Cluster Dark Matter: Comparing Two Methods of fitting Gravitational Lensing Data with an NFW Profile
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UNIVERSITY OF 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 clus- ter 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 regulariza- tion 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 con- stants..................................8 3.1.2 Fitting the data of Abell 1835 using intra-bin fluctuation regu- larization................................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 re- search 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 ve- locities [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 su- persymmetry, 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.