Gravitational lensing by galaxy clusters

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Gravitational lensing by galaxy clusters

Katja Hagedoorn June 2020

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Abstract

Galaxy clusters are some of the largest structures that act as gravitational lenses. Gravita-tional lenses can create multiple images of a source and magnify and distort these images. By observing these effects information on the cluster, for example its dark matter distribution, can be gained.

In this report, different axially-symmetric, increasingly complex lens models are discussed: the point mass lens, singular isothermal sphere and softened isothermal sphere. Several key characteristics, i.a. the magnification, shear and image positions, are derived for each model. The Navarro-Frenk-White (NFW) and Double Stretched Exponential (DSE) density profiles are discussed and used for our data analysis.

Strong lensing data from two massive, reasonably circularly symmetric galaxy clusters, Abell 1689 and Abell 1835, is analysed. The NFW, gNFW, NFW+gas, gNFW+gas and DSE profiles are fitted to the data, showing that the DSE profile best describes the data. The fits to the NFW profile give high concentration parameters for both clusters: c = 12.32 ± 0.10 for A1689 and c = 11.489 ± 0.008 for A1835. These are inconsistent with several previous studies. From the DSE fits, we obtain a mass M1= 1.75 ± 0.04 1015M¯ for the galaxy cluster and M2= 3.6 ± 0.9 1011M_¯for the brightest cluster galaxy for A1689; M1= 1.64 ± 0.04 1015M¯and M2= 7.6±0.8 1010M_¯for A1835. The cluster masses are in general agreement with previous studies and the BCG masses agree with general estimates for BCG masses.

REPORT Bachelor project report Physics and Astronomy

Conducted between 30-04-2020 and 28-06-2020 SUPERVISOR dhr. dr. T.M. Nieuwenhuizen

SECOND EXAMINER dhr. dr. C. Weniger

INSTITUTE Institute for Theoretical Physics Amsterdam (ITFA) VERSION Final version

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Introduction

In General Relativity, gravity is no longer interpreted as a force acting on a mass. Rather, it is seen as an effect of the curvature of space-time by mass or energy. An important consequence of space-time being curved is that this can also bend light. A light ray passing a massive object is deflected, as is illustrated in Figure 1.1. This image shows a light ray coming from star A and passing the Sun. The Sun, due to its mass, curves the space-time around it, and this gravitational field deflects the light ray. In this way, the light from star A still arrives at the Earth. From the perspective of an observer on the Earth, however, the light seems to have come from the position of star B. Einstein predicted that in this way the Sun would be able to bend light rays coming from more distant sources by 1.7 arcseconds [10], which was then confirmed by observations [8]. This was a very important step in General Relativity becoming the widely accepted theory of gravity.

In 1920, Eddington stated in his book Space, Time and Gravitation that there could be multiple light paths connecting the source and the observer under certain conditions [9]. This would mean that we can observe multiple images of the same source. Einstein also wrote about the splitting of images, but did not believe it had practical, astrophysical consequences, since he thought the effect would be too small to be observed [11]. However, gravitational lensing has proven to be a very useful tool in astronomy. Fritz Zwicky, a Swiss-American astronomer, was the first to postulate that not just stars, but galaxies and galaxy clusters could act as gravitational lenses [45]. These, due to their larger masses, would be able to generate multiple discernible images, separated by a large enough angle to be resolved by our telescopes.

Gravitational lenses can also deform and stretch images. By observing how sources are deformed by the lensing galaxy or galaxy cluster, Zwicky argued that it would be possible to determine the mass of the lens, irrespective of its luminosity. In this way, dark matter could be indirectly observed. Thus, we can infer the spatial distribution of dark matter and compare this to the distribution of luminous matter. In for example the Bullet Cluster, this comparison

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Figure 1.1: This figure shows how light is bend by a massive object. In the image, the massive object is the Sun, which curves the space-time around it. Therefore, light coming from star A is bend and arrives at the Earth. From the Earth’s perspective, however, it looks like the light came from the position of star B. Image credit: ScienceBlogs (2011), online:https://scienceblogs.com/startswithabang/ 2011/07/01/people-keep-making-einsteins-g.

provides an upper limit on the self-interaction cross sections of hypothetical DM particles [5]. Zwicky also predicted that gravitational lenses could function as telescopes: since the amount of photons emitted by the source does not change, but the observed size does, grav-itational lenses can magnify sources. This makes it possible to observe faint, high redshift sources that we would not be able to detect otherwise. Later, S. Refsdal proposed that grav-itational lensing could additionally be used as an independent way of measuring the Hubble constant, H₀[36]. If the luminosity of a source varies, these variations are observed at different times in the different images, since the travel times of the light in the different images are not the same. This time delay is determined by the lengths of the light paths, which is inversely proportional to the Hubble constant. All of these applications make gravitational lensing a very versatile cosmological tool.

The first observed gravitational lens was discovered in 1979 and is the galaxy YGKOW G1 which generates a double image of the quasar QSO 0957+561 [42]. The two images, A and B, are separated by 6” and can be seen in Figure 1.2. Since then, many different gravitational lenses have been discovered. We define three different regimes in gravitational lensing: strong lensing, weak lensing and microlensing. In strong lensing, the gravitational lensing effects are strong: we can observe multiple images of one source and images can be highly distorted. In weak lensing we observe just one image that is so weakly distorted that we can only

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mea-Figure 1.2: This image shows the Twin Quasar QSO 0957+561. The double imaged quasar can be seen as the two bright objects in the centre of the image. They were found to be too similar to be separate objects, and are the first observed gravitationally lensed images. Image credit: NASA/ESA, Hubble Space Telescope.

sure the effects statistically. Microlensing acts on smaller scales: microlenses are for example planets and stars. These types of lensing are discussed in more detail in Chapter 2.

This report will focus on gravitational lensing by galaxy clusters. Galaxy clusters are grav-itationally bound structures containing hundreds to thousands of galaxies. They typically have masses between 1014and 1015 M_¯, of which about 1% is accounted for by galaxies, 9% by gas and the remaining 90% is attributed to dark matter [35]. To analyse gravitational lensing by a galaxy cluster, observational data is used. From the observed lensing effects, the mass distri-bution of the cluster can be inferred. This distridistri-bution can be described using lens models, and from these certain parameters of the cluster can be derived.

This report will start by briefly describing the three main classes of gravitational lensing: strong lensing, weak lensing and microlensing (Chapter 2). Chapter 3 will then describe the basic theory of gravitational lensing. In Chapter 4 this theory will be applied to a number of different axially-symmetric lens models: the point-mass lens, the singular isothermal sphere and the softened isothermal sphere. The widely used Navarro-Frenk-White density profile and the more complex Double Stretched Exponential profile will also be discussed. These profiles were fitted to strong lensing data from two clusters, Abell 1689 and Abell 1835, to determine which profile describes the data best. This data analysis is described in Chapter 5. The results are presented and discussed in Chapter 6. Finally, Chapter 7 provides a conclusion.

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Chapter 2

Strong, weak and micro lensing

As stated before, there are three main types of gravitational lensing: weak lensing (WL), strong lensing (SL) and microlensing. These are not very strict definitions, but useful for understand-ing the different ways a gravitational lens can distort images. Some terms in this chapter will be explained in more detail further in the report but are mentioned here to give a more complete overview of the classes of lensing.

2.1 Strong lensing

Strong lensing is the most obvious form of gravitational lensing. Sources can be strongly lensed when the source, lens and observer are well aligned, so when the source is at a small angular distance from the lens’ centre [20]. Furthermore, the mass density of the lens needs to be sufficiently high (Σ ≥ Σ_c, sections 3.3 and 3.5), so strong lensing effects occur in the central, denser regions of galaxy clusters. Strongly lensed sources can produce significantly distorted images: we can observe long arcs and in extreme cases Einstein rings (an image spread out over a circle). Figure 2.1a shows an example of this. The image shows a luminous red galaxy which lenses a more distant blue galaxy. The alignment between the distant galaxy, the lensing galaxy and us as the observer is so precise that the source is distorted into an almost complete Einstein ring.

Strong lensing can also produce multiple images of a single source. As Eddington pos-tulated [9], there can be multiple light paths between the observer and source, resulting in multiple images of the same source. A famous example of this is the Einstein Cross, shown in Figure 2.1b. The lensing galaxy, which we see in the centre, splits a background quasar into four distinct images. How many images are produced is determined by the source-lens align-ment (discussed in section 3.5) and the angular separation between the images is determined by the lens’ mass distribution [20]. By measuring and analysing the spectra of the images, it

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(a) This picture shows an almost complete Einstein Ring. A distant blue galaxy is distorted into a horseshoe-shape by a luminous red galaxy. Image credit: NASA/ESA, Hubble Space Telescope.

(b) This picture, taken by the Hubble Space Tele-scope, shows the Einstein Cross, a quasar of which we can see four images. In the centre, we see the lens, which obscures the central, fifth image of the quasar. The gravitational lens is a galaxy, called G2237 + 0305. Image credit: NASA/ESA, Hubble Space Telescope.

Figure 2.1: These two HST pictures show examples of strong gravitational lensing.

can be determined if they are indeed strongly lensed images of a single source and not separate objects, since the spectral properties should remain the same despite the lensing.

2.1.1 Microlensing

Microlensing can be seen as a sub-class of strong lensing. It however occurs on much smaller scales. Microlenses are for example stars, planets or other compact objects. Since these lenses are so small, they cannot split sources into multiple discernible images that we can resolve, but they can magnify sources. This is illustrated in Figure 2.2. Here, the source is a star, which is emitting light that we observe. A microlens, in this case a MACHO, moves into our line-of-sight and lenses the star. It temporarily magnifies the source, and we observe an increased brightness. As the lens moves away, we again measure the star’s regular brightness. This effect can be used to search for MACHOs, exoplanets or free-floating planets, that we are not able to observe on their own [31]. The simplest model for microlensing, the point-mass lens, is discussed in section 4.1.

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Figure 2.2: This figure illustrates the concept of microlensing. A microlens moves into our line-of-sight and temporarily magnifies a source. As the observer, we measure a temporary increased brightness of the source. Image credit: Nature (2018), online: https://www.nature.com/articles/ d41586-018-07006-8

2.2 Weak lensing

In strong lensing, sources are highly distorted and can create multiple images or long arcs. When the angular separation between the source and lens becomes larger, these effects become smaller. This is the weak lensing regime, where sources generate only one image, which is weakly distorted. A hypothetical circular source would be distorted into an ellipse. These distortions however are so weak that they can only be measured statistically [3]. Many images need to be observed to be able to measure any statistically significant, systematic distortion among all the images, caused by the gravitational lens. This is further discussed in section 3.4.

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Chapter 3

Theory

To understand gravitational lensing we have to describe how light propagates through curved space-time. This is generally complicated to give a full description of, but thankfully in the case of gravitational lensing, we can make several assumptions that significantly simplify the calculations [24]. For one, we can describe the geometry of the universe using the FLRW metric, which describes a homogeneous, isotropic and expanding universe. The lensing objects can be seen as local perturbations. Then, to describe how light is deflected, we look at the area close to the lens, which we can treat as locally flat Minkowski space-time that is weakly perturbed by the Newtonian gravitational potential of the lens’s mass distribution.

These assumptions are valid if the Newtonian potential of the lens,Φ, is small: |Φ|/c2_{¿ 1,} and if the peculiar velocity, which is its velocity with respect to the cosmological rest frame, is small: v ¿ c. This holds well for galaxy clusters, where typical values for the gravitational potential and the peculiar velocities are |Φ|/c2_.₁₀−5 _{and v}_._{600 km s}−1_{¿ c respectively} [2, 40].

In this chapter, we will go over the general theory of gravitational lensing. We will introduce key definitions and parameters and explain how these are derived. There are many books, articles, reviews and lecture notes that describe the theory and derivations. Several were used in this chapter [2, 3, 17, 20, 24].

3.1 Deflection angle

Unperturbed space-time is described by the Minkowski metric, with the line-element

ds2_{= −c}2dt2+ d~x2. (3.1) This metric is weakly perturbed by the Newtonian potential |Φ|/c2 _{of the lens, which can be} described as

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ds2_{= −c}2 µ 1 +2Φ c2 ¶ dt2₊ µ 1 −2Φ c2 ¶ d~x2 (3.2)

where you see that gravity causes equal perturbations in the temporal as well as the spatial components of the metric.

For light, the line-element ds = 0. Using this, we can find the effective speed of the light in the weakly perturbed space-time

¯ ¯ ¯ ¯ d~x dt ¯ ¯ ¯ ¯= c 0_{= c} µ 1 +2Φ c2 ¶ (3.3) where we used the first-order Taylor expansion _1−x1 ≈ 1 + x for x = Φ_c2¿ 1. The refractive index

n can now be obtained using the relation c0₌_nc

n = 1 −2Φ

c2 . (3.4)

Conventionally, the gravitational potential is taken to vanish at infinity, makingΦ a negative quantity. Therefore, the refractive index is larger than unity. n will depend on the spatial coordinate~x, so n = n[~x(l)], where ~x(l) is the light path. However, for simplicity we shall write the refractive index simply as n in the following derivation.

An important quantity related to light deflection is the deflection angle ˆα, which is a mea-sure for how strongly the light is deflected, and is shown in Figure 3.1. We can now derive a formula for this angle, using the refractive index n and Fermat’s principle [1]. This principle states that light will always take the path between two points A and B that takes the least time. In other words, its travel time is extremal. The travel time is given by

T = Z B A dt = Z B A n cdl (3.5)

where l is the geometrical length and A and B denote the generic begin and end points, and are not specified times or places.

We need to introduce an affine parameter, λ, for the light ray, so~x =~x(λ). We then make the substitutions dl = ¯ ¯ ¯ d~x dλ ¯ ¯ ¯dλ= | ˙~x|dλwhere we define | ˙~x| =

p ˙_~x2_{. Fermat’s principle now gives} us δT =1 c δ Z B A n| ˙~x|dλ= 0 (3.6) and therefore δZ B A n| ˙~x|dλ= 0 . (3.7)

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As in classical mechanics, we can identify a Lagrange function L(~x, ˙~x,λ_{) = n| ˙}~x| = np ˙~x2_{. The} Euler-Lagrange differential equation is given by

d dλ ∂L ∂_~x˙ − ∂L ∂~x = 0 . (3.8)

For our Lagrangian we get

∂L ∂_~x˙ = n ˙ ~x p ˙_~x2 and ∂L ∂~x = ∂ ∂~x ³ np ˙~x2´_{= ~}_∇np ˙_~x2 _(3.9)

We can normalise the affine parameterλso thatp ˙~x2_{= 1 and we identify the tangent vector} to the light ray~e = ˙~x. Then equation 3.8 says

d dλ(n~e)−~∇n = 0 = µ ~∇n·d~x dλ ¶ ·~e+ n ˙~e−~∇n (3.10) which gives n ˙~e =~∇n −³~∇n·~e´·~e=~∇_⊥n (3.11) since we subtract the component along the ray from ~_{∇n, which leaves only the perpendicular} part. Thus ˙ ~e =~∇⊥n n = ~∇⊥ln n = ~∇⊥ln µ 1 −2Φ c2 ¶ ≈ −2 c2~∇⊥Φ (3.12) where we used a Taylor expansion in the final step. Now, since~e = ˙~x which is the tangent vector to the light ray, the deflection angle is given by the integrated change in~e:

ˆ ~ α= Z ˙ ~e dλ= − 2 c2 Z ~∇⊥Φ dλ (3.13)

3.2 Lens equation and lensing potential

To simplify matters, we can apply the Born approximation. This means that we can treat the light path as a straight line, and can integrate over a straight line instead of the actual light path. This is appropriate because typically deflection angles are of the order of arcseconds or smaller. This approximation enables us to easily calculate the lensing properties of the simplest lens model, the point-mass lens, which is discussed in section 4.1.

Another approximation we can use is called the thin-lens approximation. It is valid if the mass distribution of the lens is thin compared to the distances involved, which is true for galaxy clusters: the distances to the cluster and the source are typically many orders of magnitude

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Figure 3.1: This figure shows a simplified sketch of a gravitational lensing system. O is the observer, S the source and the lens is represented by the orange oval. The optical axis is represented by the dotted line going from the observer through the centre of the lens to the source plane. β is the angle

between the source and the optical axis andθ the angle between the image and the optical angle. The

deflection angle ˆα is the angle between the source and the image, measured at the lens, while the reduced deflection angleα is the angle between the source and the image, measured at the observer. DLS is the distance from the lens to the source, DL from the observer to the lens and DS from the

observer to the source.

larger then the extent of the cluster itself. We are then able to project the mass distribution of the lens along the line-of-sight and substitute a mass sheet in the lens plane, orthogonal to the line-of-sight. This means we can treat the light path as two straight lines: one from the observer to the lens and one from the lens to the source, and can say that all deflection occurs in the lens plane.

Figure 3.1 shows a sketch of a typical gravitational lens system with a thin lens. We define the angles ~_βand ~_θbetween the optical axis and the source and the optical axis and the image, respectively. Using the distances defined in Figure 3.1, we can see that

DS~β= DS~θ− DLS~αˆ . (3.14) In Euclidian space, this equation seems very logical. We can apply it in curved space-time as well because we define the distances D_S, D_LSand DLas angular-diameter distances, such that equation 3.14 holds. This also means that in general DLS6= DS− DL.

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image as measured from the observer ~α=DLS D_S ˆ ~ α. (3.15)

Substituting this in equation 3.14 and dividing by D_Swe obtain the lens equation: ~

β=~θ−~α. (3.16)

We will now define the lensing potential,ψ. This quantity describes the imaging qualities of a gravitational lens. We start by rewriting the reduced deflection angle, using equations 3.13 and 3.15, as ~α= ~∇⊥ ·₂ c2 DLS DS Z Φ dz¸ . (3.17)

It makes sense to write this gradient as a function of the angular position on the sky, ~θ, instead of distances. So, we use ~_∇_⊥_{= D}−1_L ~∇_θ to write equation 3.17 as

~ α= ~∇θψ where ψ= 2 c2 D_LS DLDS Z Φ dz (3.18)

withψthe lensing potential.

3.3 Convergence, shear and magnification

We now want to derive several properties from the lensing potential: the convergence, shear and magnification. These properties determine some key features of the image. The conver-gence changes the size of the image and the shear changes the shape of the image. Since the amount of photons coming from the source is not changed by gravitational lensing, but the size (that we observe) is, the source can be magnified or demagnified. This magnification is given by the magnification factor.

We start by taking the Laplacian of the lensing potential ~∇2 θψ= 2 c2 D_LS DLDS Z ~∇2 ⊥Φ dz (3.19) and using ~_∇_⊥_{= D}−1

L ~∇θ again to get the perpendicular Laplacian. We can replace this with the complete Laplacian ~∇2= ~∇2_⊥+_∂z∂22 because we can neglect the component along the line-of-sight,

since the lensing mass distribution is very small in comparison to the distances involved. That means we can use Poisson’s equation ~_∇2Φ = 4_πG_ρto obtain

~∇2 θψ= 8πG c2 DLS DLDS Z ρdz =8πG c2 DLS DLDSΣ (3.20) whereΣ is the surface mass density: the projection along the line-of-sight of the 3D densityρ. If we now define a critical surface mass density

Σ−1 c = 4_πG c2 D_LS DLDS (3.21)

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we can write equation 3.20 as ~∇2 θψ = 2 Σ Σc = 2κ (3.22) whereκis the convergence, or dimensionless surface mass density.

To obtain the shear and magnification, we start with the lens equation ~_β_{= ~}_θ_−~_∇_ψ, where we have written ~_∇_θ as ~_{∇ for simplicity. If we now define}δ~βas the angular distance between the centre of the source and an outer source contour and_δ~_θ as the analogous image distance, we get the following relations:

δ~β≈ A δ~θ where Ai j=∂βi

∂θj=δi j−ψi j

. (3.23)

A in this equation is the Jacobi matrix, andψi j= ∂

2_ψ

∂θi∂θj. We see here that if there is no lens,

we simply have_δ~_β₌_δ~_θ, and that the curvature of the lensing potential determines how the source is mapped to the image.

We can subtract the trace of A from A to get the shear matrixΓ Γ = − µ A −1 2(trA)I ¶ = − µ A −1 2(2 −~∇ 2_ψ₎¶ = −(A − (1 −κ)) . (3.24) We can calculate the components of the matrix, which are

Γ11= 1

2(ψ11−ψ22) =γ1 , Γ22= −γ1 , Γ12=Γ21=ψ12=ψ21=γ2 (3.25) and using these relations we obtain the Jacobi matrix

A = (1 −κ)I −Γ =   1 −κ−γ1 −γ2 −γ2 1 −κ+γ1   (3.26)

whereγ1andγ2are the components of the shear:γ2=γ2₁+γ2₂.

Now, this matrix tells us how the image is mapped on the source, but typically we cannot observe the source, only the image. Therefore, we want to know the inverse of equation 3.23, which only exists if det(A) 6= 0. The determinant is given by

det(A) = (1 −κ)2₋γ2 (3.27) For weak gravitational lensing,κ,γ¿ 1 so the det(A) ≈ 1, but for strong lensing the determi-nant can be zero. This is discussed further in section 3.5.

The inverse Jacobi matrix is then

A−1₌ 1 det(A)   1 −κ+γ1 γ2 γ2 1 −κ−γ1   (3.28)

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Figure 3.2: This figure shows the effect of convergence (dotted line) and shear (solid line) on a circular source. Image from [25].

from which we can infer that the magnification factorµ= 1 / det(A), since this factor maps the solid-angle elementδβ2toδθ2. We get

µ=δθ 2

δβ2= 1

(1 −κ)2₋_γ2 (3.29)

where for weak lensing, we can use a first-order Taylor approximation since_κ,_γ_{¿ 1 and we} getµ_{≈ 1 + 2}κ.

To make the physical meaning of the convergence and shear clearer, we can also write the Jacobi matrix as follows

A = (1 −κ)   1 0 0 1  −γ   cos 2φ sin 2φ sin 2_φ _{− cos 2}_φ   (3.30)

whereγand φ depend onθ, and the factor 2 is due to the fact that an ellipse becomes itself again after a 180◦_{rotation. Using this notation, you can see that the convergence}_κ_changes the size of the image isotropically: a circular source would become a larger, circular image. The shear_γon the other hand changes the size anisotropically: a circular source would be mapped as an ellipse. The eigenvalues of the matrix determine the semi-major and semi-minor axes of the image: (1 −κ−γ)−1_{and (1 −}κ₊γ)−1respectively. The effects ofκandγon a circular source are shown in Figure 3.2.

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3.4 Ellipticity

Using the shear and convergence, we can determine how images are changed compared to their sources. For weak lensing, the ellipticity of an image of a circular source is given by

²=a − b a + b= λ+−λ− λ++λ− = γ 1 −κ (3.31)

with a and b the semi-major and semi-minor axes, respectively. The quantity on the right-hand side is also called the reduced shear: g =_1−κγ . However, in weak lensing studies the sources are typically not observable, so we do not know their size or ellipticity. The ellipticity that is measured is then ²_{≈ g +}²S, where ²S is the intrinsic ellipticity of the source. This intrinsic ellipticity is not measurable, but when a sufficiently large amount of images are observed, we can assume that the average intrinsic ellipticity of the sources is zero. This makes weak lensing surveys statistical in nature. Figure 3.3 shows the effect of the reduced shear on the ellipticity of a group of galaxies.

3.5 Critical curves and caustics

As we have seen, the magnification factor_µ is determined by the Jacobi determinant. From equation 3.29 we can see that in the special case that det(A) = 0, the magnification becomes infinite. This is not the case for weak lensing, sinceκ,γ_{¿ 1, but for strong lensing the} conver-gence and shear are much larger and we can have det(A) = 0. This happens for 1 −κ−γ= 0 or 1 −κ+γ= 0. These conditions define respectively the tangential and radial critical curves in the lens plane [20]. On and near these critical curves you will be able to see highly magnified images. Images near the tangential curve will be distorted along the curve, whereas images near the radial curve will be radially stretched. These critical curves correspond to lines or points in the source plane, which are called caustics. Any source on or near a caustic in the source plane will be seen, highly magnified, on or near a critical curve in the lens plane.

In reality, the images are not infinitely magnified. The magnification remains finite be-cause all astrophysical sources are extended sources, they are not points. Even if it were point sources, the magnification would be finite since certain approximations then no longer hold, and they have to be described using wave-optics [38]. Still, images near critical curves can be highly distorted and magnified, which is illustrated in Figure 3.4. This figure shows image configurations for sources at different distances from a fold or cusp in the caustic.

In this figure, the multiplicity of the images is also visible. Light will always take the path where its travel time is extremal (Fermat’s principle). There could be multiple paths for the light to take, so a source could yield multiple images. The multiplicity of the images depends on where the source is located in the source plane. When a source crosses a caustic in the

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Figure 3.3: This figure illustrates the effect of the reduced shear on a group of galaxies. A patch of sky (top left) contains many galaxies, each with an intrinsic ellipticity. Their ellipticities are shown in the top right panel, and average to zero. Now, in the case of weak gravitational lensing, the ellipticity of all of these galaxies is altered by g, as you can see in the bottom left panel, where all galaxies are slightly stretched. The bottom right panel shows these ellipticities, which now no longer average out to zero but are systematically shifted. Image from [17].

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Figure 3.4: This figure illustrates the image configurations of sources at different distances from a fold or cusp in the caustics. The lens here is elliptical. The inner caustic corresponds to the outer critical line and vice versa. When a source crosses a caustic, the number of images changes with ±2, and the images are highly distorted near the critical curves. Image from [39].

source plane, two images appear or disappear near the critical curve in the lens plane. When the source is located outside of the caustics, it only generates one image. Thus we only observe multiple images inside of the outer critical line, whereΣ ≥ Σc. The number of images generated is always odd, though often we cannot observe the central image. An example of this is the Einstein Cross, where we see only four images as the lens obscures the central, demagnified image (Figure 2.1b).

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Chapter 4

Lens models

In this chapter, the theory discussed in Chapter 3 will be applied to several axially-symmetric lens models. A great advantage of using a circularly symmetric model is that a 2D problem is essentially reduced to a 1D problem, making the calculations easier. We will start with the simplest model, the point-mass lens, then discuss the singular and softened isothermal sphere and finally the NFW and DSE density profiles. Even though in reality lenses like galaxy clusters are unlikely to be axially-symmetric, these models can still be very informative. The clusters that will be analysed in Chapter 5 are both reasonably circularly symmetric and will be fitted with the models described in sections 4.3 and 4.4.

4.1 Point-mass lenses

The easiest way to model a gravitational lens is with a point-mass lens [2, 7, 20]. We assume a point mass M is situated at the origin of the coordinate system and a light ray propagates parallel to the z-axis, passing the lens at an impact parameter b. Its gravitational potentialΦ is then given byΦ = GM/pb2_{+ z}2_{and thus, using equation 3.13, ˆ}_α_is

ˆ α = −2 c2 Z ∞ ∞ ∂ ∂b G M p b2_{+ z}2d z = 4G M bc2 = 2R_S b (4.1)

where RS= 2GM/c2, the Schwarzschild radius. From this we can obtain the lensing potential

ψ, using equations 3.15 and 3.18 and writing b = DLθ:

α=4G M c2 D_LS D_LD_S 1 θ = ~∇θψ = ∂ψ ∂θ (4.2) ψ=4G M c2 D_LS D_LD_Sln |θ| . (4.3)

Now, we want to calculate where the images of the source will appear. To obtain these image positions, we start with the lens equation (3.16). Filling in equation 4.2 gives us

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β=θ−4G M c2 DLS DLDS 1 θ . (4.4)

This is a quadratic equation which will have two solutions. However, if the source is located directly behind the point-mass lens in our line of sight, i.e. if_β_{= 0, there is only one solution.} We call this the Einstein radiusθE

θE= s 4G M c2 DLS D_LD_S . (4.5)

If the source, lens and observer are perfectly aligned, a ring with this radius would form around the point-mass. We call this an Einstein ring, and Figure 2.1a shows a near-complete one. This radius is also an important scale, since the density within the radius is the critical densityΣc. Therefore, if a source is located within this radius, multiple images can be generated [7].

Plugging expression 4.5 into 4.4 we get

β=θ−θ 2 E

θ (4.6)

which we can solve forθto get the image positions:

θ±= 1 2 ³ β± q β2_{+ 4}_θ2 E ´ . (4.7)

This shows that a point-mass lens generates two images, one atθ₋and one atθ₊. In the limit where _β_{→ ∞ we see that}_θ₋_{→ 0 and}_θ₊_→_β. This tells us that when the angular separation between the source and the lens becomes large enough, as we would expect, the source is unlensed: we see an image at the position of the source. There is however still mathematically an image at θ₋= 0, but we shall see that this image is completely demagnified and hence invisible.

The image positions are plotted in Figure 4.2, along with those from the singular and soft-ened isothermal spheres. The figure shows the two images generated for 0.2 ≤β≤ 1θE. We can see that as_βbecomes larger, the image at_θ₋ gets closer to the lens’ centre.

To derive the magnification factorµwe have to determine det(A). For this purpose, it makes sense to write equation 4.7 using the adimensional parameters x =θ/θE and y =β/θE:

x_±₌1 2 µ y ± q y2_{+ 4} ¶ . (4.8)

For any axially-symmetric lens the following equation holds [24]

det(A) =β_θdβ dθ =µ

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and using equation 4.6 this gives a magnification of µ= Ã 1 −θ 4 E θ4 !−1 = µ 1 − 1 x4 ¶−1 (4.10) for a point-mass lens. By plugging in x_±we get the magnification of each separate image:

µ±= Ã 1 − 1 x_±4 !−1 = y 2 + 2 2 ypy2_{+ 4}± 1 2 . (4.11)

The total magnification is then given byµ= |µ+| + |µ−|, which becomes

µ= y 2

+ 2

ypy2_{+ 4} . (4.12)

Furthermore, since θ₋<θE, µ−< 0. This can be interpreted as the parity being inverted with respect to the source. This parity inversion means that the image of a S-shaped source would appear Z-shaped.

If we take the limit y → ∞ we see thatµ−→ 0 andµ+→ 1. So for large angular separations one image disappears because it is completely demagnified, while the other is neither magnified nor demagnified. It is completely indistinguishable from the source since it has the same flux and position: it is unlensed.

We can also derive the convergenceκand the shearγ. We know that ~∇2_θψ= 2κ(equation 3.22) and if we use circular coordinates we can writeκas

κ=1 2~∇ 2 θψ= 1 2 µ ∂2 ∂θ2ψ+ 1 θ ∂ ∂θψ ¶ . (4.13)

We can now fill in the lens potential (equation 4.3) which givesκ_{= 0. The shear can then easily} be obtained using the expressions for the determinant of the Jacobian matrix A from equations 3.27 and 4.9: det(A) = Ã 1 −θ 4 E θ4 ! = (1 −κ)2−γ2= 1 −γ2 (4.14) which gives us the value for_γ:

γ=θ 2 E

θ2 (4.15)

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4.2 SIS-models

The point-mass lens is the simplest example of a gravitational lens, but it has limited uses. Microlenses, like lensing stars or planets, can be approximated as point masses, but the point-mass lens model cannot be applied to galaxies or galaxy clusters. The simplest models to describe these larger lenses are also circularly symmetric. In this section we will discuss the singular isothermal sphere and the softened isothermal sphere, both abbreviated as SIS.

To calculate the deflection angle for a circularly symmetric model, we can generalise equa-tion 4.1 for a point mass lens. The mass M is now no longer a point mass, but a mass distribu-tion M(ξ) that depends onξ, the distance from the centre of mass. The impact parameter b no longer has a physical meaning and can be replaced with_ξ. In this way we obtain the deflection angle

ˆ

α=4G M(ξ)

c2_ξ (4.16)

in which the mass can be defined as

M(_ξ_{) = 2}_π Z ξ

0 Σ(ξ

0₎_ξ0_d_ξ0 _(4.17)

withΣ the surface density [20].

4.2.1 Singular Isothermal Sphere

A simple circularly symmetric model for galaxies and galaxy clusters is the singular isothermal sphere [15, 17, 20, 24]. In this model it is assumed that the mass components, e.g. stars and galaxies, behave like particles of an ideal gas. The equation of state of particles in an ideal gas is given by

p =ρkT

m (4.18)

in whichρand m are the mass density and the mass of the particles respectively, which can be stars in galaxies or galaxies in galaxy clusters. We will call the particles galaxies for simplicity from now on. We assume the galaxies are in thermal equilibrium so that

kT = mσ2v (4.19)

holds, whereσvis the one-dimensional velocity dispersion of the galaxies. We further assume the cluster is isothermal, so that T and therefore_σ_vare constant across the galaxy cluster. The equation of hydrostatic equilibrium is given by

p0

ρ = −

G M(r)

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where the prime stands for a derivative with respect to r. A simple solution to this equation is

ρ(r) = σ 2 v

2_πGr2 (4.21)

which we call the singular isothermal sphere. The total mass of a singular isothermal sphere is divergent, since the density profileρ_{∝ r}−2_{so M ∝ r.}

From the density profile we can calculate the rotational velocity of the galaxies, given by v2_rot₌G M

r . (4.22)

Using equation 4.20, we see that G M/r = r p0_/_ρ_{. Now, by using equation 4.19 and taking the} derivatives of 4.18 and 4.21 with respect to r, we get

p0₌ _ρ0kT m =ρ 0_σ2 v = − σ4 v πGr3 (4.23)

We can finally use this to calculate the rotational velocity v2_rot=G M(r) r = r p0 ρ = 2σ 2 v (4.24)

which is a constant. This property of the singular isothermal sphere model is one of the main arguments in its favour for modelling galaxies, since the measured rotation curves of galax-ies are flat. This flat velocity curve is plotted along with the velocity curves of the softened isothermal sphere and the NFW profile in Figure 4.3.

We can project the mass densityρalong the line-of-sight to obtain the surface mass density Σ(ξ) = σ

2 v

2G_ξ . (4.25)

This equation shows that this profile has a singularity at its centre: forξ_{→ 0,}Σ → ∞. Because of this and the fact that the total mass is divergent, the singular isothermal sphere model is not entirely physical. However, it is still used because of its simplicity and the fact that it reproduces the flat rotation curves.

Now we can use equation 4.17 to show that M =πσ2vξ/G, which gives the deflection angle ˆ

α=4πσ 2 v

c2 (4.26)

The deflection angle is independent of the distance from the centre. Using equation 3.15 and setting_β_{= 0 in the lens equation (3.16) we obtain the Einstein radius of the SIS}

θE= D_LS

D_S 4πσ2_v

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So far, we have used one-dimensional equations to describe the properties of the SIS. To calculate the image positions, we can generalise and write the lensing potential as [1]:

ψ = 4πσ 2 v c2 DLS D_S |~θ| = θE q θ2 1+θ 2 2 (4.28)

from which we can obtain the deflection angle vector: ~α = ~∇ψ = θE 1 q θ2 1+θ 2 2   θ1 θ2   = θE ~_θ |~θ| . (4.29)

Using this, we obtain the lens equation in vector form ~

β=~θ−~α = ~θ−θE ~_θ |~θ|

(4.30) which in one dimension can be written as

θ=β+θE θ

|θ| (4.31)

This equation can have one or two solutions. Forθ_{> 0 and thus}β_>θE we see one image at

θ=β+θE. Forθ< 0 andβ<θE we get the second solutionθ=β−θEand we can see two images. The image positions are thus given by

θ±=β±θE (4.32)

There are two special cases: for_β₌_θ_E we get_θ₋_{= 0 and for}_β_{= 0 we see an Einstein ring. For}

β<θE we can calculate the magnifications of the images using equation 4.9

µ±=θ_β± = 1 ±θ_βE = µ 1 ∓θ_θE ± ¶−1 (4.33) This shows us that as_β_→_θE theθ−-image gets demagnified and disappears whenβ=θE.

The image positions from equation 4.32 are plotted in Figure 4.2, along with those of the point-mass and softened sphere. We see the two images generated for 0.2 ≤β≤ 1 θE. For

β=θE, the figure shows an image at 2θE and one at the lens’ centre, which gets demagnified. Finally, we can calculate the shearγand the convergenceκin the same way we did for the point mass lens. Plugging the lensing potential (4.28) into equation 4.13 and using_α₌_θE we obtain the convergence

κ=θE

2_θ (4.34)

and using equation 3.27 we can solve for the shear

γ=θE

2θ=κ. (4.35)

Thus, the shear and convergence are equal. In section 3.5 the critical curves are defined, and we see that for the SIS the tangential critical curve is equal to the Einstein ring, while the radial critical curve is reduced to a point at the centre.

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4.2.2 Softened Isothermal Sphere

Another, more complicated, circularly symmetric model is the softened isothermal sphere [20, 24]. In this model, the singularity at the core of the singular isothermal sphere is resolved by defining a finite core radius_θcor R. The mass distribution is now flattened in the core and no longer divergent at the centre. The total mass however still diverges like the singular sphere.

The 3D Newtonian potential of the softened isothermal sphere is given by

Φ3D(r) =σ2vln (r2+ R2) (4.36) with R some core radius. We can calculate the 2D potential by integrating this expression over z and rewriting it as Φ2D(r) = Z _∞ −∞σ 2 v[ln (r2+ z2+ R2) − ln(z2)] d z . (4.37) By writing the ln (z2)-term explicitly, we obtain an infinite constant that we subtract from the integral, leaving us with a finite potential:

Φ2D(r) = 2πσ2v p

r2_{+ R}2 _(4.38)

The lensing potential can now be obtained by using equation 3.18

ψ= 2 c2 D_LS DLDS Z Φdz = 4πσ2v c2 D_LS DLDS p r2_{+ R}2 ₌ 4πσ 2 v c2 D_LS DS q θ2₊_θ2 c (4.39) where in the final step we used the small angle approximation.

To calculate the rotational velocity for this model, we start with the 3D Newtonian potential (4.36). Using the Poisson equation, we can obtain the 3D mass distributionρ_3D

∆Φ3D= 4πGρ3D (4.40)

where the Laplace operator is given by ∆ = ∇2 = µ∂2 ∂r2+ 2 r ∂ ∂r ¶ . (4.41) This gives ρ3D= σ 2 v 2πG 1 r2_{+ R}2+ σ2 v πG R2 (r2_{+ R}2₎2 . (4.42) The first term in this density profile scales as r−2for large radii while the second term scales as r−4. Thus, the second term only provides a significant contribution to the density profile for small r. We could say the first term describes the DM halo overall, while the second term describes a central "bulge" in the density.

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M(r) = Z r 0 ρ 4πr0 2dr0₌2σ 2 v G r3 r2_{+ R}2 . (4.43)

Using this relation we can calculate the rotational velocity, using equation 4.22:

vrot = σv s

2r2

r2_{+ R}2 . (4.44)

From this we recover the rotational velocity of the singular isothermal sphere (4.24) if we set the core radius R = 0.

The rotational velocity is plotted together with the velocities of the singular isothermal sphere and NFW profile in Figure 4.3. The figure shows that the singular isothermal sphere has a non-zero velocity at the centre, due to the divergent density profile, whereas the softened isothermal sphere no longer has this problem, and v → 0 as r → 0.

Now, we want to calculate the image positions. We start with the lensing potential (4.39), which we will write as ψ_{= w}pθ2_{+ b}2 _{for simplicity from now on. The deflection angle then} becomes

~

α= ~∇ψ= wp ~θ

θ2_{+ b}2 . (4.45)

Plugging this into the lens equation gives us

β=θ− wp θ

θ2_{+ b}2 (4.46)

and by settingβ= 0 we obtain w =qθ2_E+ b2_{. Solving this to obtain a formula for the image} positions can be done analytically, but we will take an easier, numerical approach. Without loss of generality, we can set_θ_E_{= 1 to get w =}p_{1 + b}2_{. We then define}_θ_{= zw,}_β_{= xw and b = yw or} b = y/p1 − y2_{. Using these definitions and equation 4.46 we get}

z = x + z/ q

z2_{+ y}2_. _(4.47)

Finally, we set z = yu to obtain

yu − x = u/pu2_{+ 1} _(4.48) which we can solve graphically. Figure 4.1 shows the LHS and RHS plotted as a function of u for y = 0.2 and x = 0.35. It shows that there are either one or three solutions. This corresponds to one or two images, because the middle solution is unstable.

We can now analytically obtain a value xmax( y) which gives the range 0 < x < xmax( y) for which there are three solutions, for a given y. By equating the slopes of the LHS and the RHS we find u = ±py−2/3_{− 1 which we can plug back into 4.48 to get}

xmax( y) = ³

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Figure 4.1: This figure shows the RHS and LHS of equation 4.48 plotted as a function of u, for y = 0.2 and x = 0.35.

Translating this back to the original variables, we get the relation

βmax(θc) = Ã 1 − µ θc q 1 +θ2c ¶2/3!3/2 q 1 +θ2c (4.50)

which for a given core radius gives you the maximum angular separation between the source and the centre for which there are two images generated. For _β_>_βmax only one image is created.

Figure 4.2 shows the image positions for the point-mass lens, the singular isothermal sphere and the softened isothermal sphere. The differences between the models are clearly visible. The softened isothermal sphere with core radius of 0.1θE (shown on the right) only produces two images if_β_<_β_max_{≈ 0.69, so in the figure we see two images for 0.2, 0.4 and 0.6}

θE and only one for 0.8θE andθE.

4.3 Navarro-Frenk-White profile

The Navarro-Frenk-White or NFW profile is a very popular model for the mass distribution in dark matter halos and one of the most widely used [28]. It is based on N-body simulations within theΛCDM cosmology. Its density profile is given by

ρ(r) = ρcritδchar r Rs ³ 1 +Rrs ´2 = ρ0 r Rs ³ 1 +Rrs ´2 (4.51)

where Rs is called the scale radius,ρc the critical density of the Universe, given byρc=3H

2

8πG,

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Figure 4.2: This figure shows the image positions for a source located at 0.2 (pink), 0.4 (blue), 0.6 (yellow), 0.8 (green) and 1 (orange) times the Einstein radius, for the point-mass lens, the singular isothermal sphere and the softened sphere. The three circles have the Einstein radius. The left, green circle with the star-shaped markers shows the Einstein circle with the image positions for the point-mass lens. The middle, grey circle with the round markers shows the image positions for the singular isothermal sphere. The right, blue circle with the triangle markers shows the image positions for the softened isothermal sphere, with a core radius of 0.1θE.

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(or_ρ₀) are the two parameters that define the NFW-profile and differ for each halo. The char-acteristic overdensity can also be written as

δchar=∆200Ωm 3

c3

ln (1 + c) − c/(1 + c) (4.52) where∆200 is an overdensity of 200; c is the concentration parameter, defined as c = c200= R200 / Rs, with R200 the radius corresponding to an overdensity∆200; andΩm is the matter density parameter, which depends on the cosmological model used. In an Einstein-de Sitter universeΩm= 1.

From the density profile (4.51) we can see that the total mass is divergent, but we can derive the mass within a certain radius Rmaxthrough integration:

M = Z Rmax 0 4πr2ρ_{(r) dr = 4}πρ0R3s · ln µ_R s+ Rmax Rs ¶ − Rmax Rs+ Rmax ¸ . (4.53) Typically the virial radius Rvir or the radius R200 is taken as Rmax. We will use R200, which gives M = 4πρ0R3s h ln (1 + c) − c 1 + c i (4.54) with c the concentration parameter.

From this mass, again, we can derive the velocity profile, using equation 4.22.

vrot= s

4_πρ0G R3s [ln (1 + c) − c/(1 + c)]

r (4.55)

We can see here that the velocity drops off as 1/pr. This velocity is plotted, along with the singular and softened isothermal sphere, in Figure 4.3. In the figure we see that the velocity of the NFW profile does not go to zero for r → 0, indicating the central cusp discussed in section 4.3.1.

4.3.1 Core-cusp problem

The NFW model generally provides a good fit to DM halos of e.g. massive galaxy clusters. However, there are discrepancies between the NFW profile and observations of low-surface brightness (LSB) and dwarf galaxies. From formula 4.51 we can see that the NFW profile has a infinite and divergent central density, so the density increases very sharply at small radii (ρ_{∝ r}−1). Thus, the NFW profile points to DM halos having central cusps. Yet, studies have shown that the observed rotation curves of LSB and dwarf galaxies suggest DM halos with central cores, i.e. a central flat DM density [4]. This is known as the core-cusp problem.

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Figure 4.3: This figure shows the velocity curves for the singular and softened isothermal spheres and the NFW profile. The parametersσv, G, Rsandρ0are set to 1, the concentration parameter c is set to

10 and the core radius is set to 0.1.

4.3.2 Concentration parameter

The concentration parameter typically has a value in the range of 4 < c < 40 for DM halos. There is a relation between the value of the concentration parameter and the formation time of a halo: in general, halos with a higher value of c formed earlier [28]. Halos that formed earlier are also generally less massive, so there is an inverted concentration-mass relation: halos that assembled later are more massive, but less concentrated [21]. This can be explained by noting that the Universe was less dense at the later time that more massive halos formed [29]. Furthermore, halos typically have a low concentration parameter if they have recently undergone a large merger. Conversely, they have a higher value of c if they have been able to grow relatively undisturbed for a longer period of time [21].

4.4 Double Stretched Exponential

While the NFW profile has been widely used to model dark matter halos, there are discrep-ancies between the model and some observations. Specifically, the core-cusp problem that is discussed in section 4.3.1 has not been resolved. Other, alternative models exist to describe DM halos. One of the most popular of these profiles is the Einasto profile [29, 30]. Its density profile is given by ln µρ α ρ−2 ¶ = −2 α ·µ _r r₋₂ ¶α − 1 ¸ (4.56) where at r = r−2 the density isρ=ρ−2. This density profile, in contrast to NFW, has a finite central slope and is not divergent, so describes observations of flat central densities better.

The Double Stretched Exponential (DSE) profile is a more generalised version of the Einasto profile. The DSE profile describes a 3D density profile, with an explicit contribution of the

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brightest cluster galaxy (BCG). Its density is given by ρDSE= 3M1 4πR3₁ exp −(r/R1)1/n1 Γ(1+3n1) + 3M2 4πR3₂ exp −(r/R2)1/n2 Γ(1+3n2) (4.57)

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Chapter 5

Data analysis

In this project strong lensing (SL) data from the clusters Abell 1689 and Abell 1835 was anal-ysed. The data was provided by M. Limousin et al. [19, 34]. Five density profiles were fitted to the data to determine which profile describes the data best, and several physical parame-ters were derived from them. This chapter will begin with briefly discussing the clusparame-ters that were analysed and then describe the data that were used in section 5.2. The different density profiles that were fitted to the data are considered in section 5.3. Finally, we will discuss the process of fitting in 5.4.

5.1 A1689 and A1835

The two clusters that were analysed in this project are the Abell clusters A1689 and A1835. They are both located in the constellation Virgo, at redshifts of z = 0.1832 and z = 0.2532 respectively [27]. They are fairly relaxed and circularly symmetric, massive clusters. A1689 is one of the largest galaxy clusters and contains the most gravitational arcs ever found in one system, which makes it very suitable for gravitational lensing studies [26].

5.2 Description of the data

To analyse the clusters A1689 and A1835 strong lensing data was used, provided by Limousin et al. [19, 34]. They used deep observations with the Hubble Advanced Camera for Surveys (ACS) combined with ground-based spectroscopy to determine the 2D mass distribution of the clusters. This data was analysed using the LensTool program [16], which produces mass maps to account for the observed arcs and arclets. There are many different mass distributions that can explain the observations. For both clusters, Limousin et al. provided 1001 maps, which were averaged and integrated over annuli to obtain the average surface mass density ¯Σ and

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Figure 5.1: This image shows A1689 with in purple the mass map of the cluster. The image shows a substructure in the mass distribution to the north-east. Image credit: NASA/ESA, Hubble Space Telescope.

the covariance matrix C. The relation between the 2D mass distribution and ¯Σ is given by ¯

Σn=

M2D(rn)

πr2_n . (5.1)

For A1689, the radii of the annuli lie in the range 3.15355 < r < 876.783 kpc and the log(rn) are all evenly spaced with 0.0380. Thus, there are 149 radii, of which 117 were used in the subsequent analysis. The LensTool program gives constant values for M2D if there are no arclets within a circular annulus. In these cases only the smaller of the r-values is physically significant, which leaves 117 values for r [19]. For A1835, the 149 radii are spaced in the same manner as A1689, and range between 4.027 < r < 1120 kpc. In the same way as in A1689, only 117 of the r_n-values contain physical information [34].

While both clusters are reasonably circularly symmetric, the mass maps of A1689 show, apart from the central dark matter clump, a clump associated with a substructure in the north-east [19]. Figure 5.1 shows the mass distribution of the cluster. The north-north-eastern substructure was masked in the analysis.

To fit the DSE profile and the modified NFW and gNFW profiles, gas data provided by Morandi et al. was used [22]. They analysed Chandra X-ray observations to determine the gas density profile for both clusters. From this data, the north-eastern sector was also masked out.

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From the ¯Σ values that were provided, I obtained values for ¯Σ−Σ, Σ and 2Σ− ¯Σ, where Σ is the surface mass density at a certain radius. To do this, I started with numerically calculating

¯ Σ−Σ, given by ¯ Σ−Σ = −1 2 d ¯Σ d log (r)≈ − 1 2 ¯ Σn− ¯Σn−1 log (rn/rn−1) . (5.2)

In this way, I obtained 58 values for ¯Σ − Σ, since two values of ¯Σ are used to calculate one value of ¯Σ − Σ. To obtain Σ, we also need 58 values of ¯Σ, to subtract from ¯Σ − Σ. We do this by averaging ¯Σnand ¯Σn+1, and the corresponding log (rn) and log (rn+1). This process of averaging over multiple values is called binning and the 58 values thus created will be called the Bin-2 data.

These Bin-2 data for ¯Σ−Σ however are not very smooth, there is a lot of variability and there are even some negative values. They are shown in grey in Figure 5.2. These values do not make physical sense, since the average density within a circle of radius r should always be higher than the density at r. They exist because the raw ¯Σ data have some fluctuations, especially at small radii, due to there being less arcs and arclets there. Therefore, I also computed Bin-4, Bin-6 and Bin-8 data for ¯Σ and ¯Σ−Σ, where I averaged over 4, 6 or 8 values respectively. Figure 5.2 compares these binned values of ¯Σ − Σ from the cluster A1689. From this, we can see that the Bin-8 data look most reliable: they are quite smooth and have less extreme outliers. A1835 showed similar results. The Bin-8 data however do consist of less values: 14 instead of 19 for the Bin-6 data. Therefore, for fitting Σ, ¯Σ − Σ and 2Σ − ¯Σ, the Bin-6 and Bin-8 data are both used and compared.

The reason the quantity 2Σ − ¯Σ is also computed and fitted, is because here the isothermal tail of the mass distribution drops out. This can be seen by filling in an isothermal density distribution_ρ_{= A/r}2 into equations 5.7 and 5.8, which gives zero.

5.3 Density profiles

Five different density models were fitted to the cluster data. First, the NFW profile was used, discussed in section 4.3. Equation 4.51 can also be written as

ρN FW= AR3_s r(r + Rs)2= 200c3ρc(1 + z)3 3[log(1 + c) − c/(1 + c)] R3_s r(r + Rs)2 (5.3) where c is the concentration parameter, z is the redshift of the cluster and ρc is the critical density of the Universe. The parameters that are fitted are A and Rs.

A generalised version of the NFW profile, called gNFW, was then fitted to the data. This profile differs from the regular NFW profile in that it has a variable exponent, n [14]:

ρgN FW=

AR3_s rn_{(r + R}

s)3−n

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Figure 5.2: This figure shows the Bin-2, Bin-4, Bin-6 and Bin-8 data for ¯Σ−Σ, in respectively grey, orange, red and cyan. The negative values for the Bin-2 data are plotted at ¯Σ₋Σ_{= 0.01, to make the} graph clearer. We see that the Bin-8 data seems the most reliable: it is smoother and has less extreme outliers. The data is from cluster A1689.

This profile has three fit-parameters: A, Rsand n.

Both the NFW and gNFW profiles were also modified by adding in a separate gas density which only depends on r, similar to the DSE profile (equation 5.6):

ρN FW+gas= AR3_s r(r + Rs)2+ρgas and ρ_{gN FW+gas}= AR 3 s rn_{(r + R} s)3−n+ρgas (5.5) and these were fitted to the data as well.

Finally, the data was fitted to a more complicated density profile: the DSE profile (equation 4.57) with an added separate contribution to the density by the gas:

ρDSE= 3M1 4πR₁3 exp −(r/R1)1/n1 Γ(1+3n1) + 3M2 4πR3₂ exp −(r/R2)1/n2 Γ(1+3n2) +ρgas . (5.6)

We will for simplicity refer to this just as the DSE profile, instead of the DSE profile with gas contribution. This profile has six fit-parameters: for both the BCG and the halo overall we fit the mass M, the scale radius R and n.

5.4 Fitting

To perform the fits, the program Mathematica was used [13]. To fit the NFW, gNFW, NFW+gas, gNFW+gas and DSE profiles to the data, the following formulae were used [33, 34]:

¯ Σ(r) = 4 r2 Z r 0 s2ρ_{(s)ds +} Z ∞ r 4sρ(s) s +ps2_{− r}2ds (5.7)

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Σ(r) =Z ∞ −∞ρ

(r2_{+ z}2)dz (5.8)

where _ρ is the 3D mass density. For _ρ the mass densities of the NFW, gNFW, NFW+gas, gNFW+gas and DSE profiles were filled in (equations 5.3, 5.4, 5.5 and 5.6).

To obtain the best fit of the models to the data, we have to minimise the value of_χ2, which for ¯Σ is given by [33]: χ2_{( ¯}_{Σ) =}XN i, j h³ ¯Σ(ri) − ¯Σobs_i ´ C0_{i, j}−1 ³ ¯Σ(ri) − ¯Σobs_i ´i . (5.9)

In this formula, the sum over i, j runs from 1 to N, the number of values of ¯Σ. ¯Σ(ri) is the value of ¯Σ at ri, computed using 5.7 and the appropriate density profile. ¯Σobs_i is the corresponding value of ¯Σ from the data. These are subtracted from each other to see how much the observed and predicted values differ. The middle term, C0−1_{, is the inverse of the modified covariance} matrix, given by C0=σ2_SLA +C.σSLis a shift and A is an N-dimensional diagonal matrix with

¯ Σ2

n on the diagonal. This method of adding a positive, constant term (σ2SL) to the diagonal of the matrix is called Tikonov regularisation. It is used because the elements of the covariance matrix differ by many orders of magnitude, with the lowest values of the order 10−9for A1689 and 10−15_{for A1835. The smallest values are unphysical, and through Tikonov regularisation} they are cut off. The value for the shift used in all the fits isσSL= 0.001.

We calculate_χ2forΣ, ¯Σ−Σ and 2Σ− ¯Σ in the same way by filling in the appropriate observed and fit values and covariance matrix. To determine the best fit, we calculateχ2/ν, where we divide equation 5.9 by the degrees of freedomν. ν is given by the number of data points N minus the number of parameters fitted. For the NFW fitsν= N −2, for gNFWν= N −3 and for DSEν_{= N − 6. Ideally, the value of}χ2/νis equal to 1.

For the NFW and gNFW profiles, the quantities ¯Σ, Σ, ¯Σ − Σ and 2Σ − ¯Σ were all fitted separately. For the NFW profile, I fitted unbinned and Bin-2 ¯Σ data and Bin-6 and Bin-8 data for the other quantities to compare them and see what data provided the best fit. The gNFW fit was only performed for the unbinned ¯Σ and Bin-8 data, since these provided the best fits for the NFW profile. The NFW+gas and gNFW+gas profiles were also fitted to the unbinned

¯

Σ and Bin-8 data. For the DSE profile, all quantities were fitted at once, obtaining one set of parameters. This was done for the unbinned ¯Σ data and for both Bin-6 and Bin-8 data.

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Chapter 6

Results and discussion

As described in the previous chapter, ¯Σ, Σ, ¯Σ − Σ and 2Σ − ¯Σ data from two galaxy clusters, A1689 and A1835, were fitted to five different density profiles: the NFW, gNFW, NFW+gas, gNFW+gas and DSE profiles. In this chapter, the results of the fits will be presented and discussed.

6.1 The average surface mass density ¯

Σ

We will start by discussing the fits to the ¯Σ data, provided by Limousin et al.. Figure 6.1 shows from left to right the NFW, gNFW and DSE fits to the ¯Σ data for A1689 with the corresponding

χ2_/_ν_{-values, Figure 6.2 shows the same for A1835. As can already be seen by eye, the DSE} profile provided the best fit to the data. Itsχ2/ν-value is also the lowest,χ2/ν_{= 0.85 for both} clusters. The NFW and gNFW profiles provide reasonable fits to the data, with_χ2/_ν-values between 1.03 and 1.85. When comparing the two fits, we see that for both clusters the gNFW fit has a lower_χ2/_ν-value than NFW, but it does not necessarily provide a better fit to the data. As you can see in the figures, the gNFW profile fits the values on the right side, for large r, better than the NFW profile does. However, the gNFW profile does not pass through the values on the left side, for small r. Since these values have larger error bars, they are less statistically significant to theχ2-test, and are all but ignored in the gNFW fit. These larger errors for small radii arise from the fact that there are less arclets in this inner region.

I have also fitted the NFW profile to Bin-2 ¯Σ data, to see how binning the data changes the fitting results. For both clusters, theχ2/ν-values were lower for the Bin-2 data: for A1689

χ2_/_ν

= 1.42 and for A1835 χ2/ν= 1.24. However, since the unbinned data was already quite smooth and had no extreme outliers, I prefer to use the unbinned to the binned data. The values of the fit-parameters of both the unbinned and binned ¯Σ data are shown in Table 6.1. This table shows that for A1835, the relative error on A and Rsis significantly greater for the

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Figure 6.1: This figure shows the NFW, gNFW and DSE fits to the ¯Σdata for A1689. Theχ2_/_ν-values

are 1.85, 1.03 and 0.85 respectively.

Figure 6.2: This figure shows the NFW, gNFW and DSE fits to the ¯Σdata for A1835. Theχ2/ν-values are 1.51, 1.24 and 0.85 respectively.

binned than for the unbinned data. For A1689, there is little difference. The parameters are discussed further in section 6.5.1.

I have also modified the NFW and gNFW profiles by adding the gas density (equation 5.5). By fitting this NFW+gas profile to the data, we obtainχ2/ν= 1.89 for A1689 andχ2/ν= 1.57 for A1835. These are slightly higher values than we obtain by fitting the unmodified NFW profile. For the gNFW+gas profile we obtainχ2/ν= 1.06 for A1689 andχ2/ν= 1.34 for A1835. These again are higher than for the unmodified gNFW profile.

I expected the fits to improve by adding the gas density contribution in separately, but they did not. This can mean that the NFW profile already describes the gas density of the cluster well. The gas density that is added to the profile has low values for small r, and only starts to play a role for larger radii. As we can see in Figures 6.1 and 6.2, both the NFW and gNFW profiles fit quite well to the data for large r. Especially for the gNFW profile, the largest differences between the model and data occur for small r, where the gas density is relatively low. The fact that the NFW and gNFW profiles provide worse fits to the data than the DSE profile is thus not due to the gas density.

Gravitational lensing by galaxy clusters