Properties of Superclusters of Galaxies from an “Adaptive Luminosity Density Field” Catalogue

The Universe at its Large:

Master Thesis University of Turku Department of Physics and Astronomy Astronomy 2012 Gerardo Lacy Mora Supervised by: Dr. Pekka Hein¨am¨aki Prof. Esko Valtaoja TURUN YLIOPISTO Department of Physics and Astronomy

Gerardo Lacy Mora: The universe at its large: Properties of Superclusters of Galaxies from an “Adaptive Lumino- sity Density Field” Catalogue

Master’s thesis, 80 pages Astronomy June 2012

At the scales of hundreds of megaparsecs, up to thousands of galaxies ensemble together into the largest, differentiated structures of the Universe: the superclusters of galaxies. Superclusters define an interconnected network of galaxy filaments and voids: the cosmic web.

In this thesis I analyzed the Adaptive Luminosity Density Field Supercluster Catalogue by Liivamägiet al.(2012). The purposes of the analysis were, first, to identify whether there are distance selection effects on the properties of superclusters. Also, to investigate the possible correlations between superclusters properties. Futhermore, this work pursue to study the relation between supercluster properties and the galaxy populations among them. In order to fulfill this objectives I used Speraman’s correlation test, Principal Component Analysis and the Luminosity Function Method.

Main results of this work are that no signicant distance selection effects affect the properties of superclusters. Properties characterizing superclusters are strongly correlated. Moreover, the various properties defining superclusters are all equally important characterizing them. It was observed that superclusters with similar characteristics contain very inhomogenoues galaxy populations. Most of the galaxies in superclusters reside within poor groups of galaxies.

Results obtained in this thesis agree very well with studies of superclusters defined at fixed density levels. Even so, it would be valuable to make a direct comparision study between the adaptive and fixed density level supercluster catalogues.

Keywords: Large-scale structures of the Universe, supercluster of galaxies, catalogues. Contents

1 Introduction5

2 Cosmological Background6

2.1 Standard Cosmological Model: ΛCDM...... 8

2.2 Formation of Structures...... 11

3 Observational Cosmology 16

3.1 The Galaxy Zoo...... 16

3.2 Large Redshift Galaxy Surveys...... 20

3.2.1 Sloan Digital Sky Survey...... 21

3.3 Superclusters of galaxies...... 24

3.3.1 General Properties...... 25

3.3.2 Superclusters’ Morphology...... 26

3.3.3 Superclusters as Galactic Environments...... 28

3.3.4 Large Scale Structures in Simulations...... 29

4 SDSS DR7 Superclusters − The Adaptive Luminosity Density Field Cat-

alogue 32

5 Analysis Methods 37

5.1 Principal Components Analysis...... 37

5.2 Luminosity Function...... 40

6 Analysis and Results 42

6.1 General Properties of superclusters...... 42

6.2 Principal Component Analysis...... 51

6.2.1 PCA with physical properties of superclusters...... 51

6.2.2 PCA with properties of galaxies and galaxy groups in superclusters. 56

6.3 Galaxy populations in superclusters...... 61

6.3.1 Selection eﬀects...... 61 6.3.2 Galaxies in superclusters in diﬀerent local environments...... 63

7 Conclusions 70

References 71

Appendices 78

A Colour Superclusters Contours 78 List of Figures

1 Hubble Tuning Fork. Image from Weil(1998)...... 17

2 SDSS cutout images galaxies of diﬀerent morphological type. From top to

bottom: Elliptical, Lenticular, Barred and Spiral galaxies. Each image is

48×48 arcsec2 ...... 18

3 Galaxy distribution from the complete 2dFGRS. Credits: Colless et al.(2003) 22

4 SDSS DR 8 sky coverage. Credits: http://www.sdss3.org/...... 23

5 Evolution of the number of particles in “high-resolution” cosmological N-

body simulations. Adapted from Suto(2003)...... 31

6 Comparison of observed and simulated large-scale galaxy distribution. Cred-

its: http://www.mpa-garching.mpg.de/millennium/...... 31

7 PCA example. Left panel: plot of 216 observations of variables x1 and x2, right panel: representation of the same data after applying PCA to it.... 39

8 The distribution of superclusters in the Representative Sample in cartesian

coordinates. Blue circles are superclusters with at least 50 groups, green

crosses superclusters with 30-49 member groups, red diamonds superclusters

with 11-29 member groups and black squares poorer superclusters...... 44

9 Principal properties of superclusters vs. distance. Horizontal blue line, in

each case marks the average value of the parameter plotted in the vertical

axis...... 44

10 Luminosity density ﬁeld in cartesian coordinates for supercluster in the

Representative Sample. Contours are in units of the mean (luminosity)

density...... 45 11 3D distribution of groups in very rich suplerclusters with more than 950

galaxies. Red diamonds indicates the position of rich groups at least 30

member galaxies. Blue dots shows positions of less populated groups. Axes

correspond to right ascension (R.A - in degrees), declination (Dec - in de-

grees) and distances (in h−1 Mpc ). In left panel, from top to bottom: SCl

001, SCl 010 and SCl 061. Right panel, from top to bottom: SCl 011, SCl

024 and SCl 055...... 46

12 (Continuation) 3D distribution of groups in very rich suplerclusters with

more than 950 galaxies. Red diamonds indicates the position of rich groups

at least 30 member galaxies. Blue dots shows positions of less populated

groups. Axes correspond to right ascension (R.A - in degrees), declination

(Dec - in degrees) and distances (in h−1 Mpc ). In left panel, from top to

bottom: SCl 060, SCl 198 and SCl 351. Right panel, from top to bottom:

SCl 094, SCl 349 and SCl 350...... 47

13 Distribution of the standardized physical parameters of superclusters in the

Representative Sample.N gal−number of galaxies in supercluster, D peak−density

peak of supercluster, L wgal−total (weighted) luminosity of supercluster,

Volume−volume of supercluster and Diameter−diameter of supercluster... 51

14 Principal planes for superclusters, PCA with physical parameters. Arrows

represent the axes where each original variable lies, and their length is

proportional to their importance within each PC. Black dots are plotted

against left and bottom axes and represent each of the original superclusters.

Arrows are plotted against right and top axes...... 54

15 Principal planes for superclusters, PCA with supercluster physical para-

meters. Red diamonds: high-luminosity superclusters with Lwgal,scl >

10 −2 400 · 10 h L ; grey dots: superclusters of lower luminosity...... 55 16 Distribution of standardized properties of galaxies (left) and galaxy groups

(right) in superclusters...... 58 17 PCA with properties of galaxies and galaxy groups in superclusters. Arrows

represent the axes where each original variable lies, and their length is

proportional to their importance within each PC. Black dots are plotted

against left and bottom axes and represent each of the original superclusters.

Arrows are plotted against right and top axes...... 60

18 Properties of galaxies in superclusters vs Distance. Notes: Fgal∈groups− Fraction of galaxies in superclusters in groups; red/blue − ratio of the

number of red and blue galaxies, Fred gal − fraction of red galaxies in superclusters, supercluster g − r − mean galaxy index colour in superclusters. 62

19 Distribution of colour index g − r for bright and faint galaxies in low and

high luminosity superclusters. Left: low-luminosity superclusters, right:

high-luminosity superclusters...... 63

20 Diﬀerential luminosity function for galaxies in superclusters of diﬀerent lu-

minosity...... 65

21 Distribution of colour index g − r for bright and faint galaxies in diﬀerent

local environments. Left: isolated galaxies, center: galaxies in poor groups,

right: galaxies in rich groups...... 66

22 Diﬀerential luminosity function for galaxies in superclusters in diﬀerent local

environments...... 67

23 Projected g − r color contours of very rich supercluster with more than 950

galaxy members. Red diamonds indicates the position of rich groups at least

30 member galaxies. Black dots shows positions of less populated groups.

Axes correspond to right ascenson (R.A - in degrees) and declination (Dec

- in degrees). In upper panel panel, from left to right: SCl 001, SCl 011,

SCl 010, SCl 024, SCl 061 and SCl 055. Right panel, from top to bottom:

SCl 060, SCl 094, SCl 198, SCl 349, SCl 351 and SCl 350...... 79 24 3D distribution of groups in superclusters without rich group members and

corresponding g − r color contour plot. First row: left: SCl 712 ,right: SCl

511. Second row: left: SCl 900, right: SCl 1117. Third row: left: SCl 1145,

right: SCl 1194. Fourth row: left: SCl 509, right: SCl 891...... 80

List of Tables

1 Particular solutions to the Friedmann equations...... 10

2 Properties of Superclusters: General Sample...... 42

3 Properties of Superclusters: Representative Sample...... 45

4 Properties of galaxy groups in superclusters...... 48

5 Results of the Spearman’s rank correlation test: r-values...... 49

6 Results of the Spearman’s rank correlation test: p-values...... 50

7 Results of the PCA of the physical parameters and distances of superclusters

in the Representative Sample...... 52

8 Results of the PCA of the physical parameters of supercluster in the Rep-

resentative Sample ...... 53

9 Results of the PCA with properties of galaxies and galaxy groups in super-

clusters in the Representative Sample...... 57

10 Galaxy populations in superclusters of diﬀerent luminosity...... 64

11 KS test results for galaxy populations in low- and high-luminosity super-

clusters...... 64

12 Galaxy populations in superclusters in diﬀerent local environments...... 66

13 KS test results for galaxy populations in diﬀerent local environments.... 67

14 Summary of the fraction of galaxy populations in diﬀerent environments.. 68

15 Group populations in superclusters of diﬀerent luminosity...... 69 1 Introduction

At the scales of hundreds of megaparsecs (Mpc)1, up to thousands of galaxies ensemble together into the largest, differentiated structures of the Universe: the superclusters of galaxies. Superclusters define an interconnected network of galaxy filaments and voids called the Cosmic Web. The road to conceive such image of the Large Scale Universe is the result of a large observational, technological and theoretical work carried on in the last century.

Given their large dimensions, superclusters are unrelaxed structures. This makes rea- sonably to argue that superclusters may keep key information about the origin of the

Universe itself. Thus, the study of the cosmic web may be one way to answer many of the fundamentals about the origin and future of the Universe.

This thesis is devoted to study the properties of superclusters of galaxies in our nearby

Universe. The document is organized as follows. Chapter2 brieﬂy covers the theoretical framework upon which modern Cosmology is build. In Chapter3 I review the main observational facts that had paved the way to picture the large-scale Universe. Chapter

4 introduces the Adaptive Luminosity Density Field Supercluster Catalogue by Liivam¨agi et al.(2012), which is the basis data analyzed in this work. A description of the methods

I used to analyze the data is given in Chapter5. The product of the analysis and the corresponding discussion is present in Chapter6. Finally, in Chapter7 I present my conclusions.

11 Mpc=106 pc = 3.086 ×1022 m. Our Galaxy has a size of ∼ 30 − 40 kpc.

5 2 Cosmological Background

“... the problem of cosmological analysis is to derive the

observed present day situation and structure of the Universe

from certain plausible assumptions about its early behaviour.”

Zeldovich(1978)

It is in the human nature to wonder about his/her position in the Universe, about its origin and its fate, about the laws that govern the motion of the heavenly bodies he/she see in the night sky. It is Cosmology, aided by hard observational work and profound mathematical concepts, the branch of science dedicated to ﬁnd answers to such questions.

The picture we have of the Universe we live in have been heavily influenced by two independent observations. The first one is the detection, first made by Arno Penzias and

Robert Woodrow Wilson in the 60’s (Penzias & Wilson 1965), that the sky is homo- geneously filled with microwave photons, called Cosmic Background Microwave (CBM) radiation. This radiation correspond to those photons that were scattered in an early age of the Universe when interactions between light and the hot plasma that filled the space became insignificant2.

The other breakthrough discovery in cosmology came from the observation that our

Universe is expanding. This was initially detected by Hubble (Hubble 1929; Hubble &

Humason 1931) based on observations of Cepheid variables in several nearby “nebulae”.

The expansion of the Universe have been conﬁrmed recently from Supernovae Type Ia (SNe

Ia)3 observations (Riess et al. 1998; Perlmutter et al. 1999), showing that the expansion rate is actually accelerating. This, together with the detection of the CBM, implies that our Universe was denser and hotter in the past.

The expansion of the Universe is associated with the existence of a vacuum energy, usually referred as Dark Energy. According to measurements of anisotropies of the CMB

2This epoch is also referred as Epoch of recombination. 3SNe Ia are the explosive phase of white dwarfs that reach the Chandrasekhar limit after accreting mass from a binary companion. Given the physical conditions of the explosion, the luminosity produced by the SN Ia can be used to measure the distance to its host galaxy, reason why SNe Ia are considered as cosmological standard candles.

6 (e.g., Smoot et al. 1992; Komatsu et al. 2011), of the SNe Ia magnitude-relation (Kowalski et al. 2008) and the galaxy clustering (for example, Tegmark et al. 2006; Reid et al. 2010), dark energy is estimated to contribute with about 73% of the total energy density of the

Universe. Of the remainder, about 22% is considered to be in the form of a non-relativistic, non-interacting form of matter, called Dark Matter (DM)4, and only a 5% to be ordinary

(baryonic) matter.

The existence of DM have been known since the early 30’s, when measurements of galactic rotation curves in the Coma and Virgo clusters by Zwicky (Zwicky 1933) and

Smith (Smith 1936) indicated that the amount of matter in those clusters is 10−100 times higher than what it is inferred from luminous matter. More recently, the existence of DM have been veriﬁed from X-rays studies of galaxy clusters and weak lensing measurements

(see for example Schneider 2003; Riemer-Sorensen et al. 2007).

Modern cosmology is build upon the theory of General Relativity (GR), proposed by

Albert Einstein (Einstein 1917), according to which the geometric properties of spacetime depend on the distribution of matter and energy in the Universe. Within GR, the large scale structures we observed today formed and evolved (and are evolving) from initial inhomogeneities in the mass distribution of the early Universe that then grow under the inﬂuence of gravity. This ideas form the basis of the Standard Cosmological Model, or as it is most commonly called, Hot Big Bang Theory.

In the next two sections I will brieﬂy outline the basic mathematical framework of the standard cosmological model (§2.1) and the theory of formation of structures (§2.2), which together describe the origin and evolution of the large scale structures we see today in the

Universe around us.

4The determination of the nature of the DM is one of the most intriguing questions in modern cosmology.

7 2.1 Standard Cosmological Model: ΛCDM

”There is nothing new to be discovered in physics now.

All that remains is more and more precise measurement”

Lord Kelvin, 1900

The Theory of General Relativity is based on Einstein’s ﬁeld equations:

1 8πG R − g R = T − g Λ, (1) µν 2 µν c4 µν µν where Rµν (the Ricci curvature tensor), R (the scalar of curvature) and gµν (the metric tensor) are terms related to the geometry of the spacetime manifold and Tµν (the stress- energy tensor) describes the density and ﬂux of energy and momentum in spacetime; c is the speed of light and G is Newton’s gravitational constant. The term Λ is called

Cosmological Constant and represents the possibility that there is a density and pressure associated with empty space and/or that there is a strange form of repulsive gravitational energy/matter, or Dark Energy. The Cosmological constant accounts to the (currently) observed accelerated expansion of the Universe5.

Supported by the observation of the CBM radiation, at large scales, the Universe can be considered to be spatially homogeneous and isotropic. For such conditions, the stress-energy tensor takes a perfect ﬂuid form,

Tµν = (ρ + p)uµuν + pgµν, (2) where ρ is the energy density and p is the pressure measured by an observer moving with four-velocity uµ. The energy density is usually expressed as the contribution of all possible ﬂuids: X ρ = ρi = ρm + ρr + ρΛ + ρκ, (3) here ρm, ρr, ρΛ and ρκ represents the contributions to the energy density budget from pressureless matter, radiation, cosmological constant and spatial curvature, respectively.

5Initially, the cosmological constant was introduced by Einstein in his ﬁeld equations to solve the problem of a static unstable Universe.

8 The radiation term is usually considered to be form by CBM photons (ργ) and neutrinos

(ρν), while the matter density has contribution from baryonic matter (ρb) and cold dark matter (ρCDM). Once the mass-energy content of the Universe is speciﬁed, a model for the geometry of spacetime has to be adopted in order to solve Einstein ﬁeld equations. The Friemann-

Roberson-Walker (FRW) metric

dr2 ds2 = −dt2 + a2(t) + r2dϑ2 + r2 sin2 ϑdφ2 , (4) 1 − κr2 is an exact solution to Einstein’s ﬁeld equations which describe all isotropic, homogeneous, uniformly expanding (or contracting)6 models of the Universe. In the FRW metric κ represents the spatial curvature of the universe7 and a(t), the scale factor, represents the relative expansion of the universe.

For a FRW universe, Einstein’s ﬁeld equations can be reduced to two independent equations a˙ 2 κc2 8πG Λc2 + = ρ + , (5) a2 a2 3 3 a¨ 4πG 3p Λc2 = − ρ + + , (6) a 3 c2 3 wherea ˙ anda ¨ denotes the ﬁrst and second time derivative of the scale factor, respectively.

These equations are collectively called Friedmann equations.

In order to ﬁnd the evolution of the scale factor a(t), a matter model has to be adopted.

In the case of FRW universe, it is suﬃcient to give the equation of state, i.e., p = p(ρ), which can be parameterized as

2 pi(ρ) = ωiρic , (7) where the subscript i indicates each of the possible contributors to the energy density

1 budget of the Universe. For the known components of the Universe, ωγ = 3 , ωm = 0, 1 1 ωκ = − 3 , ωΛ = −1 and ων is in the range [0, 3 ]. 6The coordinates (t, r, θ, φ) in equation (4) are called comoving coordinates, which indicate that they follow the expansion of the Universe, the Hubble ﬂow. 7κ can take the values −1, 0 or 1 corresponding to an open, ﬂat or closed Universe, respectively.

9 Table 1. Particular solutions to the Friedmann equations

Dominant component ω ρ a(t)

−3 3 2/3 Matter 0 a ( 2 H0t) 1 −4 1/2 Radiation 3 a (2H0t) Λ q Λc2 Λ −1 8πG exp 3 t

If each ﬂuid is decoupled, Friedmann equations can be combined to express the evolution of the energy density as

ρ˙i + 3H(1 + ωi)ρi = 0 (8) where H, the Hubble parameter, deﬁned as

da 1 H ≡ H(t) ≡ , (9) dt a gives the expansion rate of the universe. It value at the present time, called Hubble constant, is normalized to unity, i.e., H(t0) = H0 = 1. It is customary to express the Hubble constant in terms of the dimensionless Hubble parameter h, such that

−1 −1 H0 = 100 h km s Mpc (10) and all the uncertanty in the value of H0 is assigned to h. An analytical solution in the case when all ﬂuid components are present is not possible, but for many cases of interest it is possible to obtain analytic solutions to Equation (8).

A summary of such solutions is given in Table1.

The critical density is deﬁned as

3H2 ρ ≡ , (11) c 8πG is the energy density that a spatially ﬂat universe (κ=0) with no cosmological constant

(Λ = 0) that expands with rate H(t) would have (see Equation (5)).

The density parameter Ωi(t) is deﬁned as

ρi(t) Ωi(t) ≡ (12) ρc(t)

10 Following Equation (3), the density parameter can be expressed as

Ω = Ωm + Ωr + ΩΛ + Ωκ. (13)

The FRW cosmological model is deﬁned by giving the present values of the three cosmo-

−1 −1 logical parameters, H0,Ωm and ΩΛ. Observations favor H0 = 71.0 ± 2.5 km s Mpc ,

Ωb = 0.0449 ± 0.0028, ΩCDM = 0.222 ± 0.026 and ΩΛ = 0.734 ± 0.029 (Jarosik et al. 2011). FRW models, being solution to homogeneous and isotropic universes, by themselves do not account for the observed structures in our Universe. In order to reproduce the large scale structures it is needed to introduce initial density inhomogeneities that then grow under the effect of gravity and pressure gradients in an expanding universe. These “seeds” are considered to be generated from quantum fluctuations followed by an exponential expansion of the early universe, epoch known as inflation. CBM measurements provide the observational constraints to the shape of these initial density perturbations.

2.2 Formation of Structures

“Usually, instabilities of various types destroy structures, rather than create them.”

Byrd,G. and Chernin, A. and Valtonen,M(2007)

The principal idea behind the theory of formation of structures8 is that in a homogeneous and isotropic fluid, small fluctuations in density and velocity can evolve with time if the timescale for pressure effects to establish is much smaller that the timescale needed for density fluctuations to collapse under its own gravity. Thus, it is considered that the structures we observed today, such galaxies and clusters of galaxies, have grown from small initial density perturbations due to the effect of gravity. In this picture, structures grow hierarchically: smaller (dark matter) haloes merge by collisionless collapse to form larger and more massive haloes, which them become the birth place for galaxies due to the cooling of hot gas (Ratra & Vogeley 2008).

8Here we are interested in the formation of galaxies and large scale structures, although the theory of formation of structures, initially proposed by James Hopwood (1877−1946) (Jeans 1902), applies to any system driven by gravitational instabilities such as the formation of stars and planets.

11 A complete (relativistic) treatment of the growth of primordial perturbations in a expanding universe by gravitational instabilities involves the introduction of small de- partures from homogeneity and isotropy to a background FRW metric and to solve the resulting linearized Einstein’s equations. Although, if perturbations are small, i.e., much more smaller than the Hubble radius9, it is possible to treat matter as an ideal ﬂuid that follows the basic laws of hydrodynamics (continuity, Euler and Poisson equations). This approach is called linear gravitational perturbation theory10.

The perturbations can be characterized by a contrast function δ(x, t)11, which describe the deviations of the ﬁeld from homogeneity and isotropy. In terms of the contrast parameter, the dynamical quantities can be expressed as

ρ(x, t) =ρ ¯(t)[1 + δ(x, t)], (14)

p(x, t) =p ¯(t)[1 + δ(x, t)], (15) whereρ ¯ andp ¯ represents the density and pressure of the background ﬁeld, respectively.

Linear perturbation theory can be used until δ . 1. After that, evolution turns non-linear. Adiabatic perturbations are also assumed. This means that fractional enhancements are the same for both massive particles and photons. Using this approach, the density ﬂuc- tuations in a (matter dominated) FRW universe at the sub-horizon scale grows according to ∂2δ ∂δ c2 + 2H = 4πGρδ¯ − s ∇2δ, (16) ∂t2 ∂t a2 where ∂p c2 ≡ , (17) s ∂ρ is the velocity of sound and time derivatives and ∇ operates over the comoving coor- dinates12. The second term on the left-hand side in Equation (16) tends to suppress

9This is the case for a baryonic gas in local thermal equilibrium and collisionless dark matter where diﬀusion is negligible. The Hubble radius is deﬁned as RH ≡ c/H0 =∼ 4225 Mpc. 10In the non-linear regime there is no exact solution to this problem. 11Several textbooks also refer to this as the amplitude of density perturbation. 12“Normal” coordinates (r, t) are transformed to comoving coordinates (x, t) according to r = a(t)x.

1 ∂ ∂ a˙ Under this transformation ∇r → a ∇x and ∂t → ∂t − a x · ∇x.

12 perturbation growth due to the expansion of the Universe, hence it is usually called Hub- ble drag13. On the right-hand side of this equation, the ﬁrst term causes perturbations to grow due to gravitational instabilities, while the second term is a pressure term due to the spatial variations in density.

Given the linear nature of Equation (16) governing the growth of perturbations, a

Fourier series expansion of the contrast function

X ik·x δ(x, t) = δk(t)e (18) k is suitable to study the evolution of individual modes. Replacing (18) into (16) we obtain

¨ ˙ 2 δk + 2Hδk = −ωkδk, (19)

where ωk is deﬁned by the dispersion relation

kc 2 ω2 = s − 4πGρ. (20) k H

In the latter equations, k ≡ |k| = 2πa/λ is the comoving wavenumber. Note that Equation

(19) represents a damped oscillator, sometimes called the Jeans Equation. The damping comes from expansion of the Universe; while the density perturbation acts as a source.

2 The condition ωk = 0, which deﬁnes the equality between pressure and gravity forces, deﬁnes a characteristic proper length

2πa r π λJ ≡ = cs , (21) kJ Gρ¯ called Jeans length. These baryonic perturbations on scales λ > λJ remain unaﬀected by pressure forces and are able to grow; while small perturbations on scales λ < λJ feel the eﬀects of pressure gradients and oscillate as acoustic waves with an amplitude that either increases or a decreases exponentially. Therefore, for sub-Jeans scales there is no growth of structures in the linear regime. Jeans length is time dependent and it decreases rapidly after recombination, when baryons can decouple from photons. After this, baryonic perturbations can start to grow at the same rate as dark matter and catch up dark matter

13The expansion of the Universe acts as a viscous force in comoving coordinates. This drag opposes gravitational infall and as a result the growth of density perturbations is slower in an expanding universe.

13 concentrations which have started to accumulate earlier at the beginning of the matter dominated era. Indeed, while baryonic matter started to oscillate after matter-dominated equality, for dark matter only, Equation 19 gives a growing mode δk ∝ a. The description of growth of perturbation according to the linear gravitational perturbation theory accurately describe the early evolution of perturbations up to the point

δρ when density contrast reach ρ ≈ 1 and baryonic and dark matter start to detached from the Hubble ﬂow. This, extrapolated to mildly nonlinear density contrasts is called the

Zel’dovich approximation (Zel’dovich 1970; Shandarin & Zeldovich 1989). This approximation predicts a “top-down” scenario of formation: the first structures to form as the result of collapse are surfaces of high density, the so-called “pancakes”, which then brake into smaller fragments. This formalism accurately describes structure formation up to the epoch when nonlinearities become significant. Comparison with N-body simulations shows this to be very good approximation up to this stage; but once perturbations enter nonlinear regime, a network of filamentary structures seems to dominate the large scales, and not the pancake-like structures predicted by the Zel’dovich approximation. Another famous approach to describe hierarchical structure formation analytically is the Press-

Schechter Theory (Press & Schechter 1974). This approach allows to estimate the mass function of the collapse object. It is often used in theoretical studies.

In order to describe the formation of galaxy systems, like clusters and superclusters of galaxies and the cosmic web, is necessary to follow the non-linear evolution of the primordial perturbations (see for example Bernardeau et al. 2002, for a extensive treatment of the perturbation theory applied to the non-linear regime). This problem has been recently studied, for example, by Einasto(2010); Suhhonenko et al.(2011); Einasto et al.

(2011a,b). Their numerical simulations show that the non-linear evolution of density perturbations of various scales are involved in the formation of the large scale structures: galaxies and galaxy cluster form from density perturbations of the order of =∼ 8 h−1 Mpc, perturbations in the order of 8 − 64 h−1 Mpc are responsible for the ﬁlamentary galaxy distribution inside and between superclusters; while perturbations up to 100 h−1 Mpc contribute to form the network of superclusters and voids. The properties of the cosmic

14 web (richness of superclusters, spatial locations of superclusters and voids) are modulated not only by the scales of the density perturbations but by their phases; in particular, supercluster form where density waves of medium and large scales combine constructively to generate high density peaks.

15 3 Observational Cosmology

“Your observations are ruled out by the theory!!”

Anonymous

The ultimate word about whether our models and conceptions of the Universe are right or wrong comes undoubtedly from the information we gather from observations. In the ﬁeld of Cosmology in particular, this means to capture light that was emitted basically since the early ages of the Universe and had travel all across the Universe before reaching our telescopes. This chapter starts by giving a glimpse of some of the observational facts that established the diversity and complexity of the building blocks of our Universe: galaxies.

In §3.2 I give a brief description of some of the more recent and relevant redshift galaxy surveys that had paved the way into establishing the current picture of how galaxies are distributed on the large-scales in the Universe. This Chapter ends by describing the main properties of the superclusters of galaxies: the largest coherent (galactic) systems in the

Cosmos.

3.1 The Galaxy Zoo

“Let there be light.”

Genesis, 1:3

Galaxies are the basic building blocks of the Universe when looked at the largest scales.

Strictly speaking, they were ﬁrst observed by Sir Frederick William Herschel (1738 −1822), who identiﬁed a great collection of “nebulous” objects in the night sky. It was Edwin

Hubble’s (1889−1953) observations in the decade of 1920, what clearly established those objects as being well beyond the Milky Way (Hubble 1925).

From early observations, it soon became clear that galaxies have diﬀerent shapes (morphologies) and diﬀerent physical properties, situation that nowadays is referred as the

Galaxy Zoo. This diversity made necessary a “galaxy taxonomy”. The famous Hubble sequence or Hubble tuning fork was the ﬁrst consistent, morphological galaxy classiﬁca- tion scheme, gathering galaxies into 4 main categories: ellipticals, lenticulars, spirals and

16 irregulars14 (Hubble 1922). A schematic view of this classiﬁcation can be seen in Figure

Figure 1. Hubble Tuning Fork. Image from Weil(1998)

Elliptical galaxies (E) are characterized for having smooth, almost elliptical isophotes

(contours of constant surface brightness). They are further classiﬁed as En according to their ellipticity (how elongated they are), where n is the closest integer to 10(1 − b/a) and a and b are the lengths of its semimajor and semiminor axis, respectively. Empirically, n is found to vary between 1 and 7. Spiral galaxies (S) have presence of spiral arms around a thin disc structure and a central bulge. They are subdivided into normal spirals and barred spirals (SB), according to whether there is or there is not a bar-like structure in the central part of the galaxy. Each of these spiral classes is further assigned the letters a, b or c15, according to the brightness of the bulge relative to the disc, the tightness of the spiral arms around the galaxy and the degree to which it is possible to resolve stars,

HII regions and dust lanes in the spiral arms. Lenticular galaxies (S0) are intermediate between elliptical and spiral galaxies, having a disk-like structure, dominated by a bulge but without presence of spiral arms. In some cases lenticular galaxies may have a central bar, in which case they are classiﬁed as SB0. Finally, Irregular galaxies (Irr) are galaxies with almost no noticeable structure (Irr I) or no structure at all (Irr II), with a mostly patchy appearance.

14Hubble originally used the classes: spiral, elongated, globular and irregular. Elongated nebulae being further subdivided into spindle and ovate. 15For historical reasons, it is common use to refer to those objects with the a-label as of early-type and those with c-label as of late-type, with no reference to any evolutionary state whatsoever.

17 Figure 2. SDSS cutout images galaxies of diﬀerent morphological type. From top to bottom: Elliptical, Lenticular, Barred and Spiral galaxies. Each image is 48×48 arcsec2

Hubble thought that elliptical galaxies were on early stages in their evolution, while spirals in late stages, adoptiing the term early-type and late-type when moving from left to right in the Hubble diagram. So it is common to refer to elliptical and lenticular galaxies as early-type galaxies, while spiral galaxies are referred as late-type. These terms are only historical and do not have to be taken as an indication of the evolutionary state of a galaxy. (Schneider 2006; Mo, H., van den Bosch, F. C., & White, S. 2010). Figure 3.1 gives a sample of galaxies of diﬀerent Hubble type, from cutout images of Sloan Digital

Sky Survey plates.

Even when the classiﬁcation described above was based solely on the visual appearance of galaxies, it is nowadays well established that such morphological distinction is indeed connected to diﬀerent underlying physical and evolutionary processes. Especially evident from the extra-galactic surveys carried out in the late 20’s and early 30’s it was the fact that the distribution of galaxies in the sky is not uniform but that they tend to cluster (see

18 Bok 1934). Galaxies are found to belong to systems of varying richness, from binary pairs and small groups to rich clusters and superclusters (Zwicky 1942; Oort 1983); moreover, isolated galaxies have been found to be rare objects (e.g., Vettolani et al. 1986; Einasto

1990).

Not only the clustering of galaxies started to show a recognizable pattern in the sky, but also it became clear that the diﬀerent environments where galaxies reside play an important role shaping their properties. For example, diﬀerent studies in the decade of the

80’s started to ﬁnd strong correlations between the morphological type of galaxies and their environment: the fraction of star-forming spiral galaxies decreases with increasing density, while that of elliptical and S0 galaxies increases with density. This morphology-density

(environment) relation was found to vary smoothly from the less dense environment of the

ﬁeld and small galaxy groups (Bhavsar 1981; de Souza et al. 1982; Postman & Geller 1984) to the more dense environments of rich galaxy clusters (Dressler 1980) and superclusters

(Giovanelli et al. 1986). The situation is a bit diﬀerent in compact groups, where galaxies in the same group tend to be of the same type (Hickson et al. 1988). This tendency is also observed as a colour-density correlation: at ﬁx morphological type, red spiral galaxies tend to be located in denser environments than blue spirals, and red early types are found in the densest environments, like the cores of groups and clusters (e.g., Skibba et al. 2009;

Bamford et al. 2009; Deng et al. 2011).

The knowledge of the distribution of galaxies on large scales and the dependence of their properties on environment is one of the main objective of the observational cosmologist.

From these pieces of information is it possible to set constraints on the cosmological models.

This is why increasingly large and deep galaxies surveys are required.

19 3.2 Large Redshift Galaxy Surveys

“Since the ﬂedging redshift surveys during the 1970s,

the mapping of the sky has turned into a thriving industry ...”

Johnston(2011)

As it was mentioned in the previous section, from early galaxy surveys and counts of galaxies (for example, Shapley & Ames 1932; Hubble 1934; Shane & Wirtanen 1954), it became clear that the distribution of galaxies in the sky is not uniform but rather clumpy.

By the decade of the 70’s, the large number of galaxies with measured redshifts made it possible to study the spatial distribution of galaxies in greater detail. This allowed to ﬁnd that galaxies are located mainly along large ﬁlamentary structures, forming a supercluster-void network; distribution that is usually referred as the cosmic web.

Specially relevant for observational cosmology was the technological advances introduced in Astronomy in last half of the 20th century: large telescopes, multi-object spec- trometers, CCD cameras, etc. This advances had made it possible to probe deep regions of the sky. Next I describe some of the more recent and inﬂuential sky (ground-based) surveys that have contribute in this direction (for a more historical review see for example

Reshetnikov 2005).

Las Campanas Redshift Survey Las Campanas Redshift Survey (LCRS) was carried out by the Carnegie Institution’s Las Campanas Observatory in Chile, from 1988 November through 1994 October, using the Las Campanas 2.5 m Irénéedu Pont telescope’s 2◦.1 diameter field. LCRS was the first survey to implement multi-fiber-optic fed spectrographs, allowing to observe over 100 spectra simultaneously. Compiled CCD galaxy catalogue of 26000 galaxy redshfits (¯z = 0.1) over 700 deg2 along 6 long/thin strips (1.◦5 × 80◦, separated by 3◦), 3 in the North Galactic Pole and other 3 in Southern Galactic Pole. The initial goal of the project was to obtain a large and deep enough galaxy sample to carry a reliable characterization of the average properties of galaxies and their distribution. One of the main outcomes of LCRS was the statistically confirmation that the supercluster/void structure observed in previous, shallower surveys (for example, the CfA Redshift Survey

20 (de Lapparent et al. 1986)) was indeed a general property of the distribution of galaxies at the large scales (Shectman et al. 1996).

2 degree Field Galaxy Redshift Survey The 2 degree Field Galaxy Redshift Survey

(2dFGRS) was developed between 1997 and 11 April 2002, using the 4 m Anglo-Australian

Telescope (AAT) operated by the Australian Astronomy Observatory. ATT is equipped with a multiﬁbre spectrograph which is capable of observing 400 objects simultaneously over a 2◦ diameter ﬁeld. 2dFGRS measured redshifts of approximately 250 000 galaxies selected from a revised and extended version of the APM galaxy catalogue (a 2 million galaxy catalogue obtained from photographic plates of the United Kingdom Schmidt Telescope

Southern Sky Survey), covering a total area of 2000 deg2, including a strip of 80◦ × 15◦ around the South Galactic Pole (SGP strip), a 75◦ × 10◦ strip along the celestial equator (NGP strip) and 99 fields randomly distributed around the SGP strip. The survey reached a median depth ofz ¯ = 0.11 and targeted galaxies with extinction-corrected magnitudes brighter than bj = 19.45. At the effective limit of the survey, z ≈ 0.3, the strips encompass a volume of 1.2×108 h−3 Mpc3. The 2dFGRS was designed to achieve an order- of-magnitude improvement on previous redshift surveys (Colless et al. 2001, 2003). This resulted for example, in accurate measurements of the power spectrum of galaxy clustering up to 300 h−3 Mpc, in the first detection of baryon acoustic oscillations (Percival et al.

2001), and the ﬁrst direct measurements of the galaxy bias parameter (Verde et al. 2002;

Lahav et al. 2002).

3.2.1 Sloan Digital Sky Survey

The Sloan Digital Sky Survey (SDSS) is a deep, multiband, spectroscopic survey of one quarter of the entire sky. The goals of this survey is to obtain CCD imaging in ﬁve broad bands over 10, 000 deg2 of high-latitud sky, and spectroscopic of a million galaxies and

100, 000 quasars over this same region (Abazajian et al. 2009)

The SDSS telescope is located at an elevation of 2800 m at the Apache Point Observa- tory (APO) in New Mexico. It is a dedicated wide FoV 2.5 m telescope equipped with a wide-area, multiband, 120-megapixel CCD camera and a pair of 640-ﬁber-fed double spec-

21 Figure 3. Galaxy distribution from the complete 2dFGRS. Credits: Colless et al.(2003) trographs, which allows SDSS telescope to carry both imaging in u, g, r, i, z optical bands and multiobject spectroscopic observations in the range 3800 A˚ − 9200 A˚ at a resolution of ∆λ/λ ' 2000. Imaging of the sky is made by scanning along great circles at the sidereal rate; the effective exposure time per filter is 54.1 s, and 18.75 deg2 are imaged per hour in each filter. SDSS telescope began regular survey operations in April 2000 (York et al.

2000; Gunn et al. 2006; Abazajian et al. 2009).

The SDSS was originally planned for 5 years of operation, but at the end of this time the project had not reach its main objectives, so the original program (SDSS-I) was extended. The extension was named SDSS-II (the component of SDSS-II which correspond to the completion of SDSS-I is known as the Legacy Survey). The SDSS data have been made public in a series of yearly, cumulative releases (Abazajian et al. 2009). As the writing of this document (July 5, 2012) 8 major releases has been made. The latest release, the Data Release 8 (DR8) is the first release of SDSS-III, a new phase focused on studying the Galactic structure and chemical evolution, measurements of the baryon oscillation feature in the clustering of galaxies and the quasar Lyα forest, and a radial velocity search for planets around ∼8000 stars. These different objectives are addressed by four specific surveys: the Baryon Oscillation Spectroscopic Survey (BOSS), the Sloan

Extension for Galactic Understanding and Exploration 2 (SEGUE-2), the APO Galactic

Evolution Experiment (APOGEE) and The Multi-object APO Radial Velocity Exoplanet

Large-area Survey (MARVELS). SDSS-III started operation in 2008 and is planned to

22 continue operations until July 2014. DR8 adds additional data unreleased from previous campaigns and also updates matches to external catalogues. Much of the data have been reprocessed and re-calibrated in this latest release. DR8 is the largest collection of color images of the sky available (Aihara et al. 2011).

Figure 4. SDSS DR 8 sky coverage. Credits: http://www.sdss3.org/

SDSS-II and DR7. SDSS-II was completed in July, 2008. Data Release 7 (DR7) is the seventh major data release of SDSS and the ﬁnal data release of SDSS-II. Upon the completion of SDSS-II phase, the project reached the goals related to three distinct surveys:

The Sloan Legacy Survey: contiguous imaging and spectroscopy over 7500 deg2 of the Northern Galactic Cap and three stripes in the South Galactic Cap totaling 740 deg2.

The Legacy spectra consists of the so called Main Sample, a ﬂux-limited sample of bright galaxies with Petrosian r-band magnitude r ≤ 17.77, r-band Petrosian half-light surface

−2 brightness µ50 ≤ 24.5 mag arcsec and median redshift of z ≈ 0.104 (Strauss et al. 2002), the Luminous Red Galaxies (LRG) Sample, a color/magnitude based volume-limited sam-

23 ple (∼ 1 h−3 Gpc3) of ∼ 100,000 luminous intrinsically red galaxies with Petrosian r-band magnitude down to 19.5, extending the Main sample both in distance (z ≈ 0.5) and to include fainter objects (Eisenstein et al. 2001), and a Quasar Catalogue, including 105,783 spectroscopically conﬁrmed quasars and objects with luminosities Mi > −22.0, at least one emission line with FWHM > 1000 km s−1 or have interesting/complex absorbing features and are fainter than i ≈ 15.0 and have highly reliable redshifts, covering an area of

≈ 9380 deg2 and 0.065 < z < 5.46 (Schneider et al. 2010). Legacy data have supported studies ranging from asteroids and nearby stars to the large-scale structure of the universe.

Sloan Extension for Galactic Understanding and Exploration (SEGUE): imaging and spectroscopic survey of ≈ 240, 000 moderate resolution (R ∼ 1800) spectra from 3900−9000 A˚ of fainter Milky Way stars (14.0 < g < 20.3) of a wide variety of spectral types, both main sequence and evolved objects, with the aims of explore the structure; formation history; kinematics; dynamical evolution; chemical evolution and dark matter distribution of the Milky Way (Yanny et al. 2009).

The Sloan Supernova Survey: time-domain survey, involving repeat imaging of the same region of sky every other night over three 3-months campaigns. The main scientiﬁc objective was to detect and measure light curves for several hundred intermediate- redshift (0.05 < z < 0.35) Type Ia supernovae (SNe Ia) through repeat scans of a 300 deg2 stripe (about 2.5◦ wide by ∼ 120◦ long) centered on the celestial equator in the Southern

Galactic Cap. These observations aimed to set constraints on cosmological parameters and on SN determination systematics errors, to anchor the Hubble diagram, to improve the rest-frame ultraviolet data for high-z SNe and help in the determination of SN rates and rare SN types (Frieman et al. 2008).

3.3 Superclusters of galaxies

“... the dispute for the existence of second-order clusters

of galaxies already belongs to the history of extragalactic astronomy.”

Kalinkov et al.(1985)

24 3.3.1 General Properties

The existence of large conglomerates of galaxies and the irregularity of their distribution has been recognized since the early 30’s, especially from analysis of photographic plates from Lick, Mount Wilson and Harvard observatories (Shapley 1930; Hubble & Humason

1931; Shapley 1932; Mayall 1934). For example, Shapley(1930) described a “remote cloud” of 315 galaxies (between magnitudes 16 and 18), covering an area of the sky of about 2.2 deg2 in the constellation of Centaurus. It was in the study of de Vaucouleurs

(1953) where the ﬁrst description of a supercluster appeared, who used the term “a local supergalaxy” to refer to what is now known as the Local Supercluster.

First all sky redshift surveys in the mid 70’s established the inhomogeneous large-scale spatial distribution of galaxies. Early evidence of the presence of interconnected galaxy systems was given by Gregory & Thompson(1978) and J˜oeveer et al.(1978). Gregory

& Thompson(1978) demonstrated that the two rich clusters Coma (A1656) and Leo

(A1367) are embedded in a larger structure (now called the Coma Supercluster). In J˜oeveer et al.(1978) it was shown that most galaxies and clusters of galaxies around the Perseus clusters form chains (the so called Perseus Supercluster). It was the work presented by de

Lapparent et al.(1986), from data of the CfA Redshift Survey and Catalog, who ﬁnally stopped the debate and convincingly demonstrated the presence of voids and superclusters in the galaxy distribution.

It is now well accepted that the large-scale distribution of galaxies in the Universe forms a complex network of interconnected galaxy filaments, clusters and voids and walls; being superclusters the largest non-percolating (unvirialized) systems of galaxies or cluster of galaxies (Oort 1983). Superclusters, similar as in the case of clusters and groups of galaxies, have been defined as (the largest) luminosity density enhancements in the overall galaxy distribution (Einasto et al. 2003b; Liivamägiet al. 2012; Luparello et al. 2011). It is also typically defined as second-order clusters of galaxies (Jõeveer & Einasto 1978). Given their unrelaxed state, supercluster definition is not fixed, and often a given luminosity density threshold is chosen a priori such that superclusters are considered as connected density regions above that threshold (Einasto et al. 2007b; Luparello et al. 2011; Liivamägi

25 et al. 2012).

The ﬁrst supercluster catalogues were constructed in the decade of 1980, using clusters as structure tracers (Bahcall & Soneira 1984; Kalinkov et al. 1985; Bahcall 1986; West

1989). Relatively deep all-sky supercluster catalogues were ﬁrst obtained in the decade of the 90’s, based on the Abell(1958) and Abell, Corwin, & Olowin(1989) galaxy cluster catalogues (Zucca et al. 1993; Einasto et al. 1994; Kalinkov & Kuneva 1995; Einasto et al.

1997; Einasto et al. 2001). Nowadays, with the availability of deep and accurate redshift galaxy surveys (e.g. Las Campanas, 2dFGRS, SDSS), superclusters catalogues have been constructed based on the luminosity density ﬁeld (DF) of the full galaxy distribution

(Einasto et al. 2003a,b, 2006, 2007b; Costa-Duarte et al. 2011; Luparello et al. 2011;

Liivam¨agiet al. 2012).

Luparello et al.(2011), using the density ﬁeld method on SDSS DR7 galaxies, constructed a catalogue of present structures that according to the ΛCDM scenario will evolve into future virialized structures (FVS). Their analysis shows that not all DF-superclusters will collapse into virialized structures, although many known superclusters indeed coincide with one or more FVS.

More recently, Liivam¨agiet al.(2012), using the DF method, compiled the largest supercluster catalogues based on the SDSS DR7 main and Luminous Red Galaxy (LRG) samples (I will analyze and discuss more in detail the method used by the authors and the resulting catalogues in Section4).

Analysis of these superclusters catalogues have shown that the largest scale in the distribution of clusters and superclusters of galaxies is of about 100 h−1 Mpc (Oort 1983).

Richest clusters are found to be located in chains and walls, while the distribution of superclusters in void walls depends on supercluster richness (Einasto et al. 1994; Einasto et al. 1997; Einasto et al. 2007d; Einasto 2010).

3.3.2 Superclusters’ Morphology

First attempts to quantify the morphology of superclusters relied on “principal axes method” (West 1989; Jaaniste et al. 1998; Basilakos et al. 2001). This method approx-

26 imates shape and orientation of superclusters by ﬁtting triaxial ellipsoids to the data.

More recently, superclusters morpholgy have been studied mostly based on the use of

Minkowski functionals and shape finders method (see Sahni et al. 1998; Kolokotronis et al. 2002; Einasto et al. 2011e). Minkowski functionals, in short, describe the volume, superficial area and curvature of a set of points defined over a density field, which in turn serve to define the number of clumps in the sample (luminosity density superclusters in this case). The shape finders are defined in terms of the Minkowski functionals and describe the thickness (H1), width (H2) and length (H3) of the superclusters. In order to describe the planarity and filamentarity of the superclusters, two more quantities are defined: K1 = (H2 − H1)/(H2 + H1) and K2 = (H3 − H2)/(H3 + H2), respectively. On the superclusters’ morphology, several studies (e.g Einasto et al. 2003b, 2007d,

2011d,e; Tempel et al. 2011; Costa-Duarte et al. 2011) have shown that the distribution of

DF-clusters within superclusters gives information about the internal structure of the superclusters. Most superclusters morphologies can be clasiﬁed as spider (a system of several

filaments growing from one concentration - a rich cluster), multispiders (an assembly of high density regions connected by chain of galaxies), filaments or multibranching filaments.

Almost all massive superclusters are multibranching systems, while faint superclusters do not present a dominant morphological type. Regarding superclusters richness, the more rich ones appear to be multibranching ﬁlaments, less compact and more asymmetrical than poor superclusters. Also, rich superclusters are very elongated with ﬁlamentarities larger than planarities. The more elongated superclusters are more luminous, have larger diameters and contain larger number of rich clusters Einasto et al.(2011d). Similar results have been found in several studies (e.g Kolokotronis et al. 2002; Costa-Duarte et al.

2011; Luparello et al. 2011; Einasto et al. 2007d, 2011c): more luminous superclusters are richer (and contain richer galaxy clusters), larger, denser, more elongated and more internal structured than low-luminosity ones.

Recently, Einasto et al.(2011c) have analyzed a supercluster sample (comoving distances in the range between 90 h−1 Mpc and 320 h−1 Mpc and threshold density level

D = 5) drawn from the L12 catalogue using the Principal Component Analysis (PCA)

27 Method and Spearman’s correlation test to study the properties of supercluster. Regard- ing superclusters’ morphology, they have found that high-luminous supercluster (L >

10 −2 400 · 10 h L ) can be divided into two populations concerning their morphological properties: more elongated systems (shape parameter K1/K2 < 0.5) and less elongated ones

(K1/K2 > 0.5). Einasto et al.(2011d) was the ﬁrst extensive study on superclusters morphology, concluding that large-scale distribution of the superclusters is very inhomogeneous, with richest superclusters forming chains.

Superclusters inhomogeneous morphologies suggest that their evolution has been di-

ﬀerent. Thus, analyses of the morphology of superclusters may be used to distinguish between diﬀerent cosmological models (Kolokotronis et al. 2002).

3.3.3 Superclusters as Galactic Environments

The relation between superclusters’ (large scale / global) environments and galaxy clusters and galaxies properties and distribution have been analyzed especially through the Lumi- nosity Function (LF). Early in the XXI millenium, Hoyle et al.(2005); Croton et al.(2005) studied the LF dependence on density of the environment and galaxy type. They found that galaxies in higher density regions tend to be redder, of earlier type, with a low star forming rate and taht they are strongly clustered. On the other hand, voids are populated by faint, late-type galaxies. Einasto et al.(2003b) suggested that clusters and cluster filaments are formed by similar density perturbations in a way that small-scale perturbation are modulated by large-scale perturbations, making clusters and cluster filaments richer in superclusters and poorer in large voids between superclusters. Einasto et al.(2007d) were the first ones who studied the internal structure within superclusters. They found that richest superclusters are asymmetrical multibranching filaments that start to form earlier than other structures. Hence they are sites of early galaxy formation and clustering. They also found that there is a larger fraction of X-ray clusters in rich superclusters than in poor superclusters. Rich superclusters also may contain a remarkable fraction of X-ray clusters in theirs cores. Another recognized relation between the large-scale and local

(group/cluster) environments is that there are slightly more early, passive galaxies in rich

28 superclusters than in poor superclusters. These early passive galaxies are also found in excess among groups and clusters in the high-density cores of rich superclusters (Einasto et al. 2007c).

Tempel et al.(2011) analyzed the environmental dependence of galaxy morphology and color on diﬀerent density levels. They found a strong environmental dependence for the LF of elliptical galaxies, suggesting that global environmental density is an important factor for elliptical galaxy formation. On the other hand, they found that the LF of spiral galaxies is almost environmental independent, showing that spiral galaxy formation mechanisms are similar in diﬀerent environments.

Also Lietzen et al.(2009, 2011) studied the dependence between the properties of galaxies and large scale environments. They found that radio-quiet quasars and Seyfert galaxies are mostly distributed among low-density regions, while radio galaxies have higher environmental densities. BL Lac objects on the other hand, are found both in low- and very high density environments, but predominantly on low density regions. They also showed that nearby quasars prefer relatively low density regions in the supercluster outskirts.

Recent results imply that multimodal (presence of substructure) clusters of galaxies are more likely to be found in relatively dense environments and that the probability of showing substructure is higher if clusters reside within superclusters. Also superclusters morphology seems to inﬂuence clusters distribution. Multimodal clusters are more likely associated with superclusters of spider morphology and unimodal clusters are more likely to reside in superclusters of ﬁlament morphology (Einasto et al. 2012).

In general, these results tend to support the idea that supercluster environment play an important role shaping the properties of galaxies and galaxy systems.

3.3.4 Large Scale Structures in Simulations

Simulations play a very important role in the study of the formation of large-scale structures in the Universe; providing an eﬀective and essential tool to test cosmological models, make predictions and comparisons with observations. First cosmological N-body simulations started in the 60’s and 70’s (see e.g. van Albada 1961; Aarseth 1963; Peebles 1970).

29 Press & Schechter(1974) and Aarseth et al.(1979) were of the ﬁrst N-body simulations aimed to analyze the clustering and distribution of galaxies in the framework of formation of structures driven by the collapse of gravitational instabilities in an expanding Universe.

By that time, cosmological simulations followed the evolution of some 1000 collisionless dark matter particles whose only interaction was gravity.

Davis et al.(1985) were the ﬁrst who studied the large-scale clustering within the

Cold Dark Matter (CDM)16 model in diﬀerent cosmologies. They used 32 768 particles in computational boxes of . 100 Mpc, and achived to reproduced many of the observed features in the large scale structures (ﬁlamentary structures, superclusters of clumps, voids), but not in exact agreement with observations. Even though, this work emphasized the need of models with higher resolution and larger dynamical range and the fact that simulations provide a powerful tool to test cosmological models, especially within the dark matter scenario to complement both theoretical and observational work.

Since those early works, the advance in computational power, the reﬁning of codes and the implementation of diﬀerent methods (e.g., Evrard 1988; Cen 1992; Couchman et al. 1995; Quilis et al. 1996; Shapiro et al. 1996; Pearce & Couchman 1997; Shandarin et al. 2010) had made it possible nowadays, to simulate and follow the evolution of billion

(see Figure5) of particles from scales of galactic haloes (e.g., Via Lactea Project, Kuhlen et al.(2008); Aquarius Project, (Springel et al. 2008); GHALOS, Stadel et al.(2009)) to very large volumes ∼ (1h−1Gpc)3 (e.g., Hubble Volume Project, Jenkins et al.(1998);

Millenium Simulations, Lemson & Virgo Consortium(2006); Guo et al.(2011)).

One of the main achievements of cosmological N-body simulations is that they reproduce many of the observed features of the cosmic web, namely, the proper sizes and spatial distribution of ﬁlaments, the formation of wall-like structure elements surrounding over- dense regions (voids), see Figure6, favoring a model of non-linear evolution of structure in a ΛCDM cosmology (e.g., Gramann 1988, 1990; Sathyaprakash et al. 1998; Doroshkevich et al. 1999; Demia´nskiet al. 2000; Evrard et al. 2002; Shandarin et al. 2004; Ueda &

Takeuchi 2004; Einasto et al. 2007a; Yan & Fan 2011).

16The term cold in this context refers to non-relativistic.

30 Figure 5. Evolution of the number of particles in “high-resolution” cosmological N-body simulations. Adapted from Suto(2003).

Figure 6. Comparison of observed and simulated large-scale galaxy distribution. Credits: http://www.mpa-garching.mpg.de/millennium/

31 4 SDSS DR7 Superclusters − The Adaptive Luminosity

Density Field Catalogue

In this chapter, I present a description of the Adaptive Luminosity Density Field Superclus- ter Catalogue17 obtained by Liivam¨agiet al.(2012) (hereafter L12). L12 is based on the

SDSS DR7 main sample (Abazajian et al. 2009) and includes galaxies with the apparent r magnitudes 12.5 < mr < 17.77, excluding duplicated entries. The catalogue has been build using a Friedmann-Roberson-Walker cosmological model

−1 −1 with Hubble parameter H0 = 100 h km s Mpc , matter density paramater Ωm = 0.27, and the dark energy parameter ΩΛ = 0.73. Galaxies redshifts were corrected for the motion relative to the CMB. Groups and galaxy groups were extracted from the sample using a modiﬁed Friends-of-Friends (FoF) algorithm. The FoF method constructs groups of galaxies based on a linking length: a galaxy belongs to a galaxy group if this galaxy has at least one group member galaxy closer than the linking length. L12 uses a linking length that increases with distance, thus taking into account the variation in number density of galaxies with redshift (the catalogue is constructed out of a ﬂux-limited sample, the number of galaxies in superclusters depends on distance).

The forthcoming description of the L12 follows very close that given by the authors in their paper.

In order to obtain the underlying mass distribution, L12 use the (luminosity) galaxy density field from the whole galaxy distribution. The first step in calculating the luminosity density field is to k+e-correct galaxies’ absolute magnitudes, which in the r-band takes the form: Mr = mr −25−log10 dL −K, where mr is the Galactic extinction corrected apparent −1 magnitude, dL = d(1 + z) is the luminosity distance (d is the comoving distance) in h Mpc, z is the redshift and K is the k + e correction. k-correction for the SDSS galaxies were calculated from the KCORRECT algorithm (Blanton & Roweis 2007; Blanton et al.

2003a). Also an evolution-correction (Blanton et al. 2003b) is performed for each galaxy.

Then, an estimation of the amount of unobserved luminosity (from the luminosities of the

17The catalogue is available at http://atmos.physic.ut.ee/∼juhan/super/

32 galaxies that lie outside the observational window of the survey) is made according to

Lw ≡ Lgal = WL(d)Lobs, (22)

0.4(M −M) where Lobs = L 10 is the luminosity of a galaxy with the absolute magnitude

M and M = 4.64 mag is the absolute magnitude of the Sun in the r-band, and Z ∞ Lφ(L)dL 0 WL(d) = , (23) Z L2(d) Lφ(L)dL L1(d) is a distance dependent weight (the ratio of the expected total luminosity to the luminosity within the visibility window) and φ(L) is the luminosity function and L1(d) and L2(d) are the luminosity window limits at a distance d. The luminosity function is taken to be

n(L)d(L) ∝ (L/L∗)α(1 + (L/L∗)γ)(δ−α)/γd(L/L∗), (24) where α = −1.42 is the exponent at low luminosities (L/L∗) 1, δ = −8.27 is the exponent at high luminosities (L/L∗) 1, γ = 1.92 is a parameter that determines the speed of the transition between two power laws, and L∗ = −21.97 is the characteristic luminosity of the transition (Tempel et al. 2011).

To remove the cluster finger of God distortions, the distance to each galaxy is corrected according to  0 σr  dgroup + (dgal − dgroup) for groups with ≥ 3 members  σv/H0 dgal = √ , (25)  0 R/ 2  dpair + (dgal − dpair) for galaxy pairs δv/H0 0 where dgal is the initial distance of the galaxy, σr is the standard deviation of the projected distance in the sky from the group center, σv is the standard deviation of the radial velocity, R is the linking length at the distance of the group and δv is the velocity difference between √ the galaxy pair. In the case of galaxy pairs, they are moved if δv/H0 > R/ 2. The values for this distance correction were taken from Tago et al.(2010) group catalogue. Here, both σv and σr are in physical coordinates at the group location. These distance are then converted into a luminosity density field, according to

X li ≡ ρ(r) = K(r − rgal; a)Lgal (26) gal

33 where the sum is over all the galaxies, Lgal is the galaxy’s luminosity, i= (i, j, k) are grid cell indices and rgal = (xgal, ygal, zgal) are its corrected coordinates:

xgal =dgal sin λ,

ygal =dgal cos λ cos η, (27)

zgal =dgal cos λ sin η,

(λ and η are SDSS coordinates). K(r; a) is a kernel of the scale of a, which for the catalogue

(3) (1) (1) (1) KB (r; a, ∆) ≡ KB (x; a, ∆)KB (y; a, ∆)KB (z; a, ∆) (28) with a = 8 h−1 Mpc was used for the SDSS Main sample, and a = 16 h−1 Mpc fro the

SDSS LRG sample. In this latter equation

(1) KB (x; a, ∆) = B3(x/a)(∆/a), (29) deﬁnes a B3 box spline kernel of scale a and grid step ∆, and

|x − 2|3 − 4|x − 1|3 + 6|x|3 − 4|x + 1|3 + |x + 2|3 B (x) = (30) 3 12 is a B3 spline function. Before extracting the superclusters the mean density is determined as the average over all pixel values inside the mask

X li lmean = , (31) Vmask i∈mask where Vmask, the volume of the mask, is the number of cells times the cell volume. Finally, all densities are converted into the units of mean density:

li li → , (32) lmean

The extraction of the supercluster from the luminosity density ﬁeld is made in such a way that superclusters are deﬁned as connected volumes above a certain density threshold picked up from a set of density contours. Each supercluster is assigned with an individual density threshold, Dscl, dependent on the local density level, such that it includes more

34 galaxies in a larger number of superclusters, generating volume-limited supercluster samples. This procedure is referred as Adaptive (Luminosity) Density Field. This method provides an useful tool to deﬁne superclusters’ environments.

The catalogue presents a set of supercluster properties (superclusters’ identification number, richness, volume, luminosity, density, ...) for different density levels. Some of the more important definitions regarding the properties of superclusters are:

• Supercluster volume:

X 3 3 Vscl ≡ ∆ = Nscl∆ , (33) i∈scl

where ∆ is the grid size and Nscl is the amount of cells contained within each supercluster.

• Total luminosity of the supercluster:

X Lgal,scl ≡ Lgal. (34) gal∈scl

• Total (weighted) luminosity of the supercluster:

X Lwgal,scl ≡ WL(dgal)Lgal. (35) gal∈scl

In the catalogue, superclusters are identiﬁed by an unique ID number, so hereafter, names like SCl XXXX will refer to supercluster with ID number XXXX.

The supercluster catalogue provides separate ﬁles to describe the superclusters at diﬀe- rent density levels and with adaptively assigned density thresholds, referred as supercluster

files. Also it includes files with basic properties of galaxies and galaxy groups of superclusters at all density thresholds, referred as galaxy and group files. In particular, the galaxy and group files include the identification number of galaxies and galaxy groups used in

Tago et al.(2010) galaxy group catalogue.

Additional to the information in the supercluster catalogue, I use the galaxy group catalogue by Tago et al.(2010) to gather extra information about the galaxies and galaxy groups in the superclusters. This catalogue provides ﬂux-limited and volume-limited samples of groups of galaxies extracted from the SDSS DR7. The group catalogue contains

35 coordinates, distances and absolute u, g, r, i, z magnitudes of galaxies in the groups. Also it includes several properties of groups of galaxies: richness, coordinates, sizes, rms radial velocities of galaxies members, virial radii and total luminosity.

36 5 Analysis Methods

This Chapter is devoted to explain the main characteristics of the diﬀerent techniques employed in this thesis to analysis the Adaptive Luminosity Density Field Supercluster

Catalogue.

5.1 Principal Components Analysis

Principal Component Analysis (PCA) (Pearson 1901; Hotelling 1933) is a mathematical procedure that allows to analyze large data sets and re-express the original variables in terms of fewer, less noisy, uncorrelated variables called Principal Components (PCs), while retaining most of the information (variance) present in the data. This method extracts a number of PCs equal to the number of variables in the data, where each of these new variables is expressed as a linear combination of the original ones. Other than being uncorrelated (orthogonal) variables, the principal components are computed in such a way that the first component (PC1) extracted accounts for the maximum amount of variance in the data, the second component (PC2) for most of the variance that is not accounted in the first component, and so on. If the data is not too noisy, usually only the first few

PCs extracted by the method can explain most of the variance in the data, reducing the amount of variables involved in the analysis. This dimensionality reduction obtained with

PCA is the result of (some of) the original variables being redundant and/or correlated, which makes PCA an useful tool for simplifying and reducing original data sets, ﬁnding classes and correlations among the variables, identifying outliers and for modeling and prediction (Wold et al. 1987; Francis & Wills 1999; Abdi & Williams 2010; Jolliﬀe 1986).

PCA has long and widely been used to analyze large data set, specially among social sciences, but also in fields like agriculture,economy, meteorology, oceanography, ecology, genetics and image processing, just to mention some. In Astronomy, PCA has been applied in very different areas, including spectral classification of galaxies (e.g, Efstathiou & Fall

1984; Brosche 1973; Formiggini & Brosch 2004), quasars (Yip et al. 2004; Pˆariset al.

2011), and stars (Deeming 1964; Whitney 1983; McGurk et al. 2010; S´anchez Almeida et al. 2010, and references therein); light curve analysis of variable stars (Deb & Singh

37 2009, and references therein), studies of the properties of galaxies (Rogers et al. 2010;

Chen et al. 2012) and dark matter haloes (Skibba & Macci`o 2011; Jeeson-Daniel et al.

2011), and several studies constraining and parametrizing cosmological quantities (see for example Millea et al. 2012; Ishida et al. 2011).

I refer the reader to Jolliﬀe(1986) for a detailed mathematical description of the method, along with applications and examples of this technique. Here I describe just in a qualitative fashion the main features of the method needed to understand the results presented below.

Formally, we can consider PCA as a multivariate analysis method that take a set of

(n) variables and ﬁnds linear combinations of these to produce a set of (n) uncorrelated variables, PCi (i ∈ N, i ≤ n): n X PCi = cijXj, (36) j=1 Pn where {Xj} is the original set of variables and cij satisfy j=1 cijcjk = δik and −1 ≤ cij ≤ 1.

The result of a PCA can be interpreted in terms of three quantities: scores, loadings and proportions of variance of each PC. Scores are the transformed variable values corresponding to a particular data point; loadings are the weights by which each of the original variables has to be multiplied to obtain the corresponding component score (the quantities cij in Equation 36) and proportion of variance is, as the name indicates, the relative amount of variance that each PC contributes to the overall data set. Basically, for any given PC and original variable, the absolute value of the corresponding loading determines how important that variable is within that particular PC. In this sense, those properties with higher loadings in the ﬁrst PC are those that contain the largest amount of information and importance over the rest of the properties. The (cumulative) proportion of variance can be used to determine a minimum number of PCs that describe most of the variance in the data. Although there is no general rule as to what it means “most of the variance in the data”, a cumulative proportion of variance between 70% and 90% in practice preserves in the ﬁrst few PCs most of the information of the data set.

It is common use to visualize the results of a PCA in a form of a biplot, where both the

38 PC2 x_2

x_1 PC1

Figure 7. PCA example. Left panel: plot of 216 observations of variables x1 and x2, right panel: representation of the same data after applying PCA to it. scores and the projections of the original variables onto consecutive pairs of components

(usually the ﬁrst two) are included in the same plot. Samples (scores) usually are shown as points, whereas the diﬀerent variables are shown as vectors (arrows), with their directions indicating the axis along each variable increases and their length proportional to their importance within each PC. Arrows of correlated variables show as clustered groups in biplots. Biplots also make easy to identify multivariate outliers (atypical observations).

As a (trivial) example of the method, consider two strongly correlated properties, as those plotted in the left panel in Figure7. The right panel of this Figure shows the corresponding data after applying the PCA. In this case, the result of the PCA is basically an re-orientation (a rotation) of the original axes such that the larger amount of variation now lies along the PC1 axis, and there is almost no considerable dispersion along the PC2 axis. Although in this example I considered just two variables, this highlight the spirit behind the PCA: to ﬁnd a new basis (a new representation) in which to express the data in such a way that with only a few parameters (PC1 in this case) it is possible to explain the most of the variance in the original data.

The aim of applying PCA to the supercluster data is to study the correlations between the physical properties of superclusters and the presence of potential distance-dependent

39 selection eﬀects on those properties. Also I am interested in analyze the relation between superclusters properties and the properties of the galaxies and galaxy groups that populate them.

PCA to study the properties of superclusters of galaxies was applied for the first time by Einasto et al.(2011c), who used a sample of 125 superclusters drawn from the Liivamägi et al.(2012) catalogue to study the correlation between the physical and morphological properties of superclusters. Einasto et al.(2011c) considered data of superclusters at fixed density thresholds. My PCA follow their study, but is now applied to the adaptive supercluster sample. Moreover I include properties of the galaxies and groups in superclusters in my analysis.

For all the PCA calculations, I used the open source statistical software R (R Devel- opment Core Team 2011).

5.2 Luminosity Function

The Luminosity Function (LF), the observed number of galaxies per luminosity (or absolute magnitude) bin in a given survey volume, is one of the most basic observables of galaxy populations, being intrinsically related to the processes of galaxy formation and evolution itself (Benson et al. 2003). Hence, this function provides a fundamental tool to describe the distribution of galaxies at diﬀerent cosmological epochs.

The availability of large photometric and spectrometric galaxy surveys nowadays have made it possible to study the relation between galaxies luminosities and several parameters, such as star formation rate (e.g., Madgwick et al. 2002), colors (e.g., Yang et al. 2009;

Loveday et al. 2012), mass (e.g., Zandivarez & Mart´ınez 2011), redshift (e.g., Gabasch et al. 2006; Marchesini et al. 2007), morphological type (e.g., Ilbert et al. 2006; Bernardi et al. 2010), global/local environments (e.g., Croton et al. 2005; Biviano et al. 2011), just to mention some.

The LF, usually expressed as Φ(L) or n(L), is such that

dN = n(L)dLdV (37)

40 is the observed number of galaxies within a luminosity range [L, L+dL]. There are several ways in which the LF can be estimated (see, for example Johnston 2011, for a review on

−1 this topic). One of the most popular methods is the so called Vmax weighting procedure, where the diﬀerential luminosity function n(L)dL is calculated according to

X I(L,L+dL)(Li) n(L)dL = , (38) V (L ) i max i where dL is the luminosity bin width, the function I indicates the number of galaxies within certain luminosity bin, Vmax(L) is the maximum volume where a galaxy of luminosity L is found within the sample and the sum runs over all galaxies in the sample. Equation (38) gives the expected number density of galaxies with luminosities between L and L + dL.

This method depends both on the bin widths and on the bin edges locations, which can be avoid using kernel smoothing:

1 X 1 L − Li n(L) = K , (39) h V (L ) h i max i where the kernel K(x) is a distribution of zero mean and of a typical width h, and its choosing is subject of the study at hand. In this work, all LFs have been calculated taking h = 0.5 mag and a B3 box spline kernel (see Equation 28), which have shown to be an appropriate kernel (Tempel et al. 2011).

The luminosity distributions are usually approximated to analytical functions, e.g, the popular Schechter(1976) formula:

n(L)dL ∝ (L/L∗)α exp(L/L∗)d(L/L∗) (40) where α is the exponent at low luminosities and L∗ (or corresponding M ∗) is the characteristic luminosity of the turning point of the function. The double-power law:

δ−α n(L)dL ∝ (L/L∗)[1 + (L/L∗)γ] γ d(L/L∗), (41) where α is the exponent al low luminosities (L/L∗) << 1, δ is the exponent at high luminosities (L/L∗ >> 1), γ is a parameter that determines the speed of the transition between the two power laws, and L∗ is the charecteristic luminosity of the transition, has proved to give better ﬁts for the LF, especially at the bright end (see, e.g. Tempel et al.

2011, and references therein).

41 Table 2. Properties of Superclusters: General Sample

Property Average Median Standard Maximun Minimun

Deviaton Value Value

Ngal 124 54 225 3917 4

Ngr 18 8 29 428 0

−1 3 Vscl [(Mpc h ) ] 4078 2763 3987 34489 140

10 −2 Lwgal,scl [10 h L ] 565.7 391.7 546.6 5462 61.4

−1 Φscl [Mpc h ] 30.8 27.4 13.2 107.8 16

−1 Dcom [Mpc h ] 429 453 110.5 565 71.3

ρscl [mean density] 3.1 3.0 0.8 5.5 2.4

ρpeak [mean density] 6.6 6 .0 2.7 21.6 2.6

Notes: Ngal - the number of galaxies in superclusters; Ngr - the number of groups in superclusters;

Vscl - superclusters’ volume; Lwgal,scl - total (weighted) luminosity of superclusters; Φscl - diameter of superclusters; Dcom - comoving distance to the highest density peak of superclusters; ρpeak - the density of the highest peak inside superclusters; ρscl - density threshold for superclusters.

6 Analysis and Results

6.1 General Properties of superclusters

L12 contains 1313 superclusters. The main properties of these superclusters are sum- marized in Table2. Large values of the standard deviations indicate that supercluster population is very heterogeneous. In summary we may say that typical superclustres in

SDSS contain about ∼ 50 galaxies and ∼ 10 groups. Typical superclusters diameter is

∼ 30 h−1 Mpc.

In order to avoid saigniﬁcant distance selection eﬀects, I have selected a smaller sample from the entire adaptive catalogue, with superclusters’ distances in the range

−1 −1 90 h Mpc ≤ Dcom ≤ 320 h Mpc, where the luminosity-dependent selection eﬀects are the smallest (Einasto et al. 2011c). There are a total of 216 superclusters in this sample, composed of 77370 galaxies and 10344 groups. Hereafter, I will refer to this sample as the “Representative Sample”. A summary of superclusters properties for this sample is

42 shown in Table3. In general, superclusters in this sample are richer: they have more galaxies (∼ 250) and groups (∼ 40). Properties such as diameter, density threshold and density peak seem to keep fairly same variation ranges in the whole sample and in the

Representative Sample. Table4 shows the main properties of groups of galaxies in the

Representative Sample. Most of the groups in this supercluster sample are small (median value 3 members) while the largest ones have several hundred members. This agrees with results by Tempel et al.(2012) that ∼ 80% of groups have ≤ 4 members.

Figure8 shows the distribution of superclusters in this sample. Superclusters’ group richness is marked using different symbols (see Figure8 caption) and can be appreciated that the large scale distribution of superclusters, especially the fact that there are large regions totally devoid of superclusters. Figure9 shows the distribution of the principal properties of the superclusters (richness, luminosity, density peak and volume) as a function of distance. We can see that even when the sample was picked up from a distance interval where the distance selection effects are expected to be small, for distances less than 200 h−1 Mpc, there are very few superclusters with parameter values above the average. This indicates that some selection effects may still be present in the Representative

Sample. If we look at the luminosity density ﬁeld (see Figure 10) it is clear that the regions

∼ 90−180 h−1 Mpc and ∼ 250−320 h−1 Mpc are almost devoid of luminous superclusters, while in the intermediate region ∼ 180 − 250 h−1 Mpc a large chain of high-luminous superclusters can be observed: the Sloan Great Wall (SGW). SGW is the richest and largest nearby system of galaxies, at a distance ∼ 230 Mpc h−1. In Figure 10 SGW correspond to the rectangular region deﬁned by cartesian coordinates (-100,100) and (150,250). Al- together, SGW contains 13 superclusters, 1220 groups and 9360 galaxies (Einasto et al.

2011e).

Figure 11 and 12 shows 3D distribution of a sample of very rich (with at least 950

10 −2 galaxy members) high luminous (Lwgal,scl > 400 · 10 h L ) superclusters. All of them contain at least 2 rich galaxy groups (with at least 30 member galaxies).

I used the Spearman’s rank correlation test to asses possible correlations between supercluster properties in the Representative Sample. The Spearman correlation is a non-

43 Figure 8. The distribution of superclusters in the Representative Sample in cartesian coordinates. Blue circles are superclusters with at least 50 groups, green crosses superclusters with 30-49 member groups, red diamonds superclusters with 11-29 member groups and black squares poorer superclusters.

Figure 9. Principal properties of superclusters vs. distance. Horizontal blue line, in each case marks the average value of the parameter plotted in the vertical axis.

44 Figure 10. Luminosity density ﬁeld in cartesian coordinates for supercluster in the Rep- resentative Sample. Contours are in units of the mean (luminosity) density.

Table 3. Properties of Superclusters: Representative Sample

Property Average Median Standard Maximun Minimun

Deviaton Value Value

Ngal 358 253.5 351 3190 37

Ngr 48 36 43 428 6

−1 3 Vscl [(Mpc h ) ] 3274 2318 3125 25551 373

10 −2 Lwgal,scl [10 h L ] 442 343 431 4500 78

−1 Φscl [Mpc h ] 30 27 12 108 16

−1 Dcom [Mpc h ] 238 248 59 319 90

ρscl [mean density] 3.2 3.0 0.88 5.5 2.4

ρpeak [mean density] 6.1 5.44 2.64 21.6 2.6 Notes: Notation is the same as in Table2

45 Figure 11. 3D distribution of groups in very rich suplerclusters with more than 950 galaxies. Red diamonds indicates the position of rich groups at least 30 member galaxies. Blue dots shows positions of less populated groups. Axes correspond to right ascension (R.A - in degrees), declination (Dec - in degrees) and distances (in h−1 Mpc ). In left panel, from top to bottom: SCl 001, SCl 010 and SCl 061. Right panel, from top to bottom: SCl 011, SCl 024 and SCl 055.

46 Figure 12. (Continuation) 3D distribution of groups in very rich suplerclusters with more than 950 galaxies. Red diamonds indicates the position of rich groups at least 30 member galaxies. Blue dots shows positions of less populated groups. Axes correspond to right ascension (R.A - in degrees), declination (Dec - in degrees) and distances (in h−1 Mpc ). In left panel, from top to bottom: SCl 060, SCl 198 and SCl 351. Right panel, from top to bottom: SCl 094, SCl 349 and SCl 350.

47 Table 4. Properties of galaxy groups in superclusters

Property Average Median Maximun Minimun

Value Value

Ngal 5 3 883 2

10 −2 Lobs [10 h L ] 1.62 1.38 58.00 0.02

−1 Rvir [Mpc h ] 0.28 0.32 0.61 0.04

−1 Vrms [km s ] 79.00 19.53 375.00 2.50

Notes: Ngal - Richness (number of member galaxies), Lobs - observed

total group luminosity, Rvir - virial radius (projected harmonic mean),

Vrms - rms radial velocity. parametric measure of the linear relationship between two datasets. The results of the test are given in terms of two parameters: the correlation coefficient, r, and the significance level, p. A correlation coefficient r = ±1 indicates (perfect) correlation/anticorrelation, r ≈ 0 lack of correlation; p indicates up to what level the result of the test is statistically significant, generally p < 0.05 indicates that results are statiscally of very high significance.

I used logarithmic values of the parameters in order to reduce the range over which the values vary, results are given in Tables5 and6. This test shows that there is practically no correlations between superclusters’ properties and distance in this sample, other than a weak anticorrelation (r ≈ −0.5) between distance and the number of galaxies in the superclusters, as it is expected. The rest of the supercluster parameters seem to have strong correlations (|r| ≥ 0.5) between them at a high signiﬁcance level.

According to correlation analysis, we may sumarrize that: superclusters richer in galaxies are richer in groups, they are larger, more luminous and have regions with high galaxy density and luminosity.

48 Table 5. Results of the Spearman’s rank correlation test: r-values

log(ρscl) log(Ngal) log(Ngr) log(V ) log(Lwgal,scl) log(ρpeak) log(Dcom) log(Φscl)

log(ρscl) 1 0.30 0.21 0.04 0.39 0.65 -0.02 0.11

log(Ngal) 1 0.96 0.73 0.78 0.61 -0.51 0.70

log(Ngr) 1 0.8 0.81 0.55 -0.41 0.77 log(V ) 1 0.91 0.62 0.08 0.82 49

log(Lwgal,scl) 1 0.79 0.09 0.83

log(ρpeak) 1 0.07 0.43

log(Dcom) 1 0.007

Notes: Every value in the table is the rank correlation coeﬃcient r between the corresponding property on the top of its column and the corresponding property on the left of its row.r = ±1 indicates that datasets are perfect correlated/anticorrelated. r ≈ 0 indicates the absence of correlation. Notation is the same as in Table2. Table 6. Results of the Spearman’s rank correlation test: p-values

log(ρscl) log(Ngal) log(Ngr) log(V ) log(Lwgal,scl) log(ρpeak) log(Dcom) log(Φscl)

log(ρscl) 0 5.5e-06 0.0018 0.58 2.5e-09 2.6e-27 0.75 0.11

log(Ngal) 0 1.1e-118 4.3e-37 6.7e-46 2.1e-23 2.2e-15 2.7e-33

log(Ngr) 0 1.8e-48 2.9e-51 2.9e-18 3.2e-10 3e-43 log(V ) 0 1.9e-82 6e-24 0.27 6.8e-54 50

log(Lwgal,scl) 0 7e-48 0.2 2.4e-56

log(ρpeak) 0 0.28 2.9e-11

log(Dcom) 0 0.92

Notes: Every value in the table is the p-value p between the corresponding property on the top of its column and the corresponding property on the left of its row. p < 0.05 indicates that results are statiscally of very high signiﬁcance. Notation is the same as in Table2. 6.2 Principal Component Analysis

In subsection 6.2.1 I will show the results of PCA on the supercluster Representative

Sample using the same properties used by Einasto et al.(2011c). In subsection 6.2.2

I will extend the analysis by including properties of galaxies and galaxy groups in the

superclusters.

6.2.1 PCA with physical properties of superclusters

In this section, I ﬁrst apply PCA to a set of properties of superclusters and their distances,

which may identify the presence of distance-related selection eﬀects on the properties. I

include galaxy richness, total luminosity, diameter, volume and density peak of superclus-

ter in this analysis. According to standard procedure in Principal Component Analysis,

all the parameters (X) included in the diﬀerent PCAs were standardized to have zero

means (X − X¯) and unit standard deviations (divided by their standard deviantions, σX ). Also, logarithms of the all parameters were used, since for PCA the parameters should

be normally distributed. Distribution of the standardized parameters are shown in Figure

13. 0.4 0.4 0.3 0.30 0.3 0.3 0.3 0.2 0.20 0.2 0.2 0.2 Density Density Density Density Density 0.1 0.1 0.10 0.1 0.1 0.0 0.0 0.0 0.0 0.00 -2 0 2 4 -2 0 2 4 -2 0 2 4 -2 0 2 4 -2 0 2 4 N_gal D_peak L_wgal Volume Diameter

Figure 13. Distribution of the standardized physical parameters of superclusters in the Representative Sample.N gal−number of galaxies in supercluster, D peak−density peak of supercluster, L wgal−total (weighted) luminosity of supercluster, Volume−volume of supercluster and Diameter−diameter of supercluster.

Results of this ﬁrst analysis are shown in Table7, showing the scores, the standard

deviations, proportion of variance and cumulative variance within each principal compo-

nent. As a remainder, the signs of the loadings for the PCs are arbitrary and it is their

absolute value what determines the importance of each of the original parameters within

each PC.

51 Table 7. Results of the PCA of the physical parameters and distances of superclusters in the Representative Sample.

PC1 PC2 PC3 PC4 PC5 PC6

log(Ngal) -0.441 0.409 -0.024 -0.058 0.457 0.653

log(Lwgal,scl) -0.491 -0.129 -0.045 -0.069 0.559 -0.650

log(Φscl) -0.438 -0.083 0.559 -0.565 -0.412 0.015

log(Vscl) -0.465 -0.132 0.222 0.800 -0.275 0.040

log(ρpeak) -0.396 -0.079 -0.797 -0.166 -0.416 0.029

log(Dcom) 0.013 -0.887 -0.019 -0.068 0.245 0.385

Importance of components

Standard deviation 1.999 1.125 0.748 0.365 0.196 0.077

Proportion of Variance 0.666 0.211 0.093 0.022 0.006 0.001

Cumulative Proportion 0.666 0.877 0.970 0.993 0.999 1.000

Notes: Notation is the same as in Table2.

52 Table 8. Results of the PCA of the physical parameters of supercluster in the Represen- tative Sample

PC1 PC2 PC3 PC4 PC5

log(Ngal) -0.439 0.055 0.885 0.145 0.014

log(Lwgal,scl) -0.491 0.036 -0.217 -0.102 -0.837

log(Φscl) -0.439 -0.564 -0.087 -0.611 0.330

log(Vscl) -0.465 -0.232 -0.342 0.738 0.262

log(ρpeak) -0.396 0.790 -0.215 -0.225 0.349

Importance of components

Standard deviation 2.004 0.750 0.536 0.360 0.169

Proportion of Variance 0.799 0.112 0.057 0.026 0.006

Cumulative Proportion 0.799 0.911 0.969 0.994 1.000

Notes: Notation is the same as in Table2.

Table7 shows that when analyzing both the physical properties of the superclusters and their distances, the absolute values of the PC1 for the physical parameters are almost equal while for the distance, this value is very small. This result confirms that there is no correlation between superclusters’ properties and distance. This verifies that physical properties in the sample are not affected by distance selection effects.

Now, I apply PCA using only the physical parameters of the superclusters. This analysis suggests (see Table8, loadings for PC1) that the the superclusters’ physical properties consider here are all equally important characterizing the superclusters. Table

8 shows that the ﬁrst two principal components accounts for approximately 90% of the variance in the supercluster sample. This supports the idea that supercluster properties are strongly correlated and it is possible to make a good description of superclusters using a small set of parameters. From the second principal component, we can see that most of the remaining variance in the sample comes from the density of the highest peak inside superclusters and superclusters’ diameters. The third principal component shows that the

53 remaining variance in the sample is accounted by the number of galaxies in superclusters.

These trends are visualized in Figure 14. In particular, notice that in the PC1-PC2 plane arrows corresponding to supercluster luminosity and number of galaxies point to the same direction and are almost of the same length. This indicates that this two characteristics are strongly correlated. Also, in this same plane we can see that arrows corresponding to the density peak inside superclusters and supercluster diameter point almost to oposite regions in the biplot, which indicates that these to properties are anticorrelated: more dense superclusters are smaller, while larger superclusters have lowest density regions.

-3 -2 -1 0 1 2 -0.5 0.0 0.5 1.0 . N_gal . Density peak 1.5 2 ...... 0.5 1.0 ...... 0.5 1 ...... 0.5 ...... PC2 ...... PC3 ...... N_gal ...... L_wgal ...... 0 ...... 0.0 ...... 0.0 ...... 0.0 ...... Diameter...... Volume...... L_wgal. .. Density peak ...... -0.5 ...... Volume ...... -1 ...... -0.5 ...... -0.5 . .Diameter -1.0

-6 -4 -2 0 2 4 -1 0 1 2

PC1 PC2

Figure 14. Principal planes for superclusters, PCA with physical parameters. Arrows represent the axes where each original variable lies, and their length is proportional to their importance within each PC. Black dots are plotted against left and bottom axes and represent each of the original superclusters. Arrows are plotted against right and top axes.

Regarding the positions of superclusters in the principal planes (Figure 15 and 14),

10 −2 high-luminous superclusters (Lwgal,scl > 400 · 10 h L ) show slightly more scatter in their positions in the principal plane PC1-PC2 (upper left panel in Figure 15, left panel in

Figure 14) than low-luminosity superclusters. Luminous superclusters are characterized for highest negative values of PC1 (see left panels in Figure 15). The supercluster with the higher negative value of PC1 is SCl 0061, the richest member of the Sloan Great Wall.

SCl 0061 is also the richest, more luminous and bigger supercluster in the sample. High luminosity superclusters with high values of luminosity density peak show up the higher

54 2 1 0 PC2 -1 pca$scores[, 2][mask] -2

-5.0 -2.5 0.0 2.5 2

pca$scores[, 1][mask] pca$scores[,PC3 3][mask] 1 0 PC3 -pca$scores[, 3] -1 -2

-5.0 -2.5 0.0 2.5

PC1

Figure 15. Principal planes for superclusters, PCA with supercluster physical parameters. 10 −2 Red diamonds: high-luminosity superclusters with Lwgal,scl > 400 · 10 h L ; grey dots: superclusters of lower luminosity.

55 values of PC2, like SCl 0010, the Hercules supercluster. In the PC1-PC3 and PC2-PC3 planes (lower left and upper right planes in Figure 15, respectively), superclusters form elongated clouds, with similar dispersion for high- and low-luminosity superclusters.

The results reached by Einasto et al.(2011c) for fixed density thresholds are similar as the ones presented here, supporting the idea that there are not considerable selection effects related to the choice of the method (adaptive or mixed density levels) used to define the superclusters. The effect to take into account when selecting the density level is that at higher thresholds superclusters are less populated and smaller that at lower levels.

Also, at lower density levels, diﬀerent superclusters may join to form bigger systems.

Einasto et al.(2011c) have also analyze the relation between the physical parameters of the superclusters with their morphological parameters, reaching the conclusion that the morphological parameters are as important as the physical parameters shaping the properties of superclusters.

6.2.2 PCA with properties of galaxies and galaxy groups in superclusters

Here I will apply PCA to look for correlations between the properties of galaxies and galaxy groups in superclusters with the physical parameters of superclusters. I include the average virial radii, rms radial velocity deviations and total observed luminosities of groups (see Table4) in superclusters in the analysis. I also take into account the average of the index colour g − r = Mg − Mr (Mg, Mr being the absolute magnitudes in the g- and r-band of the individual galaxies), the fraction of galaxies in groups and the ratio number of bright and faint galaxies in superclusters in this analysis. Bright and faint galaxies are deﬁned in terms of their absolute magnitude in the r-band. Bright galaxies have Mr + 5 log h ≤ −20. Distributions of the standardized parameters are shown in Figure 16

In order to minimize the amount of variables involved and based on the results of the previous section that superclusters properties are strongly correlated, I took just one parameter, supercluster luminosity, as a representative characteristic of superclusters properties. Table9 and Figure 17 show the results of this analysis.

56 Table 9. Results of the PCA with properties of galaxies and galaxy groups in superclusters in the Representative Sample.

PC1 PC2 PC3 PC4 PC5 PC6 PC7

log(¯g − r¯) 0.513 -0.319 0.064 0.095 0.535 -0.150 0.560

log(Fgal∈groups) -0.499 -0.027 0.137 0.333 0.569 -0.458 -0.297

log(R¯vir) 0.035 -0.366 -0.699 0.406 -0.323 -0.327 -0.008

log(V¯rms) -0.216 -0.517 0.259 -0.576 -0.259 -0.459 0.095

log(L¯gr) -0.315 -0.565 -0.238 -0.141 0.300 0.838 -0.097 log(B/F ) 0.584 -0.204 0.049 -0.133 0.133 -0.072 -0.761

log(Lwgal,scl) 0.020 -0.368 0.603 0.588 -0.341 0.195 -0.019

Importance of components

Standard deviation 1.568 1.195 1.068 0.955 0.717 0.649 0.356

Proportion of Variance 0.351 0.204 0.163 0.130 0.073 0.060 0.018

Cumulative Proportion 0.351 0.555 0.718 0.849 0.922 0.982 1.00

¯ Notes: g¯−r¯ − mean galaxy index colour, Fgal∈groups − fraction of galaxies in groups, Rvir − mean group virial radii, V¯rms − mean rms radial velocity deviations, L¯gr − mean group total observed luminosity, B/F − ratio of the number of bright and faint galaxies, Lwgal,scl − supercluster luminosity.

57 0.5 g-r Rvir 0.5 Fraction of galaxies in groups Vrms B/F Group luminosity 0.4 0.4 0.3 0.3 Density Density 0.2 0.2 0.1 0.1 0.0 0.0

-6 -4 -2 0 2 4 -4 -2 0 2

Properties of galaxies in superclusters Properties of groups in superclusters

Figure 16. Distribution of standardized properties of galaxies (left) and galaxy groups (right) in superclusters

Table9 shows that the ﬁrst principal component accounts for about 35% of the variance in the sample. The PC1 is dominated by 3 properties related to the galaxy content in groups: average galaxy index colourg ¯ − r¯, fraction of galaxies in groups (Fgal∈groups) and ratio of bright and faint galaxies (B/F ). Thus, the largest amount of variation in this data set comes from those properties. On the other hand, supercluster luminosity

(Lwgal,scl) and the mean group virial radii (R¯vir), both have very small loadings on the PC1. These two latter properties are more important for the PC3, which accounts for

∼16% of the variance of the data. The PC2 is responsible of ∼20% of the variance, and all the parameters but the fraction of galaxies in groups have an important contribution on this component, with V¯rms and L¯gr having the highest loadings. With the 7 parameters included in this analysis, ﬁve principal components are required to explain more than 90% of the variance in the sample, an indication of the large variety of galaxy and galaxy group populations that can be found within superclusters of similar characteristics. For example, we can ﬁnd high-luminosity superclusters with either a low

Fgal∈groups such as SCl 0353 and SCl 0330 or with a high fraction, as SCl 1732 or SCl 0061. The converse is also true: there are both low-luminous superclusters with a high

Fgal∈groups, as is the case in SCl 1752 and SCl 1905, or with a low Fgal∈groups, for example SCl 3529 (the poorer, less luminous supercluster in the sample) and SCl 2734.

58 In the PC1-PC2 plane (see Figure 17) arrows of correlated parameters point to similar directions: arrows corresponding to mean galaxy index colourg ¯ − r¯ and the ratio of the number of bright and faint galaxies, B/F , point almost at right-angle to the direction of arrows corresponding to rms velocity deviations (V¯rms) and group luminosities (L¯gr); and supercluster luminosity (Lwgal,scl) and group virial radii (R¯vir) arrows point almost at right-angle to the direction of the arrow corresponding to fraction of galaxies in groups.

Also, the arrow corresponding to Fgal∈groups points in opposite direction to those of B/F andg ¯ − r¯. In summary, Figure 17 shows that there is a strong correlation betweeng ¯ − r¯ and B/F , between Lwgal,scl and R¯vir and between L¯gr and V¯rms for galaxies and galaxy groups in superclusters.

Upper left region of the PC1-PC2 plane in Figure 17 is populated by superclusters with high fraction of galaxies in groups and excess of blue and faint galaxies, such SCl

1732 and SCl 0060. Supercluster with more luminous groups and high fraction of galaxies in groups cluster around the middle and lower left part of the biplot, SCl 1113 and SCl

0362 for example. Superclusters with less luminous groups and low fraction of galaxies in groups are located in the upper right hand part of the plot, as it is the case for SCl 3010 and SCl 1250. The lower right region of the plane is dominated by supercluster where the fraction of galaxies in groups is low and there are more red and bright galaxies, for instance SCl 0327 and SCl 1364.

59 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

3 . . Rvir ...... 0.4 ...... 3 . . . . 0.6 2 ...... 0.4 ...... 0.2 2 . 1 ...... Group observed luminosity ...... 1 ...... 0.2 0 ...... 0.0 ...... Fraction galaxies in groups...... PC2 ...... PC3 ...... 0 . B/F. . . 0.0 -1 ...... g-r...... B/F -0.2 ...... Fraction. galaxies in groups ...... Rvir . g-r -1 . . . . -2 ...... -0.2 SC luminosity Vrms ...... -0.4 ...... -3 . . . . Vrms -2 -0.4 Group observed luminosity . -0.6 . -4 . -3 SC luminosity . -0.6

-4 -2 0 2 -4 -3 -2 -1 0 1 2 3

PC1 PC2

Figure 17. PCA with properties of galaxies and galaxy groups in superclusters. Arrows represent the axes where each original variable lies, and their length is proportional to their importance within each PC. Black dots are plotted against left and bottom axes and represent each of the original superclusters. Arrows are plotted against right and top axes.

60 6.3 Galaxy populations in superclusters

6.3.1 Selection eﬀects

To analyze the distribution of galaxies in diﬀerent superclusters, I classify supercluster

−2 according to their luminosity: superclusters with Lwgal,scl < 400 h L are referred as

−2 low-luminous superclusters while those with Lwgal,scl ≥ 400 h L as high-luminous superclusters. Additionally, I use galaxies absolute magnitudes in r- and g-band (Mr and Mg, respectively) to divide galaxies into red/blue and bright/faint populations. Red galaxies have galaxy colour index g − r = Mg − Mr ≥ 0.7, bright galaxies Mr − 5 log(h) ≤ −20. Now, in order to analyze the distribution of galaxy populations in superclusters based on its magnitudes/luminosities, it is important to take into account the fact that those properties are highly dependent on distance when using ﬂux limited samples: very bright and close by galaxies are excluded from the sample, as well as very far away and faint galaxies. This eﬀect can be seen in Figure 18, where several properties of the galaxy populations in superclusters are plotted as a function of distance. Also, recall from the discussion in Section 6.1 that for distances ∼ 90 − 180 h−1 Mpc and ∼ 250 − 320 h−1 Mpc there are few luminous superclusters, while in the region ∼ 180 − 250 h−1 Mpc there is a large chain of high-luminous superclusters (see Figure 10).

Table 10 shows the relations between galaxy populations in low- and high-luminosity superclusters. Kolmogorov-Smirnov (KS)18 test shows that the diﬀerences between galaxies magnitudes and colours in low- and high-luminous supercluster are statistically signiﬁcant (see Table 11).

From Table 10 we can see that all values tend to increase when going from low- to high luminous superclusters. Only Fgal∈groups shows a slightly smaller value in high-luminous superclusters then in low-luminous ones.

Distribution of galaxy colours in low- and high-luminous superclusters is shown in

18The Kolmogorov-Smirnov (KS) test is intended to determine if two data sets diﬀer signiﬁcantly. The

KS test is reported in terms of two parameters: the KS statistic, D, and the p-value. If the KS statistic is small or the p-value is high, the hypothesis that the distributions of the data sets are the same cannot be rejected.

61 Figure 18. Properties of galaxies in superclusters vs Distance. Notes: Fgal∈groups− Fraction of galaxies in superclusters in groups; red/blue − ratio of the number of red and blue galaxies, Fred gal − fraction of red galaxies in superclusters, supercluster g −r − mean galaxy index colour in superclusters.

62 Figure 1919. Here we can note that there is a small excess of blue faint galaxies among low-luminous superclusters.

Figure 20 shows the luminosity functions of the galaxies in superclusters of diﬀerent luminosity. Notice than in general, the number density of galaxies in high-luminous superclusters is always greater than in low-luminous superclusters, but around the interval

−18.5 < Mr + 5 log10(h) < −17.5 there exists an excess of galaxies in low-luminous superclusters over high-luminous ones. This is obviously the same blue faint population of galaxies seen in Figure 19.

Bright Bright Faint Faint 6 6 5 5 4 4 3 3 Density Density 2 2 1 1 0 0

0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5

g-r g-r

Figure 19. Distribution of colour index g − r for bright and faint galaxies in low and high luminosity superclusters. Left: low-luminosity superclusters, right: high-luminosity superclusters.

6.3.2 Galaxies in superclusters in diﬀerent local environments

Galaxies are very unlikely to be isolated objets, but rather belong to groups or cluster of galaxies of varying richness (Tiﬀt & Gregory 1976). About 76% of the galaxies in

L12 belong to groups of galaxies. To study the inﬂuence of the local environment on the properties of galaxies in superclusers, I analyze the distribution of the diﬀerent galaxy populations taking into account the richness of the groups they reside in. For this purpose,

I use the categories isolated galaxy for those galaxies that are not part of any group, poor

19In this ﬁgure, and similar ones, Density refers to the probability density function which describes the relative likelihood of the corresponding variable to take a given value.

63 Table 10. Galaxy populations in superclusters of diﬀerent luminosity.

Supercluster Luminosity

Population Parameter Low High

(Nscl=139) (Nscl=77)

Ngal 30567 46803 red/blue 1.22 1.48 All B/F 0.80 1.07

Fgal∈groups 0.77 0.76

Ngal 13612 24179 B red/blue 2.13 2.23

Fgal∈groups 0.77 0.76

Ngal 16955 22624 F red/blue 0.80 0.99

Fgal∈groups 0.77 0.77

−2 Notes: Supercluster richness: Low luminous − Lwgal,scl < 400 h L , High luminous − −2 Lwgal,scl ≥ 400 h L , Nscl − number of supercluster per richness class. Population: All − all galaxies, B − bright galaxies (Mr ≤ −20), F − faint galaxies (Mr > 20). Parameter: Ngal − number of galaxies in superclusters, red/blue − ratio of the number of red and blue galaxies, B/F

− ratio of the number of bright and faint galaxies, Fgal∈groups − fraction of galaxies in groups.

Table 11. KS test results for galaxy populations in low- and high-luminosity superclusters.

KS test results Parameter D p

−16 Mr 0.107 < 2.2 × 10 g − r 0.058 < 2.2 × 10−16

Notes: K-S test results: D − maximun diﬀerence and p − probability that the samples are taken from the same parent distribution. Parameter: Mr − absolute magnitude in r-band, g−r − galaxy colour index.

64 Figure 20. Differential luminosity function for galaxies in superclusters of different luminosity. group for those groups that have less than 10 galaxy members and rich group for groups with at least 10 galaxies. Results of this analysis are given in Table 12 and Figures 21 and 22. Figure 21 shows the distribution of galaxy colours for galaxies in superclusters in different local environments, Figure 22 shows the corresponding differential luminosity functions.

The KS test shows that the differences between galaxy magnitudes and colours in superclusters in the different local environments are statistically significant (see Table 13).

Lets ﬁrst analyze the distribution of red and blue galaxies in diﬀerent local environments. The ratio red/blue increases from the less dense environments of isolated galaxies to the densest regions of rich groups. This trend is also followed in both bright and faint populations. Notice that both in poor and rich groups there is an excess of red over blue galaxies. Among isolated galaxies in superclusters, there are mostly blue galaxies.

From Figure 21 we can see that the increase of the ratio red/blue when increasing group richness is largely due to a change in the distribution of faint galaxies, which turn to redder colors with increasing local density. Although in general the number density of galaxies in superclusters in rich groups is higher than in poor groups or for isolated

65 Table 12. Galaxy populations in superclusters in diﬀerent local environments.

Local environment Population Parameter Isolated Poor Group Rich Group

Ngal 17991 29479 29900 All B/F 1.00 1.12 0.78

red/blue 0.81 1.22 2.16

Ngal 9013 15599 13179 B red/blue 1.28 2.00 3.87

Ngal 8978 13880 16721 F red/blue 0.50 0.72 1.47

Notes: Local environment: Isolated galaxies − galaxies in any group, Poor groups − galaxies

in poor groups where 2 ≤ Ngal < 10, Rich Groups − galaxies in rich groups where Ngal ≥ 10).

Population: All − all galaxies, B − bright galaxies (Mr ≤ −20), F − faint galaxies (Mr > −20).

Parameter: Ngal − number of galaxies, red/blue − ratio of the number of red and blue galaxies, B/F − ratio of the number of bright and faint galaxies.

8 Bright 8 Bright 8 Bright Faint Faint Faint 6 6 6 4 4 4 Density Density Density 2 2 2 0 0 0

0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5

g-r g-r g-r

Figure 21. Distribution of colour index g − r for bright and faint galaxies in diﬀerent local environments. Left: isolated galaxies, center: galaxies in poor groups, right: galaxies in rich groups.

66 Figure 22. Diﬀerential luminosity function for galaxies in superclusters in diﬀerent local environments.

Table 13. KS test results for galaxy populations in diﬀerent local environments.

KS test results Parameter Samples D p

i p gr 0.0482 < 2.2 × 10−16

−16 Mr i r gr 0.0904 < 2.2 × 10 p gr r gr 0.1026 < 2.2 × 10−16

i p gr 0.1053 < 2.2 × 10−16

g − r i r gr 0.2376 < 2.2 × 10−16

p rg r gr 0.1353 < 2.2 × 10−16

− galaxies in rich groups (Ngal ≥ 10).

67 Table 14. Summary of the fraction of galaxy populations in diﬀerent environments.

Population Low Lum. High Lum. red blue bright faint

Isolated 0.23 0.24 0.45 0.55 0.50 0.50

Poor Groups 0.38 0.38 0.55 0.45 0.53 0.47

Rich Groups 0.39 0.38 0.68 0.32 0.44 0.56

red 0.55 0.60

blue 0.45 0.40

bright 0.45 0.52

faint 0.55 0.48

−2 Notes: Population: Low Lum. − Low luminosity supercluster (Lwgal,scl ≤ 400 h L ), High Lum. −2 − High luminosity supercluster (Lwgal,scl ≥ 400 h L ), red − red galaxies (Mg − Mr ≥ 0.7), blue

− blue galaxies (Mg − Mr < 0.7), bright − bright galaxies (Mr ≤ −20), faint − faint galaxies

(Mr > −20), Isolated − isolated galaxies, Poor Groups − galaxies in poor groups (2 ≤ Ngal < 10),

Rich Groups − galaxies in rich groups (Ngal ≥ 10). galaxies, we can see from the luminosity function in Figure 22 that between magnitudes

−19.5 < Mr + 5 log10(h) < −22 there is an abundance of galaxies in superclusters in poor groups.

Table 14 gives a summary of the fraction of galaxy populations in diﬀerent environments. Notice that the relative values of galaxies in diﬀerent local environments is pretty much the same irrespectively of whether they are located in low- or high-luminous superclusters. Also note that even when the fraction of galaxies in superclusters in both poor and rich groups is large (∼75% in total), rich groups are actually rare among superclusters, making up about 10% of the group population in superclusters (see Table 15).

68 Table 15. Group populations in superclusters of diﬀerent luminosity.

Supercluster luminosity

Group population Low High

(Nscl=139) (Nscl=77)

Fp gr 0.90 0.89

Fr gr 0.10 0.11

−2 Notes: Supercluster luminosity: Low − Low luminosity supercluster (Lwgal,scl ≤ 400 h L ), −2 High− High luminosity supercluster (Lwgal,scl ≥ 400 h L ), Nscl − number of supercluster per richness class. Group property: Fp gr − Fraction of poor groups, Fp gr − Fraction of rich groups

69 7 Conclusions

Superclusters are undoubtedly one of the most valuables cosmological laboratories from where we can learn about the general patterns behind the “design” of our Universe.

I have used the Adaptive Luminosity Density Field Supercluster Catalogue by Li- ivam¨agiet al.(2012) to study properties of superclusters in the distance interval 90 h−1 Mpc ≤

−1 Dcom ≤ 320 h Mpc. The main conclusion drawn from this study are the following:

- There are not significant selection effects affecting the properties of superclusters.

- According to correlation analysis and PCA, the properties of superclusters are strongly

correlated. Moreover, the various parameters describing their properties (luminosity,

number of galaxies, diameter, volume) are all equally important characterizing them.

- Superclusters with similar characteristics (luminosity for example) may contain very

inhomogeneous galaxy populations.

- Most of the galaxies in superclusters reside within groups of galaxies; only ∼ 25% of

the galaxies in superclusters are isolated.

- The population of red galaxies in superclusters increases steadily from the less dense

environments of the isolated galaxies to the more dense environment of rich groups.

- Among low-luminosity superclusters there is a small excess of blue faint galaxies

relative to high-luminosity superclusters.

- The population of rich groups are relatively rare among superclusters, accounting

for around ∼ 10% of the total group population.

It is important to remark that the results obtained in this work agree very well with previous studies that used superclusters defined at fixed density levels. Thus, we can conclude that results are insensitive to the method used to select superclusters: fixed or adaptive density levels. Even so, it would be valuable to make a direct comparision study between the adaptive and fixed density level supercluter catalogues.

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77 Appendices

A Colour Superclusters Contours

Figure 23 shows a sample of contour maps of the projected g − r color of very rich and luminous superclusters, along with the projected distribution of their group members (the sample shown is the same as in Figure 11 and 12, very rich suplerclusters with more than 950 galaxies). The color density levels were calculated following a similar approach as the one used to define the luminosity density field, according to equations (26)-(30). For the sake of identifying special features and compare different superclusters, all contours are given within the same limits. Contour levels limits correspond to the overall average minimum and maximum values of the g − r color of the superclusters galaxies members. In general, supercluster without rich group members appear smooth (featureless) in the g − r color contour plots, while superclusters with several rich groups show color density enhancements in the projected positions of their rich groups. A visual inspection of 3D distributions of groups within superclusters also allows to find superclusters without rich groups but with also significant structure/features in the color contour plots. Most of these superclusters seem to have morphologies resembling multibranching filaments. Examples of the latter can be seen in Figure 24.

78 79

Figure 23. Projected g − r color contours of very rich supercluster with more than 950 galaxy members. Red diamonds indicates the position of rich groups at least 30 member galaxies. Black dots shows positions of less populated groups. Axes correspond to right ascenson (R.A - in degrees) and declination (Dec - in degrees). In upper panel panel, from left to right: SCl 001, SCl 011, SCl 010, SCl 024, SCl 061 and SCl 055. Right panel, from top to bottom: SCl 060, SCl 094, SCl 198, SCl 349, SCl 351 and SCl 350. 80

Figure 24. 3D distribution of groups in superclusters without rich group members and corresponding g − r color contour plot. First row: left: SCl 712 ,right: SCl 511. Second row: left: SCl 900, right: SCl 1117. Third row: left: SCl 1145, right: SCl 1194. Fourth row: left: SCl 509, right: SCl 891