. To ﬁnd this radius we need to solve the following equation for r: Chapter 3. Results 14

AR3 = 200ρ (3.13) r(r + R)2 c Equation 3.13 is deﬁned in Mathematica where the parameters used for each cluster are the parameters found with the binning method from tables 3.2, 3.3 and 3.4. The function NSolve is used to solve the equation for r and the results are found in table 3.5. Now that we have an indication of the size of the clusters we can calculate the M200, the mass of the clusters within the r200 radius. The equation for the integrated mass within some radius Rmax reads

Rmax R + Rmax Rmax M = 4πr2ρ(r) dr = 4πAR3 ln − . (3.14) ˆ0 R R + Rmax

We calculate this mass relative to the mass of the sun, the results again are found in table 3.5.

FIGURE 3.4: The observational data in green and the best theoretical ﬁt using the intra-bin ﬂuctuation regularization for cluster A1689 of Σ in gr/cm2 as a function of the radius in kpc in blue. Chapter 3. Results 15

FIGURE 3.5: The observational data in green and the best theoretical ﬁt using the intra-bin ﬂuctuation regularization for cluster A1703 of Σ in gr/cm2 as a function of the radius in kpc in blue.

FIGURE 3.6: The observational data in green and the best theoretical ﬁt using the intra-bin ﬂuctuation regularization for cluster A1835 of Σ in gr/cm2 as a function of the radius in kpc in blue. 16

Chapter 4

Conclusions and outlook

Since equilibrated clusters have achieved some type of equilibrium, for them the history is irrelevant, so that they provide a clean, simple test for DM. For the CMB and baryon acoustic oscillations (BAO), on the other hand, the whole history of the universe is relevant. Hence for them one must make more assumptions than for equilibrated clusters. Thus these equilibrated clusters offer a great way to investigate their dark matter. The purpose of this project is to test the results of Nieuwenhuizen [16] and to compare two methods of employing a regularization procedure to fit the NFW profile to lensing data. The results of Nieuwenhuizen for the cluster A1835 were reproduced. First it didn’t seem like all the results were reproduced, so the whole program in Mathematica was checked for errors multiple times. In the end, his program wasn’t at fault, but a pre-factor for the A parameter, the amplitude of the NFW profile related to its c-parameter, was missing. With this pre-factor, the results were reproduced. As for the comparison of the methods, the method with the constant regulator seemed to be working a lot better for cluster A1689 than the intra- bin fluctuation regularization method. This can be seen if we compare the χ2/ν from table 3.1 with that from table 3.3. Also, when using the method with the constant regulator it is important that the regularization constant is chosen quite precise, if it’s too small the small eigenvalues aren’t regularized and if it’s too big the constant is fitted in stead of the data. Although the method with the constant seems to be working better, I’d advise to use the binning method. This is because of the main difference between the methods, which is the fact that the regularization of the data with the intra-bin fluctuation regularization method stems from the data itself. This seems a well-founded way of regularizing the data opposed to using an ad-hoc constant.

In trying to get better fits we could try using other models to describe the data, study different clusters, or maybe try different sized bins around the small eigenvalues. Nieuwenhuizen has been studying DM and his models show that the best fits to the clusters occur with 1.8 eV neutrinos [16]. This might shed some light on the mystery of DM. What is clear is that the search for the nature of DM is an important one that gives us insights in the universe around us and deals with all different sorts of problems ranging over multiple branches of physics. Maybe if more research like Nieuwenhuizen’s is done and better data or techniques become available we will finally know the nature of DM. 17

Appendix A

Appendix

A.1 Dutch Summary

Eén van de grote mysteries in de moderne astrofysica is de vraag waar donkere materie uit bestaat. Om donkere materie in een cluster van sterrenstelsels te kunnen beschrijven wordt vaak gebruikt gemaakt van een NFW massa profiel. Dit profiel wordt gefit op data van zogenaamde zwaartekracht lenzen om zo een model voor het cluster op te kunnen stellen. Uit deze fit kunnen we vervolgens informatie over de massaverdeling binnen het cluster halen en daarmee de massa en misschien zelfs de donkere materie van het cluster proberen te beschrijven. In dit project fitten we het NFW profiel op twee verschillende manieren op data van verschillende clusters en repliceren we de resultaten van Nieuwenhuizen. De parameters van de fit waren de A, de karakteristieke massadichtheid voor het cluster en R, de karakteristieke straal van het cluster. Via de A parameter verkregen we de c, wat een con- centratie parameter voor het cluster is. De eerste methode om de data te analy- seren is het gebruiken van een ad-hoc constante en de tweede methode maakt ge- bruik van een intra-bin fluctuatie regularisatie om de beste fit te verkrijgen. Het grootste verschil tussen de methoden is de manier waarop ze de kleine eigenwaarden van de covariantie matrix regulariseren. De intra-bin fluctuatie regularisatie methode biedt een manier om de eigenwaarden te regulariseren die vanuit de data zelf komt, in tegenstelling tot een ad-hoc constante. De methode met de ad-hoc constante leverde de volgende resultaten voor het cluster A1689 op: χ2/ν = 1.8, 3 A = 0.12 mN/cm , R = 400.0 kpc en a c = 7.38. De intra-bin fluctuatie regularisatie leverde de volgende resultaten voor het cluster A1835 op: χ2/ν = 5.5, 3 14 A = 0.43 ± 0.1 mN/cm , R = 159.0 ± 1.5 kpc, c = 9.54 en M200 = 6.32 × 10 M 2 3 en voor cluster A1703: χ /ν = 26.2, A = 0.18 ± 0.71 mN/cm , R = 245.4 ± 2.2 kpc, 14 c = 9.54 and M200 = 7.49 × 10 M . 18

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