Morphology and Kinematics of Z ∼ 1 Galaxies As Seen Through Gravitational Telescopes

Home , Abell 2667, Abell 2744, Abell 370, Cetus

Universita` degli Studi di Napoli “Federico II”

Dipartimento di Fisica “Ettore Pancini” Corso di Laurea in Fisica

Anno Accademico 2017/2018

Tesi di Laurea Magistrale

Morphology and kinematics of z ∼ 1 galaxies as seen through gravitational telescopes

Relatore: Prof. Giovanni Covone Candidato: Maria Vittoria Milo Matricola: N94000352 Se parlassi le lingue degli uomini e degli angeli, ma non avessi amore, sarei un rame risonante o uno squillante cembalo. Se avessi il dono di profezia e conoscessi tutti i misteri e tutta la scienza e avessi tutta la fede in modo da spostare i monti, ma non avessi amore, non sarei nulla. Se distribuissi tutti i miei beni per nutrire i poveri, se dessi il mio corpo a essere arso, e non avessi amore, non mi gioverebbe a niente.

(1 Corinzi 13:1-3) ii

Abstract

Observing high−z galaxies is an hard challenge for astrophysicists. As much we go back in time (and therefore far away in space), the Universe in which we live becomes smaller, denser and hotter. On the other hand, at high redshift, astronomical objects result as fainter as smaller to make their detection impossible even for the most powerful today’s telescopes. Hence, trying to determine the morphology of these high−z galaxies and even to characterize their physical properties (such as kinematics) would result just a dream of a crazy astrophysicist. Fortu- nately, Nature offers a way out, allowing us to overcome the obstacle. Telescopes able to observe regions of the Universe never before reached by any human or artificial “eye” already exist! These are galaxy clusters, the largest known gravitationally bound structures in the Universe. Through a phenomenon known as strong gravitational lensing, galaxy clusters can provide enormous magnification and amplification of distant sources that lie behind them, allowing us to study their physical and morphological properties. This is the reason why they are considered “natural telescopes” or “cosmic telescopes”. In this work of thesis, I took advantage of this extraordinary role that galaxy clusters have to reconstruct the morphology of z ∼ 1 galaxies and to characterize their kinematics . In Chapter 1, I will deal with the main characteristics of ΛCDM Model of Cosmology and General relativity. The ΛCDM model describes a flat universe in which ∼ 75 percent of the energy density is due to a cosmological constant Λ, ∼ 21 percent is due to cold dark matter (CDM) and the remaining 4 percent is due to the baryonic matter out of which stars and galaxies are made. In such a Universe, the formation of a structure (like a galaxy) could be linked to density perturbations by quantum fluctuations at early times. General relativity is a revolutionary theory of gravity formulated by Einstein in 1915, according to which the structure of space-time is determined by the mass distribution in the Universe through field equation. A direct conse- iii

quence of General Relativity is gravitational lensing, an astronomical phenomenon characterized by the bending of the paths of the light rays by gravitational field of massive bodies. This will be the principal argument of Chapter 2. In particular, I will describe the so-called strong gravitational lensing regime, i.e. lensing produced by galaxy clusters in the case in which occurs the production of giant arcs, multiple images and arclets. In Chapter 3 there will be a brief description of what are galaxy clusters and I will also introduce a way in which strong lensing can be used to build models of cluster mass distribution through a Bayesian approach. Finally, in Chapter 4 I will present my data analysis. Through the Cleanlens mode in the gravitational lensing software Lenstool, I have reconstructed the morphology of three different galaxies, at three different redshift, lensed by three different massive galaxy clusters (A370, AS1063 and A2667). Furthermore, taking advantage of the relatively new spectroscopic technique provided by MUSE, I have derived the 2D kinematics of the galaxy lensed by A370 and producing the giant gravitational arc known as the “Dragon”. In particular, I have obtained the rotational curve of this galaxy through a study of the [OII] emission line in the spectrum and I have deduced its total mass. The same procedure has been applied to get the rotational curve and the total mass of a galaxy lensed by AS1063. Finally, I have represented my small sample of three galaxies lensed by A370, AS1063 ans A2667 on a graph showing the stellar-mass Tully-Fisher Relation (smTFR) for 64 galaxies, realized by Puech et al. [1].

Contents

1 Introduction 1 1.1 The ΛCDM Model of Cosmology ...... 1 1.2 General Relativity ...... 5 1.3 Galaxy formation and evolution ...... 8 1.3.1 Galaxy formation: state of art ...... 8 1.3.2 Galaxy formation: open questions ...... 10 1.3.3 Evolution of the Tully-Fisher ...... 12

2 Gravitational lensing 15 2.1 What is gravitational lensing? ...... 15 2.2 The Fermat Principle ...... 19 2.3 The lens equation ...... 23 2.3.1 The eﬀective lensing potential ...... 29 2.4 Lens mapping and magniﬁcation ...... 30 2.5 Strong and weak lensing regimes ...... 37 2.6 Strong gravitational lensing ...... 38

3 Galaxy clusters as gravitational lenses 43 3.1 What are galaxy clusters? ...... 43 3.2 Galaxy clusters as gravitational telescopes ...... 46

v vi Contents

3.3 Strong lensing modeling ...... 47 3.3.1 Bayesian approach ...... 50 3.4 From the source to the lensed image ...... 51 3.5 Lenstool ...... 53

4 Data analysis 55 4.1 Frontier Fields ...... 55 4.2 Our sample of lensed galaxies ...... 56 4.3 Source plane reconstruction in Abell 370 ...... 57 4.4 Source plane reconstruction in Abell S1063 ...... 62 4.5 Source plane reconstruction in Abell 2667 ...... 66 4.6 Integral Field Units ...... 69 4.6.1 The integral ﬁeld spectrograph MUSE ...... 72 4.7 Kinematics of the two galaxies lensed by A370 and AS1063 . . . . . 75 4.8 Results ...... 82

5 Conclusions 85

References 87 Chapter 1

Introduction

In this Chapter, I will explain the main characteristics of the two pillars upon which modern cosmology (and hence this thesis) rest: the ΛCDM Model and General Relativity. The ﬁrst one is just one of the many possible scenarios proposed to date to describe the Universe in which we live while the second one is a theory of gravity formulated by Einstein in 1915 and published in 1916 [2] according to which the structure of space-time is determined by the mass distribution in the Universe through ﬁeld equations. Moreover, I will give a brief description of the process according to which it is believed a galaxy forms.

1.1 The ΛCDM Model of Cosmology

The ΛCDM Model of Cosmology is also called the Standard Model because, at the moment, is the simplest model that better reproduces the observations, providing reasonably good account of the following properties of the cosmos:

• the existence and structure of the cosmic microwave background (CMB);

• the large-scale structure in the distribution of galaxies;

1 2 Chapter 1. Introduction

• the abundance of elements (Big Bang Nucleosynthesis);

• the accelerating expansion of the Universe observed in the light from distant galaxies and supernovae.

The expansion of the Universe was conﬁrmed in 1929 when Edwin P. Hubble [3] observed a redshift in the spectra of distant galaxies, which indicated that these galaxies were receding from us at a velocity proportional to their distance from us. The ΛCDM model is based upon the Cosmological Principle, i.e. the hypothesis that the Universe, at large scales (I mean at scales larger than 100Mpc, that correspond about at 300 million light years), is spatially homogeneous and isotropic. The adjective “homogeneous” refers to the fact that our point of observation is neither diﬀerent nor privileged compared to any other point within our Universe. “Isotropic” means that there is no preferential direction in the Universe, that is, in any direction we observe, what we see must be in principle the same. The most general metric satisfying homogeneity and isotropy at large scales is the Friedmann-Lemaˆıtre-Robertson-Walker (FRW) metric, written here in terms

2 µ ν of the invariant geodesic distance ds = gµνdx dx :

dr2 ds2 = dt2 + a2(t) + r2(dθ2 + sin2 θdφ2) , (1.1.1) 1 − Kr2 characterized by just two quantities:

• a(t) is the scale factor of the Universe at the time t;

• K is a constant which characterizes the spatial curvature of the Universe and that can assume only three possible values:

K = +1 for a spherical geometry, that is a closed space-time,

K = 0 for a ﬂat or Euclidean geometry, that is a ﬂat and open space-time,

K = −1 for an hyperbolic geometry, that is open and divergent space-time. 1.1. The ΛCDM Model of Cosmology 3

The Standard Model describes a Universe filled by a classical and perfect fluid which satisfies Bianchi’s identity and comes in the form of radiation in the pri- mordial epoch and in the form of dust in the present epoch. The equations that define this model are:

a¨ 4πG = − , (1.1.2) a 3c2

a˙ 2 Kc2 8πG + = ε, (1.1.3) a a2 3c2

a˙ ε˙ + 3 (ε + p), (1.1.4) a

p = γε, (1.1.5)

2 cs 2 where ε = ρc is the total energy density of the Universe and γ = ( c ) is an adiabatic index describing how the fluid propagates. The first two equation come from the Einstein field equations (that I will discuss later): 1 8πG R − g R − Λg = T (1.1.6) µν 2 µν µν c4 µν

where Rµν is the Ricci tensor, describing the local curvature of space-time, R is the curvature scalar,gµν is the metric, Tµν is the energy-momentum tensor of the matter content of the Universe and Λ is the cosmological constant, which was introduced by Einstein to obtain a static Universe (the “biggest blunder” of his life). Eq.(1.1.2) is called Friedman equation and represents the spatial part of the field equations under the hypothesis of homogeneous and isotropic Universe and of a perfect fluid while Eq.(1.1.3) represents the temporal part and is called the energy equation. Furthermore, Eq.(1.1.4) is the Bianchi’s identity while Eq.(1.1.5) is a state equation which simply specifies the relation between the pressure p of the fluid 4 Chapter 1. Introduction

and its density ρ.The density ρ can be considered made up of 3 components: a non- relativistic matter component (dark matter + baryons), a radiation component, and a possible vacuum energy (cosmological constant) component. I will denote their energy densities (written in terms of mass densities) at the present time t0 by ρm,0 , ρr,0 and ρΛ0, respectively. 3H2(t) These quantities can be expressed in term of a critical density ρcrit = 8πG

ρm,0 through other three quantities: Ωm,0 = that, from current observational ρcrit,0 ρr,0 −5 −2 constraints, is equal to 0.27 ± 0.05, Ωr,0 = that is equal to 4.2 × 10 h ρcrit,0 ρΛ,0 and ΩΛ,0 = . The present day energy density provided by the cosmological ρcrit,0 constant is constrained rewriting the Friedmann equation in this way:

2 8πG 2 Kc ρΛ,0 = H0 [1 − Ωm,0 − Ωr,0 + ] (1.1.7) 3 a0 and, setting K = 0 (as the observations of the microwave background suggest), we obtain:

ΩΛ,0 = 1 − Ωm,0 − Ωr,0. (1.1.8)

Data from WMAP [4] combined with other observations give ΩΛ,0 = 0.75±0.02. Therefore, the ΛCDM model describes a flat universe (K = 0) in which ∼ 75 % of the energy density is due to a cosmological constant Λ, ∼ 21 % is due to cold dark matter (CDM) and the remaining 4 % is due to the baryonic matter out of which stars and galaxies are made. The concept of dark matter has been introduced to explain some gravitational effects observed in very large-scale structures that otherwise cannot be accounted for considering only the quantity of observed matter. For example,the “flat” rotation curves of spiral galaxies or the gravitational lensing of light by galaxy clusters can be fully understood supposing this “additional” mass. This kind of matter is non-baryonic, i.e. it is not formed by protons, neutrons or electrons (even if electrons are fermions), is dissipationless, i.e. it cannot cool by radiating photons and collisionless, i.e. dark matter particles 1.2. General Relativity 5

interact with each other and with other particles but only through gravity . In particular, cold dark matter means the hypothetical form of dark matter formed by “slow” and therefore “cold” particles. According to this theory, the structures grow hierarchically through the collapse of small objects under their self-gravity at ﬁrst and then through their merging. Although dark matter and dark energy are responsible for more than 95 % of the energy density of the Universe, their nature is still unknown. This fact represent the most important challenge in Modern Cosmology 1

1.2 General Relativity

The second pillar on which my thesis rests is General Relativity, an extension of special relativity to non-inertial systems. In 1905, with the theory of special relativity, Einstein radically revised the concepts of space and time of classical physics. Brieﬂy, what Galileo and Newton said is that:

1. space and time are absolute, i.e. the measurement of spatial lengths and time intervals provides the same results in any reference system;

2. space is ﬂat, i.e. the geometry of Euclid is valid;

3. the laws of mechanics always have the same form in inertial reference systems: there is no experiment that can distinguish a reference system from another one in uniform rectilinear motion with respect to the ﬁrst (Galilean relativity);

1In this regards, the quote by A. Einstein appears appropriate: ”One thing I have learned in a long life: that all our science, measured against reality, is primitive and childlike-and yet it is the most precious thing we have.” Source: Letter to Hans Muehsam (9 July 1951), Einstein Archives 38-408, quoted in The Ultimate Quotable Einstein 2010 by Alice Calaprice, p. 40. 6 Chapter 1. Introduction

4. the universal gravitation law explains the attraction of bodies due to their mass.

However, classical physics is in contradiction with Maxwell’s theory of electromagnetism. In fact, Maxwell’s equations are not invariant in form with respect to the group of Galileo transformations and, therefore, according to the Galilean Relativity principle, two inertial observers should have used different equations to describe the same electromagnetic phenomena. Moreover, Maxwell’s theory pre- dicts that the electric and magnetic field propagate in empty space at a finite and constant speed: this is impossible according to the Galilean Relativity since the speed measured by a moving observer must respect the law of transformation of the velocities. To solve these problems, Einstein introduced 2 postulates that are the basis of the special relativity.The first postulate establishes the covariance of the laws of electromagnetism and of mechanics in all inertial reference systems, while the second states that the speed of light in vacuum is the same in all reference systems. Space and time are no longer absolute entities but rather a “continuum”, united under the concept of space-time which is flat and is described by the Minkowsky metric gµν = (1, −1, −1, −1). In order to describe the Universe as a whole, such a metric is not suitable and there is a need to formulate a theory of gravity that special relativity does not provide. For these reasons, Einstein, about 10 years later, published the theory of General Relativity that confirmed him as a “star” of the scientific panorama. The basis of General Relativity is the assumption, known as the Equivalence Principle, that an acceleration is indistinguishable locally from the effects of a gravitational field and, therefore, that the inertial mass

Gmgm F = mia is equal to the gravitational mass F = r2 . Einstein explained this fact through a mental experiment: a person in a box isolated from the outside world can not tell if he is in a stationary box, under the eﬀect of Earth’s gravity, or in a box moving upwards with an acceleration ~a whose intensity is equal to that 1.2. General Relativity 7

Figure 1.1: Artistic representation showing the distortion of space-time due to the presence of three massive bodies, represented as coloured spheres. The warping caused by each sphere is proportional to its mass. Credit: ESA of ~g; at the same time, if we suppose that the box is an elevator and that its rope suddenly breaks,the same person can not distinguish whether he is in free fall or in absence of gravity. Space-time can be compared to an elastic sheet held in tension (see Fig.1.1): when a mass is placed on it, the sheet is deformed and therefore any body that will move on it will no longer follow rectilinear trajectories. Hence, for Einstein the cause of the motion of objects (for example, the planets around the Sun) is the modification of the geometry of the space in which the object moves. The space-time in which the object moves is curved due to the presence of large masses and this curvature determines the trajectory of the object. There is no force acting remotely in the Newtonian sense: gravity is explained in geometrical terms. A direct consequence of this new point of view is that the intensity of a gravitational field affects the way in which the time flows: time flows more slowly in those areas of space-time in which the gravitational field is more intense, i.e those areas characterized by a major curvature (for example, this situation is verified near a black hole). According to General Relativity, therefore, space- 8 Chapter 1. Introduction

time is a curved space of 4 dimension t, x, y, z. The trajectories followed by an object in presence of a mass are called “geodetics” that are particular curves that describe the shortest paths between points in a particular space. The equation that describes the link between the presence of matter / energy at the metric, i.e. the geometry of the curved space-time is the Einstein ﬁeld equations:

1 8πG R − g R − Λg = T (1.2.1) µν 2 µν µν c4 µν The meaning of this equation is that matter produces the curvature of space and the curvature determines, in turn, the motion of matter in space. And light? How does light behave in presence of a mass, i.e of a gravitational ﬁeld? The answer at this question will be the starting point of Chapter 2.

1.3 Galaxy formation and evolution

In this thesis we aim at studying the physical properties of disk galaxies when the Universe was about 7.5 Gyrs old, by means of the so-called “gravitational telescopes” (see Chapter 3). This observational study is important because it can shed light on the physical mechanism driving galaxy evolution. In this Section I brieﬂy describe our current knowledge in this ﬁeld.

1.3.1 Galaxy formation: state of art

According to recent numerical simulations, three dimensional distribution of matter in space is described in terms of the so-called “cosmic web” [5]. This is a network formed by knots and links comparable to the human nervous system (see Fig.1.2). Knots are those regions characterized by the highest gravitational force that, therefore, collapse forming galaxies which are arranged into clusters and super 1.3. Galaxy formation and evolution 9

Figure 1.2: Representation of the large-scale distribution of matter obtained through a digital simulation. Credit: V. Springel, Max-Planck Institut fur Astrophysik, Garching.

clusters. The connection between several knots is ensured by filamentary structures that are made up of gas and dark matter which keeps the galaxies bound together under the force of gravity. How can be explained the formation of galaxies in a spatially homogeneous and isotropic Universe? One possible solution to this problem is to assume some deviations from perfect uniformity. In this case, the formation of the structures observed in today’s Universe could be linked to density perturbations by quantum fluctuations at early times when Universe was smaller, denser and therefore hotter. These perturbations grow with time giving rise to gravitational instabilities: a region with an initial density higher than the mean will attract its surroundings more strongly than the average becoming denser. Viceversa, regions characterized by a lower initial density become more δρ rarefied. When the perturbation reaches over-density ρ ∼ 1, it starts to collapse. What happens after this collapse depends essentially by the matter content of the perturbations (if it consists of baryonic gas or cold dark matter). According to 10 Chapter 1. Introduction

CDM models, a perturbation contains both baryonic gas and collisionless dark matter. In these cases, the collapse of an object is characterized by the violent relaxation of dark matter that leads to the formation of a dark matter halo and by the rising of strong shocks in the baryonic gas located in the potential well of this halo. The shocked and hot gas can undergo different processes that cause it to cool down.The cooling and the flowing inwards a dark matter halo of the gas involves its segregation from the dark matter: this occurs when gas self-gravity dominates over the halo gravity. At the same time, the collapse makes the gas denser and hotter and could provoke the fragmentation of the gas cloud into small,high-density cores that may eventually form stars giving rise to a protogalaxy at the center of the dark matter halo. An important role in galaxy formation theory is played by feedback processes an by mergers. The first ones are processes that prevent the gas from cooling, or reheat it after it has become cold: this would explain the observational fact that only a relatively small fraction of all baryons are in cold gas or stars. Mergers are an important feature of currently popular CDM cosmologies according to which dark matter halos grow hierarchically (as I have mentioned in the previous section). This means that the coalescence (merging) of smaller progenitors leads to the formation of larger halos. This kind of formation process is usually called a hierarchical or “bottom-up” scenario. Therefore, the formation history of a dark matter halo is described in terms of a “merger tree” that includes all its progenitors, as shown in Fig.1.3

1.3.2 Galaxy formation: open questions

So far, I have described a possible explanation of how galaxies might form, but a lot of questions about this process are still without answer and many details are still unclear. For example, the mass fraction of, and the time scale for, a self-gravitating cloud to be transformed into stars can not be still predicted. At 1.3. Galaxy formation and evolution 11

Figure 1.3: A schematic “merger tree”, illustrating how a dark matter halo forms through several merging of smaller halos starting from an epoch t1 to an epoch t4. Source: [4]. present, even the initial mass function (IMF), i.e. the empirical function that describes the initial distribution of masses for a population of stars, has to be assumed ad hoc checking its validity by comparing model predictions to observations. This is a tough issue because, as we know, the evolution of a star (in particular its luminosity as function of time and its eventual fate) is largely determined by its mass at birth. Thus, in order to predict observable quantities for a galaxy, its IMF must be known. According to observations, there are two modes of star formation: quiescent star formation in rotationally supported gas disks, and starbursts. The latter are characterized by much higher star-formation rates and require the accumulation of large amounts of gas in a small volume (the nucleus of a galaxy). Relatively to these two diﬀerent modes of star formation, we don’t know what is the fraction of stars formed in the quiescent and in the starburst mode or if both modes produce stellar populations with the same IMF. Another 12 Chapter 1. Introduction

important question is whether the sizes and morphology of galaxies were set at formation, or are the result of later dynamical processes such as ram-pressure and tidal stripping. In this context bulges, for example, may be interpreted as rem- nant of the first stage of galaxy formation, may reflect an early merger which has grown a new disk, or may result from the buckling of a bar. Moreover, is the famous Hubble sequence still appropriate in an high−z Universe? In other words, could early times galaxies be classified according to this morphological classifica- tion scheme? An exhaustive answer to these questions can be given through an accurate study of the high−z galaxies morphology and of their physical properties. At first sight, this may seem impossible since, at high redshift, the galaxies are smaller and fainter and, therefore, difficult to detect even from the most powerful telescopes we have today. However, a phenomenon, called gravitational lensing (see Chapter 2 for details), allows us to overcome the obstacle: as a magnifying glass would do, it enlarges high−z galaxies that will result brighter in the core of massive galaxy clusters. Physical properties of these galaxies can be investi- gated through use of integral field spectroscopy (see Chapter 4), a new technique to obtain spectra of an astronomical object. In my thesis both these aspects are present and I have focused my attention on galaxies with redshift z ∼ 1, when the Universe was about 7,5 Gyr old. The method that I will describe in Chapter 4 may be used even to better understand the evolution of galaxies - another complex problem - matching, for example, the morphology of galaxies at different redshift.

1.3.3 Evolution of the Tully-Fisher

Unlike elliptical galaxies that are almost similar to each other, spiral galaxies show great diversity in luminosity, size, rotation velocity and rotation-curve shape. What the spiral galaxies have in common is a scaling relation between luminosity, L and rotation velocity (usually taken as the maximum of the rotation curve, Vmax). 1.3. Galaxy formation and evolution 13

Figure 1.4: The logarithm of the zero-point, b, as a function of the redshift z. Source: [6].

This relation is known as the Tully-Fisher relation and, typically, is expressed as:

α L = kVmax, (1.3.1) where k is the zero-point while α is a parameter that varies between 2.5 and 4 and is larger in redder bands. In terms of the stellar mass, M∗, the Tully-Fisher relation becomes: M V α = k max , (1.3.2) M V0 where V0 is a reference velocity usually taken equal to 220 km/s. If we consider the logarithm of Eq.(1.3.3), we obtain:

M V log = b + α log max , (1.3.3) M V0 where b = log k is an important parameter because it it expect to depend on the redshift of the galaxy [6], (see Fig. 1.4). 14 Chapter 1. Introduction

In particular, b results quite constant up to z ' 1 but, beyond this cosmic epoch, it rapidly decreases. For this reason, the Tully-Fisher relation is very useful in order to understand the evolution of galaxies. In Chapter 4, I will analyze the kinematics of three galaxies located at z ∼ 1, hence in the “critical” epoch for the parameter b. My sample is small therefore it is not possible to draw a robust conclusion based on a statistical study, at moment. However, we performed a pilot study that is a starting point for future research. Chapter 2

Gravitational lensing

Light, just like massive particles, is deflected in a gravitational field. This is one of the specific predictions by Einstein’s theory of gravity, General Relativity. The astronomical phenomenon produced by the bending of the paths of the light rays by the gravitational field of massive bodies is called gravitational lensing. In this Chapter I will describe in detail this phenomenon and its fundamental characteristics. In particular, I will deal with lensing produced by galaxy clusters in the case in which occurs the production of giant arcs, multiple images and arclets: the so called strong lensing phenomenon.

2.1 What is gravitational lensing?

In his “Optyks” (1704), Newton was the first to discuss the problem of the deflection of light by astronomical objects but Cavendish, around 1784, was the first to address this problem quantitatively. He started from some assumptions: the principle of equivalence, Newton’s law of gravity and the corpuscular nature of light. Considering that the orbit described by a light corpuscle, emitted at infinity with velocity c, in presence of a gravitational field produced by a mass M, could only

15 16 Chapter 2. Gravitational lensing

Figure 2.1: Deﬂection of a light ray in passing near the Sun. be an hyperbole, he computed the deﬂection angle of its trajectory with impact parameter b 2GM ∆θ = . (2.1.1) c2b

If the lens is the Sun (M = M ) and the “light particle” grazes its surface

(b = R ), the deflection angle is about 0”.875. However, his value is not correct for a factor of two! In 1919, in fact, a total solar eclipse allowed Eddington to measure the deflection suffered by the light of the stars in passing near the Sun and this was in good agreement with that computed theoretically by Einstein through General Relativity. Stars were shifted by their “unperturbed position” (i.e., the position at night) of an angle

4GM ∆θ = 2 ' 1”.75, c R as shown in the Fig.2.1. This was the second of many checks of General Relativity (the ﬁrst one was the precession of the perihelion of the orbit of Mercury). According to this theory, light propagates along lines, called “null-geodesics”, in a space-time which is curved by the mass of the so-called “Aˆ gravitational lens”. The bending of a light ray is a direct consequence of the Equivalence Principle. To demonstrate this statement let’s consider again the mental experiment elaborated by Einstein that I have described in chapter 1 supposing, this time, that the box, in which the observer 2.1. What is gravitational lensing? 17

Figure 2.2: In the left side of the image, (a), there is a woman with a torch in a box accelerated upward; in the right side, the same woman is in a stationary box on the Earth, (b). In both cases light is curved according to equivalence principle. is, has a hole on its left side. Imagine that the box is accelerated upwards with an acceleration ~a whose intensity is equal to the intensity of ~g and that a light ray entering the hole propagates towards right. The observer will see that the ray hits the right wall of the box at a lower point than it enter. Therefore, since the box is accelerated, the light ray appears curved (see Fig.2.2 (a)). Now, let’s consider the opposite experiment: the box is not accelerated but is in the Earth’s gravitational ﬁeld (see Fig.2.2 (b)). If light will not result deﬂected by gravity, then the observer has the possibility to discriminate between gravity and acceleration, violating the principle of equivalence. The bending of light by massive bodies gives rise to several important phenomena:

• observation of multiple images of a single source, due to the fact that multiple paths around a single mass become possible;

• distortion of the image, caused by the diﬀerent way in which two neighboring 18 Chapter 2. Gravitational lensing

rays may be deﬂected (for example, a ray which passes closer to the lens will be bent more than another one). This implies that sources might appear stretched, larger or smaller than they originally are;

• magnification (or demagnification) of the image, that is a consequence of the conservation of the photon number, i.e, of the surface brightness. Flux amplification (magnification) is proportional to image size: if the source is enlarged it will appear brighter, otherwise fainter;

• relative time delay between two images that has two components:

t = tgeom + tgrav.

tgeom is due to the diﬀerent path length of the deﬂected light rays compared

to the unperturbed ones while tgrav comes from the slowing down of photons traveling through the gravitational ﬁeld of the lens.

What kind of lens are the gravitational lenses? Compared to an ideal thin convex lens, gravitational lenses are very bad, as Fig.2.3 shows. In this ﬁgure, rays coming from a point-like source placed at great distance arrive at the lenses forming a parallel beam. In the case of the convex lens (Fig. 2.3 (a)), the ray passing through the center of the lens is not bent while rays displaced far from the center are bent through increasingly greater angles so that all rays are brought to a single focus at a “real” image Io. For a point-like gravitational lens (Fig.2.3 (b)) the situation is exactly the opposite: it provides the greatest bending for rays

4GM at small impact parameter b according to c2b . Since the bending decreases with distance from the center, the outgoing rays diverge and do not create a single “real” image: gravitational lenses, thus, do not focus! In particular, when source, point-mass lens and observer are perfectly aligned (see Fig.2.3 (b)), a ring, called “Einstein ring”, is formed. 2.2. The Fermat Principle 19

Figure 2.3: Comparison of (a) an ideal convex thin lens and (b) a gravitational point-mass lens, each for incident parallel rays from an inﬁnitely distant source. Source: [7].

2.2 The Fermat Principle

An equivalent way to describe light deﬂection is through Fermat’s principle which states that, among all the possible paths connecting 2 points A and B (see Fig.2.4), light travels the one that requires the shortest travel time.

Figure 2.4: Some Possible paths connecting two points A and B.

In geometrical optics, when light traverse a medium with refractive index n, its

1 velocity is reduced by a factor n with respect to its velocity in vacuum, c. Thus, the time required for light to travel some distance in such a medium is n times 20 Chapter 2. Gravitational lensing

the time light takes to go the same distance in vacuum. The computation of the minimum time required for light to go from a point to another leads to Snell’s law. An analogy between the propagation of light in a gravitational ﬁeld and the propagation of light in a medium with refractive index n can be established: a gravitational ﬁeld behaves like a medium with refractive index n slowing down the light. To compute the gravitational refractive index n, it is necessary to start from the Fermat’s principle: light will follow a path along which the travel time

Z n dl c will be extremal. So, the path ~x(l) followed by light is that for which:

Z B δ n[~x(l)] dl = 0, (2.2.1) A where the starting point A and the end point B are kept ﬁxed. Now, it’s necessary to assume that the lens has two characteristics:

• the lens must be weak, that means Φ/c2 1, where Φ is the Newtonian gravitational potential;

• the lens must be small compared to the overall dimensions of the optical system composed of source, lens and observer.

When there is no lens, the metric that describes space-time is the Minkowski metric:

ηµν = (1, −1, −1, −1) 2.2. The Fermat Principle 21

If a weak lens perturbs space-time, the metric will be:

  2Φ 1 + c2 0 0 0    2Φ   0 −(1 − c2 ) 0 0  gµν =   , (2.2.2)  2Φ   0 0 −(1 − c2 ) 0    2Φ 0 0 0 −(1 − c2 ) for which the line element is

2Φ 2Φ ds2 = g dxµdxν = (1 + )c2dt2 − (1 − )(d~x)2 µν c2 c2 For light ds2 = 0, so:

2Φ 2Φ (1 + )c2dt2 = (1 − )(d~x)2 c2 c2 The speed of light in the gravitational ﬁeld is thus:

s 2Φ |d~x| 1 + 2 2Φ c0 = = c c ≈ c(1 + ) dt 2Φ c2 1 − c2 where we have used that Φ/c2 1 by assumption. The index of refraction is thus

c 1 2Φ n = = ≈ 1 − . (2.2.3) c0 2Φ c2 1 + c2 With Φ ≤ 0, n ≥ 1, and the light speed c0 is lower than in vacuum. Therefore, a gravitational ﬁeld acts like a medium of refractive index n which slows down the light at a speed c0. Moreover, from Eq.(2.2.3), it is clear that n depends only on the local gravitational potential and not on the wavelength λ of the light ray that arrives at the lens so, despite an ideal lens, we have no formation of rainbows. 22 Chapter 2. Gravitational lensing

The solutions of the variational problem (2.2.1) are the well known Euler equations that can be written in this way:

d~e 2 = ∇ n(~x) = − ∇ Φ(~x) (2.2.4) dl ⊥ c2 ⊥ where vector ~e is a tangent vector to the light path. Hence, a light ray passing near a weak lens is deﬂected by an angle:

Z λB ˆ 2 ~ ~α = ~ein − ~efin = 2 ∇⊥Φ(~x)dλ (2.2.5) c λA where λ is a curve parameter. Since ∇~ Φ points away from the lens centre,the deflection angle ~αˆ points towards it. The last integral is non-trivial because it has to be computed on the actual light path that depends on the deflection. Since Φ/c2 1, ~αˆ is expected to be small so, the so-called Born approximation can be used and the integral can be performed above the unperturbed light path. For example, a light ray that is initially traveling along the z-direction and passes a lens (M) at z = 0, with impact parameter b, undergoes a deflection of an angle:

Z +∞ ˆ 2 ~ ~α(b) = 2 ∇⊥Φdz . (2.2.6) c −∞ This formula tells us that the deﬂection is caused by the projection of the Newtonian gravitational potential on the lens plane. With “lens plane” I refer to the fact that, since the physical size of the lens is generally much smaller than the distances between observer, lens and source, it is possible to approximate the lens to a planar distribution of matter which is well described by a surface mass density. This assumption holds even in the case in which the lens is a massive galaxy cluster. For example, if a source at redshift z ∼ 1 is lensed by a galaxy cluster at redshift z ∼ 0.3, the distances between source,lens and observer are of several Gpc, while the typical size of a galaxy cluster is of order of a few Mpc. 2.3. The lens equation 23

Even the sources are assumed to lie on a plane, called the source plane. This is the so called thin-screen approximation.

2.3 The lens equation

Let us consider a mass concentration placed at redshift zL, corresponding to an angular diameter distance DL, which behaves as a lens deflecting the light rays coming from a source at redshift zS, corresponding to an angular distance DS. There is a simple geometrical relationship between the real position of the source,S (that observed in absence of lens) and its observed position on the sky, I (see Fig.2.5.) To deduce this relationship, let’s consider the geometrical representation of a lensing event shown in Fig.2.6, obtained keeping in mind the thin screen approximation. In this figure,the dashed line perpendicular to the lens and source planes and passing through the observer is the optical axis with respect to all the angular positions are measured. β~ is the angular position of the source, which lies on the ~ ~ ~ source plane at a distance ~η = βDS from the optical axis. ξ = θDL is the impact parameter of the light ray coming from that source deflected of an angle ~αˆ and θ~ is the position of the lensed image. θ,~ β~ and ~αˆ (in the case in which they are small) are related in the so-called lens equation or ray tracing:

~ ~ ˆ θDS = βDS + ~αDLS .

Dividing both members of this equation for DS and deﬁning the reduced de- ﬂection angle D ~α = ~αˆ LS , DS 24 Chapter 2. Gravitational lensing

Figure 2.5: Lensing geometry 2.3. The lens equation 25

Figure 2.6: Lensing geometry through thin-screen approximation the lens equation becomes:

β~ = θ~ − ~α(θ~) (2.3.1)

The quantities DS and DL,as mentioned at the beginning of this section, are respectively the angular diameter distance of the source and of the lens while DLS is the angular diameter distance between lens and source. It is important to keep in mind that these distances are not additive, i.e in a curved space DL +DLS 6= DS 1. In the definition of the reduced deflection angle ~α, the term ε = DLS is a measure DS of the efficiency of a lens at redshift zL,i.e of its power in deflecting and distorting

1In an a Euclidean space, given a source with physical size x and angular size ξ, the angular- x diameter distance D is defined by the relation D = tan(ξ) , which, considering that for small x angles, ξ 1 (condition always verified in astrophysics situations), tanξ ≈ ξ, becomes D = ξ . In a curved space, the last relation is not always verified and occurs defining distances that hold x D = ξ even in a non Euclidean space. 26 Chapter 2. Gravitational lensing

DLS Figure 2.7: Lensing efficiency ε = as a function of zS for different DS cosmologies.The two sets of curves correspond to two different lens redshift, zL = 0.3 and zL = 0.9, while the dotted curves correspond to Ωm = 0.1 and

ΩΛ = 0.9, the dashed curves correspond to Ωm = 1 and ΩΛ = 0,the solid curves correspond to Ωm = 0.1 and ΩΛ = 0.

a light ray . It depends on the redshift of the lens zL, on the redshift of the source zS and on the cosmological parameter Ωm and ΩΛ [8]. Fig.2.7 shows the dependence of ε from the redshift of the source, zS : when zS increases, it increases too. Therefore, this means that for a source with a higher redshift, deflection and distortion effect are stronger. Eq.(2.3.1) can be also written using the so-called ”dimensionless notation” dividing both members of the equation by a reference angle θ0 defined as follows:

ξ0 η0 θ0 = = DL DS

where η0 is a reference scale on the source plane and ξ0 is a reference scale on the lens plane. Deﬁning the dimensionless vectors: 2.3. The lens equation 27

ξ~ ~η ~x ≡ , ~y ≡ ξ0 η0 and the scaled deﬂection angle:

DLDLS ˆ ~α(~x) = ~α(ξ0~x), (2.3.2) ξ0DS

the equation (2.3.1) becomes:

~y = ~x − ~α(~x) . (2.3.3)

From a mathematical point of view, the lens equation is an application from the lens plane to the source plane. This is the exact opposite of what happens physically because the light rays start from the source plane and reach the lens plane. Assuming thin-screen approximation, the deﬂection angle by a system of point masses can be computed using the superposition principle (see Fig.2.8). It results the vectorial sum of the deﬂection angles of the single lenses:

~ ~ ˆ ~ X ˆ ~ ~ 4G X (ξ − ξi) ~α(ξ) = ~αi(ξ − ξi) = Mi . (2.3.4) c2 ~ ~ 2 i i |ξ − ξi| In the case of a continuum distribution of mass (see Fig.2.9)

Z Σ(ξ~) = ρ(ξ,~ z)dz , the deﬂection angle is given by:

4G Z (ξ~ − ξ~0)Σ(ξ~0) ~αˆ(ξ~) = d2ξ0. (2.3.5) c2 |ξ~ − ξ~0|2

The gravitational deﬂection depends only on the mass distribution Σ(ξ~) projected along the line of sight. 28 Chapter 2. Gravitational lensing

Figure 2.8: Deﬂection angle ~αˆ by a system of point masses.

Figure 2.9: Deﬂection angle ~αˆ by a continuum distribution of mass 2.3. The lens equation 29

2.3.1 The eﬀective lensing potential

The projection of the 3D potential on the lens plane deﬁnes the eﬀective lensing potential:

Z ˆ ~ DLS 2 ~ Ψ(θ) = 2 Φ(DLθ, z)dz, (2.3.6) DLDS c which scales with distances ( DLS ) and its dimensionless counterpart is given DLDS by:

2 DL ˆ Ψ = 2 Ψ. (2.3.7) ξ0 Ψ satisﬁes two important properties:

• the gradient of Ψ gives the scaled deﬂection angle:

~ ∇xΨ(~x) = ~α(~x) (2.3.8)

• the Laplacian of Ψ gives twice the convergence:

∆xΨ(~x) = 2k(~x) (2.3.9)

which is a “lensing version” of the Poisson equation.

k(~x) is the convergence and is deﬁned as:

Σ(~x) k(~x) = (2.3.10) Σcr

where Σ(~x) is the surface mass density of the lens and Σcr is the critical surface density, given by:

2 c DS Σcr = (2.3.11) 4πG DLSDL 30 Chapter 2. Gravitational lensing

Figure 2.10: A lens with Σ = const = Σcr.

To understand the meaning of Σcr, we can consider the following situation: a lens with a constant surface mass density Σ and equal to Σcr. This kind of lens will focus perfectly a source on β~ = 0 on the observer as shown in Fig.2.10

Moreover, the critical surface density Σcr is an important parameter because it allows to distinguish between strong and weak gravitational lenses: the ﬁrst ones are “supercritical” lenses, i.e Σ(~x) ≥ Σcr while the second ones are “under critical” lenses , i.e Σ(~x) < Σcr

2.4 Lens mapping and magniﬁcation

Fig.2.11 shows SDSS J103842.59+484917.7, a beautiful cluster of galaxies at z=0,431 observed by the Chandra X-Ray Observatory and Hubble Space Telescope. This cluster is a clear example of gravitational lens.The smiling face that appears in the image (reminiscent of the “Cheshire Cat” in the novel “Alice in Wonderland”) is nothing but a surprising combination of gravitational arcs. These are the images, distorted and amplified by the gravitational field of the cluster, of galaxies further away in the background. To have an idea of the physical size of these arcs, we have to take into account that tipically their angular sizes are about 20”. So, if the lens is, for example, at a redshift z ∼ 0.4 (as the “Cheshire Cat” cluster), considering that the angular diameter distance is a function of redshift (as illustrated in Fig.2.12) and assuming a ΛCDM cosmological model, physical size of arcs will 2.4. Lens mapping and magnification 31

Figure 2.11: A composite image of the Cheshire Cat obtained through the combination of optical images from NASA’s Hubble Space Telescope and of X-ray images from Chandra X-Ray Observatory. Credit: NASA. result ∼ 100 kpc. But how do these arcs form? The main responsible of the formation of arcs in a galaxy cluster is the distortion introduced into the shape of the sources that lie in background and have no negligible apparent sizes. This distortion is mainly due to the diﬀerential deﬂec- tion of light bundles coming from the sources. Ideally, the shape of the images can be determined by solving, for all the points within the extended source, the lens equation which is nothing but an application between lens plane and source plane. Since, in many cases, the angular sizes of lensed sources are smaller than the typical angular scale of the lens, it is possible linearize this mapping considering its local properties which are described in terms of the “lensing Jacobian”, A. This is obtained supposing a small perturbation (θ~0, ~α0) of the position (θ,~ ~α) on the lens plane that corresponds,through lens equation 2.3.1, to a position β~ = θ~ − ~α on the source plane. Hence, θ~0 and ~α0 are respectively: 32 Chapter 2. Gravitational lensing

Figure 2.12: Angular diameter distance DA as a function of redshift z, assuming

Ωm= 0.3, ΩΛ=0.7 and H0=70 km/s/Mpc.

θ~0 = θ~ + dθ~

d~α ~α0 = ~α + dθ~ dθ~ The position β~0 on the source plane, corresponding to (θ~0, ~α0), is given by β~0 = θ~0 − ~α0 (see Fig.2.13). Subtracting β~0 and β~, it is obtained:

d~α (β~0 − β~) = (I − )(θ~0 − θ~) = A(θ~0 − θ~), dθ~ where A is equal to:

~ ~ 2 ~ ∂β ∂αi(θ) ∂ Ψ(θ) A ≡ = (δij − ) = (δij − ) , (2.4.1) ∂θ~ ∂θj ∂θi∂θj 2.4. Lens mapping and magniﬁcation 33

Figure 2.13: Illustration of the way in which a position (θ,~ ~α) on the lens plane and its small perturbation (θ~0, ~α0) are mapped into the source plane. where, in the last equality, Eq.(2.3.8) has been taken into account. A is a symmetric second rank tensor describing the ﬁrst order mapping between lens and source planes and can be decomposed into an isotropic part and an anisotropic part. Splitting oﬀ the isotropic part from the Jacobian A, the so-called shear matrix is obtained:

1 1 A − trA · I = δij − Ψij − (1 − Ψ11 + 1 − Ψ22)δij = 2 ij 2   1 1 − 2 (Ψ11 − Ψ22) −Ψ12 −Ψij + (Ψ11 + Ψ22)δij =   = 2 1 (2.4.2) −Ψ12 2 (Ψ11 − Ψ22)   γ1 γ2 −   γ2 −γ1 where I have used the shorthand notation for the second derivatives of the potential:

∂2Ψ(θ~) ≡ Ψij , ∂θi∂θj while γ1 and γ2 are the components of a pseudo-vector called shear ~γ = (γ1, γ2) 34 Chapter 2. Gravitational lensing

whose direction can be written as:

2∂xyΨ tan 2θshear = . (2.4.3) ∂yyΨ − ∂xxΨ Instead of the intensity, the direction of the shear is independent of the source redshift, zS [8], because it is a ratio of the components of the lensing potential and, therefore, doesn’t depend on the eﬃciency,ε. The eigenvalues of the shear matrix are:

q 2 2 ±γ = ± γ1 + γ2 . (2.4.4)

Thus, there exists a coordinate rotation by an angle φ such that:

    γ1 γ2 cos2φ sen2φ   = γ   . (2.4.5) γ2 −γ1 sen2φ −cos2φ Since

1 1 1 trA = 1 − (Ψ + Ψ ) δ = 1 − 4Ψ δ = (1 − k)δ , (2.4.6) 2 2 11 22 ij 2 ij ij

the Jacobian matrix ca be written as follows:

  1 − k − γ1 −γ2 A =   (2.4.7) −γ2 1 − k + γ1 where k is the convergence (see 2.3.10). In order to understand the meaning of γ and k,in Fig.2.14, I have shown how they act on the shape of a circular source. It is evident that k is responsible for isotropic expansion or contraction while γ is responsible for anisotropic distortion. This becomes clear diagonalizing A 2 and

2A can be diagonalized because is real and symmetric. The simmetry of A is also the reason why an image will never result a pure rotation of the source that produces it. 2.4. Lens mapping and magniﬁcation 35

Figure 2.14: How a circular source is distorted considering several values of k, γ1 and γ2 . re-writing it in its principal axes as follows:

    1 0 γ 1 0 A = (1 − k)   +   . (2.4.8) 0 1 1 − k 0 −1 One of the main feature of lens mapping at first order is that it distorts areas: the eigenvectors of A give the distortion directions while the eigenvalues give the distortion amplitudes in these directions (see Fig.2.15). In particular,distortion amplitude or magnification, µ, is quantified by the in- verse of the determinant of the Jacobian matrix:

1 1 1 1 µ = = = (2.4.9) detA 1 − k + γ 1 − k − γ (1 − k)2 − γ2 Ideally, the magniﬁcation diverges when:

λr = 1 − k + γ = 0, or

λt = 1 − k − γ = 0.

where λr and λt are the eigenvalues of the Jacobian matrix. The conditions 36 Chapter 2. Gravitational lensing

Figure 2.15: Illustration of how a regular grid and a circle (on the left) are distorted by a lens with constant value of the convergence κ and the shear γ over the region (on the right).Source: [8]

λr = 0 and λt = 0 define two curves in the lens plane, called the radial and the tangential critical line, respectively. When an image forms along the tangential critical line, it is strongly distorted tangentially to this line while, when an image forms close to the radial critical line, it is stretched in the direction perpendicular to the line itself. As I said, critical lines are characterized by the condition detA = 0, hence lens mapping is locally invertible on these curves. Thus, via the lens equations, the critical lines are mapped into the caustics lines in the source plane. As critical lines, caustic lines are closed lines but, contrary to the critical lines, they can intersect each other. Essentially, magnification (or demagnification) derives from the fact that surface brightness Iν, defined as:

dE I = , (2.4.10) ν dt dA dν dΩ is conserved, so:

I ~ S ~ ~ Iν (θ) = Iν [β(θ)] , (2.4.11)

where I and S stand for “image” and “source”. What lensing change is the 2.5. Strong and weak lensing regimes 37

solid angle δβ2 (or equivalently the surface element δy2) under which the source is seen that, through lens equation, is mapped into the solid angle δθ2 (or in the surface element δx2 ).This implies a change in the amount of photons received (ﬂux) by the observer. In fact, the magniﬁcation can be also written as:

δθ2 δx2 1 µ = = = , (2.4.12) δβ2 δy2 detA

I so, the ﬂux of the image, Fν , is:

Z Z I I ~ 2 S ~ ~ 2 S Fν = Iν (θ)d θ = Iν [β(θ)]µd β = µFν . (2.4.13) I S If µ > 1, then we are in presence of a magniﬁcation of the image with respect to the source; if 0 < µ < 1 the image is demagniﬁed while if µ < 0 the image will result ”reversed” with respect to the source.

2.5 Strong and weak lensing regimes

There are two types of lensing that can occur when a light ray meets a lens along its path:

• strong lensing regime, characterized by eﬀects readily seen by eyes such as the production of multiple images, of giant arcs and arclets.

• weak lensing regime, characterized by small deformations in the shapes of background galaxies into small ellipses only detectable statistically. A weak lensing application can be found in [9].

Another diﬀerence between strong and weak lensing regime consists in the alignment observer-cluster-background source as it is evident in Fig.2.16. 38 Chapter 2. Gravitational lensing

Figure 2.16: What happens to a wave front of light emitted by a background galaxy in the strong,intermediate and weak lensing regimes. Source:[8]

In strong lensing regime the observer, the lens and the source are well aligned along the line of sight while in weak lensing regime this alignment is less perfect, i.e the angular separation between source and lens is large.

2.6 Strong gravitational lensing

The main feature of strong lensing regime is the production of multiple images of the same background source. This event occurs in massive clusters cores (the inner one arc-minute region around the cluster center) whose surface mass density is close to or larger than the critical surface mass density, Σcrit. In this case, the wave front of a distant source is literally broken in several pieces as shown in Fig.2.16. So, when a distant galaxy is strongly lensed, it will appear distorted and highly magniﬁed: its image will have a typical elongated shape and will be aligned 2.6. Strong gravitational lensing 39

Figure 2.17: Image conﬁgurations produced by a compact source moving away from the center of an elliptical lens and crossing a fold caustic (left panel) and a cusp caustic (right panel). Source: Narayan & Bartelmann, 1995 around the cluster center in a preferential tangential manner forming spectacular

1 “gravitational giant arcs”. Since Σcrit is proportional to , only distant objects DLS can produce strong lensing eﬀects, so it is necessary to have powerful telescopes to detect them and to obtain their spectra. The number of multiple images of a single source in a strong lensing event is obtained solving the lens equation and can be also estimated considering the “catastrophe theory” according to which each time one crosses a caustic line in the source plane, two additional lensed images are produced and, each time one crosses a cut, one additional lensed images is produced. In the case of a non-singular mass distribution, the number of multiple images is always odd but quite often some images are de-magniﬁed and for this reason they are not observable [8].

In Fig.2.17, for example, are illustrated the image conﬁgurations produced by a compact source moving away from the center of an elliptical lens and crossing 40 Chapter 2. Gravitational lensing

a fold caustic (left panel) and a cusp caustic (right panel).On the right of each panel are portrayed the caustics lines and the diﬀerent positions occupied by the source while on the left there are the critical lines with the corresponding (same color) positions of the images [10].An important feature of multiple images is their parity,the symmetry of the image with respect to the source, which is deﬁned by the signs of the eigenvalues of the Jacobian matrix, (λr,λt). The parities can assume the following values:

• (+, +) = the eigenvalues of the Jacobian matrix are both positive, hence detA > 0 and trA > 0. Therefore,the images have positive magniﬁcation;

• (+, −) = the eigenvalues have opposite signs. Since detA < 0, the images have negative magniﬁcation which means that they are ﬂipped compared to the source;

• (−, −) = the eigenvalues are both negative, hence detA > 0 and trA < 0. These images, therefore, have positive magniﬁcation.

Since the critical lines are deﬁned by the condition detA = 0, each time one crosses these lines the parity of the image changes (look an example in Fig.2.18). If the lens is a galaxy cluster, a way to “visualize” these critical lines (and also caustics) is to realize a mass distribution model of the cluster. Moreover, this is very useful to identify and conﬁrm the existence of a multiple-image system, testing if a set of images, having similar morphology and colors, can actually be multiple images of the same source 3. Cluster modeling is an argument dealt with Chapter 3.

3Giant arcs, for example, are often composed by images of sources at diﬀerent redshift as their diﬀerent colors suggest. 2.6. Strong gravitational lensing 41

Figure 2.18: A NASA/ESA Hubble Space Telescope photograph (left) showing a pair of L-shaped images with mirror-symmetry due to a change in parity. The 2 images arise from a galaxy at z = 1.867 seen through AC114 galaxy cluster located in foreground (right). Credit: NASA & ESA

Chapter 3

Galaxy clusters as gravitational lenses

In this Chapter I deal with galaxy clusters and their special role as cosmic telescopes. Through their enormous magniﬁcation power, galaxy clusters allow us to detect and characterize distant, intrinsically small and/or faint sources, such as galaxies at high redshift. Moreover, I will discuss the way in which strong lensing can be used to build models of cluster mass distribution through a Bayesian approach. This is the main aim of the software Lenstool.

3.1 What are galaxy clusters?

Galaxy clusters are the largest known gravitationally bound structures in the Uni- verse. They contain from hundreds to thousands galaxies which move within the cluster potential well with a velocity dispersion in the range of 500-1000 km/s, see Fig. 3.1.

13 15 Galaxy clusters have total masses from 10 up to 10 M [11] thus they strongly deform space-time locally. For this reason, they are the most power-

43 44 Chapter 3. Galaxy clusters as gravitational lenses

Figure 3.1: Representation of potential wells of a galaxy, a galaxy group, a massive cluster and a super cluster. Under each structure it is shown the typical velocity dispersion of stars (in a galaxy) and of galaxies (in a group, in a cluster and in a super cluster).

ful gravitational lenses in the Universe and are called “natural telescopes”. In- deed, light rays coming from a distant source (for example, a background galaxy), traversing through clusters, are deflected and the resulting images of these distant objects appear distorted and magnified. Galaxy clusters consist essentially of two components: a baryonic component and a non-baryonic component. The first one includes the galaxies (which form at the minimum points of the cluster potential) and a cloud of hot gas which fills the space between and around these galaxies, the so-called intracluster medium (ICM). The temperature of this gas, that emits X-ray radiation by thermal bremsstrahlung, can be obtained starting from the scalar virial theorem which states that the time-averaged kinetic energy K of the particles that make up a physical system is equal to half the average potential energy W [12]:

2K + W = 0 (3.1.1)

The kinetic energy of a particle (for example, a proton) which forms the ICM 3.1. What are galaxy clusters? 45

is totally converted in thermal energy, so:

3 K = K T , (3.1.2) 2 B g

−16 −1 where KB is the Boltzmann constant (∼ 10 erg K ) and Tg is the temperature of the ICM. The total potential energy is given by:

G M m W = − N C p , (3.1.3) RC 47 where MC is the mass of the cluster (∼ 10 g), mp is the mass of the proton −24 24 (∼ 10 g) and RC is the radius of the cluster(∼ 10 cm). Substituting Eq.(3.1.2) and Eq.(3.1.3) in Eq.(3.1.1) and solving for Tg, the temperature of the ICM results of the order of tens of millions of degrees,corresponding to a typical photon energy of 1−10keV , so that the gas is expected to be fully ionized. The principal and most massive component of a galaxy cluster (about 80% of the total mass of a cluster) is dark matter,as first pointed out by Fritz Zwicky in the 1933 [13] computing the total mass of Coma Cluster. Galaxy clusters are young structures that have formed quite recently (if it is assumed a bottom-up galaxy formation scenario) and may not be fully virialized. At the center of the most compact and regular 1 clusters there is often an huge elliptical galaxy that dominates the dynamics of the cluster. This galaxy is called Brightest Galaxy Cluster (BCG) and, often, has an extraordinarily diffuse and extended outer envelope (in this case it is called a cD galaxy - the “D” stands for “diffuse”). It is believed that its formation is linked to cooling flows, i.e. the cooling of the intracluster medium (ICM) due to its rapid energy loss through the emission of X-rays. Clusters are in general rich in early- type galaxies: the fraction of E+S0 galaxies is about 80 % in regular clusters,

1Regular galaxy clusters are spherically shaped,usually have thousands of galaxy members, most of which are ellipticals or irregulars. Irregular clusters have no speciﬁc shape and have only about a hundred members or less (more spirals than the regular clusters). 46 Chapter 3. Galaxy clusters as gravitational lenses

and about 50 % in irregular clusters. This fact suggests that galaxies undergo morphological transformations in clusters and that interesting galaxy evolution processes happen in these dense environments . An important feature of elliptical galaxies is that they are concentrated in a plane, known as the fundamental plane, that, in mathematical form, can be written as:

log Re = a log σ0 + b loghIei + constant (3.1.4) where hIei is the mean surface brightness within Re and σ0 is the central velocity dispersion. The scaling relations for the ellipticals are just two-dimensional pro- jections of this fundamental plane. In particular, on the plane surface brilliance against velocity dispersion there is the Faber-Jackson relationship according to which ellipticals with a larger (central) velocity dispersion are both brighter:

4 L ∝ σ0

Another evidence of the rapid evolution of the galaxy population in clusters is the so-called Butcher-Oemler eﬀect: there is an increase in the fraction of blue galaxies in clusters at intermediate redshifts (0.3 ≤ z ≤ 0.5) compared to present- day clusters [4]. Although some suspect it may be only a selection eﬀect, this phenomenon could be real and is probably related to the greater availability in the past of material from which new stars could form.

3.2 Galaxy clusters as gravitational telescopes

Galaxy clusters provide enormous magniﬁcation and ampliﬁcation of distant sources that lie behind them. This is the reason why they are considered “natural telescopes” or “cosmic telescopes”, allowing us to study very high-z galaxies that formed during the infancy of the Universe. As I said in the previous chapter, if a 3.3. Strong lensing modeling 47

source is compact enough and is located exactly behind a caustic line, it will have an infinite magnification. This event is really rare but, in strong lensing clusters, amplification factors larger then 40× (∼ 4 magnitudes) have been measured and amplification factors larger than 4× (∼ 1,5 magnitudes) are quite common. The regions closest to critical lines in the image plane and to caustic lines in the source plane are the regions characterized by the largest magnification. Cluster lenses are exploited as cosmic telescopes for two purposes:

• to discover the most distant objects and low-luminosity objects that would otherwise remain undetected,

• to study the morphology of distant galaxies which otherwise would not be resolved and explore their physical properties that would otherwise be impossible to characterize.

In this thesis project, I focused my attention precisely on this second aspect as I will show in chapter 4.

3.3 Strong lensing modeling

Strong lensing allows to build good models of the cluster mass distribution that are able to reproduce observed data (image positions and shapes, fluxes, , time delays, multiplicities of images...) which are known with great accuracy.To build a good model of a cluster one has to take into account the lensing effects of cluster member galaxies (and their associated individual dark matter halos that contribute about 10% of the total cluster mass), of dark matter and of Intra Cluster Medium(ICM). The traditional way of doing this is to use “parametric models” that describe the mass distribution in terms of a finite number of clumps in order to represent these three components. In turn, each clump is described by a mass profile 48 Chapter 3. Galaxy clusters as gravitational lenses

characterized by a finite number of parameters (for example,position on the sky, a projected ellipticity of the mass distribution, a position angle, velocity dispersion). The three main mass distribution profile that are often used in lensing analysis are the Singular Isothermal Sphere (SIS), the Navarro-Frenk-White (NFW) and the Pseudo Isothermal Elliptical Mass Distribution (PIEMD).The latter is very common as a lensing model because, instead of SIS, it is characterized by a finite mass and a finite central density, Σ0, therefore is more adequate to match true mass distributions. The projected surface mass density for this model is given by:

! Σ r r 1 1 Σ(R) = 0 core cut − (3.3.1) p 2 2 p 2 2 rcut − rcore rcore + R rcut + R where rcore is a core radius and rcut is a cut radius, rcut rcore.It is possible to generalize the simple circular model to the elliptical case by re-deﬁning the radial coordinate in this way:

x2 y2 a − b R2 = + ; = (3.3.2) (1 + )2 (1 − )2 a + b This model is very versatile because it can reproduce a large range of mass distributions, from cluster scales to galaxy scales by varying the ratio:

rcut

rcore .When a galaxy within a cluster is described adopting a PIEMD model, it is possible assume that the parameters that deﬁne this model (the core radius rcore, the cut-oﬀ radius rcut and the central velocity dispersion σ0) satisfy the following scaling relations:

L 1/2 r = r∗ (3.3.3) core core L∗

L α r = r∗ (3.3.4) cut cut L∗ 3.3. Strong lensing modeling 49

L 1/4 σ = σ∗ (3.3.5) 0 0 L∗ where L is the luminosity of the galaxy and L∗ is the typical luminosity of a galaxy at the cluster redshift. Eq.(3.3.5) establishes a relation between the velocity dispersion and the total luminosity in agreement with Faber-Jackson relation for elliptical galaxies (for spiral galaxies there is the Tully-Fisher relation that relates the total luminosity with the rotation velocity but this kind of galaxies is a minority in cluster cores). In this framework, the cluster gravitational potential can be written as the sum of two contributions [8]:

X X φtot = φci + φpj (3.3.6) i j where φci represents large-scale potentials (dark matter and intra-cluster gas) while

φpj represents the sub-halo potentials that are associated with all the massive cluster member galaxies that are roughly within two times the Einstein radius of the cluster. Obviously, the number of sub-halos to include in a model needs to be quantified, including those that can increase significantly the deflection angle at its associated galaxy position. For this reason, only galaxies within the cluster red sequence and brighter than a given luminosity limit are selected. The strategy to obtain an accurate and detailed mass modeling of massive clusters is to use multiple images (with preferably measured spectroscopic redshift) as constraints.These are available thanks to the recent deep images of cluster cores from Hubble Space Telescope (HST). In general, a likelihood L for the observed data D and parameters p of the model is defined in the following way:

N χ2 Y 1 − i L = P r(D|p) = √ e 2 (3.3.7) Qni i=1 j=1 σij 2π where N is the number of multiple-image systems and ni is the number of multiple 50 Chapter 3. Galaxy clusters as gravitational lenses

images for the system i. To model the mass distribution of a cluster, a χ2 ﬁt statistic is adopted. This can be computed both in the image and in the source plane. The best ﬁt model will be the one characterized by the minimum χ2. In the image plane, the contribution of the multiple-image system i to the overall χ2 is:

ni j X [θ − θj(p)]2 χ2 = obs (3.3.8) i σ2 j=1 ij j where θ (p) is the position of image j predicted by the model and σij is the error position of image j whose determination depends on the signal-to noise of the image. Thus, minimize the χ2 means making the scatter between the positions of the observed and the estimated images as small as possible. The computation of χ2 can be also performed in the source plane and, in this case, χ2 is written in terms of:

ni j j X [θ (p) − hθ (p)i]2 χ2 = S S (3.3.9) Si µ−2σ2 j=1 j ij j j where θS(p)is the corresponding source position of the observed image j, hθS(p)i is the barycenter position of all the ni source positions and µj is the magniﬁcation for image j. The calculation of χ2 in the source plane is very fast because there is no need to solve the lens equation repeatedly but is also biased because mass models with large ellipticity are preferred. Both the source and the image plane χ2 methods have been implemented in the software Lenstool that I will describe later.

3.3.1 Bayesian approach

Strong lensing modeling is a problem characterized by the fact that the data by themselves do not suﬃciently constrain the lens mass model. In these cases, instead 3.4. From the source to the lensed image 51

of regression techniques, the best approach is the Bayesian one. Bayes theorem can be written as follows:

P r(D|p,M)P r(p|M) P r(p|D,M) = , (3.3.10) P r(D|M) where P r(p|D,M) is the posterior Probability Density Function (PDF), P r(D|p,M) is the likelihood of getting the observed data D given the parameters p of the model M, P r(p|M)is the prior PDF for the parameters and P r(D|M) is the evidence, i.e the probability of getting the data D given the assumed model M. The set of parameters p which gives the best ﬁt is the one that maximizes the posterior PDF consistently with the prior PDF.

3.4 From the source to the lensed image

Suppose we have an elliptical source, whose semi-axes are a and b:

β2 β2 1 + 2 = 1, (3.4.1) a2 b2 ~ where β1 and β2 are the components of the vector β on the source plane while θ1 ~ and θ2 are the components of the vector θ on the image plane. Which is the ﬁrst order distortion of this kind of source? In the reference frame where the Jacobian A is diagonal:

      β1 1 − k − γ 0 θ1   =     . (3.4.2) β2 0 1 − k + γ θ2

So:

β1 = (1 − k − γ)θ1 (3.4.3) 52 Chapter 3. Galaxy clusters as gravitational lenses

Figure 3.2: First order distortion of an elliptical source, with semi-axes are a = 7

◦ and b = 5,whose semi-major axis forms an angle of 30 with β1 axis, considering a convergence k = 4 and a shear ~γ = (0, 1).

and

β2 = (1 − k + γ)θ2 (3.4.4)

Substituting β1 and β2 in 3.4.1, we have:

(1 − k − γ)2θ2 (1 − k + γ)2θ2 1 + 2 = 1 (3.4.5) a2 b2 This is the equation of an ellipse with semi-axes a0 and b0 given by:

a a0 = 1 − k − γ and b b0 = . 1 − k + γ If we choose a = 1 − k − γ and b = 1 − k + γ, on the image plane we obtain a circumference of radius r = 1. 3.5. Lenstool 53

In the Fig.3.2, I have shown what happens when an elliptical source with semi-

◦ axes a = 7 and b = 5, whose semi-major axis forms an angle of 30 with β1 axis, is distorted at ﬁrst order,considering a convergence k = 4 and a shear ~γ = (0, 1).

3.5 Lenstool

Lenstool is a software tool [14] [15] [16] which allows to model the mass distribution of galaxies and galaxy clusters, both in strong and in weak lensing regime. It works giving in input a set of parameters: those related to the choice of the density proﬁle for each mass component of the cluster the user wants to model and those related to the positions of a set of multiple images he has to use as constraints. At this point, Lenstool explores the full parameters space sampling the posterior PDF and searching for the sample characterized by the higher one. To progressively converge the posterior PDF to the prior PDF, Lenstool uses a Markov Chain Monte Carlo (MCMC) method whose sampling strategy consists in the construction of an aperiodic and irreducible Markov chain2 for which the stationary distribution is exactly the posterior distribution of the parameter of interest. The MCMC convergence to the posterior PDF is performed as follows: at each optimization step [15], 10 new PDF samples are drawn randomly from the posterior PDF of the current step and are weighted according to their likelihood. If the sample has a “bad” likelihood, then it is deleted; if the sample has a ”good” likelihood, then it is used for the next step to draw new PDF samples. Obviously, at each step,the computation time increases because more samples are generated and the exploration of the parameters space become more complicated. For each posterior

2A Markov chain is a Markov process that is a stochastic process in which the probability that a system which is in a state moves to another state depends only on the state of the system immediately preceding (Markov’s property) and not on the“way” in which it has come to this state. 54 Chapter 3. Galaxy clusters as gravitational lenses

PDF drawn, the values of the free parameters with the higher probability in the PDF are estimated and in correspondence to these values the χ2 in Eq. (3.3.8) is computed.Lenstool can not only determine the best mass model for a given set of input parameters but, through the identiﬁer CleanLens, is also able to reconstruct the shape of a source reading a CCD pixel-frame of an image located in the core of a massive galaxy cluster. The source morphology of an image is obtained by ray tracing the image pixel by pixel through the modeled mass distribution. It is possible that several points of the image plane correspond to the same point on the source plane. In this case, the intensity of a pixel on the source plane that S corresponds to nij pixels on the image plane, Iij, is computed as the mean of the I intensity of the corresponding image pixels, Iijk:

k=nij S 1 X I Iij = Iijk . (3.5.1) nij k=1 In the source plane, the error of this reconstruction at position ij is given by:

k=nij X I S 2 eij = (nij − 1) (Iijk − Iij) . (3.5.2) k=1 Furthermore, Lenstool gives also the opportunity to choose diﬀerent cosmologies by changing the cosmological parameters Ωm,ΩΛ and the Hubble constant

H0. For my data analysis (see Chapter 4), I have chosen a cosmology characterized by Ωm= 0.3, ΩΛ=0.7 and H0=70 km/s/Mpc [17]. Chapter 4

Data analysis

In this Chapter I describe the three source reconstructions I made by means of the software Lenstool. In particular, I used the Cleanlens mode that allows to compute, for each point of the image plane, the corresponding position in the source plane through a lens model. I have analyzed three massive galaxy clusters (Abell 370, Abell S1063 and Abell 2667) that contain three luminous giant gravitational arcs produced by the distortion and deﬂection of light rays coming from three sources beyond them. Abell 370 and Abell S1063 are two of the six galaxy clusters used as “natural telescopes” in the Frontier Fields research program.

4.1 Frontier Fields

“Frontier Fields” is a research program whose purpose is to observe the “frontiers” of the visible Universe, that is, to explore regions of the Universe never before reached by any human “eye”, allowing even the study of the first galaxies, born after the Big Bang [18]. At first sight this goal can seem unreachable but the strength of this program is in the combination of the power of the Hubble Space Telescope with the gravitational lensing effect produced by massive galaxy

55 56 Chapter 4. Data analysis

clusters. The Hubble Space Telescope has provided a view of the Universe about 435 million years after the Big Bang. Frontier Fields program overcomes this limit exploiting the natural telescopes function of galaxy clusters that, according to general relativity, curve the shape of the space-time around them causing the spreading out of background galaxies light along multiple paths. The gravity of these clusters distorts but also magnifies the faint light of distant galaxies behind them forming a series of streaks and arcs around the clusters cores. The program started in 2013 and ended in 2017. During this period, Hubble Space Telescope has made 560 orbits around the Earth (that correspond to about 630 hours) to observe 12 “frontier fields”: six of them are centered on regions with known galaxy clusters, chosen for their peculiar features (like, for example, the strength of their lensing); the remaining six additional fields, known as parallel fields, are located near each one of the galaxy cluster fields and are used to perform deep field observations. The galaxy clusters that have been imaged in great detail and that allow a proper ”time travel” are: Abell 2744, MACS J0416.1-2403,MACS J0717.5+3745, MACS J1149.5+2223, Abell 370 and Abell S1063. The Frontier Fields program has produced the deep- est observations ever made of galaxy clusters and of the magnified galaxies behind them. This gives a great help to scientists to understand, for example, the evolution of stars and galaxies during the period when a dark Universe transformed in a transparent to light Universe or to understand the nature of dark matter, the major component of galaxy cluster.

4.2 Our sample of lensed galaxies

In order to choose my sample of lensed galaxies, at ﬁrst, I have used the scientiﬁc data publicly available regarding all the six galaxy clusters belonging to the Fron- 4.3. Source plane reconstruction in Abell 370 57

Object MUSE programme α δ Exp.(h) z Size (arcsec2) A370-arc 094.A-0115, 096.A-0710 02:39:53 -01:35:05 6.0 0.725 30 AS1063-arc 060.A-9345 22:48:42 -44:31:57 3.25 0.611 33 A2667-arc 094.A-0115 23:51:39 -26:04:50 2.0 1.033 89

Table 4.1: List of gravitational arcs I have used for my analysis. Coordinates α and δ are the celestial coordinates of the gravitational arc. tier Fields program to ﬁnd the ones with the most interesting giant arcs inside them. Then, I have chosen to focus my attention on A370 and AS1063. In the following sections, I will explain the procedure I have adopted to reconstruct the morphology of sources “lensed” by these 2 clusters and by A2667, starting from the giant arcs they produce. In Table 4.1 are summarized the main characteristics of these arcs. The HST images I have used in my data analysis are all produced through the ”MosaicDrizzle” pipeline. This algorithm takes as input ﬁles which have been produced by calacs software that processes the raw exposures, performing initial bias and dark current subtraction. The output consists in a combination and alignment of sets of images into a large mosaic [19] (I have employed the one at 30 mas/pixel that ensures a better spatial resolution). This technique is very powerful to remove, for example, cosmic rays from the images.

4.3 Source plane reconstruction in Abell 370

Abell 370 is a galaxy cluster located in the constellation Cetus, at redshift z = 0.375 and was the ultimate goal of the Frontier Fields observational campaign1. As Fig.4.1 shows, the core of A370 is essentially made of yellow-white, massive, elliptical galaxies that are the brightest and largest galaxies in the cluster.

1www.nasa.gov 58 Chapter 4. Data analysis

Figure 4.1: Image of Abell 370 galaxy cluster obtained by Hubble Space Telescope. Credit: NASA, ESA/Hubble, HST Frontier Fields.

The bluish ones are spiral galaxies characterized by younger populations of stars. Among these two types of galaxies, several arcs of blue light and about 30” long appear but they are not “real”, they are a sort of mirages. A370 acts like a natural telescope, curving space and deflecting light traveling through the cluster toward Earth [20]. Arcs are nothing but the distorted,stretched and magnified images of distant galaxies located behind the cluster which are too faint to be seen directly by Hubble. The most amazing example of these arcs in Abell 370 is “the Dragon” (see Fig.4.2), a composition of 5 images of a single background spiral galaxy at z = 0.7251, with the appearance of a dragon. In order to reconstruct the morphology of this galaxy, I have divided the “Dragon” in 5 “pieces” corresponding to the 5 images (one for the head, three for the body and one for the tail) through use of a function in the software DS9 that allows to build polygons (see Fig.4.3). At this point, I have written a file in which I put a mass model of A370 4.3. Source plane reconstruction in Abell 370 59

Figure 4.2: Zoom up view of the so-called “Dragon” in Abell 370.

Figure 4.3: Division of the Dragon in 5 ”pieces” corresponding to the 5 images of the source galaxy. 60 Chapter 4. Data analysis

elaborated by the CATS2 team characterized by 99 PIEMD potentials. In the CleanLens identifier, I have specified the HST image of A370 that I have used to make the division of the Dragon and an ASCII file containing a list of points that define the contours of the polygons that hold the single images of the background galaxy. In particular, I have performed the reconstruction of the background galaxy in 4 cases:

• I have chosen an HST image of Abell 370 obtained using the ﬁlter F475W and,to make the inversion, I have considered, at ﬁrst, only the polygon holding the head of the Dragon and then all the 5 polygons holding head,body and tail .

• I have performed the same procedure described above but choosing an HST image of Abell 370 obtained using the ﬁlter F814W.

The source reconstructions obtained in the first case are shown in Fig.4.4, while the ones obtained in the second case are illustrated in Fig.4.5. These reconstructions show quite well that the background galaxy that has been lensed by A370 is a spiral galaxy whose bulge is very evident in Fig. 4.5. In Fig.4.4 the bulge is not visible because it is formed by an old stellar population so,considering that the galaxy is at z = 0.7251, in the filter F475W I’m observing the UV light emitted by it (about 3000 A˚ ). At this wavelength it is possible only to appreciate the spiral arms that are populated by young, blue stars. Source reconstructions obtained using head,body and tail (see right images in Fig.4.5 and Fig.4.4) are “noisier” : I think that this is due to the fact that the images forming the body of the Dragon are not counter-images, i.e each of these images contains only a part of the lensed galaxy and is highly distorted. Both this factor make difficult to trace polygons that hold the single images. Finally, using a γ

2https://archive.stsci.edu/prepds/frontier/lensmodels/ 4.3. Source plane reconstruction in Abell 370 61

Figure 4.4: Source plane reconstruction choosing an HST image of Abell 370 obtained using the ﬁlter F475W and considering only the polygon holding the head of the Dragon (left) and those holding the head, the body and the tail (right).

Figure 4.5: Source plane reconstruction choosing an HST image of Abell 370 obtained using the ﬁlter F814W and considering only the polygon holding the head of the Dragon (left) and those holding the head, the body and the tail (right). 62 Chapter 4. Data analysis

and a κ model elaborated by CATS team, I have also obtained, through a script python, the magnification map of A370 for a source located at z = 0.7251 (see Fig.4.6). Taking advantage of this map, I have deduced that the magnification in the “head” of the “Dragon” reaches factors of almost 9 while, in the images forming the “body”, the background galaxy is lensed with an average magnification factor of 30 (see Fig.4.7). Moreover, I have also computed the magnification along the radial direction, µr, and the magnification along the tangential direction, µt, both in the “head” and in the “body” and I have obtained the following values:

(µr)head ∼ 0.5

(µt)head ∼ −1.1

(µr)body ∼ 5

(µt)body ∼ −0.8

In the “head”, µr and µt are ∼ 1 therefore in this region of the cluster the magniﬁcation is quite isotropic while, in the “body”, µr and µt diﬀers for a factor of ∼ 5.

4.4 Source plane reconstruction in Abell S1063

Abell S1063 is a galaxy cluster located in the direction of the constellation Grus and is approximately 4 billion light-years away from Earth, at z = 0, 3475 (see Fig.4.8). It belongs to the group of six galaxy cluster that have been studied in detail during the Frontier Field observational campaign. So,as A370, it acts as a lens, magnifying the image of the objects located behind it.“S” stands for “The Southern Survey”, the name of the second part of the catalog written by George Abell [21] in which are described the clusters visible in the southern hemisphere. 4.4. Source plane reconstruction in Abell S1063 63

Figure 4.6: Magniﬁcation map of A370 for a source located at z = 0.7251 using a γ and a κ model elaborated by CATS team.

Figure 4.7: Zoom up view on the position of the ”Dragon”. 64 Chapter 4. Data analysis

Figure 4.8: Image of the cluster Abell S1063. Credit: NASA, ESA/Hubble, HST Frontier Fields.

Abell S1063 is characterized by a high luminosity in high-energy X-ray light, as witnessed by NASA’s Chandra X-Rays Observatory. This suggests that we are in the presence of an event involving the merging of multiple galaxy clusters. Indeed, when clusters merge due to gravity, there is the collision of the gases. This provoke shocks that heat the gas which then emits X-rays. Near the core of Abell S1063 there is a distorted and stretched image of a background galaxy that is at z = 0.611 [22]. In order to reconstruct the morphology of this galaxy, I have used an HST image of AS1063 obtained in the ﬁlter F814W and a lens model elaborated by the CATS team. I have enclosed the image in a polygon (see Fig.4.9) and I have obtained the reconstruction illustrated in Fig.4.10 through Lenstool.

Also for AS1063, I have realized a magnification map for a source located at z = 0.6111 using a γ and a κ model elaborated by CATS team (see Fig.4.11). From this map, I have deduced that the galaxy whose morphology I have reconstructed has been magnified of a factor of 5 (see Fig.4.12). The values of the magnification 4.4. Source plane reconstruction in Abell S1063 65

Figure 4.9: Polygon around the image of the background source in AS1063.

Figure 4.10: Source plane reconstruction of the image in AS1063. 66 Chapter 4. Data analysis

Figure 4.11: Magniﬁcation map of AS1063 for a source located at z = 0.6111 using a γ and a κ model elaborated by CATS team. along the radial and the tangential direction are respectively:

µr = 2.2

µt = 1.3

4.5 Source plane reconstruction in Abell 2667

Abell 2667 is a galaxy cluster located in the constellation of the Sculptor at redshift z = 0.233 (see Fig.4.13). It is one of the richest and brightest clusters in the X-ray band and, because of its conspicuous mass, is also a well-known gravitational lens. This cluster is characterized by a giant arc that is formed by three images of a source at redshift z = 1.0334 [24]. Following the same procedure described above, 4.5. Source plane reconstruction in Abell 2667 67

Figure 4.12: Zoom up view on the position of the image of the galaxy whose morphology I have reconstructed.

Figure 4.13: Image of Abell 2667 from Hubble Space Telescope. Credit: NASA, ESA/Hubble. See Covone et al (2006) [23]. 68 Chapter 4. Data analysis

Figure 4.14: Polygons around the three images forming the arc in Abell 2667.

I divided this arc in three “pieces” through three polygons (see Fig. 4.14). Using an HST image of A2667 obtained through the ACS WFC1 F850LP ﬁlter and a lens model elaborated by Covone et al. [23], I have obtained a reconstruction of the source exploiting all the three images (see Fig.4.15 ).

I have reconstructed the source even exploiting each of the three images indi- vidually. The results are shown in Fig.4.16.

Image 1 in Fig.4.14 is a counter-image so it contains the whole image of the source while image 2 and 3 are not counter-images because they contain only a part of the image of the source and appear mirrored with respect to the critical line that passes through them. 4.6. Integral Field Units 69

Figure 4.15: Source plane reconstruction of the arc in A2667 exploiting all the three images.

4.6 Integral Field Units

Integral Field Spectroscopy is a relatively new spectroscopic technique that provides spectra of the sky over a two-dimensional field-of-view 3. This is done em- ploying particular instruments, called Integral Field Units (IFUs),which divide the field-of-view into many cells or segments. The signal from each cell or pixel of the field is fed into a spectrograph which then generates a spectrum for each individual pixel.The final result of this operation is a datacube that consists in the decom- position of an astronomical image in its spatial dimensions (right ascension and declination) and its spectral dimension (λ). There are essentially three ways 4 in which an IFU can divide the 2D spatial plane into a continuous array:

3https://www.eso.org/public/ireland/teles-instr/technology/ifu/ 4https://jwst-docs.stsci.edu/display/JPP/Introduction+to+IFU+Spectroscopy 70 Chapter 4. Data analysis

Figure 4.16: Source plane reconstruction of the source given by the image 1 (top left), image 2 (top right) and image 3 (bottom). 4.6. Integral Field Units 71

• Lenslet array: An array of lenslets splits up the input image. Light from each element of the observed object is focused onto a single diﬀraction grating.To prevent overlap, the grating is arranged such that the spectra are dispersed at an angle on the CCD.

• Lenslets + fibres: this is the most used technique. A 2D bundle of optical fibres transfer the light from the lenslets to the slit of the spectrograph. These fibres are flexible enough to reformat the round/rectangular field-of- view into one (or more) “slits”:the light is then dispersed perpendicular to those slits, providing the spectrum for each spaxel (spatial element).

• Image-slicer: the light coming from the telescope is sent onto a slicing mirror that is a mirror characterized by several rows which reflect light into different directions.In this way “image slices” are formed: these meet other mirrors that direct them onto the same spectograph slit. The light is then diffracted by a grating, providing a spectrum for each row.

The power of IFUs is that they make possible the study of extended objects, such as nebulae, galaxies or galaxy/stars clusters, just in one shot allowing to produce eﬃcient spatial maps of spectroscopic quantities such as the stellar kinematics or the movement of galactic gases. Moreover, Integral Field Spectroscopy try to overcome the main disadvantages of traditional long-slit spectroscopy related, for example, to the wavelength-dependent slit-losses due to the diﬀerential atmospheric refraction (DAR). Nowadays the equipment of the most largest and advanced telescopes includes IFUs. 72 Chapter 4. Data analysis

Figure 4.17: Basic parameters of MUSE. Source: ESO website.

4.6.1 The integral ﬁeld spectrograph MUSE

On Yepun (UT4), the fourth Unit Telescope of the Very Large Telescope at the Paranal Observatory, is installed MUSE (Multi Unit Spectroscopic Explorer)5. It is an integral field spectrograph (IFS) that works splitting the field of view into 24 individual image segments or channels which are each split further into 48 slices or ”mini slits”, for a total of 1152 mini slits. Each set of 48 mini slits is fed into a spectrograph: in this way each pixel of the image has a full spectrum of the light. MUSE operates in the visible wavelength range with a mean resolution of 3000. and is characterized by a Wide Field Mode (WFM) with a field of view (fov) covering 1x1 arcmin and a Narrow Field Mode (NFM) with a f.o.v. of 7.5x7.5 arcsec. The basic parameters of the instrument are summarized in the Fig. 4.17. MUSE is employed in several scientific goals such as the study of galaxy for-

5https://www.eso.org/sci/facilities/develop/instruments/muse.html 4.6. Integral Field Units 73

mation, of nearby galaxies, of stars and resolved stellar populations. One of the targets of MUSE was A370, which has been observed in UT on November 20, 2014, as part of the GTO (Guaranteed Time Observing) Programme. Four 30-min exposures in WFM-NOAO-N mode, centred at (α = 2h 39m 53s.111, δ = −1◦ 340 55.0077) have been observed. In order to average out systematics on the detector, a small (∼ 0.5 arcsec) dither pattern has been applied between these exposures that have been taken alternating between PA = 0◦ and PA = 8◦. The typical seeing during the observations was 0.75 arcsec, as measured by stars in the field. In order to flux-calibrate A370 data a combination of several standard stars have been used. The individual exposures have been aligned to a common world coordinate system (WCS) shifting each frame relative to a reference image. Then, the re-aligned images have been transformed into data cubes resampling all pixels on to a common three-dimensional grid with two spatial and one spectral axis. The final spectral resolution of the cubes varies from R = 2000 and R = 4000, with a spectral range between 4750 and 9350 A.˚ The wavelength grid has been set to 1.25 Apixel˚ −1 and the final spatial resolution is 0.2 arcsec pixel−1. Next, each cube has been processed with a software package to remove known systematics from the sky model, further improving sky-subtraction residuals. As final step, all cubes have been merged together to create a combined master cube on which a process has been run to eliminate low-level sky residuals that can only be seen in the improved signal-to-noise ratio of the combined data. The result of these operations are shown in Fig.4.18. 74 Chapter 4. Data analysis

Figure 4.18: A colour image of the A370 ﬁeld of view, using the F435W, F606W and F814W observations from the Hubble Space Telescope Frontier Fields (HFF) project. The region of the cluster covered by the MUSE GTO programme is shown in white. The positions of the multiply imaged systems used as constraints by Lagattuta et al. to elaborate a new mass model of A370 are shown as coloured circles: previously known systems are in red, while in green there are newly identiﬁed objects. Images predicted by the model but not detected in either imaging or spectroscopic data are shown in yellow. Source: [20]. 4.7. Kinematics of the two galaxies lensed by A370 and AS1063 75

4.7 Kinematics of the two galaxies lensed by A370 and AS1063

One of the most important aspects of gravitational lensing is that it allows us to investigate not only the morphology but also the detailed kinematics of high-z galaxies. In this section I explain the procedure that I have used to obtain the rotational curve of the spiral galaxy, located at z = 0.7251, lensed by Abell 370 and producing the set of 5 images forming the giant arc known as the “Dragon”. The first step of this procedure was to download a MUSE cube of Abell 370 containing the whole arc from the ESO website 6. Matching the downloaded cube with the HST images of Abell 370 that I have used in the previous analysis, I realized that there was an offset and that the cube had to be aligned in order to exploit again the lensing mass model of Abell 370 elaborated by the CATS team. In fact, the publicly available lensing mass models on the Frontier Fields website are all perfectly aligned with the HST images. I have performed the alignment “manually” with the software DS9 through the following steps. I have chosen a bright star in the MUSE cube and I have measured its center (in WCS), then I have identified the same star in the HST image and, again, I have measured its center. I have computed the difference (offset) between these two centers and I have adjusted the keywords CRVAL1 and CRVAL2 in the header of the MUSE cube subtracting the offset through the hedit task in IRAF. At this point, I have used the imcopy task in IRAF to extract the MUSE cube’s first extension which is a sort of “primary header” and contains no data. In order to obtain a cube in the source plane relative only to the “head” of the Dragon, I have written a file .par with the lensing mass model of Abell 370 elaborated by the CATS team and with the keywords “cubeframe” (in which I have specified the MUSE cube’s

6Web site: http://archive.eso.org/scienceportal/home 76 Chapter 4. Data analysis

Figure 4.19: A reconstructed white-light image of the galaxy obtained by summing each spatial pixel of the cube in the source plane along the wavelength axis. This converts the 3D cube into a 2D image.

first extension) and “contour” (in which I have specified an ASCII file containing a list of points that define the contours of the polygon holding the “head” of the Dragon) in the Cleanlens mode. I have run this paremeter file with Lenstool and I have obtained a cube showing the lensed galaxy, producing the “head” of the Dragon, at different wavelength (see Fig. 4.19).

Through a python code, I was also able to extract the spectrum of this galaxy in the visible wavelength range and to make a zoom in the region between 6400-6450 A˚ where there is the [OII] emission line doublet (λλ 3726.1-3729.8 A˚ rest-frame). They are shown respectively in Fig. 4.20 and in Fig. 4.21.

As we known, the [OII] doublet is an important SFR tracer, especially for galaxies at redshift beyond z = 0.2−0.3 for which the Hα Balmer line of hydrogen 4.7. Kinematics of the two galaxies lensed by A370 and AS1063 77

Figure 4.20: Spectrum of the galaxy in the region between 4750-7300 A.˚

Figure 4.21: Zoom of the spectrum in the region between 6400-6450 A˚ where there is the [OII] emission line doublet. 78 Chapter 4. Data analysis

Figure 4.22: Subcubes that I have considered in order to compute the observed redshift in diﬀerent parts of the galaxy. The center of each subcube is represented by a cross while the diamond point is the center of the galaxy. becomes redshifted to the near infrared where many strong sky emission lines are present. The 3726.1A˚ and 3729.8A˚ emission lines are due respectively to the

3/2 3/2 5/2 3/2 2D 99K 2S and 2D 99K 4S transitions. The [OII] doublet is also a prime candidate for tracing the internal kinematics of the ionized gas as I will show. In order to obtain the rotational curve of the galaxy lensed by A370, I started from the following considerations [25]. In ﬁrst approximation, the total velocity v and the observed redshift z of a galaxy are given by:

v =v ¯ + vp (4.7.1) and

z =z ¯ + zp , (4.7.2) wherev ¯ is the velocity of the system due to the expansion of the Universe, vp is the peculiar velocity respect to the cosmic background,z ¯ is the cosmological redshift and zp = cvp is the “peculiar redshift”. Hence, in order to measure the observed redshift z in diﬀerent part of the galaxy, I have considered 9 subcubes of my cube in the source plane (see Fig.4.22) and, for each subcube, I have extracted 4.7. Kinematics of the two galaxies lensed by A370 and AS1063 79

the spectrum in the region 6400 − 6450A˚. I have performed a Gaussian ﬁt to the

[OII] emission line (see Fig.4.23) and I have considered the λpeak corresponding to the peak to compute the observed redshift z that is given by:

λ z = peak − 1 , (4.7.3) λem

where λem is the wavelength emitted at rest frame. At this point, through Eq.(4.7.2),

I have deduced zp and I have computed the peculiar velocity, correcting it for a factor sini, where i is the inclination of the galaxy [22]. Finally, I have computed the distance, R, from the center of the galaxy (that I have obtained through a visual inspection) to the center of each subcube, primarily in pixels, then in arcsec and, considering that 1 arcsec = 7.245 kpc at z = 0.7251 (H0=70 km/s/Mpc, Ωm=

0.3 and ΩΛ=0.7), at the end, in kpc. In order to obtain the rotational curve of the galaxy, I have plotted the corrected peculiar velocity vp versus R as shown in Fig.4.24.

I have computed even the total mass of this galaxy obtaining a value of Mtot = (1.1 ± 0.6) · 1011M . I have adopted the same procedure to deduce the rotational curve of the galaxy lensed by AS1063 whose morphology I have reconstructed in the previous sections. This galaxy is located at z = 0.611, therefore the [0II] doublet is in the region of the spectrum around 6000 A.˚ This time I have considered 6 subcubes along the major kinematics axis of the galaxy (as shown in Fig.4.25) and I have obtained the rotational curve illustrated in Fig.4.26.

In this case, for the mass of the galaxy I have obtained a value of Mtot = (1.8 ± 0.5) · 1011M . The masses have been computed in the simplest case of orbital circular motion (approximation of spherical galaxy) and refer to the whole content of matter in the galaxy, i.e Mstell + MDM . 80 Chapter 4. Data analysis

Figure 4.23: Gaussian ﬁt to the [OII] emission line at the center of the lensed galaxy.

Figure 4.24: Rotational curve of the galaxy lensed by A370. 4.7. Kinematics of the two galaxies lensed by A370 and AS1063 81

Figure 4.25: Subcubes that I have considered in order to compute the observed redshift in diﬀerent parts of the galaxy lensed by AS1063. The center of each subcube is represented by a circle while the diamond point is the center of the galaxy obtained through visual inspection.

Figure 4.26: Rotational curve of the galaxy lensed by AS1063. 82 Chapter 4. Data analysis

4.8 Results

The rotational curves in Fig.4.24 and Fig.4.26 have the typical shape of a rotational curve of a spiral galaxy: up to about 2 − 3 kpc, the peculiar velocity, increases linearly with the distance from the center of the galaxy, R, (as in a rigid disk occurs) then follows a plateaux. The ﬂat trend of the peculiar velocity after 2 − 3 kpc is discordant with the Keplerian predictions (according to which the peculiar velocity would scale as √1 ) and can be explained only assuming the existence of R 2 rM dark matter in the halo of the galaxy. Indeed, if v = G holds, then M has to increase proportionally with R to keep the quantity v2 constant. I have used the values of Vmax that I have obtained for the galaxies lensed by A370 and AS1063 in the previous section to “update” a graph realized by Puech et al. [1] showing the stellar-mass Tully-Fisher Relation (smTFR) for 64 galaxies (see Fig. 4.27). For the stellar masses of the galaxies belonging to my sample, I have considered the values reported by V.Patr´ıcioet al. in their paper [22] as well as for the Vmax of the galaxy lensed by A2667. In this plot, there are three types of galaxies:

• “rotating disks”, virialized galaxies;

• “perturbed rotators”, galaxies with a disk morphology but characterized by irregular rotational curves due to the fact that probably they are not at the equilibrium, or they are still assembling or they are dynamically young,

• “complex kinematics”, galaxies with a disturbed morphology and a complex kinematics due, for example, to a merging in progress.

As I have shown in the previous sections, the galaxies lensed by A370 and AS1063 have quite regular morphology and rotational curves hence I believe that they could belong to the class of “rotating disks”. For the galaxy lensed by A2667, 4.8. Results 83

Figure 4.27: The smTFR in the sample of 64 galaxies considered by Puech et al. and in my sample of three galaxies represented by a star-shape. The black line is the local smTFR while the dash-line represents a linear ﬁt to the z ∼ 0.6 smTFR. 84 Chapter 4. Data analysis

I did not get the rotational curve but the morphology seems that of a disk so I think that the case of “complex kinematics” can be reasonably excluded. The study of the rotational curve of the galaxy lensed by A2667 could be a starting point for a new work. Chapter 5

Conclusions

In this thesis I have reconstructed the morphology of three z ∼ 1 gravitationally lensed in the massive galaxy clusters: A370, AS1063 and A2667. The first two clusters belong to the “Frontier Fields” research program, whose purpose was to observe the “frontiers” of the visible Universe, combining the power of the Hubble Space Telescope with the gravitational lensing effect produced by massive galaxy clusters. In order to perform the reconstructions, I have used the software Lenstool in the Cleanlens mode. I have specified an HST image of the cluster containing the luminous giant gravitational arc I wanted invert and an ASCII file containing a list of points defining the contours of polygons holding the single images of the background galaxy. In particular, I have obtained different reconstructions of the lensed galaxies in two ways:

• choosing clusters’s HST images produced by diﬀerent ﬁlters.

• selecting both all the images and a single image of the background galaxy.

The three lensed galaxies turned out to be spiral galaxies with a prominent bulge and several arms. Moreover, using a MUSE cube of Abell 370 containing the gravitational arc known as the “Dragon”, I have obtained the rotational curve and

85 86 Chapter 5. Conclusions

the total mass of the lensed galaxy producing this arc. This has been possible through a study of the [OII] emission line in the spectrum of the galaxy. Adopting the same procedure I have obtained even the rotational curve and the total mass of a galaxy lensed by AS1063. Finally, I have “updated” a graph realized by Puech et al. [1] showing the stellar-mass Tully-Fisher Relation (smTFR) for 64 galaxies with my sample of three galaxies lensed by A370, AS1063 and A2667. References

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