Properties of Galaxies and Groups: Nature versus Nurture
TTURUN YLIOPISTON JULKAISUJAANNALES UNIVERSITATIS TURKUENSISSARJA – SER. A I OSA – TOM. 419ASTRONOMICA-CHEMICA-PHYSICA-MATHEMATICA
PROPERTIES OF GALAXIES AND GROUPS:NATURE VERSUS NURTUREbySami-Matias Niemi
TURUN YLIOPISTOTurku 2011 a r X i v : . [ a s t r o - ph . C O ] S e p rom the Department of Physics and AstronomyUniversity of TurkuTurku, Finland Supervised by
Dr. Pekka Heinämäki and Dr. Pasi NurmiDepartment of Physics and AstronomyTuorla ObservatoryUniversity of TurkuTurku, Finland
Reviewed by
Prof. Heikki SaloDivision of AstronomyDepartment of PhysicsUniversity of OuluOulu, Finland and
Dr. Antti TammTartu ObservatoryTõravere, Estonia
Opponent
Prof. Volker MüllerLeibniz-Institut für AstrophysikPotsdam (AIP)Potsdam, Germany ne could not be a successful scientist without realizing that,in contrast to the popular conception supported by newspapersand mothers of scientists, a goodly number of scientists arenot only narrow-minded and dull, but also just stupid.
Dr. James Dewey WatsonNobel Laureate cknowledgments
Countless warm and well deserved thanks are due to all the people who contributed, andmore importantly, forced me to push through the hard times when quitting felt like theonly right thing to do. These thanks are intended especially for the benefit of those whoare actually not interested in the rest of the thesis, but are driven to read this sectionand this section only to find out if their names are listed. To their convenience, I havedeliberately minimised the number of names spelled out.First words of appreciation must go to my supervisors Pekka Heinämäki and PasiNurmi who exposed me to the mysteries of cosmological simulations, formation of large-scale structure, and cosmology. I may not have been a good student - I was hardly everpresent during my years of studies - thus, warm thanks for sticking with me. I also wantto express my gratitude to Mauri Valtonen for his ideas in galaxy group studies. I amin debt to Henry Ferguson and Rachel Somerville for guiding me through the last paperof the thesis. Your guidance truly convinced me that galaxy formation and evolution isworth spending countless long, but also fruitful, hours when others sleep.For financial support, that made this work possible, I thank the following contrib-utors: the Väisälä foundation, Nordic Optical Telescope Science Association, SpaceTelescope Science Institute, and G.J. Wulff foundation. Special thanks go to my motherLiisa Misukka for always providing extra funds when times weren’t so great.I would like to thank the warm staff of Nordic Optical Telescope for making my stayin la Palma so enjoyable. I did not only enjoy eating pata negra and drinking wine (bothof which I did more than I care to admit), but I also learned invaluable lessons aboutobservational astronomy. For a hardcore theoretician this might sound odd, but astron-omy is actually observation driven science and we should all make sure that we have atleast seen the night sky. I would also like to thank all the individuals who made me feelwelcome to STScI and taught me more about observations. Special thanks go to StefanoCasertano, Danny Lennon, Harry Ferguson, and Rachel Somerville for countless careeradvices and extremely useful science discussions. I would also like to thank MichaelWolfe for reading the early draft of this manuscript and standing for my Finglish. Lastbut most importantly I would like to thank Carolin Villforth for the countless sciencearguments and all the wonderful moments I have been able to share with you.I thank the official reviewers of this thesis, Drs. Heikki Salo and Antti Tamm, fortimely reading and useful criticism. I am greatly honoured that Prof. Volker Müller hasagreed to act as my opponent in the public disputation of this dissertation. Finally, tothose who wish to know the tips and tricks of accomplishing a PhD, I confidently saythat all the people mentioned here are part of it and to them I owe my deepest gratitude. ami-Matias Niemi ontents
Acknowledgments 5List of publications 91 Introduction 102 Formation of Structure 14
Bibliography 99 ist of publications
I Are the nearby groups of galaxies gravitationally bound objects?
Niemi S.-M. , Nurmi P., Heinämäki P. and Valtonen M.
MNRAS , 1864 (2007)
II The origin of redshift asymmetries: how Λ CDM explains anomalousredshift
Niemi S.-M. and Valtonen M.
A&A , 857 (2009)
III Formation, evolution and properties of isolated field elliptical galaxies
Niemi S.-M. , Heinämäki P., Nurmi P. and Saar E.
MNRAS , 477 (2010)
IV Physical properties of
Herschel selected galaxies in a semi-analyticalgalaxy formation model
Niemi S.-M. , Somerville R.S., Ferguson H.C., Huang K.-H., Lotz J. andKoekemoer A.M. submitted to MNRAS
HAPTER Introduction
How did the Universe begin? How will it evolve? How did all the structures such asgalaxies, groups, and clusters we observe in the night sky begin to form? How did theygrow and how will they evolve? These are some of the most profound questions themankind have and shall seek to answer.Observations of the cosmic microwave background (CMB) radiation from the timewhen the Universe was only ∼ , years old, and significantly smaller than today,have shown that the early Universe was extremely smooth containing only small ∼ − temperature anisotropies (Komatsu et al., 2009; Hinshaw et al., 2009). Even so, wecan observe a vast amount of different type of structure in visible light and in otherfrequencies (e.g. Blanton & Moustakas, 2009, and references therein) when looking atthe night sky. The smooth early Universe must therefore have gone through radicalchanges when evolving from the smooth primordial gas and dark matter density fieldto inhomogeneous structures such as dark matter haloes, galaxies, groups and clustersobserved, directly or indirectly, today.In physical cosmology the global evolution of the temperature, pressure, and densityfields can be studied using Friedmann’s world models. These models can describe theevolution of an homogeneous and isotropic universe. However, as we do not live in anempty universe, a proof for that is obvious - you are reading this write up - we must alsodescribe how the small anisotropies, seen in the CMB, evolve. During the early times, inthe so-called linear regime, the evolution of the density field can be followed using linearperturbation theory and Newtonian gravity. However, after the density perturbationsgrow enough they start to collapse and their evolution turns nonlinear. Their evolutioncan still be followed using analytical approximations, however, their validity is limited.Fortunately, more accurate methods have also been developed.Due to the inherently nonlinear nature of gravity cosmological N -body simulationshave become an invaluable tool when the growth of structure is being studied and mod-elled closer to the present epoch. Large simulations with high dynamical range (e.g.Springel et al., 2005; Boylan-Kolchin et al., 2009) have made it possible to model the for-mation and growth of cosmic structure with unprecedented accuracy. Moreover, galax-ies, the basic building blocks of the Universe, can also be modelled to good accuracy incosmological context and studied from their initial formation down to the present time Λ Cold Dark Matter model has been assumed, see Section 2.4.1. (e.g. Naab et al., 2007). For example, semi-analytical models of galaxy formation (e.g.Somerville & Primack, 1999; Somerville et al., 2001, 2008) allow us to populate darkmatter haloes with galaxies that are formed from baryonic matter when lacking hydro-dynamical simulations (e.g. Springel et al., 2005; Croton et al., 2006; Lucia & Blaizot,2007). Reassuringly, both semi-analytical models as well as hydrodynamical simula-tions, which both model for example an inflow of gas, how gas can cool and heat upagain, how stars are formed within galaxies, and how stellar populations evolve, are inreasonable agreement with observational data at lower ( z < ∼ redshifts (e.g. Kitzbich-ler & White, 2007). Instead, at high redshifts ( z > ∼ the small number of candidategalaxies (e.g. Yan et al., 2010; Labbé et al., 2010; Bouwens et al., 2011) still compli-cates more detailed comparisons to simulations (e.g. Dayal et al., 2011; Razoumov &Sommer-Larsen, 2010). Even so, simulations can be used to make predictions for differ-ent observables and aid when interpreting observational results. The growth of cosmicstructures, cosmological N -body simulations, and formation of galaxies are briefly re-viewed in Chapter 2.Despite all the simulations and successes in recent years and decades, there are stillmany unanswered questions in the field of galaxy formation and evolution. One of thelongest standing issues in galaxy evolution is the significance of the formation place andthus initial conditions to a galaxy’s evolution in respect to environment, often formulatedsimply as “nature versus nurture” like in human development and psychology. We aretherefore left to ponder if the galaxies we see today are simply the product of the pri-mordial conditions in which they formed, or whether experiences in the past change thepath of their evolution. Unfortunately, our understanding of galaxy evolution in differentenvironments is still limited, albeit the morphology-density relation (e.g. Oemler, 1974;Dressler, 1980) has shown that the density of the galaxy’s local environment can affect itsproperties. For example, on average, luminous, non-starforming elliptical and lenticulargalaxies have been found to populate denser regions than star forming spiral galaxies.Consequently, the environment should play a role in galaxy evolution, however, despitethe efforts, the exact role of the galaxy’s local environment remains open.A group of galaxies is the most common galaxy association in the Universe (e.g.Holmberg, 1950; Humason et al., 1956; Turner & Gott, 1976; Huchra & Geller, 1982;Ramella et al., 1995; Zabludoff & Mulchaey, 1998). As such, more than half of all galax-ies are found in groups and small clusters. They are therefore important cosmologicalindicators of the distribution of matter in the Universe. Moreover, groups and clusterscan also provide important clues for galaxy formation and evolution physics as the prop-erties of galaxies can be studied as a function of the local environment. As a result,the environmental dependency in galaxy evolution can be understood to some extent interms of the group environment (e.g. Moore et al. 1996, 1998; Mihos 2004; Fujita 2004,but see also Kauffmann et al. 2004; Blanton 2006). Note, however, that the debate over CHAPTER 1. INTRODUCTION the exact role of group environment is far from over. Groups of galaxies, their propertiesand the group environment in the context of galaxy evolution is discussed in Chapter 3.One fundamental question concerning groups of galaxies, and closely related togalaxy evolution in groups, is whether the identified systems are gravitationally boundor not. This is a valid concern because from the observational point of view, groupsand their member galaxies are not well defined. Early studies based on simulations gavehints that not all observed systems of galaxies are dynamically relaxed, but might bein the process of formation (e.g. Diaferio et al., 1994; Frederic, 1995a,b). Despite this,many observational studies, even today, treat all identified groups like they were gravita-tionally bound structures. The work presented in Papers I and II tackles this issue and issummarised in Chapters 3.5 and 3.6, respectively. Results of this work show that a sig-nificant fraction of systems of galaxies are gravitationally unbound when the most oftenused grouping algorithm, namely Friends-of-Friends, is being applied to simulated data.This result has several important implications for the studies of galaxy groups and forthe evolution of galaxies in groups. For example, Paper II shows that groups with a largeexcess of positive redshifts are more often gravitationally unbound than groups that donot show any significant excess. Fortunately, this prediction can be used, together withthe methods developed for and presented in Paper I, to assess whether observed groupsare likely to be gravitationally bound or not.Elliptical galaxies are most often found in dense environments like the cores ofgroups and clusters (e.g. Dressler, 1980). Yet, observations have shown that there is asignificant population of isolated elliptical galaxies that are found in under-dense regionswith no bright nearby companions (e.g. Aars et al., 2001; Reda et al., 2004; Smith et al.,2004; Denicoló et al., 2005; Collobert et al., 2006). Whether these galaxies originallyformed in under-dense regions or if the local environment has impacted their evolution,in form of a collapsed group, is a profound question with long reaching implications.Key observations of galaxy evolution and the significance of the environment for galaxyevolution are briefly reviewed in Chapter 4.The results of a theoretical case study of isolated field elliptical galaxies, Paper III,are also presented and summarised in Chapter 4. These results show that three differ-ent yet typical formation mechanisms can be identified, and that isolated field ellipticalgalaxies reside in relatively light dark matter haloes excluding the possibility that all ofthem are collapsed groups as suggested earlier. Additionally, also another case studyconcerning luminous infrared galaxies, Paper IV, is discussed. The results of this studyimply a strong correlation, such that more infrared-luminous galaxies are more likely tobe merger-driven. However, the results also imply that a significant fraction (more thanhalf) of all high redshift infrared-luminous galaxies detected by
Herschel
Space Obser-vatory are able to attain their high star formation rates without enhancement by a merger.These and other results discussed in Chapter 4 imply that both “nature” and “nurture” play a role in galaxy evolution. HAPTER Formation of Structure “. . . the biggest blunder of my life.” Albert Einstein
The study of the structure formation of the Universe dates back to 1610 and GalileoGalilei who realised that the Galaxy can be resolved into stars when observed through atelescope. Galilei’s observations can also be considered as a starting point for early ob-servational cosmology. However, it was René Descartes and Thomas Wright who werelikely the first ones to speculate and publish their cosmological views. Already around1760s Immanuel Kant and Johann Lambert developed the first hierarchical model ofthe Universe, although since these early models it took almost two hundred years beforecosmology developed into a physical science and before the formation of structure in theUniverse could be truly appreciated and studied in a physical context. Below I brieflymention a few key moments from the history that have lead to the structure formationtheory, as we know it today. However, many important events are not mentioned, thusI refer the interested reader to more comprehensive reviews, see e.g. Ratra & Vogeley(2008).It was Albert Einstein and his General Theory of Relativity (GR) in 1915 (Einstein,1915) that gave birth to the physical cosmology as we know it today. GR is a frameworkthat explains one of the four fundamental forces of the Universe, namely gravity, andenabled cosmologists to predict the behaviour of a model universe. As a result, it becamepossible, for the first time in the history of mankind, to formulate self-consistent modelsthat describe the Universe and large-scale structure. As a consequence, in 1917 Einsteinderived the first fully self-consistent model of the Universe (Einstein, 1917). However,soon after he realised that without modifications his field equations predicted that a staticUniverse was not stable. At the time the Universe was assumed to be static, thus, Einsteinintroduced a cosmological constant Λ , that enabled a static universe, to solve the issue.In 1924 Aleksander Friedmann and in 1927 Georges Lemâitre derived solutions forexpanding universes, paving the way for evolving universe models. Later, Friedmann’smodels became the standard models describing the dynamics of the Universe. In 1927Lemâitre first proposed what has come to be known as the Big Bang theory of the origin .2. OBSERVATIONAL BACKGROUND OF COSMOLOGY of the Universe, albeit the name was introduced by Fred Hoyle. The framework for theBig Bang model relies on the Einstein’s GR, the Cosmological Principle, Friedmann’sequations, and it is also the standard theory for the origin of the Universe.In 1935 Howard Percy Robertson and Arthur Geoffrey Walker derived independentlythe space-time metric for all isotropic, homogeneous, uniformly expanding models of theUniverse. However, the Friedmann world models are isotropic and homogeneous, thus,all observable structures such as galaxies and groups are absent. Consequently, the nextstep towards developing more realistic models of the Universe was to include small den-sity perturbations and to study their development under gravity, namely the formation ofstructure. Fortunately, already in 1902, well before GR, sir James Hopwood Jeans hadshown that the stability of a perturbation depends on the competition between gravityand pressure (Jeans, 1902): gas pressure prevents gravitational collapse on small spatialscales and gives rise to acoustic oscillations. Jeans showed that density perturbations cangrow only if they are heavier than a characteristic mass scale, while below this scaledissipative fluid effects remove energy from the acoustic waves, which dampens them.The application of the Jeans criterion and the growth of spherically symmetric perturba-tions in an expanding universe were worked out by Lemâitre and Richard Tolman in the1930s. Albeit it was not before 1946 when Evgenii Lifshitz worked out relativistic per-turbation theory and started applying it to the linear growth of cosmic structure. Finally,a general scheme for structure formation was first outlined by Lev Davidovich Landauand Lifshitz in the 1950s, and developed further by Phillip James Edwin Peebles duringthe 1970s.However, to truly appreciate the study of the formation of structure of the Universe,observational constrains on the initial density perturbations were required. The serendip-itous discovery of Cosmic Microwave Background (CMB) radiation by Arno Penziasand Robert Wilson in (Penzias & Wilson, 1965), predicted already in byRalph Alpher, Robert Herman, and George Gamow (Gamow, 1948; Alpher & Herman,1948), paved the way for understanding the initial conditions of large-scale structure for-mation. Measurements of the CMB describe the initial conditions after recombinationof the very early Universe and led way to the standard model of cosmology. Further-more, large galaxy surveys of recent years have helped to set constrains for the modelsof structure formation at low redshift. According to the Big Bang model, the background radiation from the sky measured to-day comes from the so-called last scattering surface. As the name implies the surface oflast scattering is a spherical surface where the Cosmic Microwave Background photons Now referred to as the Jeans’ mass. CHAPTER 2. FORMATION OF STRUCTURE were scattered for the last time before arriving at our microwave detectors. This decou-pling of photons from matter happened t (cid:63) ∼ , years after the Big Bang duringthe epoch of recombination when the rate of Thomson scattering became slower than theexpansion of the Universe (e.g. Weymann, 1966; Peebles, 1968). At that moment, pho-ton interactions with matter became insignificant, leading to the CMB radiation. Thismoment also defines the “optical” horizon; the largest volume from which we can receiveinformation via photons.On the largest scales the most robust evidence for the isotropy of the Universe comesfrom the CMB measurements, while galaxy surveys compliment the CMB informationby probing later epochs and smaller angular scales. Therefore, in the next two SectionsI will briefly review what the CMB and galaxy survey observations can tell us about theformation of structure. The epoch when the ionisation state of the intergalactic gas changed from being a fullyionised plasma to a neutral gas is known as the epoch of recombination. This is theredshift ( z ∼ when the detailed anisotropy structure of the early Universe wasimprinted onto the Cosmic Microwave Background (CMB). It can therefore provideinformation on the initial density perturbations and describe the Universe on the epochwell before galaxies, groups and the formation of large-scale structure.Ever since the discovery of the CMB radiation, observations have played a key rolein shaping and constraining the standard model of cosmology (for a review, see e.g. Hu &Dodelson, 2002; Bartelmann, 2010). The CMB observations probe the earliest observ-able Universe and hence the initial conditions of the Universe, such as its homogeneity,isotropy, and flatness. Based on the CMB observations it has been established, first byCosmic Background Explorer (COBE) and later by Wilkinson Microwave AnisotropyProbe (WMAP), that the electromagnetic spectrum of the CMB is extremely close to athermal blackbody with a temperature ∼ . K. Moreover, COBE and WMAP hasestablished that the largest temperature anisotropies ∆ TT in the CMB are of the order of ∼ − (Smoot et al., 1992; Hinshaw et al., 2009), as shown in Figure 2.1. The absenceof ∼ − K fluctuations alone show that the matter in the Universe must be dominantlysomething that does not interact electromagnetically (Peebles, 1982), i.e., “dark”.The angular temperature fluctuations associated with the primordial density pertur-bations are assumed to originate in a rather narrow range of redshifts. If this holds thenthe pattern of the angular temperature fluctuations in the CMB map (Fig. 2.1) gives usa direct snapshot of the distribution of radiation and energy at the moment of recombi-nation. The angular scale Θ ∼ ◦ corresponds to the Hubble radius at recombination, This however is not true for all CMB photons: some have scattered from free electrons that havebecome available due to reionization. .2. OBSERVATIONAL BACKGROUND OF COSMOLOGY Figure 2.1:
The foreground-reduced Internal Linear Combination map based on the five yearWMAP data. Image from Hinshaw et al. (2009). which can be taken as a dividing line between the small-scale perturbations that havebeen substantially modified by gravity and the large-scale inhomogeneities that have notchanged much. The fluctuations on large angular scales (Θ (cid:29) ◦ ) arise from inhomo-geneities with wavelengths exceeding the Hubble radius at recombination. As a result,they provide pristine information about the primordial inhomogeneities. On the otherhand, sub-horizon perturbations are formed by primordial sound waves.The anisotropy of the CMB can be divided into two types: primary anisotropy, due toeffects which occur at the last scattering surface and earlier; and secondary anisotropy,due to effects such as interactions of the CMB photons with hot gas or gravitationalpotentials (for example, the Sunyaev-Zel’dovich and integrated Sachs-Wolfe effects),between the last scattering surface and the observer. The structure of the CMB primaryanisotropies is mainly determined by two effects: acoustic oscillations and, on smallangular scales, photon diffusion (also known as Silk damping (Silk, 1968)). The photondiffusion damping arises from the fact that the photon-baryon-electron (PBE) fluid isnot tightly coupled and the photons can diffuse through the fluid, while the acousticoscillations result from the constructive and deconstructive interference. Overdensitiesin the dark matter compresses the fluid due to their gravity until the rising pressure inthe coupled PBE fluid is able to counteract gravity. The cosmological importance ofthis is that the PBE fluid underwent acoustic oscillations, while the dark matter, beingdecoupled, did not. CHAPTER 2. FORMATION OF STRUCTURE
The density fluctuations in the early Universe are assumed to be critical for structureformation, because they can provide the seeds from which the structures within the Uni-verse can grow and eventually collapse to form the first stars and galaxies (e.g. Brommet al., 1999; Barkana & Loeb, 2001; Abel et al., 2002; Bromm et al., 2009, and refer-ences therein). The density perturbations of the early Universe are thought to have avery specific character when inflation is assumed: they form a Gaussian random field(Bardeen et al., 1986), which is nearly scale-invariant according to the spectral index n measured by WMAP (Hinshaw et al., 2009; Komatsu et al., 2009; Dunkley et al., 2009;Jarosik et al., 2011). I will return to this in Section 2.5 where a more detailed discussionof the initial density perturbations is presented. Power Spectrum of the CMB
To maximise the information a CMB map (such as Fig. 2.1) can provide, the CMBinformation is most often presented in the form of a power spectrum P in terms of theangular scale or multipole moment l as shown in Figure 2.2. The power spectrum, whichis a spherical harmonic transform of the CMB map, and polarisation of the CMB radia-tion provide a wealth of information (see e.g. Tristram & Ganga, 2007, for data analysismethods) for both constraining cosmological parameters (Table 2.2) and for understand-ing the formation of the large-scale structure in the Universe. The CMB anisotropy is apowerful cosmological probe because the parameters which determine the spectrum canall be directly related to the basic cosmological parameters such as the energy densities Ω i , the dark energy equation of state w , and the Hubble parameter H .The general shape of the power spectrum (Fig. 2.2) - a plateau at large angular scales(small l ) and acoustic peaks at small angular scales (large l ) - confirms that the spectrumis predominantly nearly scale-invariant and adiabatic in agreement with the basic pre-dictions of the Big Bang and inflationary paradigm. The dominant acoustic peaks inthe CMB power spectra are caused by the collapse of dark matter over-densities and theoscillation of the photon-baryon fluid into and out of these over-densities (Lineweaver,2003). The underlying physical notion is that the pressure of photons can erase anisotropies,whereas the gravitational attraction of baryons makes them to collapse and to formdense haloes. As a result, these effects can create acoustic oscillations, which give theCMB its characteristic peak structure (see Fig. 2.2). The first acoustic peak is associ-ated with perturbations on the scale of the sound horizon at the last scattering surface ( l ∼ ∼ ◦ ∼ Mpc ) , while following peaks are on the scales less than the soundhorizon. Combining the information about the heights and locations of the peaks, manycosmological parameters can be determined with good accuracy independent of otherobservations such as galaxy surveys. Fourier transformation is not possible on a sphere, thus, spherical harmonics which are analogous areused instead. .2. OBSERVATIONAL BACKGROUND OF COSMOLOGY Figure 2.2:
The temperature power spectrum for the seven-year WMAP data. The solid lineshows the predicted spectrum for the best-fit flat Λ CDM model (Section 2.4.1). The error bars onthe data points represent measurement errors while the shaded region indicates the uncertaintyin the model spectrum arising from cosmic variance. Image from Jarosik et al. (2011).
The large galaxy surveys of today such as the 2dF Galaxy Redshift Survey (2dFGRS;Colless, 1999; Colless et al., 2001; Percival et al., 2001) and the Sloan Digital Sky Sur-vey (SDSS; York et al., 2000; Stoughton et al., 2002; Abazajian et al., 2003) have pro-vided large, statistically significant samples of different types of galaxies. The ability toexplore many dimensions of galaxy properties and scaling relations simultaneously andhomogeneously has been greatly beneficial. Moreover, spectroscopic observations andmulti-wavelength imaging allows galaxies to be sorted in classes and sub-populations by,e.g., morphology, environment and luminosity, while large sky coverage allows galaxiesto be grouped in groups and clusters enabling studies of galaxy evolution as a functionof environment (e.g. Blanton & Berlind, 2007; Mateus et al., 2007, see also Chapter 4).However, large galaxy surveys not only provide good statistics for galaxy properties butthey can also be used to study the large-scale structure and cosmology.Despite the fact that the CMB radiation is very smooth, the visible Universe thatis dominated by the light from galaxies looks highly inhomogeneous and consists ofstructures from the scales of isolated galaxies and voids, through groups (Chapter 3) and CHAPTER 2. FORMATION OF STRUCTURE clusters to superclusters and to large filaments between them (see Fig. 2.3). One aimof large galaxy redshift surveys is therefore to map the three dimensional distributionof galaxies, in order to understand the properties of this distribution and what it impliesabout the contents and evolution of the Universe. Because the topology of the distri-bution of galaxies is closely related to the initial conditions of the Universe and to theassumption that the initial perturbations were Gaussian fluctuations with random phaseson large scales this mapping can also provide information concerning the conditions inthe early Universe independent from the CMB measurements. Unfortunately, the spatialdistribution of galaxies, groups and clusters, depends not only on the matter distributionin the Universe, but also on how they form in the matter density field. It is therefore im-portant to understand galaxy formation (Section 2.8) and evolution (Chapter 4) in detailwhen studying the clustering of galaxies.A representation of the large-scale distribution of galaxies on the sky in the 2dFGRSis shown in Fig. 2.3. From this figure alone it is obvious that galaxies form largerstructures such as clusters and filaments and that the visible light is unevenly distributedin the Universe even on relatively large scales. Even though the distribution of galaxiesbecomes smoother and smoother when larger and larger scales are considered, non-random structure is still present in forms of superclusters and filaments between them.This non-random structure is sometimes called the cosmic web, as the long filaments ofdark and baryonic matter seem to form a “threaded" structure.The filaments seen in large galaxy surveys are the largest known structures in theUniverse and can be up to ∼ − h − Mpc long (Einasto et al., 1980; Batuski& Burns, 1985; White et al., 1987; Bahcall, 1988). Filaments are important structuresfor galaxy formation as it is assumed that they are the channels that carry baryons tothe nodes of the filaments where clusters of galaxies are formed. Filaments can alsohelp cool gas to avoid shock-heating (e.g. Dekel & Birnboim, 2006; Dekel et al., 2009a;Kereš et al., 2009), while over-densities in filaments can form gravitationally bound darkmatter haloes. There is however still a debate in how gas can enter and coalesce into darkmatter haloes and how it cools down to form stars and eventually galaxies (see e.g. Kerešet al., 2005; Kaufmann et al., 2006, and references therein). The exact role of filamentsin the formation and evolution of galaxies is therefore currently unclear.While filaments are the largest known structures, superclusters (e.g. Araya-Meloet al., 2009) are the largest non-percolating galaxy systems (Oort, 1983; Bahcall, 1988;Einasto et al., 2007, 2008). Unlike super clusters, the scales of the largest voids are ingeneral ∼ to times the scale of a regular relaxed cluster, i.e., up to ∼ h − Mpc,although the size measurements vary greatly (Zeldovich et al., 1982; Rood, 1988; Vo-geley et al., 1994; Lindner et al., 1995; El-Ad & Piran, 1997; Hoyle & Vogeley, 2004; Here h refers to the dimensionless Hubble parameter defined such that H = 100 h km s − Mpc − .Also, see equation 2.13 for a definition of the Hubble parameter. .2. OBSERVATIONAL BACKGROUND OF COSMOLOGY Ceccarelli et al., 2006; von Benda-Beckmann & Müller, 2008; Tinker & Conroy, 2009).Because of all the structure in the cosmic web, the galaxy distribution seems to havea sponge-like topology, with both high- and low-density regions forming an intercon-nected network, where voids are separated from high-density regions by flattened struc-tures called “walls”. This raises an obvious question: how can we quantify the clusteringof different types of objects and what does this tell us about the formation of the large-scale structure?
Figure 2.3:
The projected distribution of galaxies in the nearby Universe as a function of redshiftand Right Ascension. Earth is at the centre, and each blue point represents a galaxy. Courtesyof the 2dFGRS website.
Two-point correlation function and the power spectrum
Among the simplest methods to measure clustering properties of galaxies (or of other ob-jects like quasars, groups, clusters, etc.) is with the spatial two-point correlation function ξ ( r ) . It describes the excess probability above Poisson of finding an object at distance r from another object selected at random over that expected in a uniform, random distri-bution (see Peebles, 1980, for a complete discussion). We can now write the probabilityto find galaxies in infinitesimal small volumes d V and d V as follows P = (1 + ξ ( r ))¯ n d V d V , (2.1)where ¯ n is the mean galaxy density. In practise, it is however often more convenient toderive the two-point correlation function ξ ( r ) using, for example, the Landy & Szalay CHAPTER 2. FORMATION OF STRUCTURE (1993) estimator.Large datasets provided by galaxy surveys have been used to study, for example,the spatial correlation functions (e.g. Connolly et al., 2002; Scranton et al., 2002; Zehaviet al., 2004; Masjedi et al., 2006), clustering of matter, galaxies (e.g. Coil et al., 2007) andgroups (e.g. Coil et al., 2006a) in the Universe. Consequently, allowing to set constrainsfor the structure formation and cosmological parameters (Lahav & Suto, 2004; Tegmarket al., 2004b). In galaxy surveys, redshifts of galaxies are usually used as distances, thusthe correlation function is said to work in redshift-space. However, because of peculiarvelocities , an isotropic distribution in real-space will appear anisotropic in redshift-space and vice versa. The redshift-space correlation function therefore differs from thereal-space correlation function. This effect is known as the redshift-space distortion (forobservational studies, see e.g. Hamilton, 1998; Tegmark et al., 2004a). It is important tonote that redshift-space distortions due to peculiar velocities along the line of sight willintroduce systematic effects to the estimate of ξ ( r ) . For example, at small separations,random motions within a virialized overdensity cause an elongation along the line ofsight (dubbed as “fingers of God"). On the other hand, on large scales, coherent infallof galaxies into forming structures causes an apparent contraction of structure along theline of sight (dubbed as the “Kaiser effect").On scales smaller than ∼ h − Mpc the real-space correlation function is wellapproximated by a power law ξ ( r ) = (cid:18) rr (cid:19) − γ , (2.2)where the slope γ ∼ . and r ∼ h − Mpc is the correlation length. This shows thatgalaxies are strongly clustered on scales < ∼ h − Mpc, and the amplitude of clusteringbecomes weak on scales much larger than > h − Mpc. Note, however, that the exactvalues of γ and r are found to depend on the properties of the galaxies. Particularly,brighter and redder galaxies are more strongly clustered than fainter and bluer ones (Nor-berg et al., 2001; Zehavi et al., 2005; Coil et al., 2006b; Wang et al., 2008). Additionally,early-type galaxies have been found to be much more clustered on small scales, leadingto a morphology-density relation (Peacock, 2002, see also Chapter 4).The galaxy correlation function is a measure of the degree of clustering in either thespatial ( ξ ( r )) or the angular distribution ( w ( θ )) of galaxies. The spatial two-point corre-lation (or autocorrelation) function ξ ( r ) and the power spectrum P ( k ) forms a Fourier- The peculiar velocity is the velocity that remains after subtracting off the contribution due to the Hubbleexpansion. .2. OBSERVATIONAL BACKGROUND OF COSMOLOGY transform pair, i.e. P ( k ) = 1 V (cid:90) ξ ( r ) e − i k · r d r (2.3) ξ ( r ) = V π (cid:90) P ( k ) e i k · r d k . (2.4)Here V is the volume within which ξ ( r ) is defined. Assuming isotropy and that thetwo-point correlation function is spherically symmetric (d k = 4 πk d k ) leads to P ( k ) = 4 π (cid:90) ∞ ξ ( r ) sin krkr r d r (2.5) ξ ( r ) = 12 π (cid:90) ∞ P ( k ) sin krkr k d k . (2.6)Note that the function sin kr ( kr ) − allows only wave-numbers k ≤ r − to contributeto the amplitude of the fluctuations on the scale r .Figure 2.4 shows a matter power spectrum at the present time. According to thefigure on large scales (small wave-numbers k < . h − Mpc) the current matter powerspectrum still has its primordial shape. This shape corresponds to a power law depen-dence on scale; P ( k ) ∝ k n , where n is the so-called spectral index, assumed to be closeto unity (for theoretical background, see Section 2.5.2 and for the reference value, seeTable 2.2). The horizon scale at the epoch when the matter and radiation densities areequal (the matter-radiation equality, z eq , see Table 2.2 for a value) is imprinted upon thepower spectrum as the scale at which the spectrum turns over. Hence, the peak positionin the spectrum corresponds to the Jeans length (Eq. 2.26) at matter-radiation equality (Iwill return to this in Section 2.5). The position of this turnover corresponds to a physicalscale determined by the matter (Ω m h ) and radiation densities (Ω r h ) . Moreover, theshape of the observed power spectrum P ( k ) depends on the amount and the nature ofthe matter in the Universe, providing constrains for cosmology. For example, if all ofthe dark matter were hot then the matter power spectrum would fall off sharply to zeroto the right of the peak.Large redshift surveys can be used not only to study the power spectrum of galaxyclustering but also the presence of the acoustic oscillations of baryons (e.g. Tegmarket al., 2004a; Cole et al., 2005; Eisenstein et al., 2005; Percival et al., 2007). The physicsof these oscillations are analogous to those of the CMB acoustic oscillations. The am-plitude of baryonic acoustic oscillations (BAOs) is however suppressed in comparisonto CMB because not all the matter in the Universe is composed of baryons. In practise,many studies of BAOs have taken advantage of the clustering of luminous red galaxies(LRGs; e.g. Padmanabhan et al., 2007; Sánchez et al., 2009). The clustering of LRGsalso allows the values of cosmological parameters to be derived independently from the CHAPTER 2. FORMATION OF STRUCTURE
Figure 2.4:
The matter power spectrum P ( k ) as a function of wave-number k at the present time.Note the turnover at k eq ∼ . h − Mpc. The plot combines data from different scales: CMB,large galaxy surveys, weak lensing and Ly α forest, in order of decreasing co-moving wavelength.In addition, there is a single data point for galaxy clusters. Figure from Tegmark et al. (2004a). CMB measurements. Furthermore, the observed power spectrum allows to constrainon both the amplitude and the scale dependence of the galaxy bias (e.g. Padmanabhanet al., 2007). This ultimately links the galaxy power spectrum to the matter power spec-trum. Despite this connection, large galaxy surveys can provide constrains for structureformation models independent of the CMB measurements.A viable structure formation model has to therefore simultaneously explain both thesmoothness of the CMB and the clear evidence from the galaxy surveys that the assump-tions of isotropy and homogeneity do not hold on smaller scales, but galaxies and otherinhomogeneities such as groups do form. Moreover, large galaxy surveys together with .3. DYNAMICS OF AN EXPANDING UNIVERSE the CMB observations can be used to set strict constrains for cosmological parameters(see Section 2.4.1 and Table 2.2). Importantly, these observations can provide informa-tion about structures of different sizes and times from the recombination to the presentepoch. A successful structure formation model must therefore be able to explain all thecurrent observations. However, before we start looking into structure formation modelsin more detail, some tools to study the dynamical evolution of an expanding universethat is isotropic and homogeneous must be introduced. In the following Sections I shallgive a minimalistic overview of the dynamics of an expanding universe on top of whichthe structure formation theory can be build upon. For more detailed treatment, I refer theinterested readers to great textbooks of e.g. Peebles (1980); Dodelson (2003); Longair(2008); Mo et al. (2010). Being able to model the dynamical evolution of an expanding universe is a basic require-ment for any structure formation model. Because space-time can be curved and is notstatic, we must rely on General Relativity when deriving the equations that govern theevolution of the background universe.
Einstein’s General Relativity (GR) enabled self-consistent models of the Universe to beconstructed as it relates matter and energy to the geometrical properties of the Universe.In GR, the gravity field is described by Einstein’s field equation: G αβ = 8 πGT αβ , (2.7)where G αβ is the Einstein tensor, G is Newton’s gravitational constant and T αβ is theenergy-momentum tensor. Note that matter is incorporated in Einstein’s equation throughthe energy-momentum tensor. On large scales, matter can be approximated as a perfectfluid characterised by an energy density (cid:15) , pressure P and four velocity u α . Now theenergy-momentum tensor may be written as T αβ = ( (cid:15) + P ) u α u β − P δ αβ , (2.8)where the equation of state P = P ( (cid:15) ) depends on the properties of matter. Often incosmologically interesting cases P = constant × (cid:15) (or more general as in Eq. 2.16).Equation 2.8 shows that in GR the strength of the gravitational field depends not only onthe energy density (cid:15) , but also on the pressure P . CHAPTER 2. FORMATION OF STRUCTURE
It is also possible to write the Einstein equation in a form that explicitly shows thecosmological constant Λ , now G αβ = R αβ − δ αβ R − Λ δ αβ , (2.9)where R αβ is the Ricci tensor, R is the scalar curvature, Λ is the cosmological constant,and δ is the unit tensor defined by the metric such that g αγ g γβ = δ αβ . As the cosmo-logical constant can be interpreted as the contribution of vacuum energy to the Einsteinequation it can also be included in the energy-momentum tensor.According to the Cosmological Principle (Milne, 1933) the Universe is homoge-neous and isotropic, at least on large enough scales, thus space-time can be described bythe Robertson-Walker metric: d s = c d t − a ( t ) (cid:18) d r − Kr + r (d θ + sin θ d φ ) (cid:19) , (2.10)where the spatial positions are described by spherical coordinates ( r, φ, θ ) , c is the speedof light and a ( t ) is the scale factor. The scale factor and the changing scale of theUniverse causes the cosmological redshift z , which can be defined as z = a ( t ) a ( t em ) − . (2.11)Here a ( t ) is the present value of the scale factor and a ( t em ) is the value of the scalefactor at the time when the light was emitted. As a result, the redshift z can be usedto parameterise the history of the Universe; a given z corresponds to a time when ourUniverse was z times smaller than now. The importance of the metric 2.10 is thatit allows to define the invariant interval d s between events at any epoch or location inan expanding universe, and thus determines the metric, Riemann, and Ricci curvaturetensors.The Einstein equation describes the geometry of space which is curved by matter andenergy. It is also the basic equation of GR that the dynamical variables characterisingthe gravitational field must follow. The cosmological evolution of relativistic matter cantherefore be derived from the Einstein equation (2.7) when the metric of the space-timeand the energy-momentum tensor are fixed. Alexander Friedmann was the first to derive a pair of equations that can describe theexpansion rate of a homogeneous and isotropic universe. The first Friedmann equation Note that later in this Chapter δ will refer to the density perturbation field. .3. DYNAMICS OF AN EXPANDING UNIVERSE can be derived directly from Einstein’s field equation (2.7), which can be simplifiedusing the Robertson-Walker metric (Eq. 2.10). Now, Friedmann’s first equation can bewritten as H + Kc a = 8 πG (cid:15) + Λ c , (2.12)where G is Newton’s gravitational constant, K is the spatial curvature ( ± or ), (cid:15) isthe sum over all energy densities (e.g. baryons, photons, neutrinos, dark matter and darkenergy), and H is the Hubble parameter that describes the rate of the expansion: H ( t ) = ˙ a ( t ) a ( t ) . (2.13)Here ˙ a denotes the time derivative of the scale factor. Note that the dynamical contentof the metric is encoded in the function H ( a, t ) . Friedmann’s second equation can bederived from the trace of Einstein’s field equation and can be written as: ¨ aa = − πG (cid:18) (cid:15) + 3 Pc (cid:19) + Λ c . (2.14)Now the ¨ a is the second time derivative of the scale factor and P is the pressure.The importance of the two Friedmann equations (2.12 and 2.14) is that they de-termine the two unknown functions; the scale factor a ( t ) and the energy density (cid:15) ( t ) .Consequently, the Friedmann equations can describe the dynamics of a homogeneous,isotropic and expanding universe, because the scale factor completely describes the timeevolution of such a universe. The Friedmann equations therefore determine the expan-sion rate of the Universe, based on the density of material within it and the curvatureof space. Note that, a Robertson-Walker metric (2.10) whose scale factor a satisfiesFriedmann’s equations is called the Friedmann-Lemaiˆtre-Robertson-Walker metric andthe cosmological standard model upholds that the Universe at large is described by sucha metric.Construction of a cosmological model requires solving the Friedmann equations,resulting to the expansion rate as a function of time, and hence the size of the Uni-verse. However, solving the Friedmann equations alone is not enough for a cosmologi-cal model, it is also essential to know how the energy density (cid:15) changes as a function oftime. The evolution of energy density in an expanding universe can be described with theso-called fluid equation. The fluid equation, which holds only for adiabatic processes, CHAPTER 2. FORMATION OF STRUCTURE can be obtained from the Friedmann’s equations when equation 2.14 is rewritten usingequation 2.12 resulting in: ˙ (cid:15) + 3 ˙ aa (cid:18) (cid:15) + Pc (cid:19) = 0 , (2.15)where (cid:15) denotes the energy density and P is the pressure, as defined earlier. The fluidequation expresses the conservation of mass-energy, thus it is also known as the energyconservation or the continuity equation. The second term in the fluid equation corre-sponds to the loss in energy because the pressure of the material has done work as thevolume of the Universe increased. However, because the energy is conserved the energylost from the fluid via the work done goes into gravitational potential energy. To gen-eralise, for an adiabatically expanding volume the entropy per unit co-moving volumeis conserved, and the expansion of the Universe causes an increase or decrease of itsinternal energy depending on whether the pressure P is smaller or larger than zero.For a given equation of state, P ( (cid:15) ) , the fluid equation gives the density and pressureas a function of the scale factor a . The equation of state, which describes the energycontent of the Universe, is often parametrized with w i in the following way: P i ( (cid:15) ) = w i (cid:15) i c . (2.16)Here the subscript i denotes the species of the material. Note, however, that this so-called perfect fluid hypothesis holds only for material whose pressure is directly relatedto its density. Finally, if w is time-independent, then substituting Eq. 2.16 into 2.15 givesthe time evolution of the mean energy density of the Universe as follows: (cid:15) i ∝ a − w i ) . (2.17)The fluid equation together with the equation of state allow the derivation of the meanenergy density, temperature and pressure of the Universe at any redshift from their valuesat the present time. At early times the Universe is assumed to be radiation dominated thuswe can approximate that the energy content of the Universe is dominated by an ultra-relativistic radiation fluid for which w rad = . As a result, the mean energy densityevolves proportional to a − . After the nucleosynthesis, but before the recombination,at z eq ∼ , the densities of non-relativistic matter and relativistic radiation areequal. However, after this point the Universe turns into a matter dominated. Now, a non-relativistic gas can be approximated with a fluid of zero pressure and we take w mat = 0 .As a consequence, the mean energy density evolves ∝ a − . Finally, at recent epochs The exact time depends on the matter density of the Universe and the Hubble constant. The redshiftgiven is for the reference values, see Table 2.2. sometimes referred to as dust .4. INGREDIENTS OF STRUCTURE FORMATION the energy density seems to have become dominated by vacuum energy. In order tokeep a constant energy density as the Universe expands, the pressure must be negative,therefore, for vacuum energy we take w vac = − . Note that w = − can also be takenfor the cosmic inflation.Table 2.1 summarises the evolution of energy density, pressure, and temperature asa function of the scale parameter a . Note, however, that these scaling relations only holdif the equation of state remains constant with respect to time, while in reality such asimplification may not hold on all times. It should also be kept in mind that although thecontribution of baryons and photons to the present day energy budget is small, they makean important contribution to shaping the matter power spectrum. Moreover, in realisticcases the Universe is not made out of a single material component as presented above.Fortunately, each material, baryons, photons, dark matter, neutrinos, dark energy, etc.,obey their own fluid equation containing the appropriate expression for its pressure if thefluids are non-interacting. One can therefore take a linear combination of the terms andsubstitute that into the Friedmann equations in case of more realistic models. Table 2.1:
Thermodynamics of a homogenous and isotropic universe.
Dominant component w Energy density (cid:15)
Pressure P Temperature T Radiation a − a − a − Matter a − a − a − Vacuum energy − − The ultimate goal of a structure formation theory is to describe how the phase transi-tion progressed from almost perfectly homogeneous initial fields to all the structure weobserve. To get closer to achieving this goal - to model the formation of structure inan evolving background universe - we must next concentrate on small density perturba-tions. A realistic structure formation model must be able to describe the evolution of thedensity field in the Universe with time when the field contains small fluctuations. Theusual approach is to model the fluctuations as a perturbation to a smooth backgroundwhich we assume is homogeneous and isotropic.The key idea of any structure formation model is that if there are small perturbations,i.e., fluctuations in the energy density of the early Universe, then gravitational instabilitycan amplify them leading to virialized structures such as the galaxies, groups, and clus-ters we observe today. To model the formation of structure in a realistic, self-consistent,and physical way several ingredients are required (Coles & Lucchin, 2002): CHAPTER 2. FORMATION OF STRUCTURE
I. a background cosmology (Section 2.3),II. an initial fluctuation spectrum (Section 2.5.2),III. a choice of fluctuation mode and a statistical distribution of fluctuations,IV. a Transfer function (Section 2.5.3),V. a recipe for the nonlinear evolution (Sections 2.6 and 2.7), andVI. a prescription to relate mass fluctuations to observable light (Section 2.8).As can be seen from the comprehensive list above, detailed modelling of structure for-mation involves several different ingredients. All of which interact in a complicatedmanner. To complicate the matter even further, most of the above items involve one ormore assumptions (see e.g. Coles & Lucchin, 2002). It should, however, be kept in mindthat most of the assumptions are physically motivated, albeit this does not exclude thepossibility that they are inaccurate or even incorrect.Figure 2.5 shows a logical flow chart describing the formation of structure. Thechart shows how structures start to form from small initial density field fluctuations afterthe Big Bang and proceed through various steps to galaxy formation and to the mat-ter power spectrum observed, for example, in large galaxy surveys (Section 2.2.2). Atpresent, the physical mechanism that can best describe the initial density field is infla-tion (Guth, 1981; Mukhanov & Chibisov, 1981; Linde, 1982; Narlikar & Padmanabhan,1991). The initial conditions of the Universe are thought to arise from the scale-invariantquantum-mechanical zero-point fluctuations of the scalar field that drove the inflation inthe very early Universe (Guth & Pi, 1982; Hawking, 1982; Baumann, 2007). Inflationpredicts, for example, that the initial fluctuations are adiabatic (i.e. perturbations arein thermal equilibrium) and behave as a Gaussian random field with a nearly scale in-variant spectrum. In the following Sections I will therefore concentrate on adiabaticfluctuations with a Gaussian random phase and leave out isocurvature perturbations andnon-Gaussian random fields (this is the structure formation ingredient III).The first ingredient of the structure formation describes the global evolution of thebackground universe (Section 2.3), and is usually done using Friedmann equations (Sec-tion 2.3.2). The rest of the ingredients are related to the formation and evolution ofdensity perturbations under gravity in an expanding background universe. These will bedescribed in the following sections in more detail: the spectrum of the initial fluctuationsand the Transfer function are discussed in Section 2.5, while the nonlinear evolution us-ing both analytical methods and cosmological N − body simulations is explored in Sec-tions 2.6 and 2.7, respectively. Finally, to form realistic galaxies that can be compared tothe galaxies observed in large galaxy surveys at least hydrodynamics, baryonic physics,and star formation (e.g. McKee & Ostriker, 2007) must be considered. One realisationfor modelling gas, star formation, and feedback processes is the semi-analytical models .4. INGREDIENTS OF STRUCTURE FORMATION Big Bang
Quantum fluctuationsInflation
Initial Power Spectrum
Formation of dark matter haloesDamping Expansion of the UniversePhoton diffusion Gravitational instability Infall of gasGalaxy formation and evolution
Observed matter power spectrum
Transfer functionCosmological N-body simulationsComparison LinearregimeNon-linearregimeExponentialexpansion TheoryRadiationdomination Nucleosynthesis Initial Conditions GastrophysicsMatterdominationDark energydomination
Figure 2.5:
A logic flow chart for the formation of structure. The exponential expansion refers tothe expansion of the Universe during inflation. The radiation, matter and dark energy dominationrefers to the dominant energy content during a given time period, while the linear and nonlinearregimes refer to the evolution of the density perturbation field. The gastrophysics indicate theregime where hydrodynamics and also forces other than gravity play a significant role. Redhexagons show the initial and the observed power spectrum. Galaxy formation is described inmore details in Figure 2.8. of galaxy formation, which are discussed in Section 2.8.4. However, before exploringthe theory of structure formation, lets introduce the most successful structure formationmodel so far in more detail. CHAPTER 2. FORMATION OF STRUCTURE Λ Cold Dark Matter model
The Λ Cold Dark Matter ( Λ CDM) -model is nowadays accepted by the majority ofastronomers as a standard model of Big Bang cosmology and cosmological structureformation. The success of Λ CDM is widely recognised and is due to its simplicity,yet at the same time, it has the capability to simultaneously explain several profoundobservations of the Universe. Λ CDM can explain the structure and existence of thecosmic microwave background, the large-scale structure of galaxy groups and clusters,weak and strong gravitational lensing, and the accelerated expansion of the Universeinferred from type Ia supernovae observations (e.g. Narlikar & Padmanabhan, 2001).The statistical analysis of observations (Section 2.2) strongly support the flat Λ CDMcosmological model with the total energy density equal to the critical density. Table 2.2summarises the energy budget and the values of the basic Λ CDM parameters based onWMAP results (Hinshaw et al., 2009; Komatsu et al., 2009; Dunkley et al., 2009; Jarosiket al., 2011).The Λ term of the standard cosmology stands for the Einstein’s cosmological con-stant, often assumed to be the vacuum energy of space, and dubbed as dark energy due toits unknown origin (Narlikar & Padmanabhan, 2001; Frieman et al., 2008). The energybudget of the model is dominated by this unknown dark energy; the Λ -term constitutesalmost per cent of the total energy composition. The remaining ∼ per cent ofthe energy budget is matter, however, about per cent of the matter is assumed to bein form of non-baryonic dark matter. Because the dark matter particles are assumed tobe non-relativistic, the dark matter is said to be “cold”. The term cold dark matter wasintroduced by Peebles and Richard Bond in to cover the wide range of particlesthat were (and have been) suggested for the origin of this unknown gravitating material.Note that the coldness of the dark matter particles is actually required by the large-scalestructure: hot dark matter, i.e., relativistic particles, does not predict enough structure onsmall scales.The shape of the Λ CDM power spectrum is such that structures form from smallerto larger structures, i.e., “bottom-up”, with galaxies forming first followed by the for-mation of groups and clusters. More general, in all CDM-models, independent of the Λ -term, the initial density fluctuations have larger amplitudes on smaller scales, thus theCDM-models are hierarchical; larger structures form by clustering of smaller objectsvia gravitational instability (e.g. Davis et al., 1985; Frenk et al., 1988; White & Frenk,1991; Bullock et al., 2001). In CDM-models the collapse of matter happens when alocal perturbation starts to turn around while the Universe is expanding. The process ofcollapsing continues until the internal velocity of system’s components are large enoughto hold the system against more collapse. As a result, a dark matter halo is formed. Notethat unlike baryonic matter, the behaviour of dark matter does not depend on the scale ofthe system since dark matter only interacts gravitationally. Thus, in the Λ CDM-model .4. INGREDIENTS OF STRUCTURE FORMATION dark matter haloes with different sizes and masses are scaled versions of each other. Table 2.2:
Values of the basic Λ CDM parameters based on the WMAP 7-year results (Jarosiket al., 2011).
Quantity Symbol Value ± ErrorTotal Density Ω t . +0 . − . Dark Energy Density Ω Λ . ± . Matter Energy Density Ω m . ± . Dark Matter Density Ω dm h . ± . Baryonic Matter Density Ω b h . ± . Hubble Parameter h . +0 . − . Power Spectrum Normalisation σ . ± . Scalar Spectral Index n s . ± . Redshift of Matter-radiation Equality z eq +134 − Redshift of Decoupling z (cid:63) . +0 . − . Age of Decoupling t (cid:63) +5187 − yrSound Horizon at Decoupling r s ( z (cid:63) ) 146 . +1 . − . MpcRedshift of Reionization z re . ± . Reionization Optical Depth τ . ± . Age of the Universe t . ± . GyrEven though the Λ CDM-model is widely accepted, there are still many unknowns.For example, several particle candidates exist for dark matter (see e.g. Baltz, 2004;Muñoz, 2004; Bertone et al., 2005), however, none has been observed thus far. Oneof the leading candidates for the dark matter particle is the lightest stable supersymmet-ric particle called neutralino, which is weakly interacting and massive, but several othercandidates, for example Axions, exist. Hence, the type of the dark matter particles is yetto be confirmed. The nature of dark energy is even more mysterious, though the leadingcandidate is the vacuum energy of space (Peebles & Ratra, 1988; Frieman et al., 2008).As both dark matter and energy reveal themselves only via gravity all attempts to detectthem directly have been unsuccessful thus far. A significant amount of work remainstherefore to be done before the formation of structure and cosmology can be consideredas fully understood. CHAPTER 2. FORMATION OF STRUCTURE
The present consensus in cosmology, as seen in the previous Sections, is that the ob-served structures developed from small initial perturbations of the physical fields (den-sity, velocity, gravitational potential, etc.) resulting from the instability of the Friedmannmodels for small perturbations. Hence, to model structure formation we must model howthe density perturbation field evolves. A full treatment would require GR and wouldproceed by perturbing the background metric and the energy-momentum tensor (see e.g.Weinberg, 2008). However, a fully relativistic treatment is beyond the scope of this in-troduction to the formation of structure, thus I will describe a Newtonian method, whichgives an excellent approximation. Even so, it should be kept in mind that the followinganalysis holds only for perturbations on scales much smaller than the Hubble radius (i.e.on sub-horizon scale), because the Newtonian description assumes instantaneous gravity(i.e. the speed of gravity has been assumed to be infinite).
The hierarchy of cosmic structures is assumed to have grown from primordial densityseed fluctuations, which can be described with the density contrast δ ( x , t ) . The densitycontrast, as a function of the co-moving coordinates x , motivates the study of the densityperturbation field. It can be defined as δ ( x , t ) = ρ ( x , t ) − ¯ ρ ( t )¯ ρ ( t ) , (2.18)where ¯ ρ ( t ) is the mean density. A critical feature of the δ field is that it inhabits auniverse that is isotropic and homogeneous in its large-scale properties.In order to describe the structure formation in an expanding universe we must followthe evolution of the initial perturbation field as a function of time, while gravitationmagnifies the perturbations in both the baryonic and dark matter distribution. Afterrecombination the amplitude of the density fluctuations is ∼ − . Thus, the onset ofstructure formation happens well within the linear regime where δ (cid:28) . Consequently, alinear perturbation theory can be used as long as the density field does not turn nonlinear ( δ (cid:38) (for a comprehensive presentation, see e.g. Coles & Lucchin, 2002).On large scales matter can be described with a perfect fluid approximation. Thus,at any given time matter can be characterised by the energy density distribution (cid:15) ( x , t ) ,the entropy per unit mass S ( x , t ) , and the vector field of three-velocities v ( x , t ) . Thesequantities satisfy the hydrodynamical equations that allow the study of the behaviour ofsmall perturbations in a homogeneous, isotropic background. The equations of motionfor a non-relativistic fluid are the continuity, Euler, and Poisson equation. The continu-ity equation states that the change in the mass inside an element of the fluid equals to .5. EVOLUTION OF INITIAL PERTURBATIONS: STRUCTUREFORMATION the mass convected into the element, i.e., it defines the conservation of mass. On theother hand, the Euler equation states that the acceleration of a small fluid element isdue to the difference in pressure acting on opposite sides of the element, while the Pois-son’s equation describes the relation between the potential fluctuations and the densityperturbations causing them.In their basic form (see e.g. Coles & Lucchin, 2002; Longair, 2008) the continuity,Euler and Poisson equation hold for a smooth background. However, in this Sectionwe derive the evolution of the density perturbations and hence we must consider smallperturbations to the background. All the quantities involved must be then written asa sum of the smooth background and the perturbed quantity, e.g., in case of pressure: P = ¯ P + δP . Here, ¯ P corresponds to the smooth background pressure field, while δP is a small perturbation to this smooth component. The perturbed quantities can thenbe substituted to the basic continuity, Euler and Poisson equation. After ignoring termshigher than the first order in the perturbation and subtracting the zero order equationsone obtains the linear perturbation forms of these equations. Thus, it is possible to de-scribe the evolution of the density, velocity, potential and pressure fields in an expandinguniverse with Newtonian gravity by using the perturbed continuity, Euler, and Poissonequations, together with the conservation of entropy (for a full derivation, see e.g. Pee-bles, 1980; Longair, 2008; Mo et al., 2010).The linear perturbation theory, briefly described above, holds that, during the matter-dominated era, the density field δ of sub-horizon perturbations can be described with thegrowth equation as follows ∂ δ ( x , t ) ∂t + 2 H ( t ) ∂δ ( x , t ) ∂t = 4 πG ¯ ρδ ( x , t ) + c s a ( t ) ∇ δ ( x , t ) . (2.19)Here H ( t ) is the Hubble parameter (defined in Eq. 2.13), c s is the speed of sound, ∇ denotes the differentials with respect to co-moving coordinates, and ∂∂t is a partial timederivative. The above second-order growth equation has been written in a general butsingle-fluid form as a function of cosmic time t and co-moving coordinate x . This givesa linear approximation for the growth of density perturbations in an expanding universe.Note that the second term on the left-hand side is the so-called Hubble drag term, whichtends to suppress perturbation growth due to the expansion of the Universe. On theother hand, the first term on the right-hand side is the gravitational term, which causesperturbations to grow via gravitational instability, while the last term on the right-handside is a pressure term and is due to the spatial variations in density.The solution to the growth equation (2.19) apply to the evolution of a single Fouriermode of the density field. However, in the linear regime, the equations governing theevolution of the perturbations are all linear in perturbation quantities. It is then useful Setting c s = 0 gives the often shown form of the growth equation. CHAPTER 2. FORMATION OF STRUCTURE to expand the perturbation fields in chosen mode functions. If the curvature K of theUniverse can be neglected, the mode function can be chosen to be plane waves. Now theperturbation fields can be represented by their Fourier transforms. We therefore seek awave solution for δ of the form δ ( x , t ) = (cid:88) k δ k ( t ) e i k · x . (2.20)We can now write a wave equation for δ after taking a Fourier transform of equation2.19. Because each k mode is assumed to evolve independently, we can write ¨ δ k + 2 H ˙ δ k + (cid:18) k c s a ( t ) − πG ¯ ρ (cid:19) δ k = 0 . (2.21)The equation 2.21, sometimes called the Jeans equation, describes the evolution of eachof the individual modes δ k ( t ) , corresponding to δ ( x , t ) . Note that because x (in Eq.2.19) is in co-moving units, the wave-vectors k are also and that the derivatives (denotedby dots) are time derivatives, because δ k does not explicitly depend on a spatial position.If the dark matter is cold and collisionless we can neglect the pressure term in Eq.2.21. This allows us to write a general solution in a form of two linearly independentpower laws, i.e., δ k ( t ) = A D + ( k , t ) + A D − ( k , t ) , (2.22)where A and A are constants to be determined by initial conditions. The growth (orJeans) equation therefore has two solutions: a growing (+) and decaying ( − ) mode. Thelatter is hardly interesting for structure formation, thus, hereafter we concentrate on thegrowing mode. The growing mode is described by the growth factor D + defined suchthat the density contrast at the time t is related to the density contrast today δ ( t ) by δ ( t ) = δ ( t ) D + ( t ) D + ( t ) . (2.23)It is useful to note that the solution to equation 2.21 can either grow or decreasedepending on the sign of the k J = (cid:18) k c s a ( t ) − πG ¯ ρ (cid:19) . (2.24)The density perturbations can grow only if the second term in the above equation dom-inates, while the transition takes place at the wave-number for which the two terms areequal, at k J = a ( t ) c s (cid:112) πG ¯ ρ . (2.25) .5. EVOLUTION OF INITIAL PERTURBATIONS: STRUCTUREFORMATION We can now write 2.25 in terms of the physical wavelength using the simple relation k = 2 πaλ − , and by doing so obtain the Jeans length: λ J = c s (cid:114) πG ¯ ρ , (2.26)which defines a scale length on which structures can grow. In general, on scales smallerthan the Jeans length, i.e., λ < λ J ( or k > k J ) , the solution to the Jeans equationcorresponds to a sinusoidal sound wave; so pressure can counter gravity. Due to thedamping caused by the Hubble drag term, there is no growth of structure for sub-Jeansscales, but the solution is oscillatory. Instead, on scales longer than the Jeans length, butsmaller than the horizon, the pressure can no longer support the gravity and the solutioncan grow. As the Jeans length is time dependent in an expanding universe, for example,before the recombination λ J ∼ Mpc, while after the Jeans length is only ∼ kpc, a given mode λ may switch between periods of growth and stasis governed by theevolution of λ J . But what does all this mean for the growth of perturbations?The growth rate of a density perturbation depends on epoch or more precisely onwhat component dominates the global expansion, whether a perturbation k -mode issuper- or sub-horizon, and the Jeans length. As already noted in the case of the evolu-tion of the energy density (Section 2.3.3), the early Universe was assumed to be radiationdominated until the time of matter-radiation equality z eq . During the epoch of radiationdomination the Universe can be taken as flat and for k → the growth equation canbe solved by δ ∝ t n (for a detailed derivation, see e.g. Coles & Lucchin, 2002). Thus,for all perturbations, the growing mode (here we ignore the decaying mode) outside thehorizon (on super-horizon scales) grows as δ ∝ t ∝ a . Instead, on the sub-horizon scales, the cold and collisionless dark matter, which has no pressure ( c s = 0) of its ownand is not coupled to photons, grows at most logarithmically δ dm ∝ ln ( a ) . After thematter-radiation equality, matter begins to dominate the dynamics. On the super-horizonscale all perturbations (dark matter, baryons, and photons) grow as δ ∝ a . Dark matter,being pressureless, grows with the same rate ( δ dm ∝ a ∝ t ∝ (1 + z ) − ) (e.g. Coles& Lucchin, 2002) also on sub-horizon scales. However, baryons are still coupled to theradiation until the time of decoupling z (cid:63) . As a result, the sub-horizon perturbations inthe baryons cannot grow but instead oscillate. Finally, at the time of decoupling z (cid:63) , therate of collisional ionization does not dominate any longer and the baryons can decouplefrom the photons. At this point the baryonic perturbations can start to grow as ∝ a ∝ t on scales λ > λ J . On the smaller scales they instead continue to oscillate. Finally, atthe latest times when the Λ -term is assumed to be dominant, the growing mode solutionis ∝ constant . Note that when the universe was radiation dominated c s = c ( √ − and thus the Jeans length isalways close to the size of the horizon. CHAPTER 2. FORMATION OF STRUCTURE
Note, however, that in general the presentation in this Section applies only to adi-abatic perturbations in a non-relativistic fluid with a single component. Fortunatelythough, the Newtonian perturbation theory is valid even with the presence of relativisticenergy components, such as radiation and dark energy, as long as they can be consideredsmooth and their perturbations can be ignored. In this case they contribute only to thebackground solution.
In the previous Section an equation (2.19) describing the evolution of the density pertur-bation field (Eq. 2.18) was presented. However, as was noted, it is often convenient toconsider the density perturbation field by the superposition of many modes. The naturaltool for achieving this is via Fourier analysis in case the co-moving geometry is flat orcan be approximated as such (Eq. 2.21). In such case the power spectrum can be definedas P ( k ) = (cid:104)| δ k | (cid:105) , (2.27)where the angle brackets indicate an average. As a result, the power spectrum describeshow much the density field varies on different scales. Note that in an isotropic uni-verse, the density perturbation spectrum cannot contain any preferred direction. Thus,we must have an isotropic power spectrum and we can write it simply as a function ofwave-number k rather than a vector. Even with such simplification, the power spectrumprovides a complete statistical characterisation of a particular kind of stochastic process:a Gaussian random field (Bardeen et al., 1986).Thus, the power spectrum can characterise the statistical properties of the cosmologi-cal perturbations. This is highly useful in order to be able to relate theory to observations.For example, results of large galaxy surveys (e.g., correlation functions, Section 2.2.2)suggest that the spectrum of the initial fluctuations must have been very broad with nopreferred scales. Because power spectrum P ( k ) is related to a two-point correlationfunction by a Fourier transform (Eqs. 2.3 and 2.4), it is then natural to assume that thepower spectrum of the initial fluctuations generated in the early phases of the Big Bangis of a power-law form. Thus, P ( k ) = Ak n , (2.28)where A is the amplitude, k is a wave-number in physical units h − Mpc and n is a freeparameter. Hence, the power spectrum also describes the normalisation of the spectrumof density perturbations on large physical scales.Because there is yet no theory for the origin of the cosmological perturbations, theamplitude A of the power spectrum has to be fixed by observations. The amplitude A .5. EVOLUTION OF INITIAL PERTURBATIONS: STRUCTUREFORMATION is therefore set using either the so-called COBE normalisation (e.g. Bunn & White,1997) or by the variance of the density fluctuations within spheres of h − Mpc radius, σ (for the reference model value, see Table 2.2). The σ parameter can be measured, forexample, using the cosmic-shear autocorrelation function, the abundance and evolutionof the galaxy-cluster population, the statistics of Lyman- α forest lines (Seljak et al.,2006), or by counting the number of hot X-ray emitting clusters in the local universe(see e.g. Bartelmann, 2010, and references therein).According to equation 2.6 the power-law form of the power spectrum P ( k ) ∝ k n corresponds to a two-point correlation function of form ξ ( r ) ∝ r − n − . (2.29)The mass M within a fluctuation is ∝ ρr , thus, the spectrum of a mass fluctuation is ξ ( M ) ∝ M − n − . (2.30)Finally, the root-mean-square (rms) density fluctuation at mass scale M can be writtenas δ rms ∝ M − n − . (2.31)As can be seen from the equations above, the spectral index n has a significant role inthe structure formation. If n (cid:54) = 1 , the power spectrum is called tilted: a tilted spectrumis called “red” if n < and “blue” if n > (for a review, see e.g. Abbott & Wise, 1984;Lucchin & Matarrese, 1985). A red spectrum shows that there is more structure at largescales, while a blue spectrum describes that there is more structure at small scales. Thespecial case is found when the spectral index n equals unity. The Harrison-Zel’dovich power spectrum
Simple inflationary theories predict that right after inflation the matter power spec-trum would have a simple power-law form. Consequently, the primordial, or Harrison-Zel’dovich (Harrison, 1970; Zeldovich, 1972), power spectrum can be written as P ( k ) = Ak n with the spectral index n equals unity. The simple power-law form, P ( k ) = Ak ,now results the spectrum of density perturbations to have a following form δ rms ∝ M − , (2.32) In the COBE normalisation the amplitude of the large scale temperature anisotropies in the CMB areused to constrain the amplitude. CHAPTER 2. FORMATION OF STRUCTURE while the two-point correlation function takes the form: ξ ( r ) ∝ r − . (2.33)The importance of the Harrison-Zel’dovich spectrum is the property that it is scale-invariant: the density contrast ∆( M ) had the same amplitude ( ∼ − ) on all scaleswhen the perturbations came through their particle horizons during the radiation dom-inated era. Interestingly, the scale-invariant spectrum corresponds to a metric that isa fractal, leading to a fractal nature of the Universe (e.g. Jones et al., 1988; Balian &Schaeffer, 1989).The current large-scale observations (see Section 2.2.2 and the citations therein) arereasonably well fit by an n = 1 scale-invariant primordial spectrum of perturbations.Theoretically the spectral index n would be precisely unity if inflation lasts forever. Asthis is obviously not the case, the spectral index must however deviate slightly from theunity. It can be shown (for a detailed discussion, see Liddle & Lyth, 2000) that n mustbe slightly smaller than unity, in agreement with the Λ CDM -model value ( n s in Table2.2). It can also be shown that the spectral index of the temperature fluctuations (Fig.2.1) as a function of angular scale depends only upon the spectral index n of the initialpower spectrum. Thus, for the Harrison-Zel’dovich power spectrum, the amplitude isindependent of the angular scale. If we wish to model how the form of the power spectrum evolves as a function of time,the statistical description of the initial fluctuations described by the initial power spec-trum P ( k ) must be evolved. During the evolution the form can be modified by severalphysical phenomena. For example, radiation and relativistic particles can cause kine-matic suppression of growth of the initial perturbations. Moreover, the imperfect cou-pling of photons and baryons may also cause dissipation of perturbations. On the otherhand, gravity will amplify the perturbations and eventually leads to collapsed and boundstructures. Thus, real power spectra result from modification of any primordial powerby a variety of processes: growth under self-gravity, the effects of pressure, and dissi-pative processes. In general, however, modes of short wavelengths have their amplitudereduced relative to those of long wavelengths.A possible way to quantify how the shape of the initial power spectrum is modifiedby different physical processes as a function of time is to use a simple function of awave-number, namely the Transfer function T ( k ) . For statistically homogeneous initialGaussian fluctuations, the shape of the original power spectrum is changed by physicalprocesses and the processed power spectrum P ( k ) is related to its primordial form P ( k ) via the Transfer function as follows P ( k, t ) = (cid:104)| δ ( k, t ) | (cid:105) = P ( k ) T ( k ) D ( t ) . (2.34) .5. EVOLUTION OF INITIAL PERTURBATIONS: STRUCTUREFORMATION Here D + ( t ) is the solution of the linearised density perturbations equation (2.21), i.e.,the growth factor. Hence, once the Transfer function is known, one can calculate thepost-recombination power spectrum from the initial conditions.The form of the Transfer function is a function of the amount and type of the darkmatter particles. As Section 2.4.1 described, the currently favoured dark matter particlesare non-relativistic. Damping processes can also effect T ( k ) during the linear evolution.As noted already, the cold dark matter does not suffer from strong dissipation, but onscales less than the horizon size at matter-radiation equality there is a kinematic sup-pression of growth on small scales. Additional complication to the form of the Transferfunction arises from having a mixture of matter (both collisionless dark matter and bary-onic plasma) and relativistic particles (collisional photons and collisionless neutrinos).One more complication for the shape of the Transfer function arises from the fact thatsub-horizon perturbations grow differently during the radiation and matter dominatederas (Section 2.5.1). Due to the complicated form of the Transfer function it there-fore must, in general, be calculated using an approximation formula, e.g., by Bond &Efstathiou (1984); Bardeen et al. (1986) or more precisely numerically using publiclyavailable programs such as CMBfast (Seljak & Zaldarriaga, 1996). Section 2.2 showed that the current matter power spectrum is far from its initial form,even though at scales λ > h − Mpc a Harrison-Zel’dovich power law is a goodapproximation. Thus, if one assumes that the initial power spectrum has a Harrison-Zel’dovich form after the inflation, it must have evolved significantly. As describedabove, the Transfer function can describe how the shape of the initial power spectrumevolves through the epochs of horizon crossing and radiation-matter equality. For thelargest scales ( (cid:29) h − Mpc ) the perturbations are still small even today, and one canuse the Transfer function. However, for smaller scales such as galaxies, groups andclusters, the inhomogeneities have become so large at later times ( z < that thephysics of structure growth has become nonlinear.The caveats in mind, we can still provide an approximation for the form of thepresent-day power spectrum. If the mass fluctuations inside the co-moving horizon ra-dius at matter-radiation equality are independent of time, and assuming that the darkmatter is cold, the present day power spectrum can be approximated with a functionalform P ( k ) ∝ (cid:26) k n when k (cid:28) k eq k n − ln k when k (cid:29) k eq , where the co-moving wave-number k eq ∼ . h − Mpc. By and large this form is in CHAPTER 2. FORMATION OF STRUCTURE agreement with Figure 2.4, however, the real power spectrum has a smooth maximum,which is caused by the different rates of growth before and after the matter-radiationequality. Note that the co-moving wave-number describes the location of the peak andis set by the co-moving horizon radius at matter-radiation equality. The steep declinefor structures smaller than the horizon radius reflects the suppression of structure growthduring radiation domination.The evolution of the matter power spectrum from its initial to the current form (seeFig 2.4) can be summarised as follows. Before the matter-radiation equality, the mat-ter distribution followed mostly that of the radiation. Because radiation has significantpressure perturbations were forced to oscillate on sub-horizon scales. Instead, the largestperturbations were too large for radiation pressure to be able to hold back the collapse,thus, perturbations on scales larger than the horizon scale were able to grow (with therate given in Section 2.5). This caused the matter power spectrum to increase in heightand started to induce a bump at small scales (larger k ). As the times went on, the hori-zon scale increased and larger scales were able to oscillate, causing the bump to shiftto larger scales (smaller k ). Instead, after matter-radiation equality, z eq , the dark matterwas able to grow with a rate given in Section 2.5. Because the dark matter is assumedto be cold and collisionless it does not have significant pressure, and hence there werepractically no more acoustic oscillations. During the matter dominated epoch the lackof pressure in dark matter allows it to continue to collapse, causing the whole of thematter power spectrum to increase. As a result, the turn-over point of the matter powerspectrum is frozen into the power spectrum, which corresponds to the co-moving hori-zon radius at matter-radiation equality. Finally, closer to the current time, the small scaleperturbations (high k ) have turned nonlinear. Consequently, their growth is fast causingthe matter power spectrum to rise on smaller scales. Quantitatively, such an evolutionhas been noted to lead to the current form of the matter power spectrum.Unfortunately, the power spectrum can only describe the statistical properties of thedensity contrast, not the evolution of the individual density fluctuations. As a result, theevolution of individual perturbations as well as the nonlinear evolution on small scalesmust be studied using some other techniques. In such a case one must resort to, forexample, analytical techniques or cosmological N -body simulations. As the density contrast δ (Eq. 2.18) approaches unity, the evolution of the density fluc-tuations becomes nonlinear. In the course of nonlinear evolution, overdensities contract,causing matter to flow from larger to smaller scales causing the power spectrum to de-form. Even though the linear perturbation theory (Section 2.5.1) fails for δ (cid:38) , theonset of nonlinear evolution can still be described analytically. .6. NONLINEAR EVOLUTION: I. ANALYTICAL METHODS The simplest analytical model for the nonlinear evolution of a discrete perturbation iscalled the spherical top-hat model (for a textbook review, see e.g. Padmanabhan, 1993).In this approximation the perturbation evolves according to Birkhoff’s theorem; in aspherically symmetric situation, matter external to the sphere will not influence its evo-lution. An evolving density perturbation will therefore eventually stop expanding, turnsaround, and collapses. Because in this model the perturbation has no internal pressure, itcollapses to infinite density. Interpreting this literally leads to a conclusion that all spher-ical perturbations would result to black holes. In reality, this however has obviously nothappened, and thus, one should be aware of its limitations.In the spherical top-hat model a perturbation reaches a maximum size r max at thetime of a turnaround t turn . According to the model, the perturbation will then collapseat t turn . In a realistic case during the collapse the gravitational potential energy mustbe converted into kinetic energy of the particles involved in the collapse because a col-lisionless system cannot dissipate energy. This can be achieved, for example, via theprocess of violent relaxation (Lynden-Bell, 1967). The collapsed object will thereforeeventually relax to a structure supported by random motions and satisfy the virial theo-rem, in which the internal kinetic energy of the system is equal to half of its gravitationalpotential energy (see Section 3.5.2 and equation 3.14).If one assumes that the relaxed object virialises at t vir = 2 t turn , then a mean over-density within a virial radius r vir = r max / can be derived using the virial theorem.The mean overdensity ∆ vir within the virial radius at the time of virialisation can nowbe written as ∆ vir = ρ ( t turn )¯ ρ ( t turn ) ¯ ρ ( t turn )¯ ρ ( t vir ) (cid:18) r max r vir (cid:19) − , (2.35)where ¯ ρ ( t ) is the background density at time t . In the case of a non-zero cosmologicalconstant and for a flat universe, an approximation for the mean overdensity can be written ∆ vir ≈ π + 82 x − x Ω m ( t vir ) , (2.36)where x = Ω m ( t vir ) − (Bryan & Norman, 1998). This simple approximation canbe used to derive an average density of a virialized object formed through gravitationalcollapse in an expanding universe. In the simplest approximation (Ω m = 1) , this leadsto ∆ vir ∼ π ∼ . Note also that the virial theorem and the spherical collapsemodel can also be used to estimate the redshift at which the object became virialized.One of the shortcomings of the spherical top-hat model is however the assumptionthat the perturbations were exactly spherically symmetric. Hence, a more general ap-proximation should be considered. CHAPTER 2. FORMATION OF STRUCTURE
Given the fact that fluctuations of early times were small, it is reasonable to assume thatat later epochs only the growing mode has a significant amplitude. Now, if one assumesthat the density field grows self-similarly with time, the onset of nonlinear evolutioncan be described by the so-called Zel’dovich approximation (Zel’Dovich, 1970). TheZel’dovich approximation is a form of the linear perturbation theory and is applicableto a pressureless fluid. The basic assumptions of the Zel’dovich approximation are asfollows: the scales of interest are much smaller than the size of horizon; the uni-verse is dominated by the matter component; and the curvature of the universe is zero.The Zel’dovich approximation does therefore not assume spherical symmetry, like thetop-hat model.The Zel’dovich approximation is a Lagrangian description for the growth of pertur-bations and it specifies the growth of structure by giving the displacement and the pe-culiar velocity of each mass element in terms of the initial position (Zel’Dovich, 1970;Shandarin & Zeldovich, 1989). Furthermore, the Zel’dovich approximation is a kine-matic approximation by nature: particle trajectories are straight lines. The first nonlinearstructures to form in this approximation will be two-dimensional sheets, also called theZel’dovich pancakes. However, the approximation is not valid after the formation of thepancakes when shell crossing will start to occur. Thus, other techniques to follow theevolution of density perturbations further into the nonlinear regime are required. The Press-Schechter (PS; Press & Schechter, 1974) theory, which was derived heuris-tically using the linear growth theory and the spherical top-hat model (Section 2.6.1)provides an analytical description for the evolution of gravitational structure in a hierar-chical universe. In the PS formalism, an early universe is assumed to be well-describedby an isotropic random Gaussian field of small density perturbations. Moreover, thephases of fluctuations are assumed to be random so that the field is entirely defined byits power spectrum (Bower, 1991). The basic idea of the PS theory is to imagine smooth-ing the cosmological density field at any epoch z on a given scale R so that the massscale of virialized objects of interest satisfies M = π ¯ ρ ( z ) R . However, as noted ear-lier, the growth of the density perturbations can only be followed with simple analyticaltechniques until they become nonlinear. The PS formalism circumvents this difficulty byassuming that the region collapses rapidly and independently of its surroundings onceit has turned nonlinear. As a result, the collapsed region can be described as a singlelarge body to the rest of the universe. This simplification allows the linear equations tobe applied, however, one must still take into account the nonlinear single body objectswhen modelling the formation of large-scale structure. .7. NONLINEAR EVOLUTION: II. COSMOLOGICAL SIMULATIONS The PS formalism allows the modelling of the growth of cosmic structure in a highlysimplified universe (Cole, 1991). Perhaps more importantly, it also allows to estimatethe mass function of the collapsed objects. The PS formalism leads to a halo massfunction, which has the form of a power law multiplied by an exponential. While thePS formalism gives a reasonable approximation to the numerical data, it has been shownto underestimate the number of massive systems, while over-predicting the number of“typical” mass objects (e.g. Governato et al., 1999; Sheth & Tormen, 1999; Jenkinset al., 2001). Thus, a more realistic and detailed modelling of the nonlinear growth ofstructure in an expanding universe is required for more vigorous comparisons. This canbe achieved, for example, by using numerical simulations.
Cosmological N -body simulations provide a robust method to study large-scale struc-ture and the formation and growth of cosmic structure of a universe on the nonlinearregime. This is possible because the equations of motions are integrated numerically(e.g. Springel et al., 2001, and the references therein). The basic idea of cosmologicalsimulations was founded in the 1960s and it owes its existence to early few body simu-lations. The first cosmological simulations were run with a modest number of particlesand only the formation of a few galaxies were followed. Instead, today large cosmo-logical simulations use billions of particles, however, many simulations still use puredark matter and no baryons. To overcome the issue that in dark matter only simulationsgalaxies must be placed by hand using, e.g., semi-analytical methods (Section 2.8.4) hy-drodynamics can also be modelled. Hydrodynamical simulations (Katz & Gunn, 1991;Navarro & Benz, 1991; Katz, 1992; Cen, 1992; Rosswog, 2009) that contain gas parti-cles are gaining popularity with increasing computing power, but their volumes are stillmodest compared to dark matter only simulations (Frenk et al., 1999; Teyssier, 2002;Sommer-Larsen et al., 2003; Springel & Hernquist, 2003; Davé et al., 2010; Razoumov& Sommer-Larsen, 2010; Tantalo et al., 2010). Moreover, even in hydrodynamical sim-ulations some key (sub-grid) physics, such as star formation, is not modelled directlyfrom the first principles but using similar prescriptions as in semi-analytical models.Observations of the cosmic microwave background, discussed in Section 2.2.1, showthat the perturbations of the gravitational potential are caused by non-relativistic material(e.g. Hu & Dodelson, 2002). Therefore, cosmological N -body simulations can usuallyoperate on Newtonian limit without the framework of General Relativity. Even so, theexpansion of the Universe must be taken into account. This is often done using a co-moving coordinate system, which moves as a function of time as the Universe ages andexpands. Consequently, cosmological N -body simulation codes essentially follow the CHAPTER 2. FORMATION OF STRUCTURE evolution of the density field δ in an expanding background by following the motions ofparticles caused via gravity by integrating the equations of motions numerically.The basic assumptions of a modern cosmological N -body simulation code can besummarised as follows: • mass content of the Universe is build up mainly from dark matter; • gravity is the only notable force on large scales; • each dark matter particle in the simulation volume represents several particles ofthe real Universe and they are collisionless; • a simulation starts from initial conditions with all modes well within the linearregime; • periodic boundary conditions are adopted, resulting in that no particle can be lostduring the simulation.These notions provide the basic assumptions that most simulations obey. It is also note-worthy that the assumption that the dark matter particles are collisionless also meansthat the evolution of the Universe is driven by the mean gravitational potential ratherthan two-body interactions. In what follows, I briefly describe a general idea of generat-ing initial conditions and how to follow the motions of particles. For most galaxy formation and large-scale structure problems, setting up the initial con-ditions of a cosmological N -body simulation can be split into three parts:I. generating a power spectrum;II. generating a Gaussian random density field using the power spectrum;III. imposing density perturbation field on the particle distribution.The first step, generation of a power spectrum, defines the dark matter model. The powerspectrum can be generated, for example, by taking a primordial power spectrum (Sec-tion 2.5.2) and then multiplying it with the Transfer function (Section 2.5.3) of a chosencosmology. The second step sets up a “smooth” distribution of particles (for a technicaldescription, see e.g. Martel, 2005) by generating a single realisation of the density fieldin k -space. The third step is to impose density perturbations with the desired characteris-tics, i.e., the assignment of displacements and velocities to particles. A suitable particledistribution, i.e., a linear fluctuation distribution can be generated using, for example, theZel’dovich approximation (Section 2.6.2, but see also Efstathiou et al., 1985). Note that .7. NONLINEAR EVOLUTION: II. COSMOLOGICAL SIMULATIONS when following this technique the matter density and velocity fluctuations are initialisedat the starting redshift chosen usually such that all modes in the simulation volume arestill within the linear regime.After the initial conditions have been set and the starting redshift has been chosen,the simulation can be evolved towards the current epoch by using a cosmological N -body code. In general, the code allows the time evolution of the simulated particles tobe followed by integrating the equations of motions. Several different techniques to follow the gravitational evolution of the density field incosmological N -body simulations exist. In this Section the basics are briefly introduced,while more detailed descriptions of different techniques can be found from the literature(see e.g. Efstathiou et al., 1985; Barnes & Hut, 1986; Couchman, 1991; Cen, 1992; Xu,1995; Kravtsov et al., 1997; Teyssier, 2002; Aarseth, 2003; Springel & Hernquist, 2003;Bagla & Padmanabhan, 1997, and references therein).In usual cases, dark matter (and stars if applicable) can be modelled as a self-gravitating collisionless fluid in cosmological simulations. Since the number of darkmatter particles is large, two-body scattering events are assumed to be seldom. As aresult, it is convenient to describe the system in terms of the single particle distributionfunction f = f ( x , ˙x , t ) in phase space. Now, if we make a reasonable assumption thatthere are no collisions between particles, the evolution of the distribution function f ofthe fluid follows, in the co-moving coordinates x , the collisionless Boltzmann equation: ∂f∂t + ˙ x ∂f∂ x − ∂ Φ ∂ r ∂f∂ ˙x = 0 , (2.37)where the self-consistent potential Φ is the solution of Poisson’s equation ∇ Φ( r , t ) = 4 πG (cid:90) f ( r , ˙x , t )d ˙x . (2.38)Here f ( r , ˙x , t ) is the mass density of the single-particle phase space. Unfortunatelythough, the coupled equation pair consisting of the collisionless Boltzmann and Poissonequation is difficult to solve directly. Thus, simulations often follow the so-called N -body approach, where the smooth phase fluid f is represented by N particles which areintegrated along the characteristic curves of the collisionless Boltzmann equation. Con-sequently, the problem is conveniently reduced to a task of following Newton’s equationsof motion for a large number of particles under their own self-gravity (Springel et al.,2001). CHAPTER 2. FORMATION OF STRUCTURE
The dynamics of N particles can be described by the Hamiltonian: H = (cid:88) i p i m i a ( t ) + 12 (cid:88) i,j m i m j φ ( x i − x j ) a ( t ) , (2.39)where x i and x j are the co-moving coordinate vectors, the corresponding canonicalmomenta are given by p i = a m i ˙x i , and the φ ( x ) is the interaction potential. Note thatthe time dependency of the Hamiltonian H is caused by the time dependency in thescale parameter a = a ( t ) . Before the equations of motions of simulated particles can bederived the interaction potential has to be solved. When periodic boundary conditionsare assumed the interaction potential can be solved from equation: ∇ φ ( x ) = 4 πG (cid:34) − L + (cid:88) n ˆ δ ( x − L n ) (cid:35) , (2.40)where L is the side length of the simulation volume, n = ( n , n , n ) , and ˆ δ is theparticle density distribution function. Finally, after the interaction potential has beensolved the Hamilton’s equations of motions ∂ H ∂ x = − ˙p (2.41) ∂ H ∂ p = ˙x (2.42)can be derived. To follow the time evolution of the simulated particles the derived equa-tions of motions must be integrated, after making a small variation δt to time, via e.g.the “leapfrog integration scheme” (e.g. Dolag et al., 2008). Note that in most cases, theparticle motion integrals are time-integrals and require integrating the scale parameter a ( t ) . Far in the nonlinear regime, towards the end of a simulation run, bound structures startto form (for an example, see Fig. 2.6). After their formation they grow in mass eitherby accretion or by merging with other bound structures. These bound structures can beidentified from simulations by using the so-called halo finders that search for collectionsof dark matter particles that are gravitationally bound. The bound structures of particlesare called dark matter haloes due to their relatively spherical nature.The most popular algorithm to identify virialized haloes is likely the so-called Friends-of-Friends halo finder (Davis et al., 1985). This simple algorithm links all particles withdistances less than a linking length to a single halo. In general, the linking length is .7. NONLINEAR EVOLUTION: II. COSMOLOGICAL SIMULATIONS set to correspond to the mean virialisation overdensity ∆ vir (Eq. 2.36) derived usingthe spherical top-hat model (Section 2.6.1). The linear linking length of the Friends-of-Friends halo finder is a free parameter, often taken to be a fraction, e.g. . , of the meanparticle separation. For other halo finding algorithms, see for example Eisenstein & Hut(1998); Neyrinck et al. (2005); Kim & Park (2006); Knollmann & Knebe (2009) andreferences therein.Figure 2.7 shows an extraction from the Millennium-II simulation (Boylan-Kolchinet al., 2009). The Figure shows a zoom sequence from to . h − Mpc into the mostmassive halo in the simulation at redshift zero visualising the dynamical range of thesimulation. Figure 2.6 shows the structure formation as a function of time. This set of images shows the growth of the most massive halo over the cosmic time. The leftcolumn is h − × h − Mpc, the centre column is h − × h − Mpc, and theright is h − × h − Mpc in co-moving units. From top to bottom, the regions plottedare at redshifts , , , and . The simplified scheme of collisionless particles used in cosmological N -body simula-tions leads to finite mass- and force-resolution. In an ideal case the number of simulationparticles should be as large as possible to enable a detailed study of formation and growthof cosmic structure in all scales ranging from the smallest dwarf galaxies to the largestclusters and filaments. However, in reality this choice is in general limited by the avail-able computing resources. Moreover, the number of particles must also be balanced withthe choice of a simulation volume size to compete against the cosmic variance.The mass resolution R m of a simulation can be derived when the simulation volumeand the number of dark matter particles have been chosen. The mass resolution, in unitsof solar mass M (cid:12) , describes the mass of a single dark matter particle and can be derivedfrom R m = C L N . (2.43)Here L is the side length of the simulation volume in Mpc, N is the number of particlesin the volume L , and C ∼ . × is a constant. Note that the mass resolution setsa hard limit: no object below the mass resolution can form in a simulation. In reality,however, the smallest structures to form and which are identifiable must be made out ofseveral tens of particles. This renders the effective mass resolution at least an order ofmagnitude worse than indicated by equation 2.43. Furthermore, a finite mass resolutionalso limits the ability to study the internal structures of dark matter haloes with massesclose to the resolution limit.On the other hand, a finite force resolution arises from the fact that the gravitationalforce between two particles diverges as their distance approaches to zero. In reality, CHAPTER 2. FORMATION OF STRUCTURE
Figure 2.6:
Example of structure formation as a function of time in the Millennium-II simulation.Courtesy of Michael Boylan-Kolchin and the Millennium-II. however, the gravitational force between two extended objects is finite. The force reso-lution, which is more subtle effect than mass resolution, is more complicated to quantifybecause it depends on the simulation code used. In a mesh-based simulation code, the .7. NONLINEAR EVOLUTION: II. COSMOLOGICAL SIMULATIONS Figure 2.7:
A zoom sequence from to . h − Mpc into the most massive halo in theMillennium-II simulation volume at redshift zero. Courtesy of Michael Boylan-Kolchin and theMillennium-II. force is automatically softened on the scale of a chosen mesh. Instead, in particle-particlealgorithms, the force softening is often applied artificially by modifying Newton’s grav-ity law by writing it as: F = G M r + (cid:15) , (2.44) CHAPTER 2. FORMATION OF STRUCTURE where M is the mass of a single particle, r is the distance between the two particles, and (cid:15) is the gravitational softening length. Non-zero softening length now guarantees thatthe force does not diverge even when r → . However, by doing so the (cid:15) simultaneouslysets a limit for the highest density contrast that can be resolved. So far I have briefly shown how the formation and growth of large scale dark matterstructure can be modelled from the CMB down to the current epoch. However, observa-tions such as large galaxy surveys (Section 2.2.2) cannot yet directly probe dark matter,thus one should also try to model the luminous component or baryonic matter, i.e. thegalaxy formation and evolution. A detailed discussion of the theory of galaxy formation,is however beyond the scope of this introduction. Instead, in what follows I try brieflyto summarise the main concepts of galaxy formation, illustrated in Figure 2.8, to pro-vide background for the following chapters. For more detailed presentations, I refer theinterested reader to the great text books of Longair (2008) and Mo et al. (2010).The previous sections showed how to model dark matter haloes. If we now assumethat galaxies form and reside in dark matter haloes, it becomes obvious that the propertiesof the galaxy population are related to the cosmological density field and to the darkmatter halo population. One can therefore try to link the properties of dark matter haloesto the properties of observed galaxies by using statistical arguments. As a result, itcan be shown that the properties of the galaxy population depend on the properties ofthe dark matter halo and subhalo populations (see e.g. Mo et al., 2010, for detaileddiscussion). This allows, for example, the galaxy luminosity function to be compared tothe dark matter halo mass function. The correspondence of light and mass is important,for example, when the mass power spectrum is being derived from observations (Section2.2.2), because a correlation between the observed light and the underlying mass mustbe assumed.
To overcome the difficulty of linking dark matter haloes to luminous galaxies, Vale &Ostriker (2004) (see also Oguri, 2006) proposed that a galaxy’s luminosity can be relatedto its host dark matter halo’s virial mass M vir via a simple relation as follows: L ( M vir ) = 5 . × h − L (cid:12) M p (cid:104) q + M s ( p − r )11 (cid:105) /s , (2.45) .8. GALAXY FORMATION AND EVOLUTION where q , p , r , and s are free parameters and M is scaled such that M = M vir h − M (cid:12) . (2.46)However, Cooray & Milosavljevi´c (2005) showed that the relation between the mass ofa dark matter halo and the luminosity of the galaxy it hosts is not straightforward due tothe complicated baryonic physics involved.The baryonic content of a dark matter halo becomes dynamically important in thenonlinear regime when dark matter haloes are forming. Consequently, to model realisticgalaxies, such as the Galaxy we live in, baryons must be modelled, albeit they do notcontribute to the structure formation as much as dark matter. Hydrodynamical effectssuch as heating and cooling processes of gas, shocks, star formation, and feedback pro-cesses must all be taken into account when the formation of baryonic structure is beingconsidered. One solution is to use hydrodynamical simulations to model gas directly,but the lack of fundamental theories for physical processes involved in the formationand evolution of galaxies, such as star formation, render them less than optimal. As aresult, the hydro-simulations also require simple prescriptions for the so-called sub-gridphysics, which cannot be modelled directly. What can be modelled then? Galaxy formation (for a comprehensive view, see e.g. Longair, 2008; Mo et al., 2010) isexpected to proceed via a two-stage process originally outlined by White & Rees (1978),but see also Hoyle (1953); Binney (1977); Rees & Ostriker (1977); Silk (1977) for earlydevelopment. In this paradigm, the gravitational instability acting on collisionless darkmatter results in the formation of self-gravitating dark matter haloes (as already notedearlier). Because baryons, initially well mixed with the dark matter, are assumed to“feel” the dark matter via gravity, they also participate in this collapse after the darkmatter haloes have started to form. However, unlike the dark matter, the gas is notcollisionless, but can dissipate. As a result, in a very simplified picture, the gas can beassumed to be heated by shocks to the virial temperature of the dark matter halo duringthis infall. After which, the hot gas can cool radiatively, on a time scale set by atomicphysics.During the collapse and cooling, the gas is assumed to condense to the cores of col-lapsed dark matter haloes. However, it is assumed that this process is not the same in allhaloes. In smaller structures such as galaxy host haloes, the dominant physical processis cooling, which allows baryons to be more centrally concentrated than dark matter. Incontrast, in larger structures, baryons experience a deeper gravitational potential and cantherefore gain potential energy as they fall to the centre of a halo. This process heatsbaryonic matter and increases its temperature via shocks (e.g. Birnboim & Dekel, 2003). CHAPTER 2. FORMATION OF STRUCTURE
As a result, baryonic matter experiences pressure forces which do not let them to be asconcentrated as its host dark matter. In large-scale structures the process of cooling willtherefore be highly important in the dense cores of dark matter where the gas can cool.Finally, the cold gas can fragment into stars, and a galaxy is born.To simplify, in hierarchical models, such as the Λ CDM (Section 2.4.1), the galaxyformation involves at minimum the following three stages:I. the hierarchical formation of dark matter haloes,II. the accretion of gas into the haloes, andIII. the cooling and fragmentation of the hot gas into stars.Figure 2.8 shows a logic flow chart for galaxy formation . The paths leading to theformation of various galaxies (green ellipses) are drawn from the initial conditions setby the cosmological framework, discussed in earlier Sections and outlined in Figure 2.5.Note that the flow chart does not include any feedback effects, which have been foundto be significant and will be discussed later. Due to the complicated physics related to galaxy formation and evolution (as seen inFig. 2.8), simple rules that can be easily varied to study the importance of differentphysical processes are highly useful. Semi-analytical models (SAMs) of galaxy forma-tion try to fill this void by encoding simplistic rules for the formation and evolution ofgalaxies within a cosmological framework. A SAM is a collection of physical recipesthat describe an inflow of gas, how gas can cool and heat up again, how stars are formedwithin galaxies, how stellar populations evolve and how black holes grow using simpli-fied physics (Cole, 1991; White & Frenk, 1991; Lacey & Silk, 1991; Kang et al., 2005;Baugh, 2006; Lucia & Blaizot, 2007). SAMs can also easily include different feedbackeffects: stellar winds, active galactic nuclei (AGN) or supernovae (SNe) feedbacks (e.g.Croton et al., 2006; Somerville et al., 2008; Ricciardelli & Franceschini, 2010), for ex-ample. Hence, SAMs try to describe all the gas physics that goes into galaxy formationand evolution, but is not modelled in the dark matter only simulation.Due to their nature SAMs can be used to explore ideas of galaxy formation andevolution and to understand which physical processes are the most important in the lifeof a galaxy by changing the recipes describing the physics. SAMs can also be appliedto the so-called sub-grid physics that operates below the resolution of a simulation. Asit is not yet possible to simulate all star formation processes (McKee & Ostriker, 2007; The flow chart is by no means a complete description of all gas physics that may play a role in galaxyformation, but rather tries to capture the main aspects. .8. GALAXY FORMATION AND EVOLUTION Figure 2.8:
A logic flow chart for galaxy formation. The dominantly gravitational processeshave been coloured with blue, while mainly hydrodynamical processes are coloured with yellow.The red hexagons indicate a binary choice, while dark green ellipses note the end products of thesimplified processes described in the flow chart. The figure is an adaptation of Fig. 1.1 from Moet al. (2010).
Krumholz, 2011) in a cosmological context, sub-grid physics must be modelled withsimplified physics even when hydrodynamics is involved.The backbone of a SAM is the evolution of dark matter haloes. Often, this evolution CHAPTER 2. FORMATION OF STRUCTURE is parameterised with dark matter halo merger trees (Fig. 2.9) that allow the hierarchi-cal nature of gravitational instabilities to be explicitly taken into account (Baugh et al.,1998). Dark matter merger trees describe how the dark matter haloes form via merg-ers of smaller haloes. They provide the backdrop for the introduction of the baryoniccomponent which reacts gravitationally to the growing network of dark matter potentialwells. Even though modern studies derive merger trees directly from simulations, this isby means not necessary as they can be derived also by using Monte Carlo techniques. Insuch case the extended Press-Schechter (EPS) theory (Bond et al., 1991; Lacey & Cole,1993) is often used. The Monte Carlo techniques provide a fast method of generatingmerger trees, however, they have been found to be less than reliable in some cases (e.g.Cole et al., 2008).
Figure 2.9:
Example of a merger tree as a function of the lookback time. Symbols are colour-coded as a function of the B − R colour and their area scales with the stellar mass. Onlyprogenitors more massive than h − M (cid:12) are shown. After a cosmological model has been chosen and the merger trees have been gener-ated the baryonic processes must be taken into account. A SAM typically consists of thefollowing steps: 1) follow the three baryonic components: hot and cold gas, and starsand adopt a recipe for disk formation; 2) specify a recipe for the conversion rate betweenthe three components, including star formation and feedback effects; 3) keep track of .8. GALAXY FORMATION AND EVOLUTION the metallicity of each component; 4) convert the star formation history and metallic-ity of the stellar populations into luminosities; and 5) adopt a recipe for galaxy-galaxymergers. In what follows, I briefly describe these steps in turn. Heating and cooling of gas
Modern galaxy formation theories assume that the gas density profile follows that ofdark matter. When gas falls into the potential well of a dark matter halo it is assumed tobe shock-heated (White & Rees, 1978) to the virial temperature of the halo, given by T vir = µm H GM halo kr vir . (2.47)Here µ is the mean molecular mass of the gas, m H is the mass of a hydrogen atom, k is the Boltzmann’s constant, M halo is the mass of the halo and r vir is the virial radius,within which the mean density is 200 times the critical density.Before stars can form from the cold ( T < K ) molecular clouds the shock-heatedhot gas must cool radiatively (e.g. Helly et al., 2003). The cooling time can be defined,for example, as the ratio of the thermal energy density and the cooling rate per unitvolume. In this case, the cooling time can be written as t cool ( r ) = µm p kT vir µm H ρ gas ( r )Λ( T vir , Z gas ) , (2.48)where ¯ µm p is the mean particle mass, k refers to the Boltzmann constant, ρ gas ( r ) is thehot gas density, and Λ( T vir , Z gas ) is the cooling function. Note that the cooling rate isa function of metallicity of the gas Z gas and the virial temperature of the halo T vir , andthus, the cooling is in general more effective in higher density regions. Additionally, inhighly simplified scenarios, the more metal-enriched gas tends to cool faster.Different cooling mechanisms (inverse Compton scatter, molecular and atomic cool-ing, and bremsstrahlung) can however be dominant at different times and tempera-tures complicating the gas cooling modelling significantly. For example, in massivehaloes, where T vir (cid:38) K, gas is fully collisionally ionised and cools mainly throughbremsstrahlung emission from free electrons. In the temperature range K < T vir < K excitation and de-excitation mechanisms dominate, while in haloes with T vir < K gas is mainly neutral and the cooling processes are therefore suppressed. How-ever, in a simplified scenario the gas is assumed to be able to cool if cooling time isshorter than some characteristic timescale, which is model dependent.As the cold condensed gas accumulates in the central regions of the dark matterhaloes it can be identified as the ISM of the protogalaxy. What is assumed to followafter the cold gas has settled down is a disk formation. CHAPTER 2. FORMATION OF STRUCTURE
Disk formation
The disk formation may be modelled after, for example, the amount of cold gas andthe properties of the host halo. The underlying physical notion is that when structuresgrow and collapse in the early universe, they exert tidal torques on each other. Thisprovides each collapsing dark matter halo with some angular momentum. As a result,SAMs typically assume that the cold gas will form a disk with the same specific angularmomentum as the dark matter halo, while the size and rotation velocity are determinedby the spin parameter:
Γ = J | E | GM . (2.49)Here J is the angular momentum of the halo, E refers to the total energy of the halo, and M is the mass of the dark matter halo. When adopting this simple prescription, whatfollows is that the mass and spin of the disk is tightly coupled to the mass and spin ofthe dark matter halo.The dimensionless angular momentum Γ is a measure of the degree of rotationalsupport of the galaxy. Note, however, that the typical values ( ∼ . of Γ of collapseddark matter haloes have been found to be significantly smaller than that of the largelyflattened centrifugally supported disk galaxies we observe today ( with Γ ∼ . − . .Hence, a considerable amount of dissipation must have occurred to produce the observeddisks. Star formation
In a typical SAM star formation (for a general review, see Kennicutt, 1998a) is assumedto take place in the disks of galaxies, while the actual onset of star formation is assumedto occur once the surface density of cold gas exceeds a critical density (Kennicutt, 1998b,1989) in a molecular cloud (Krumholz, 2011). Ideally, the star formation law should bederived from the first principles as a function of the physical conditions, such as density,temperature, metallicity, radiation and magnetic fields of the ISM, however, the detailedphysics involved in the fragmentation of the cold gas, collapse and onset of a protostar,and the physical conditions of the ISM are not yet well understood. I will, however,return to the importance of the ISM in Section 4.5.Due to the complicated physics involved, SAMs often derive the star formation rate(SFR) of a galaxy using a simple relation ˙ ρ (cid:63) = (cid:15) SF ρ cold τ (cid:63) , (2.50)where ρ cold is the density of the cold gas, τ (cid:63) is the characteristic timescale, and (cid:15) SF is a measure for the efficiency of star formation. Note, however, that several different .8. GALAXY FORMATION AND EVOLUTION forms of the above equation have been developed. These recipes for the star formationefficiency range from simple models that assume that τ (cid:63) is a constant to models that areproportional to the dynamic time of the galaxy and take into account, for example, thecircular velocity and/or the radius of the disc. In any event, the above star formation lawis a variant of the empirical Schmidt (1959) law, for which it has been assumed that theSFR is controlled by the self-gravity of the gas.Closely tied to the star formation in galaxies is the number of stars of a given massthat forms, that is the the initial mass function (IMF; e.g. Kroupa, 2001; Chabrier, 2003).SAMs typically assume that the IMF of stellar populations is universal when modellingstar formation. Note, however, that theoretical arguments (e.g. Davé, 2008) and indi-rect observational evidence suggest that the stellar IMF may evolve with, e.g., time (vanDokkum, 2008, and references therein) or environment, casting a shadow over the as-sumption of universality. Feedback effects
The early SAMs, using recipes similar to the above ones, were however, not able toreproduce the observed form of the galaxy luminosity function (LF; Eq. 3.8). They oftenover-predicted the number of both faint and bright galaxies, especially in the infrared(e.g. Benson et al., 2003; Croton et al., 2006; Benson & Devereux, 2010). To alleviate thediscrepancy i.e. to limit the number of faint and bright galaxies a feedback mechanismwas introduced. To regulate the star formation in both light and massive dark matterhaloes the feedback was divided to two separate mechanisms that operate on differentmass regimes.Modern SAMs model active galactic nuclei (AGN) feedback, which can suppressthe cooling flow in high mass systems (e.g. Silk & Rees, 1998; Croton & Farrar, 2008;del P Lagos et al., 2008). The AGN feedback (Sijacki et al., 2007) provides additionalenergy that can suppress the cooling of hot gas generating a sharp cut-off to the high-luminosity end of the LF. In many models the strength of the AGN feedback dependsdirectly on the mass accretion of the black hole ˙ M BH , thus, the modified cooling ratecan be written, for example, as ˙ M modcool = ˙ M cool − η ˙ M BH c V . (2.51)Here, η refers to the black hole accretion efficiency. The additional energy from theAGN can prevent gas from cooling and is more important in later times when galaxieshave more massive black holes. The AGN feedback can therefore help to regulate theSFR at later epochs and to prevent the overproduction of very massive galaxies. Anothereffect of AGN feedback is seen on the ages of high stellar mass systems, which aresignificantly older for AGN feedback models (Khalatyan et al., 2008). CHAPTER 2. FORMATION OF STRUCTURE
While the AGN feedback affects mainly massive galaxies, the supernovae (SN) feed-back is important for lighter galaxies and for the metal enrichment of the inter-stellar andpossibly even the inter-galactic medium (Vecchia & Schaye, 2008). The SN feedbackhelps to self-regulate the process of star formation throughout the galaxies’ history. Inthe absence of the SN feedback star-formation rates (SFRs) are extremely high in earlytimes and fairly low in more recent epochs. However, with the SN feedback, SFRsare initially lower so more gas is available for later periods of star formation (see alsoSection 4.2.3).The SN feedback can function because in the models it is assumed that a SN canblow gas out of a star forming disc. Moreover, it is also assumed that the rate of massejection is proportional to the total mass of stars formed. As a result, the re-heating ofgas due to the feedback can then be modelled as follows ˙ M reh = (cid:15) η SN E SN V c ˙ M (cid:63) , (2.52)where E SN is the energy injected by the SN, η SN is the number of SN per solar mass, and V c is the circular velocity of the dark matter halo. In dwarf galaxies with low circularvelocities the energy from SNe can efficiently heat the inter-stellar media (ISM) whichcan then escape the halo through a cool wind. Note that hot metals expelled in SNeexplosions are much less bound to the galaxy than the cold ISM and can therefore escapeto the intra-galactic media (IGM), causing the enrichment of the IGM. Hence, the energyfrom SNe not only suppresses the star formation in light galaxies, but is also partiallyresponsible for the chemical evolution in galaxies (see also Section 4.2.4). Galaxy mergers
The above definitions for the amount of cool gas, disk formation, star formation, metalenrichment, and feedback effects have assumed a static dark matter halo. Dark matterhaloes are not, however, static, but can interact and merge with other haloes (e.g. Kauff-mann & White, 1993; Fakhouri & Ma, 2008) as described by their merger trees. Themergers are assumed to be important to the extent that, for example, Li et al. (2007)showed that each dark matter halo, virtually independent of its mass, experiences about ± major mergers since its main progenitor has acquired one per cent of the finalhalo mass. Consequently, a realistic SAM must also take mergers into account.When dark matter haloes merge, galaxies share the same potential well, but do notimmediately merge (e.g. Stewart et al., 2009). Instead, the two galaxies are expected toorbit in a common halo. However, with enough time tidal interactions and dynamical Note that Li et al. (2007) defined a major merger as a merger with a progenitor mass ratio greater than1:3. .8. GALAXY FORMATION AND EVOLUTION friction remove orbital energy and cause the orbit of the subhalo (and its galaxy) to decay,transporting it towards the centre. The orbit decay time has been shown to depend on themass ratio of the merging haloes, eccentricity of the orbits, and the mass loss due to tidalstripping (Colpi et al., 1999; Boylan-Kolchin et al., 2008). The details of dark matterhalo (and galaxy) mergers are therefore less than simple. For example, if the angularmomentum is high and the orbital energy is not low enough, the merger cannot happenin a Hubble time.Despite the complications, SAMs often apply simple calculations for the dynamicalfriction time. If however N − body merger trees are used instead of the EPS formalism,the spatial information of the simulated haloes can be used. In any event, after thehaloes merge SAMs usually assume that the hot halo gas is shock-heated to a new virialtemperature, while the cold gas is attached to the centre. The hot gas that has not escapedfrom the new halo is assumed to form a reservoir in the halo, while galaxies residinginside the halo will merge within a dynamical friction time. It is obvious that the physicsof such description of mergers is highly simplified, but even so, the simplified mergerscenario can lead to galaxies that can statistically match observations reasonably well(see, for example, Paper IV).Several different types of mergers have been identified: major (e.g. Springel & Hern-quist, 2005) and minor (e.g. Bournaud et al., 2007) that are related to the mass ratio ofthe merging pair, and the so-called wet (e.g. Lin et al., 2008) and dry mergers (e.g. Bellet al., 2006a; Khochfar & Silk, 2009), which are related to the gas richness of the merg-ing pair. The remnants of mergers between two galaxies are therefore assumed to dependprimarily on four properties: 1) the progenitor mass ratio; 2) the gas mass fraction of theprogenitors; 3) the orbital properties; and to some extent 4) the morphologies of the pro-genitors. It is then usual to assume that the different types of mergers require differentphysical prescriptions (see the lower part of Fig. 2.8).Unfortunately, the relevant mass and gas ratios are often rather arbitrarily definedin SAMs, rendering the current treatment of the different types of galaxy mergers lessphysical. In case of a minor merger, the simplified schemes of mergers often transferstars from the lighter galaxy to the bulge of the more massive one, generating a sphericalcomponent to the merger remnant. Instead, in a case of major merger, the models mayalso take into account the fact that major mergers can induce rapid star formation, astarburst, changing the stellar population and the integrated colour of the newly formedgalaxy rapidly. The possible starburst is however often tied up to the gas fraction of themerging pair and to the final surface density of cold gas. Thus, the treatment of starburstsis often model dependent. Mergers may also induce AGN activity, which has been takeninto account in some models (e.g. Somerville et al., 2008), while some have implementeda variable IMF (e.g. Lacey et al., 2010). Independent of the adopted model, it is howeverclear that due to the mergers of dark matter haloes and galaxies, a detailed analysis of CHAPTER 2. FORMATION OF STRUCTURE the formation history and the possible environmental effects must be taken into accountwhen studying galaxy formation and evolution. I will return to this in Chapter 4 whendiscussing the evolution of galaxies.
HAPTER Groups of Galaxies “Someone told me that each equation I included in the book would halve the sales. . . ”Stephen Hawking
Chapter 2 briefly described the formation of structure and how observable structuressuch as galaxies form. It was also mentioned that large galaxy surveys have shownthat galaxies (Fig. 4.1) can be considered to be the basic building blocks of the visibleUniverse. However, as Figure 2.3 implies, galaxies are found to be located in largerstructures such as groups (Fig. 3.1) and clusters more often than in isolation. To be moreprecise, more than half of all galaxies are found to be part of larger structures. This hasbeen found to be true to the extent that most galaxies with luminosities less than thecharacteristic luminosity L (cid:63) (for a definition, see Eq. 3.8) have been found to be locatedin an environment comparable to the Local Group. Thus, to understand galaxy formationand evolution - and to understand the Universe - one must understand groups and clustersof galaxies as the evolution of most galaxies takes place in these systems. Additionally,groups of galaxies are important cosmological indicators of the distribution of matterin the Universe, making them important also for cosmology. But what defines a galaxygroup?A galaxy group is a concentration of galaxies, assumed to be embedded in an ex-tended dark matter halo . Ideally, the galaxies forming a group are physically boundtogether due to their mutual gravitational attraction and the presence of the dark matterhalo. However, from the observational point of view, group members are not easy todefine because dark matter haloes cannot be observed directly. Thus, not all observedgroupings of galaxies are real physical and gravitationally bound systems as they can bea result of chance superpositions of galaxies at different distances (see, e.g., Fig. 3.1) orgalaxies within filaments that are viewed edge-on. Such systems are gravitationally un-bound (sometimes phrased as spurious or pseudo-groups) rather than real gravitationallybound groups of galaxies (e.g. Hernquist et al., 1995; Ramella et al., 1997). This definition applies also to galaxy clusters. CHAPTER 3. GROUPS OF GALAXIES
Groups of galaxies typically contain fewer than ∼ members and are often dom-inated by spiral galaxies. When larger groups are considered, the main constituent ofgalaxies usually shifts from spirals to lenticulars, but no clear cut-off in number of mem-bers exists between groups and clusters. The number of member galaxies is in generala problematic property (see e.g. Paz et al., 2006), especially in magnitude limited obser-vations, where deeper observations can reveal new members who were previously toofaint to be detected. This is true even for the Local Group (LG). Thus, better quantitiesto discriminate between groups and clusters are mass and size, though, these do not pro-vide clear cut-off values either. Typical groups are ∼ − Mpc in a diameter and theirtotal mass M DM ∼ . − . h − M (cid:12) (Huchra & Geller, 1982), while typical clustersare about an order of magnitude more massive and a few times larger (e.g. Einasto et al.,2003a, 2005; Koester et al., 2007). Typical groups are therefore smaller and less massivethan clusters of galaxies, but larger and more massive than binary galaxy systems.Most of the stellar mass in the present Universe is in groups similar to the LG withmasses a few times M (cid:12) and only ∼ per cent is in clusters with total mass > × M (cid:12) (Eke et al., 2006). Groups have been found to be present already at redshifts z > (e.g. Francis et al., 1996; Moller & Warren, 1998; Andreon et al., 2009; Bielby et al.,2010) and their environment density is intermediate between that of isolated galaxiesand that of the cores of rich clusters. The study of groups may therefore provide clues tothe processes that create the observed dependency of galaxy morphology on environment(Postman & Geller, 1984; Allington-Smith et al., 1993; Whitmore et al., 1993; Zabludoffet al., 1996; Hashimoto et al., 1998). I return to this question in the next Chapter.The birth of the study of galaxy groups (and clusters) can be dated back to and to Fritz Zwicky who was the first to apply the virial theorem to the Coma cluster(Zwicky, 1933). In his early work Zwicky derived a dynamical mass for the cluster thatseemed to be significantly larger than if all of the mass came from visible galaxies. Thiswas interpreted as a first sign of the yet-to-be-seen invisible matter holding the clustertogether. Note that the early interpretation of Zwicky’s dynamical mass statement holdsonly when the object is at least gravitationally bound, if not in virial equilibrium, but thiswas taken for granted at the time.Since several authors have studied groups of galaxies and concluded that themajority of galaxies in the Universe lie in groups (Holmberg, 1950; Humason et al.,1956; Turner & Gott, 1976; Huchra & Geller, 1982; Geller & Huchra, 1983; Nolthenius& White, 1987; Ramella et al., 1989; Karachentsev, 2005). Large astronomical redshiftsurveys such as the 2dFGRS (2dF Galaxy Redshift Survey) and SDSS (Sloan DigitalSky Survey) have also produced large group catalogues (Eke et al., 2004a,b; Baloghet al., 2004a; Merchán & Zandivarez, 2005; Tago et al., 2006; Eke et al., 2006; Berlindet al., 2006; Yang et al., 2007; Knobel et al., 2009) with up to ten thousand groups (e.g.Eke et al., 2004a). Even so, the physical processes operating in groups of galaxies are .2. GROUP IDENTIFICATION still poorly understood, albeit groups are known to be important for several reasons. Forexample, as Chapter 2 described, dark matter haloes and even galaxies merge together(Section 2.8.4). This is obviously more probable in denser environments where relativevelocities are lower and dynamical friction is higher as in groups. Moreover, mergers areonly effective in systems with a velocity dispersion smaller than or comparable to theinternal velocities of galaxies (for detailed discussion, see e.g. Mo et al., 2010). Hencegalaxy mergers are assumed to take place effectively in groups of galaxies.Before groups can be used to study the evolution of galaxies or even before anygroup properties can robustly be measured, groups of galaxies must be correctly iden-tified and discriminated from other forms of structures like binary galaxy pairs, largeclusters of galaxies and most importantly from chance alignments. Interestingly, thishas been found to be not as simple as one might naïvely first assume (Papers I and II andreferences therein). There are generally three basic pieces of information available in observations for thestudy of galaxy distribution: position, luminosity and the redshift of each galaxy. Al-though the apparent luminosity is important as a measure of the object’s visibility, it isusually a poor criterion for group membership. Moreover, the apparent luminosity canalso be misleading when applied in group studies as will be shown in Section 3.6. Ob-servations are therefore usually left with only information describing the position of thegalaxies in redshift-space (see, however, the caveats mention in Section 2.2.2).In recent years a number of different grouping algorithms have been developed andapplied (e.g. Turner & Gott, 1976; Materne, 1978; Huchra & Geller, 1982; Botzler et al.,2004; Goto et al., 2002; Kim et al., 2002; Bahcall et al., 2003; Gerke et al., 2005; Koesteret al., 2007; Yang et al., 2007) to identify groups using the redshift-space information.Despite the vast number of grouping algorithms, the Friends-of-Friends (FoF; Huchra& Geller 1982) percolation algorithm remains the most frequently applied one. The FoFalgorithm, or slightly modified versions of it (see e.g. Knobel et al., 2009), are the mostwidely used grouping algorithms, even for modern day galaxy surveys like SDSS andCOSMOS. Hence, it is important to understand its functionality and limitations. The Friends-of-Friends (FoF) group finding algorithm (Huchra & Geller, 1982) takesadvantage of two often available quantities in observational galaxy catalogues: the pro- Note that although the name is unfortunately the same as in case of the most popular halo finder, thesetwo algorithms are not identical. CHAPTER 3. GROUPS OF GALAXIES jected separation in the sky and the velocity difference in the redshift space. The group-ing method itself begins with the selection of a galaxy, which has not been previouslyassigned to any of the existing groups. After choosing a galaxy the next step is to searchfor companions with the projected separation D smaller or equal to the separation D L : D = 2 sin (cid:18) θ (cid:19) VH ≤ D L ( V , V , m , m ) , (3.1)where the mean cosmological expansion velocity V = V + V , (3.2)and the velocity difference V is smaller or equal to the velocity V L : V = | V − V | ≤ V L ( V , V , m , m ) . (3.3)Here, V and V refer to the velocities (redshifts) of the galaxy and its companion, m and m are their magnitudes, and θ is their angular separation in the sky. If no compan-ions are found, the galaxy is entered on a list of isolated galaxies, while all companionsfound are added to the list of group members. The surroundings of each companionare then searched by using the same method. This process is repeated until no furthermembers are found and all potential group members have been searched.There is a variety of prescriptions for D L and V L in the literature (for references,see Paper I). However, the original method assumes that the luminosity function (LF) isindependent of distance and position and that at larger distances only the fainter galaxiesare missing. For each pair we therefore take D L = D (cid:32) (cid:82) M −∞ Φ( M )d M (cid:82) M lim −∞ Φ( M )d M (cid:33) − , (3.4)where the integration limits can be calculated from equations: M lim = m lim − − D F ) (3.5)and M = m lim − − V ) . (3.6)In Eq. 3.4 Φ( M ) is the differential galaxy luminosity function for the sample, and D is the projected separation in Mpc chosen at some fiducial distance D F . The limitingvelocity difference can be scaled in the same way as the distance D L , i.e., V L = V (cid:32) (cid:82) M −∞ Φ( M )d M (cid:82) M lim −∞ Φ( M )d M (cid:33) − , (3.7) .3. DIFFERENT SPECIES OF GROUPS: A BRIEF OVERVIEW where the fiducial velocity V is often taken to be ∼ − km s − and the integrationlimits are given by Eqs. 3.5 and 3.6.For simplicity, it is often convenient to assume that the differential galaxy luminosityfunction can be described in form of a Schechter (1976) LF: Φ( M ) = 25 Φ (cid:63) ln 10 (cid:16) ( M (cid:63) − M ) (cid:17) α +1 e −
25 ( M (cid:63) − M ) , (3.8)where M is the absolute magnitude of the object and α , M (cid:63) , and Φ (cid:63) parametrize theSchechter luminosity function. In the Schechter formalism the M (cid:63) and L (cid:63) refer tothe characteristic absolute magnitude and luminosity, respectively, and mark a point inwhich the luminosity function exhibits a sudden change in the slope. For example, thecharacteristic luminosity in B -band L (cid:63)B ∼ × L (cid:12) is comparable to the brightnessof the Galaxy (Driver et al., 2007), but the exact value depends on the environment andon the dark matter halo mass (Cooray & Milosavljevi´c, 2005). Note also that, the galaxyLF has been found to evolve significantly as a function of redshift, especially at earlycosmic epochs (see e.g. Bouwens et al., 2011). Nevertheless, the luminosity functionis a powerful tool and provides information on the relative frequency of galaxies with agiven luminosity.The theory of gravitational instability (see Section 2.5) predicts that the number ofvirialized systems that have formed at any given time depends on their mass, with moremassive systems being less abundant than less massive ones. Therefore, by coupling to-gether the information on the luminosity and the number of galaxies, the LF can provideinformation on the formation and evolution of both the structural and visible componentsof galaxies. However, for identifying galaxy groups, the exact shape of the luminosityfunction is less important than the adopted values of D and V , as was noticed duringthe study of Paper I. Hence, I will not concentrate more on the LF in the context ofgroups, I will however return to it in Chapter 4. Grouping algorithms, such as the FoF algorithm, produce catalogues of galaxy systemswith a vast amount of different properties. This has resulted in a classification of systemswith similar properties. For example, systems of galaxies can be classified as loose (e.g.Ramella et al., 1995, 1997; Tucker et al., 2000; Einasto et al., 2003b), poor (e.g. Zablud-off & Mulchaey, 1998; Mahdavi et al., 1999), compact (e.g. Shakhbazyan, 1973; Hick-son, 1982; Hickson et al., 1989; Diaferio et al., 1994; Barton et al., 1996; Tovmassianet al., 2006) or fossil groups (e.g. Ponman et al., 1994; Jones et al., 2003; Santos et al.,2007; von Benda-Beckmann et al., 2008) depending on common properties. However,the vast number of different classes can also be interpreted as a sign that the relationshipamong systems of galaxies is not yet well understood. CHAPTER 3. GROUPS OF GALAXIES
The key idea behind the above classification rises from the diversity of evolutionarystages within which a galaxy group can be found. For example, compact groups areassumed to be observed briefly before they are about to merge, while fossil groups, whichare often dominated by a large central elliptical galaxy, are possibly the end product ofsuch a merging. Therefore, a general unification of different classes might be possiblein the future when the evolution of galaxy groups is better understood. However, due tothe historical reasons I briefly summarise each class and how they have been defined inthe following Sections. When appropriate, I will also describe some selected propertiesof each class, their evolutionary stage and the possible significance for galaxy evolutionin general.
Loose groups of galaxies with a space density of ∼ − Mpc − (Nolthenius & White,1987) represent the most common class of groups and are often simply referred to asgroups. They comprise ∼ members, including large number of faint dwarf galaxies,and the whole group typically extends to a diameter of up to ∼ . Mpc. Thus, loosegroups are, as the name implies, an intermediate in scale between compact groups (Sec-tion 3.3.3) and rich clusters. Consequently, it has been argued that their dynamics isimportant for the study of the distribution of dark matter (Oemler, 1988).Tucker et al. (2000) finds a median line-of-sight velocity dispersion of 164 km s − and median virial mass ∼ . × h − M (cid:12) for loose groups. However, Einasto et al.(2003b) argue that loose groups in the neighbourhood of a rich cluster are typically . times more massive and . times more luminous than loose groups on average.Furthermore, Einasto et al. (2003b) find that these groups have velocity dispersions ofabout . times larger than the loose groups on average. The immediate neighbourhoodof a loose group can therefore have a significant impact on the group properties anddynamics. Hence, a nearby large cluster can enhance the evolution of the neighbouringloose group making it difficult to draw common values for the properties of loose groupsthat would apply to all of them.Due to their relatively small velocity dispersions and intermediate sizes, loose groupscan be important when galaxies that may merge in the future are being identified andstudied. For example, Mamon (1986) was the first to suggest that compact groups mightbe transient unbound cores of loose groups. Indeed, further studies have confirmed thatloose groups are associated with compact groups, and that loose groups are often thebirth places of compact groups (Vennik et al., 1993; Ramella et al., 1994; Diaferio et al.,1994). Thus, loose groups can host some sort of an association with a compact groupproviding a link between their evolutionary stages. .3. DIFFERENT SPECIES OF GROUPS: A BRIEF OVERVIEW Most galaxies in the local Universe, including the Galaxy, belong to a poor groupof galaxies (Zabludoff & Mulchaey, 1998). The historical definition of a poor groupdictates that poor groups typically contain fewer than five bright ( L ∼ L (cid:63) ) galaxies(Zabludoff & Mulchaey, 1998) and that the total number of group members is less thanin a typical loose group. Deeper redshift surveys have however identified new and faintgalaxies around poor groups bringing them closer to the definition of loose groups. Forexample, surveys such as the SDSS have helped to identify new faint dwarf galaxies thatbelong to the Local Group (LG) (e.g. Willman et al., 2005; Irwin et al., 2007; Walshet al., 2007; Belokurov et al., 2008; Belokurov et al., 2010), raising the total number ofLG members to ∼ . Obviously the number of brighter L (cid:63) -galaxies in the LG has notchanged and satisfy the criterion of less than five, but as the total number of members isreaching it is questionable whether one can talk about a poor group any longer.As mentioned earlier, the total number of companions may however not be all thatimportant for groups. Instead, the fraction of early-type galaxies in the poor groupshas been found to vary significantly, ranging from that characteristic of the field ( ∼ per cent) to that of rich clusters ( ∼ per cent) (Zabludoff & Mulchaey, 1998) indisagreement with many loose groups. The relatively high early-type fraction in poorgroups is indeed surprising because most poor groups have low galaxy number density,thus, the effects of disruptive mechanisms such as galaxy harassment (Moore et al.,1996) are assumed to be weaker than in rich or compact groups. In contrast, however,the kinematics of poor groups makes them preferred sites for galaxy-galaxy mergers,which may alter the morphologies and star formation histories of some group members(Zabludoff & Mulchaey, 1998), providing a possible explanation for the relatively largenumber of elliptical galaxies.The dynamical status of poor groups has also been questioned in several studies (e.g.Zabludoff & Mulchaey, 1998; Mahdavi et al., 1999, and references therein). The highergalaxy densities than in the field and lower velocity dispersions than in cluster cores,make them favourable sites for galaxy-galaxy mergers (Barnes, 1985). Consequently,one would assume that if poor groups are old structures several mergers should havetaken place. Thus, if some or even all poor groups are gravitationally bound, why do weobserve them at all? One possible explanation is that bound poor groups are collapsingfor the first time, and in such, they will eventually face the same destiny as loose groups,that is being eventually associated with compact groups. A compact group (CG) of galaxies can be loosely defined as a group of galaxies witha small number of members in which the typical intergalactic separation is of the order CHAPTER 3. GROUPS OF GALAXIES of the scale of the galaxies. Historically, CGs have been studied by several authors(e.g. Shakhbazyan, 1973; Rose, 1977), however, probably the most well-studied CGcatalogue is the Hickson Compact Groups (HCGs) (see e.g. Hickson, 1982; Hicksonet al., 1988, 1989).HCGs are compact configurations of relatively isolated systems of typically four orfive galaxies in close proximity to one another (see Fig. 3.1, and Hickson et al., 1989,and references therein). They have also been found to show peculiarities in terms ofmorphology or kinematics, starbursts or even AGN activity (for a complete review, seeHickson, 1997). Furthermore, HCGs have been found to contain large quantities ofdiffuse gas and to be dynamically dominated by dark matter (for predictions of X-rayproperties, see Diaferio et al., 1995). They have also been found to trace the large-scale structure, but to prefer low-density environments. As already mentioned in Section3.3.1, HCGs may form as subsystems within looser galaxy associations and evolve bygravitational processes. Thus, while compact groups are associated with loose groupsand filaments, these tend to be low-density and sparsely populated systems. Even so,Walker et al. (2010) speculate that due to their mid-infrared colours the compact groupenvironment fosters accelerated evolution of galaxies.The fraction of late-type galaxies has been found to be significantly lower in compactgroups than in the field. However, a given CG is also more likely to contain galaxies of asimilar type than would be expected for a random distribution (Hickson, 1997, and refer-ences therein). Zepf & Whitmore (1993) found that elliptical galaxies in compact groupstend to have lower internal velocity dispersions than do ellipticals of similar propertiesin other environments. Moreover, elliptical galaxies of CGs have been found to lie offthe fundamental plane defined by ellipticals in other environments. This suggests thatthe velocity dispersion is of greater physical relevance to the formation and evolution ofgalaxies in CGs, than is the apparent physical density.The strong galaxy interactions in CGs, see Fig. 3.1, are expected to induce mergersof group members (for early simulations, see Mamon (1987)). The dynamical timescalesof CGs have however been argued and range from relatively short ( ∼ . H − ) (e.g.Diaferio et al., 1994) to longer ones ∼ H − (e.g. Governato et al., 1996; Athanassoulaet al., 1997). If however the merger timescales are somewhere between the two extremes,it is likely that CGs are rather short lived. To explain this, it has been suggested that CGsmay be continuously replenished through dynamical evolution of loose groups (Diaferioet al., 1994; Ramella et al., 1994). This provides a reasoning for the existence of CGs, butwhat will they become in the course of evolution? One potential scenario was describedby Borne et al. (2000) who suggest that the evolutionary progression from CGs canlead first to pairs followed by ultra-luminous infrared galaxies (ULIRGs) and finally toelliptical galaxies. Thus, it is possible that CGs are the progenitors for ULIRGs (see alsoSection 4.5). Note, however, that it has also been suggested that the eventual demise of .3. DIFFERENT SPECIES OF GROUPS: A BRIEF OVERVIEW Figure 3.1:
The Hickson Compact Group 92 or Stephan’s Quintet, as the name implies, is agroup of five galaxies. Note, however, that NGC (at upper left) is actually a foregroundgalaxy not a real group member. Courtesy of NASA, ESA and the Hubble SM4 ERO Team. the CG due to mergers could lead to the formation of a fossil group (e.g. Vikhlinin et al.,1999; Mulchaey & Zabludoff, 1999). CHAPTER 3. GROUPS OF GALAXIES
The definition of a fossil group is often based on the following criteria of Jones et al.(2003). A fossil system is defined as a spatially extended X-ray source with an X-ray luminosity from diffuse, hot gas of L X , bol ≥ h − erg s − , while the opticalcounterpart is a system of galaxies with ∆ m ≥ . , where ∆ m is the magnitudegap between the brightest and the second brightest galaxy in the R -band within half theprojected virial radius of the group centre. The reasoning for the optical criterium is thatit is supposed to guarantee that the system is dominated by an E or cD type galaxy andthat other members of the system can only cause small perturbations to the total potentialwell of the system. Consequently, rendering fossil groups as systems of galaxies that aredominated mainly by a single massive galaxy.The first fossil group was discovered by Ponman et al. (1994) using Röntgensatellit(ROSAT) X-ray data. Since the discovery of fossil groups Khosroshahi et al. (2007)compiled a list of seven fossil groups based on Chandra X-ray Observatory data andSantos et al. (2007) used SDSS data to identify 34 candidates. Fossil groups have alsobeen studied theoretically using cosmological N -body simulations (e.g. D’Onghia et al.,2005; D’Onghia et al., 2007; Sales et al., 2007; Díaz-Giménez et al., 2008). Interestingly,based on the results of different studies fossil groups have been interpreted in differentways.Jones et al. (2003) describe fossil groups as old, undisturbed systems which haveavoided infall into galaxy clusters, but where galaxy merging of most of the L (cid:63) galaxieshas occurred. Khosroshahi et al. (2007) suggest that fossil groups have formed early,while Vikhlinin et al. (1999) and Mulchaey & Zabludoff (1999) suggest that fossils canbe the result of galaxy merging within a compact group. In contrast, it has also been sug-gested that fossil groups are the remnants of what was initially a poor group of galaxiesthat has been transformed to this old stage of galaxy evolution in low density environ-ments with compact groups acting as likely way station in this evolution (Eigenthaler& Zeilinger, 2009). In cosmological N -body simulations fossil groups represent undis-turbed, early forming systems in which large and massive galaxies have merged to forma single dominant elliptical galaxy (Dariush et al., 2007; von Benda-Beckmann et al.,2008). Given the different interpretations, it is clear that more work is required beforefossil groups can be considered as fully understood. The earlier discussion about the properties of different group classes implied that the dy-namical state and properties can be helpful when studying the effects of the environmentinvolved in galaxy evolution. The first basic property of a galaxy group to consider istherefore the velocity dispersion σ v . The velocity dispersions of groups usually range .4. DYNAMICAL PROPERTIES OF GROUPS up to a few hundred km s − , while clusters of galaxies can show dispersions up to abouta thousand km s − . Due to the difficulties in measuring the true three dimensional mo-tions of galaxies, the velocity dispersion is however often measured in radial direction.Moreover, the velocity dispersion of a group is only a meaningful quantity if the group isa gravitationally bound system, otherwise some of the group members are participatingin the pure Hubble flow and their radial velocities are biased (e.g. Baryshev et al., 2001;Macciò et al., 2005; Teerikorpi et al., 2008; Chernin et al., 2009). The assumption ofboundness is however not straightforward, and there is no guarantee that, e.g., the FoFalgorithm has identified only groups that are bound structures. This assumption and itsimplications will be discussed in more detail as described in the next Section. Despitethe potential complications, studies (Paper I, and references therein) have shown thatobserved velocity dispersions are in general in agreement with those of cosmological N -body simulations when a Λ CDM cosmology is adopted.The velocity dispersion of a group is obviously connected to its internal dynamicsand to the depth of the potential well of the host dark matter halo. Thus, for a gravita-tionally bound system, the velocity dispersion can be used to derive the total dynamicalmass of the group. For example, one simple yet often applied method links the group’svelocity dispersion and size to its mass in a following way M obs ∝ σ v R H . (3.9)Here R H is the mean harmonic radius, i.e., the size of the group. When this simplerelation is applied to an observed group the assumption of boundness is often taken forgranted or argued based on the small value of the virial crossing time t c = 3 R H / σ v , (3.10)which is written in units of the Hubble time H − .The virial crossing time is usually assumed to describe the group’s dynamical sta-tus. If a crossing time is short compared to the Hubble time, the group must be bound,otherwise it would have dispersed long ago. However, as one may choose various def-initions for the velocity dispersion and for the group size, which in combination definethe crossing time, these choices introduce significant and systematic biases in the finalvalue of the crossing time. Moreover, the inclusion of non-members and the existenceof galaxy pairs, both of which increase the mean projected velocity, can systematicallybias the velocity dispersion. The crossing time is therefore not a robust indicator fordefining a group’s dynamical status and to discriminate between gravitationally boundand unbound groups (e.g. Diaferio et al., 1994, Paper I). CHAPTER 3. GROUPS OF GALAXIES
Due to the uncertainties, for example, in the virial crossing times, it is unclear, whichobserved groupings of galaxies, if any, are gravitationally bound structures. However, asChapter 2 described, galaxies reside in large dark matter haloes. Thus, if group membersare required to belong to the same dark matter halo, most groups of such type can readilybe taken as gravitationally bound structures. However, as observations cannot directlyobserve dark matter haloes, and therefore identify galaxies that belong to the same halo,the problem of identifying real group members remains. Moreover, it is possible thatsome substructure of a given halo has higher velocity than the required escape velocity,complicating the matter even further.Fortunately, cosmological N -body simulations provide a tool to study whether group-ing algorithms, such as the FoF, can identify groups of galaxies that are gravitationallybound systems. In simulations it is simple to mimic observations by placing the observerinside the simulation volume, either to an arbitrary location or a specially selected envi-ronment that mimics the observed surroundings of the Local Group. After choosing theorigin it is then relatively straightforward to use simulation data to generate mock groupcatalogues that mimic observations (for a detailed description, see Papers I and II). Fi-nally, such catalogues can be used to make theoretical predictions about the propertiesof groups. As cosmological N -body simulations provide detailed dynamical information of eachdark matter halo, and even each particle that forms the halo, the virial theorem can beused to measure the dynamical status of a galaxy group by relating its kinetic energy tothat of the potential. In general, the total kinetic energy of a galaxy group may be writtenas T = 12 M (cid:88) i An excess of higher redshift galaxies with respect to the group centre was discoveredby Arp (1970, 1982) and it was studied in detail by Jaakkola (1971). Since then manyauthors have found a statistically significant excess of high redshift companions relativeto the group centre. Bottinelli & Gouguenheim (1973) extended the study of Arp (1970)to nearby groups of galaxies, and Sulentic (1984) found a statistically significant excessof positive redshifts while studying spiral-dominated (i.e. the central galaxy is a spiralgalaxy) groups in contrast to the E/S0-dominated (i.e. the central galaxy is an ellipticalor lenticular galaxy) groups that showed a minor blueshift excess. Also Girardi et al.(1992) found discordant redshifts while studying nearby small groups identified by Tully(1988) in the Nearby Galaxy Catalogue. However, the conventional theory holds that thedistribution of redshift differentials for galaxies moving under the gravitational potentialof a group should be evenly distributed. Even systematic radial motions within a groupwould be expected to produce redshift differentials that are evenly distributed. To solvethis discrepancy between observations and the conventional theory even new physicswas suggested (see e.g. Arp, 1970).Since the first observations of discordant redshifts in the 1970s, multiple theorieshave been suggested to explain the observed redshift excess. Sulentic (1984) listed somepossible origins for the observed redshift excess, while Byrd & Valtonen (1985) andValtonen & Byrd (1986) argued that this positive excess is mainly due to the unboundexpanding members and the fact that the dominant members of these groups are some-times misidentified. Opposite to this, Girardi et al. (1992) argued that the positive ex- .6. DISCORDANT REDSHIFTS cess may be explained if groups are still collapsing and contain dust in the intragroupmedium. Hickson et al. (1988) ran Monte Carlo simulations and concluded that the ef-fects caused by the random projection can explain discordant redshifts, and Iovino &Hickson (1997) similarly found that projection effects alone can account for the highincidence of discordant redshifts. However, studies by Hickson et al. (1988) and Iovino& Hickson (1997) dealt only with Hickson Compact Groups that have only a few mem-bers in close proximity (as noted in Section 3.3.3). Zaritsky (1992) studied asymmetricdistribution of satellite galaxy velocities with Monte Carlo simulations and concludedthat observational biases partially explain the observed redshift asymmetry, but cannotaccount for the whole magnitude of it. Despite, and partially because of, the vast numberof explanations none of the explanations were found to be truly satisfactory and someeven contradict one another. As a result, the problem of discordant redshifts remainedopen. Byrd & Valtonen (1985) and Valtonen & Byrd (1986) were the first ones to proposethat redshift asymmetries should arise in nearby groups of galaxies, if a large fraction ofthe group population is unbound to the group. They argued that the redshift asymmetryexplains the need for "missing matter", the dark matter that was at the time supposedto exist at the level of the closing density of the Universe in groups of galaxies. Theseauthors argued that if the group as a whole is not virialized, there is no need for ex-cessive amounts of binding matter and it might also lead to the observed asymmetries.Despite their efforts, a confirmation was never obtained. Fortunately, cosmological N -body simulations (Section 2.7) provide an invaluable tool to study projection effects asthe origin of the observer can be chosen freely. Simulations also provide informationabout the dark matter halo dynamics and substructure, while the group dynamics couldbe studied with the tools developed and introduced in Paper I and briefly summarised inthe previous Section. Thus, cosmological simulations provided an excellent tool to testthe explanation of Byrd & Valtonen (1985) and Valtonen & Byrd (1986). This was themain topic of Paper II.Paper II finds that gravitationally bound groups of simulated galaxies do not showany statistically significant redshift excess. This result is in agreement with the conven-tional theory, where it is expected that the distribution of redshift differentials are evenlydistributed. A simulated group catalogue that includes only gravitationally bound groupsof galaxies show an equal number of galaxies relative to the brightest member withinstatistical fluctuations. It is therefore important for galaxy group studies to be able toaccurately identify gravitationally bound structures and exclude change alignments andspurious groups from the group catalogue (as noted already in Section 3.5 and in PaperI). CHAPTER 3. GROUPS OF GALAXIES A detailed study of simulated groups shows that when the dominant members ofgroups are identified by using their absolute B -band magnitudes a small blueshift ex-cess arises. This is mainly due to the magnitude limited observations that can miss thefaint background galaxies in groups. Moreover, B -band is also problematic because ofanother reason; it tends to make star-forming spirals, which have a larger number of hotO and B-type stars and which do not have much dust, brighter than regular red ellipticalgalaxies. Thus, the brightest galaxy in B -band may not accurately mark the centre of thepotential well of a group and can create an artificial imbalance between the front and theback part of the group. This, together with the current inability to accurately measurethe relative distances inside groups of galaxies, except for a few of the nearest ones (e.g.Karachentsev et al., 1997; Jerjen et al., 2001; Rekola et al., 2005; Rekola et al., 2005;Teerikorpi et al., 2008) complicates the identification of the group centre if no X-raydetection of possible hot IGM is available.The misidentification of the group centre can lead to a redshift excess, since it ismore likely that the apparently brightest galaxy is in the front part of the group than inthe back part of it. Paper II shows that when the group centre is not correctly identified,it can cause the majority of the observed redshift excess. If simultaneously the group isalso gravitationally unbound, the level of the redshift excess becomes as high as in ob-servations. The explanation of Byrd & Valtonen (1985) and Valtonen & Byrd (1986) forthe origin of the redshift excess was therefore verified in Paper II. This further renderedthe need to introduce any “anomalous" redshift mechanism futile in order to explain theredshift excess first noted by Arp (1970). The result also underlines the importance ofrobust group centre and member identification. Interestingly though, this fact, togetherwith the method of Paper I, can also be used as an advantage; that is, to estimate if agiven group is likely to be gravitationally bound system. The ability to identify gravitationally bound groups of galaxies from observational datais of great importance (as discussed in this Chapter). Paper I shows that a large fractionof groupings are gravitationally unbound when the most often used grouping algorithm(Friends-of-Friends) is applied to simulated data. This result was found to be mostly in-dependent of the cosmology and values of the free parameters adopted as all tested casesled to large fractions of gravitationally unbound systems. We can therefore conclude thatthe results of Paper I imply that all group catalogues based on the FoF are likely to holda significant fraction of spurious groups.To mitigate the difficulty of identifying gravitationally bound groups a method wasdeveloped in Paper I that can be used to estimate the probability that an observed group .7. SUMMARIES OF PAPERS I AND II is a gravitationally bound system. This method however requires an estimate for the totalgroup mass, which may not be straightforward to obtain from the available observationaldata. Even so, this method has been successfully applied, for example, in Mendel et al.(2008). Paper II extended the study and the results of Paper I. It explains a long-standing phe-nomenon of discordant redshifts that was found from observational galaxy group cata-logues. The explanation of redshift asymmetries was found to be two fold: 1) gravita-tionally unbound groups and 2) the misidentification of the group centre. Together thesetwo difficulties can comprise a redshift asymmetry that is as large as found from obser-vational group catalogues. Thus, observational bias together with the Λ CDM cosmologycan naturally provide a satisfactory explanation for the discordant redshifts without theneed for a new physics or more complicated explanations, which had earlier been sug-gested in literature. HAPTER Galaxy Evolution “I hate tennis, hate it with all my heart, and still keep playing, keep hitting all morning, andall afternoon, because I have no choice.” Andre Agassi Chapter 2 briefly described the theory of structure formation, while Section 2.8 con-centrated on the theory of galaxy formation and evolution. It was noted that galaxyformation and evolution are the results of a complex sequence of events that occurredduring the structure formation. However, how does the theory relate to observations andwhat can the properties of galaxies tell us about their evolution? Thanks to the finitespeed of light, galaxies at earlier cosmic epochs can be studied simply by observingdistant galaxies. Hence, observations can help to set constraints for evolution of galax-ies. In this Chapter I therefore briefly summarise the observational constraints on galaxyevolution and present two theoretical case studies of the evolution of galaxies.The galaxies we observe in the Universe at the current epoch (Fig. 4.1) exhibit anenormous variety of properties such as morphologies, colours, luminosities, masses anddynamics (e.g. Blanton & Moustakas, 2009, and references therein). For example, mostregular elliptical galaxies have very low atomic gas content, albeit they often have sig-nificant hot ionised gas (Mathews & Brighenti, 2003). Instead, spiral galaxies frequentlyshow star formation, which is fuelled by cool molecular hydrogen. Thus, the galaxy pop-ulation is vast in their properties, but why do galaxies show such a variety? Is a givenproperty an indication of an evolutionary phase in the life of a galaxy, or have galaxieswith different properties formed and evolved in completely different ways?A goal of galaxy evolution studies is to reconstruct back in time the physical mecha-nisms that led to the present-day galaxies and to explain them. One of the main questionsin the study of galaxy evolution is whether initial conditions of the formation place (andtherefore time) will govern the galaxy evolution or if the surrounding environment willshape galaxies when they mature. This question is often dubbed shortly as ’nature ver-sus nurture’ dichotomy and has been a long standing puzzle in modern astrophysics. Iwill return to this important question in Section 4.3, but before delving into it, I shallbriefly describe the basic properties of galaxies and how they have evolved as a functionof cosmic time. .2. PROPERTIES OF GALAXIES The observed galaxy population, both locally and out to redshift ∼ , is found to beeffectively described as a combination of two distinct galaxy types: red, early-type (el-liptical or lenticular) galaxies lacking much star formation and blue, late-type (spiralor irregular) galaxies with active star formation (e.g. Strateva et al., 2001; Balogh et al.,2004b; Baldry et al., 2004; Bell et al., 2004; Kauffmann et al., 2004; Croton et al.,2005). In general, bluer galaxies are dominated by emission from young hot stars, whilered galaxies contain old stars and/or more attenuating dust. Note, however, that the in-tegrated colour of a galaxy does not necessarily correlate directly with age or the rate inwhich a given galaxy is forming stars, but can also be due to other effects such as dustor chemical evolution. I will return to this important matter in the following Sections.The two sequences of galaxies can also be identified in the luminosity function (LF;for a definition, see Eq. 3.8): the faint end of the luminosity function is dominated bythe blue cloud galaxies, while the bright end is dominated by the red, passively evolvinggalaxies (e.g. Bell et al., 2004). It has also been noted that while red galaxies constituteroughly one-fifth of the population, they produce about two-fifths of the total cosmicgalaxy luminosity density (Hogg et al., 2002). More importantly, observations implythat the total stellar mass density of galaxies on the red sequence has roughly doubledover the last − Gyr while that of blue galaxies has remained almost constant (e.g.Bell et al., 2004; Faber et al., 2007). If we assume that new stars form mostly in bluegalaxies, this suggests that galaxies are being transformed from the blue cloud to the redsequence. This transformation is assumed to take ∼ Gyr after the star formation ofthe blue cloud galaxy has been truncated (Mo et al., 2010). Thus, significant evolutionbetween the two sequences is assumed to take place as a function of cosmic time. Asa result, it should be kept in mind that even though the galaxy distribution of manyproperties is bimodal, all galaxies, irrespectively of morphology or colour, show greatvariations in the amounts and spatial distribution of gas, dust, stars, and metals as wellas their luminosities, surface brightnesses, and masses. This is illustrated in Fig. 4.1,where a small sample of local galaxies is being presented. The galaxy population is frequently phrased today in terms of the morphology-densityor colour-density relation, while the morphological diversity of galaxies is often dubbedas the galaxy zoo. As first noted by Hubble & Humason (1931) and later quantified The nomenclature of early- and late-type is historical and does not reflect the current understanding ofgalaxy evolution. CHAPTER 4. GALAXY EVOLUTION Figure 4.1: Galaxies are arranged in bins of increasing ultraviolet colour (the difference be-tween far and near ultraviolet flux). Those with relatively strong far ultraviolet emission appearblue, toward the left, and those with relatively strong near ultraviolet emission appear red, to-ward the right. Each colour bin is sorted vertically by far ultraviolet luminosity, with the mostluminous objects at the top. Courtesy of NASA/JPL-Caltech. by several other authors (e.g. Oemler, 1974; Davis & Geller, 1976; Dressler, 1980), themorphology-density relation holds that star-forming, disc-dominated spiral galaxies tendto reside in regions of lower galaxy density relative to those of ’red and dead’ elliptical .2. PROPERTIES OF GALAXIES galaxies (Croton & Farrar, 2008). This implies that the environment can affect at leastgalaxy morphologies and star formation history. Moreover, in disk galaxies, which arerotationally supported, the galaxy sizes are a measure of their specific angular momenta,while in case of elliptical galaxies, which are supported by random motions, the sizes area measure of the amount of dissipation during their formation. Hence, the morphologiesand dynamical differences of galaxies have been interpreted as evidence for differentevolutionary histories (Kennicutt, 1998a).In general, it is assumed that the environment local to a galaxy is most fundamental indetermining its morphology. The fact that the morphology of a galaxy is closely relatedto the density of galaxies in its vicinity, also implies that the local environment is of keyimportance in the formation and evolution of a galaxy. Because groups (see Chapter 3)form on time scales of gigayears, the morphology-density relation implies that at leastsome elliptical and lenticular galaxies are likely to be the result of mergers of spirals.Mergers are therefore probably responsible for a lot of the enhanced early-type fractionin groups and clusters. However, secular evolution (Sellwood & Merritt, 1994; Normanet al., 1996; Kormendy & Kennicutt, 2004) and disk instabilities can also change themorphology even in absence of mergers (e.g. Raha et al., 1991; Dekel et al., 2009b, butsee also Figure 2.8). Disk instabilities form when the disk of the galaxy grows largeenough and becomes unstable. Thus, morphology alone does not describe the formationhistory of a galaxy well, though, it can give hints from the possible past merging activity.The origin of the morphology-density relation can be a combination of several pro-cesses (i.e., ram-pressure and tidal stripping, strangulation, galaxy harassment, etc., seefor example Tasca et al. (2009)). Thus, whether the influence of the local environmentis felt at the time of formation or when the galaxy evolves, or during both phases, is notyet clear. I will, however, return to the question of environment in Sections 4.3 and 4.4. One of the main questions in the study of galaxy evolution is related to the star formationrate and history of galaxies and to the physics that triggers star formation (e.g. Kennicutt,1989; Madau et al., 1996, 1998; Kennicutt, 1998a; Bouwens et al., 2010). Section 2.8showed that as the gas in a dark matter halo cools its self-gravity will eventually domi-nate over that of the dark matter. It was noted that in general star formation of a galaxycan be assumed to be limited by the availability of hydrogen gas that can cool, fragmentand form new stars. Interestingly though, the efficiency of the conversion of moleculargas into stars has been argued to be nearly independent of the galaxy type, its large-scaleenvironment, or the particular local conditions within the galaxy (e.g. Rownd & Young,1999; Leroy et al., 2008). So, what controls the star formation?Based on observations (e.g. Kennicutt, 1998a, and references therein), it has beenargued that star formation takes place in two modes: quiescent star formation in gas CHAPTER 4. GALAXY EVOLUTION disks and circumnuclear starburst. In the disks of spiral galaxies star formation has beenobserved to proceed at a relatively low pace but in a continuous fashion. Since gas-richspirals are relatively common in the nearby Universe, disks must be replenished with in-fall of gas, or otherwise the present time must mark the end of the epoch of star formingdisk galaxies. It is also likely that the galaxies that are forming stars at present epochhave obtained cool molecular hydrogen relatively recently, because the typical star for-mation timescales are ∼ − Gyr. Indeed, observations (e.g. Solomon & Sanders,1980; Sanders et al., 1984; Frayer et al., 1998; Papadopoulos et al., 2001; Frye et al.,2008) suggest that many spiral galaxies, such as the Galaxy, have a hydrogen gas reser-voir around them. This reservoir is likely to consist of three parts: 1) gas that is justfalling onto the halo; 2) gas that has already fallen in and been shock heated, but whichhas not yet cooled radiatively; and 3) gas that has been reheated and expelled from thegalaxy due to feedback processes. As a consequence, galaxies with such reservoir canform new stars from the gas that infalls from the reservoir to the disk (Kormendy & Ken-nicutt, 2004). Additional to infall, also gas rich (the so-called wet) mergers can bringnew gas to a galaxy and induce star formation, even at late cosmic times. Moreover,mergers of galaxies of a roughly equal size (shortly major mergers), may also trigger anexceptionally high rate of star formation or starburst (e.g. Mihos & Hernquist, 1996).Unfortunately though, the exact role of mergers, the composition of molecular gas reser-voir, the form and potential evolution of the IMF (briefly mentioned in Section 2.8.4),and consequently the detailed physics of star formation are not yet well understood. Ifthe exact physical conditions for star formation are not well known, can the observationstell something about the global star formation rate evolution instead?The cosmic star formation rate density (SFRD) of the Universe has been observedto evolve significantly with redshift and to peak at z ∼ (e.g. Madau et al., 1998; Gi-avalisco et al., 2004; Bouwens et al., 2010, 2011). This is therefore the epoch when themajority of galaxies were growing most vigorously and forming stars fastest. This islikely true for today’s elliptical galaxies that show little star formation at current epochsand are therefore often considered to be ’red and dead’. However, this is also true glob-ally as the current SFRD is about a factor of ten smaller than at z ∼ (Bouwens et al.,2011) as Fig. 4.2 shows. Unfortunately, the obscuring effects of dust cast serious un-certainty over the interpretation of data. As a result, the star formation rate density asa function of cosmic time is somewhat uncertain as illustrated by the blue and orangeregions in the Figure. Despite the complication due to dust, the decline between z ∼ and z ∼ in the global trend of SFRD is obvious. On the other hand however, the rateof decline at high redshifts ( z > is more uncertain. This is mainly due to the factthat currently there are only a handful of galaxy candidates at such high redshifts (seee.g. Yan et al., 2010; Labbé et al., 2010; Bouwens et al., 2011). Moreover, usually thesecandidates have not been confirmed spectroscopically, but rely on the dropout technique, .2. PROPERTIES OF GALAXIES which may suffer from contamination. Thus, the SFRD of the early Universe will likelyremain an active area of research for years to come.Despite the uncertainties in SFRD, it has been argued that the observations indicatethat disk galaxies at higher redshifts have higher specific star formation rates (SSFRs).This can be interpreted such that the SFRD is not primarily driven by evolution in thefrequency of starburst, but rather reflects a decline in the typical SSFRs of star-forminggalaxies (for a detailed discussion, see e.g. Mo et al., 2010). I will return to this matterin Section 4.5. Figure 4.2: The star formation rate density (left axis) and luminosity density (right axis) as afunction of cosmic time. The lower set of points (and the blue region) shows the SFR densitydetermination inferred directly from the UV light, and the upper set of points (the orange region)shows what one would infer using dust corrections inferred from the UV-continuum slope mea-surements. The conversion from ultraviolet luminosity to star formation rate assumes a Salpeterinitial mass function. Image from Bouwens et al. (2011). As already pointed out, blue galaxies are dominantly made out of hot, young, blue stars.As these stars are short lived, one could naïvely assume that blue galaxies are on averageyounger than red galaxies. Unfortunately though, another element closely related to starformation, which is not related to the age of the stellar populations, can affect the colourof a galaxy, namely the chemical evolution of stellar populations (e.g. Tinsley, 1980; Fall& Pei, 1993; Pei & Fall, 1995; Pagel, 1997; Pipino & Matteucci, 2004; Tantalo et al., CHAPTER 4. GALAXY EVOLUTION . As a result, galaxiesthat are forming stars at the current epoch, late in cosmic time scales, have on averagestellar populations that are more metal rich (Population I stars) than stellar populationsof non-starforming galaxies (Population II stars). Note, however, that if a galaxy canobtain pristine gas, that is gas that has not yet been polluted by metals, and such gas isnot mixed with the existing ISM, the galaxy can form metal poor stars also at late cosmicepochs. In general though, this may not be so simple, because the IGM is also assumedto be metal-enriched.Usually Population I stars have been found to populate the plane of the disk in spiralgalaxies, while Population II stars are located in the spheroidal component of galaxies(for a review of stellar populations in the Galaxy, see Mould, 1982). What does this tellabout the evolution of galaxies? Stellar evolutionary models show that typical Pop IIstars are old and of low metallicity. As Pop II stars are usually older than ∼ Gyr, theymust have formed early in the formation history of a galaxy. Consequently, ellipticalgalaxies that are mainly made out of Pop II stars, must have formed most of their starsearly. Note, however, that this does not necessarily imply that elliptical galaxies havealso assembled at high redshifts. I will return to the formation and assembly times ofelliptical galaxies in Section 4.4.4. For now, we can conclude that it is likely that thestellar population that forms an elliptical galaxy has formed a long time ago. How aboutdisk galaxies then?A fraction of Pop I O and B stars are as young as ∼ yr, implying that the disksof spiral galaxies assemble by relatively continuous infall of gas. The existence of PopI stars could also lead us to conclude that spiral galaxies are predominantly blue andcontain a young stellar population. While the latter is at least partially true, the formerstatement is more complicated due to the chemical evolution. One should note that aconfusion in the colours and ages of galaxies arise from the fact that metal rich stars, i.e.,Population I stars, have lower temperatures than metal poor stars of the same age andmass, and thus, they end up looking redder. This results in the so-called age-metallicitydegeneracy that complicates the age estimates of all galaxies.The chemical evolution of the gas and stars in galaxies is important also for otherreasons. The cooling efficiency of gas has been found to depend strongly on its metal- In Astronomy every element more massive than helium is called a metal. .3. ENVIRONMENTAL DEPENDENCE: NATURE VERSUS NURTURE licity, as noted in Section 2.8.4. As the metal-enriched gas can cool faster, the chemicalevolution of galaxies may offset the star formation rates in these galaxies. Additionally,the extinction by dust (Section 4.5.2), which is believed to be produced in the envelopesof AGB stars and injected into the ISM through stellar winds, is assumed to depend on,for example, the chemical composition of dust grains (Mo et al., 2010). Consequently,the amount of dust in the ISM is assumed to scale with its metallicity. I will return tothe importance of ISM and dust when discussing the case study of luminous infraredgalaxies in Section 4.5. From the theoretical perspective the environmental dependency can be divided into twoparts: those related to the host dark matter halo and those of larger scales (filaments,sheet, etc.). As discussed in Section 2.8.3, the properties of a galaxy are assumed todepend strongly on the properties of the host halo, however, the effect of larger scalesare less clear. One complication is that the dynamical times on scales significantly largerthan dark matter haloes are large. As a result, the gravitational processes have had onlylittle time to make an impact. It is, however, possible that non-gravitational effects mayplay a role, so we should not exclude the option lightly. Independent of the scale, thecomplication of studying environmental dependencies arises from the fact that a cor-relation between galaxy properties and an environmental property does not necessarilyimply causality.Chapter 2 described how dark matter haloes form from the initial quantum densityfluctuations of the early Universe. It is therefore natural to assume that the formationplace in space affects the halo collapse. As a consequence, the initial conditions andthe formation place of the dark matter halo should also have an effect on the galaxy thatforms in the centre of the halo. The part of galaxy evolution, which is imprinted to theinitial conditions of dark matter halo formation is often dubbed as ’nature’. Observationshave shown that on average more massive dark matter haloes host more massive galaxies(e.g. Guzik & Seljak, 2002; Prada et al., 2003; Hoekstra et al., 2004; Conroy et al., 2007;Cacciato et al., 2009). One simple explanation for such a correlation can be providedby the argument that more massive haloes form from the larger volumes, which containmore baryons. It has also been noted that, on average, central galaxies in more massivehaloes are also redder and more centrally concentrated (Mo et al., 2010). These findingsprovide support for the idea that the immediate environment, i.e. the dark matter halo,can affect the evolution of a galaxy residing in it. Thus, ’nature’ should be consideredan important part of galaxy formation and evolution.In Section 2.8.3 it was noted that dark matter haloes and galaxies are not static nor dothey remain unaltered throughout the cosmic evolution of Universe. Observations (e.g. CHAPTER 4. GALAXY EVOLUTION van Dokkum, 2005; Bell et al., 2006b,a; Lin et al., 2008; Tacconi et al., 2008; Bundyet al., 2009; Kormendy et al., 2009), theory, and simulations (e.g. Hausman & Ostriker,1978; Lacey & Cole, 1993; Hernquist & Mihos, 1995; Springel & Hernquist, 2005;Boylan-Kolchin et al., 2006; Naab et al., 2006; Bournaud et al., 2007; Fakhouri & Ma,2008; Khochfar & Silk, 2009; Stewart et al., 2009) have shown that galaxies and darkmatter haloes can merge in dense environments. The galaxy properties can therefore beaffected by environment through physical mechanisms acting on galaxies. As a result,the surrounding environment of a galaxy and a dark matter halo it resides in should alsohave an effect on their evolution. The effects directly linked to the physical mechanismscaused by the environment (briefly mentioned in Chapter 3) are often described in acollective manner with a word ’nurture’. It therefore seems that also ’nurture’ plays arole at least in the formation and evolution of some galaxies.It is obvious that the properties of galaxies are correlated with their environment,as was noted when discussing the morphology-density relation, but what physical pro-cesses are likely to drive these correlations? Interestingly, much of the environmentaldependency can be understood in terms of the group environment, discussed briefly inChapter 3. For example, observations seem to favour an important role for preprocessingof galaxies in groups, possibly by mergers (e.g. Mihos, 2004), by gas-dynamic interac-tions with warm or hot gas (e.g. Fujita, 2004), or by tidal harassment or ram pressurestripping (e.g. Moore et al., 1996, 1998). However, some evidence (e.g. Kauffmannet al., 2004; Blanton, 2006) have accumulated lately that seem to support the oppositeconclusion that the large-scale density field appears to be less important than what masshalo hosts the galaxy and what its position is within the halo (e.g. Blanton & Berlind,2007, and references therein). Thus, it has been argued that environmental effects maybe relatively local even at low density (Blanton & Moustakas, 2009). Hence, the debateover the roles of nature and nurture and their physical mechanisms continues. The envi-ronmental effects will be discussed next in a context of a case study of isolated ellipticalgalaxies. The merger hypothesis (Toomre & Toomre, 1972) suggests that the product of the mergerof two spiral galaxies will be an elliptical galaxy. If this holds, then the probability to findan elliptical galaxy is larger in environments with high densities and low velocity disper-sions, i.e., groups of galaxies. This is in agreement with the morphology-density relationthat has shown that majority of elliptical galaxies are located in dense regions. However,observations (e.g. Smith et al., 2004; Reda et al., 2004; Hernández-Toledo et al., 2008;Norberg et al., 2008, and references therein) have shown that elliptical galaxies can also .4. A CASE STUDY: ISOLATED FIELD ELLIPTICAL GALAXIES be found from the field. These galaxies are often dubbed as isolated field elliptical galax-ies (IfEs) and are considered an unusual class among the galaxy zoo, because of their"misplacement". Hence, a study that identifies and describes their properties in detailis important if we are to understand the big picture of the formation and evolution ofelliptical galaxies and the effects of the environment and formation place. In the past two decades several studies based on observations have identified and studiedthe properties of isolated field elliptical galaxies (e.g. Smith et al., 2004; Reda et al.,2004; Hernández-Toledo et al., 2008; Norberg et al., 2008, and references therein). Inmany of these studies different possible formation scenarios have been proposed (see e.g.Mulchaey & Zabludoff, 1999; Reda et al., 2004, 2007), ranging from a clumpy collapseat an early epoch to multiple merging events. Also equal-mass mergers of two massivegalaxies or collapsed groups have been suggested.Observational studies have also shown that several IfEs reveal a number of featuressuch as tidal tails, dust, shells, discy and boxy isophotes and rapidly rotating discs (e.g.Reduzzi et al., 1996; Reda et al., 2004, 2005; Hau & Forbes, 2006; Hernández-Toledoet al., 2008) indicating recent merger and/or accretion events. Some observational evi-dence suggests that some isolated elliptical galaxies may have suffered late dry mergers(Hernández-Toledo et al., 2008), while others could have formed via a major merger oftwo massive galaxies (Reda et al., 2004, 2005). A collapsed poor group of a few galaxieshas also been suggested as a possible formation scenario (Mulchaey & Zabludoff, 1999),but other studies (e.g. Marcum et al., 2004) have concluded that isolated systems are un-derluminous by at least a magnitude compared with objects identified as merged groupremnants. In general, studies based on observations (e.g. Reda et al., 2007) have con-cluded very broadly that mergers at different redshifts of progenitors of different massratios and gas fractions are needed to reproduce the observed properties of IfEs.Despite the evidence, some IfEs do not show any signs of recent merging activity(e.g. Aars et al., 2001; Denicoló et al., 2005) complicating the picture of IfE formationeven further. It is possible that the merging events have happened in distant past, and allsigns of these events have been wiped out. For some IfEs this is even likely as mergerremnants usually appear morphologically indistinguishable from a “typical” elliptical ≤ Gyr after the galaxies merged (Combes et al., 1995; Mihos, 1995). In reality, however,the merger observability time-scales depend on the method used to identify the mergeras well as the gas fraction, pericentric distance and relative orientation of the merginggalaxies (Lotz et al., 2008), rendering the time-scale in which a merger can still beobserved uncertain. As a consequence, the lack of observable evidence does not excludethe possibility of mergers. On the other hand, it is also possible that some or even all ofthese galaxies have initially formed in underdense regions and developed quietly without CHAPTER 4. GALAXY EVOLUTION any major mergers and disturbances. It is therefore unclear from the observational pointof view how IfEs form and evolve and whether nature or nurture, or both, are importantfor the evolution of IfEs. These questions were the main driver of Paper III, which tookadvantage of a semi-analytical galaxy formation model to study the physical properties,evolution, and formation of IfEs. Before discussing the evolution of IfEs in detail, letsbriefly look into their basic properties and what hints these properties may give aboutthe evolution of IfEs. It was noted in Paper III that the number density of isolated field elliptical galaxiesis as low as ∼ . × − h Mpc − . That is a few orders of magnitude lower thanthat of local luminous spiral galaxies, however, as Paper III shows the number densityof IfEs is tightly coupled to the identification criteria adopted. For example, up to anorder of magnitude more IfEs can be identified if the criteria are relaxed. This howeverwill affect the immediate environment from which the IfEs are identified, resulting ina sample of less isolated galaxies that can have relatively massive companions. Hence,one can readily conclude that IfEs are relatively rare objects: more isolated ones beingthe rarest.Observational studies (e.g. Marcum et al., 2004) have found that IfEs show globalblue colours compared to other elliptical galaxies. Blue colours can imply that IfEshave formed on average later than group and cluster elliptical galaxies, however, a re-cent merging event or a gas reservoir could have supplied new gas and induced starformation rendering the integrated colours of the IfEs bluer. Moreover, due to the colour-metallicity degeneracy it is unclear whether the blue global colours should be interpretedas a measure of age or metallicity or both. Hence, from the observational point of viewit is unclear if blue colours mean that IfEs are younger than cluster ellipticals, if theirmetallicities are different, or both.The median mass weighted age of simulated IfEs was found to be ∼ . Gyr (PaperIII). The number density of young (mass weighted age < Gyr) IfEs is extremely low,giving a lower limit for the age of the IfE population. These age estimates are in agree-ment with some observations (e.g. Collobert et al., 2006), while others (e.g. Reda et al.,2005; Proctor et al., 2005) have quoted age estimates around Gyr. Note, however, thatthe observational age estimates use a different definition (luminosity weighted age) thanthe values quoted for simulated IfEs, rendering a direct comparison less conclusive. Inany event, Collobert et al. (2006) derived a broad range of stellar ages for their IfEs;ranging from ∼ to Gyr in a modest agreement with the simulated IfEs of Paper III.The big scatter in age estimates suggest that the formation of IfEs is not concentrated ata fixed epoch, which further implies that there may also be a large scatter in the basicproperties of IfEs. .4. A CASE STUDY: ISOLATED FIELD ELLIPTICAL GALAXIES Paper III shows that on average IfEs have a few ( ∼ - companion galaxies.This result alone indicates that IfEs should reside in relatively low density environments,unlike groups (see Chapter 3) and clusters, which usually contain ∼ and ∼ to galaxies, respectively. Moreover, when the number of IfE companions is (cid:46) as in the case of the most IfEs, most of the surrounding galaxies were found within ∼ . h − Mpc from the IfE. If IfEs were to reside in cluster-type dark matter haloesand environment, companion galaxies should also be found further away from the IfE.Even though for a given IfE the number of companions is small, their properties maystill provide some information for the evolution of IfEs and their dark matter haloes.For example, observations (e.g. Reda et al., 2004) have found that only the very faintdwarf galaxies ( M R ≥ − . ) appear to be associated with isolated ellipticals. On thecontrary, simulated IfEs show companion galaxies with a broad range of magnitudes.Deeper observations and larger samples are therefore required to identify more dwarfcompanions that should surround IfEs as predicted by simulations based on the Λ CDMcosmology.The simulated isolated field elliptical galaxies are mainly found to reside in darkmatter haloes that are lighter than × h − M (cid:12) , while even the most massive darkmatter halo hosting an IfE is lighter than . × h − M (cid:12) . These masses are compa-rable to a dark matter halo of a small group thus rendering it impossible that IfEs werecollapsed rich groups. Memola et al. (2009) calculated the total masses of two of theirisolated ellipticals NGC 7052 and NGC 7785 from X-ray observations and quote values ∼ × M (cid:12) and ∼ . × M (cid:12) , respectively. These mass estimates agree wellwith the masses of dark matter haloes of simulated IfEs, giving support for the idea thatIfEs reside inside relatively light dark matter haloes. In general, galaxies in massivedark matter haloes tend to form the bulk of their stars already at a very early cosmicepoch, rendering it likely that some IfEs contain younger stellar populations than thosein cluster ellipticals. The large scatter in the ages of IfEs suggest that they have formed at different times andunder different circumstances. Is this interpretation correct? To quantify the evolutionof IfEs one must first define a general set of times related to the formation and evolutionof dark matter haloes and galaxies that reside in the haloes. Briefly, the different timesthat are of interest can be defined as follows (followed by Lucia & Blaizot, 2007): • Assembly time ( z a ) is the redshift when per cent of the final stellar mass isalready present in a single galaxy of the merger tree; • Identity time ( z i ) is the redshift when the last major (the two progenitors bothcontain at least per cent of the stellar mass of the descendant galaxy) merger CHAPTER 4. GALAXY EVOLUTION occurred; • Formation time ( z f ) is the redshift when per cent of the mass of the stars in thefinal galaxy at z = 0 have already formed; • Last merging time ( z l ) is the redshift when the last merger occurred.Using these definitions, it can be shown (for details, see Paper III) that isolated fieldelliptical galaxies have assembled at lower redshifts than other elliptical galaxies. Onaverage, the stars of an IfE assemble to the main halo later than the stars of group andcluster ellipticals. This is in agreement with the conventional theory, which states thathigher density areas collapse earlier than less dense areas. Note, however, that in thehierarchical merger scenario, the star formation history (related to the formation time)and assembly history of an elliptical galaxy can be very different. For example, in somecases the stars that will eventually form a simulated IfE galaxy are present already athigh redshifts (high z f ), in agreement with some observational findings (e.g. Reda et al.,2005), even though the assembly of the final galaxy will happen late (small z a ). Theresults of Paper III also show that IfEs undergo their possible major merging events atsignificantly lower redshifts than group and cluster ellipticals. This gives further supportfor the conclusion that IfEs assemble late.The mass accretion of IfEs differs from the mass accretion of other elliptical galax-ies according to the findings of Paper III. In general, IfEs seem to form (or accrete) starsmore efficiently than group and cluster ellipticals. However, oppose to this, IfEs seem toaccrete dark matter slightly slower than the comparison galaxies. Nevertheless, simula-tions confirm that IfEs form the bulk of their stars at z > , as suggested in Reda et al.(2005), and show that at a redshift of one IfEs have formed over half of their stars (stellarmass) and have gathered as much as per cent of their final dark matter. Similarly, thegroup and cluster ellipticals have, on average, accreted roughly the same fraction of darkmatter as IfEs at z ∼ , however, IfEs continue to accrete dark matter till z = 0 , whilethe group and cluster ellipticals have gathered ∼ per cent of their final dark matteralready at z ∼ . . All these results point towards a different formation mechanismfor isolated and cluster elliptical galaxies and also suggest that late merging or accretionevents are likely to be a significant part of the evolution of an IfE. Formation classes Early theoretical studies predicted that isolated ellipticals are formed in relatively re-cent mergers of spiral galaxy pairs, while large isolated ellipticals may be the resultof the merging of a small group of galaxies (e.g. Jones et al., 2000; D’Onghia et al.,2005). However, these results are in disagreement with some observations (Section4.4.2). Moreover, the result that the majority of IfEs reside in light dark matter haloes .5. A CASE STUDY: LUMINOUS INFRARED GALAXIES disagrees with the hypothesis of merging of a group. Hence, the obvious question is ifsome IfEs are descendants of merged groups while others have developed more quietly,and what are the possible formation mechanisms.In Paper III three different yet typical formation mechanisms were identified. Inthe first formation mechanism, named “solitude”, the IfE develops quietly without anymajor or minor mergers. Solitude IfEs start to form later than other IfEs and their darkmatter haloes are usually lighter than haloes of IfEs that form via different mechanisms.The second formation scenario, called “coupling”, comprise of IfEs that have undergoneat least one “equal” sized merger during their evolution. The third and the last formationmechanism identified in Paper III is called “cannibalism”. IfEs of this formation classundergo several minor and also possibly major mergers during their evolution. Theevolution of the cannibal IfEs is significantly impacted by the merging events and theirdark matter haloes are usually the most massive ones among all IfEs. Consequently,they can be collapsed poor groups as suggested by, e.g., Mulchaey & Zabludoff (1999).Interestingly, all three formation scenarios are in agreement with selected observations.This shows that the formation of an IfE is not a simple process that could be quantifiedusing one or two simple quantities. Most importantly, Paper III also shows that an IfEcan be formed without major mergers. This clearly implies that disk instabilities shouldbe significant in the evolution of some isolated field elliptical galaxies.Paper III also discusses possible observational techniques that could distinguish be-tween each formation class. Future observational studies can therefore try to identifywhich formation and evolution process might have been the driving force behind theevolution of an observed IfE. Identification of a formation class can further aid in thequest of understanding the galaxy evolution in different environments and help to attackthe fundamental dichotomy of nature vs nurture. It was noted in Section 2.8 that galaxies are not made solely of stars. Instead, a pic-ture was painted in which the stars that form a galaxy are more likely embedded inan interstellar medium (ISM) consisting not only of hot and cold gas, but also dust.Consequently, one of the most important discoveries from extragalactic observations atmid- and far-infrared has been the identification of luminous and ultra-luminous infraredbright galaxies (LIRGs; L IR > L (cid:12) and ULIRGs; L IR > L (cid:12) , respectively) andthey have been studied extensively in the literature (e.g. Rieke & Low, 1972; Harwitet al., 1987; Sanders et al., 1988, 1991; Condon et al., 1991; Sanders & Mirabel, 1996;Auriere et al., 1996; Duc et al., 1997; Genzel et al., 1998; Lutz et al., 1998; Rigopoulouet al., 1999; Rowan-Robinson, 2000; Genzel et al., 2001; Colina et al., 2001; Colbert CHAPTER 4. GALAXY EVOLUTION et al., 2006; Dasyra et al., 2006; Hernán-Caballero et al., 2009; Magdis et al., 2011).These objects emit more energy in the infrared ( ∼ − µ m) than at all other wave-lengths combined. Even though the (ultra-) luminous infrared galaxies are reasonablyrare objects in the Universe, reasonable assumptions about the lifetime of the infraredphase suggest that a substantial fraction of all galaxies pass through a stage of intenseinfrared emission (Sanders & Mirabel, 1996, and references therein). Consequently, themajority of the most luminous galaxies in the Universe emit the bulk of their energy inthe far-infrared, rendering IR extremely interesting wavelength regime to study, espe-cially in the context of galaxy formation and evolution.Understanding the IR bright galaxies is especially important because light frombright, young blue stars is often attenuated by dust. The rest-frame ultra-violet (UV)light of a galaxy may therefore provide a biased view on, for example, star formationrate in the galaxy. For example, the global star formation rate required to explain thefar-infrared and sub-millimetre background appears to be higher than that inferred fromthe data in the UV-optical (e.g. Mo et al., 2010). The dust attenuated light from bluestars is however not lost, but assumed to be re-radiated thermally at IR wavelengths. Asa result, a large fraction of radiation of the cosmic star formation is radiated not at UVbut at IR rest-frame wavelengths. This has been found to be true to the extent that themajority of a luminous star-forming galaxy’s energy is emitted in the IR (for a review ofIR bright galaxies, see Sanders & Mirabel, 1996). It is therefore important to understandthe properties and evolution of IR bright galaxies when trying to understand the cosmicstar formation history, and galaxy formation and evolution. However, before we canconcentrate on these questions, the modelling of dust attenuation and emission shouldbe briefly discussed. The ISM is assumed to be complex structure (see, for example, the textbooks of Kaplan& Pikelner, 1970; Osterbrock, 1989), complicating an accurate modelling of it and itsinteractions with the interstellar radiation. It is usually assumed, however, that most ofthe dust in the ISM is produced by AGB stars and injected into the ISM through stellarwinds. It is therefore likely that the extinction depends on the physical properties of dustgrains, which may vary even within a galaxy. Obviously, it is possible to try to quantifythe dust extinction by using a statistical description and simplified physics. For example,in semi-analytical models of galaxy formation (Section 2.8.4) the dust extinction byISM is often assumed to follow simply from the diffuse dust in the disc of a galaxy andfrom a second component, which is associated with the dense ‘birth clouds’ surroundingyoung star forming regions (Charlot & Fall, 2000). Such simplification now allows thecomputation of the total fraction of the energy emitted by stars that is absorbed by dust,over all wavelengths. If one then assumes that all of this absorbed energy is re-radiated .5. A CASE STUDY: LUMINOUS INFRARED GALAXIES in the IR (hereby neglecting scattering), one can thereby compute the total IR luminosity L IR of each galaxy. Finally, it is then possible to make use of dust emission templatesto determine the spectral energy distribution (SED) of the dust emission, based on thehypothesis that the shape of the dust SED is well-correlated with L IR . The underlyingphysical notion is that the distribution of dust temperatures is set by the intensity of thelocal radiation field; thus more luminous or actively star forming galaxies should have alarger proportion of warm dust, as observations (e.g. Sanders & Mirabel, 1996) seem toimply.There are two basic kinds of approaches for constructing these sorts of templates.The first is to use a dust model along with either numerical or analytic solutions to thestandard radiative transfer equations to create a library of templates, calibrated by com-parison with local prototypes. This approach was pioneered by Desert et al. (1990) whoposited three main sources of dust emission: polycyclic aromatic hydrocarbons (PAHs),very small grains and big grains. The latter are composed of graphite and silicates,with small and big grains probably dominated by graphite and silicate respectively. Thethermal properties of each species are determined by the size distribution and thermalstate. Big grains are assumed to be in near thermal equilibrium, and their emission canbe modelled as a modified black-body spectrum. However, small grains and PAHs areprobably in a state that is intermediate between thermal equilibrium and single photonheating. They are therefore subject to temperature fluctuations and their emission spec-tra are much broader than a modified black-body spectrum. In the Desert et al. (1990)type approach, the detailed size distributions are modelled using free parameters, whichare calibrated by requiring the model to fit a set of observational constraints, such as theextinction or attenuation curves, observed IR colours and the IR spectra of local galaxies.The second approach is to make direct use of observed SEDs (e.g. Chary & Elbaz, 2001;Dale et al., 2001; Rieke et al., 2009) for a set of prototype galaxies and to attempt tointerpolate between them, allowing to determine the SED of the dust emission. Finally,after a proper SED of the dust emission has been derived, it can be used together withthe total IR luminosity of a given galaxy to compute the flux of the galaxy at any givenIR band. Although the brightness of a galaxy at a given band is relatively simple to derive fromobservations, the physical properties are often more important quantities for galaxy for-mation and evolution. However, when deriving physical properties from observationaldata, several assumptions are usually required. A priori predictions are therefore of-ten useful when interpreting observational results and drawing conclusions. Predictionsfor physical properties such as sizes of late-type galaxies, stellar masses, star formationrates, and merger activity of IR bright galaxies are therefore briefly discussed. CHAPTER 4. GALAXY EVOLUTION The simulated galaxies of Paper IV imply that on average more massive disk galax-ies have larger disks. This trend, however, seems to be driven mainly by the galaxieswith less massive stellar populations (log ( M (cid:63) /M (cid:12) ) < . . Consequently, the trendmostly disappears when limiting to only IR bright galaxies (with µ m flux S > mJy). The late-type high-redshift galaxies contain stellar disks which on average are ∼ . kpc in size. This is significantly larger than the mean disk size ( ∼ . kpc) of alllate-type galaxies in the same redshift range (2 ≤ z < . This small disk size for allhigh-redshift late-type galaxies is however driven by the galaxies with the lightest stellardisks (as their number density is highest), which, on average, contain the smallest disks.However, even if the stellar masses are matched the average size of all disk galaxiesis almost a factor of two smaller ( ∼ . kpc ) than the mean disk size of the IR brightgalaxies. This implies that at high-redshift IR bright galaxies contain on average largerstellar disks than their IR faint counterparts. Interestingly thought, the largest disks arenot always associated with galaxies with the highest stellar masses. Instead, they seemto be distributed rather equally for all stellar masses. Thus, the formation of stellar ma-terial, while important in general, is not the only quantity important for sizes of stellardisks of IR bright galaxies. Even so, Paper IV shows that at high redshift (2 ≤ z < the IR observations are likely to probe the galaxies with the highest stellar masses. Thiswill obviously bias the observational results if not taken into account, as only the tip ofthe iceberg is being probed.Paper IV also shows that at lower redshifts ( z < . the IR bright galaxies canbe found to reside in dark matter haloes as light as log ( M DM /M (cid:12) ) ∼ . . Instead,at higher redshifts (2 ≤ z < Paper IV predicts that all galaxies with S > mJy reside in relatively massive dark matter haloes. Even so, it should be noted that themasses of dark matter haloes cover a broad range from log ( M DM /M (cid:12) ) ∼ . to . at higher redshifts. Despite the broad range the bulk of IR bright galaxies can be foundto reside in dark matter haloes with masses log ( M DM /M (cid:12) ) ∼ . . Interestingly,the simulated galaxies show evidence for a weak correlation between the dark matterhalo mass and the 250 micron IR flux. Statistically speaking galaxies residing in moremassive dark matter haloes emit a higher median IR flux, however, this trend was notedto be weak. Moreover, for the high-redshift galaxies (2 ≤ z < with S > mJy nostatistically significant correlation can be confirmed. Thus, something else than what isdirectly linked to the mass of a dark matter halo must drive the IR flux.If dark matter halo masses and stellar disk sizes show only weak correlations withthe IR fluxes at best, how about star formation? It is hardly surprising that on average the250 micron IR flux correlates well with a star formation rate (SFR): the higher the me-dian S flux the higher the star formation rate (e.g. Paper IV and references therein).When moving towards earlier cosmic times it becomes clear that higher and higher starformation rates are required for galaxies to be detected in currently available IR observa- .5. A CASE STUDY: LUMINOUS INFRARED GALAXIES tions. At z > the galaxies directly detectable, for example, with Herschel have medianstar formation rates > M (cid:12) yr − , while the average SFR is ∼ M (cid:12) yr − . This issignificantly higher than for all galaxies in the same redshift range, for which Paper IVpredicts a mean SFR of merely ∼ M (cid:12) yr − . For high-redshift IR bright galaxies thehighest star formation rates predicted are as high as a few thousand M (cid:12) yr − , raising aninteresting question: what can cause and fuel such high star formation rates? Paper IV shows that at high redshift (2 ≤ z < the majority of IR bright ( S > mJy) galaxies has experienced a merger during their formation process. To be precise,out of all high redshift IR bright galaxies ∼ per cent have merged with another galaxyat some point in their formation history. This alone, however, does not yet imply that amerger activity would be behind the high star formation rates. When only concentratingon major mergers that are assumed to trigger starburst (Section 2.8.3), it was noted thatabout half of the IR bright galaxies had experienced a major merger. These results showthat a large fraction of all IR bright galaxies have experienced a major merger and as aconsequence experienced a starburst. Could this be the reason for high SFRs?As starbursts are usually relatively short lived phenomenon, more important quantitythan the fraction of mergers to look for, is the time since the last merger. Paper IVshows that about 84 (53) per cent of high-redshift galaxies with S > mJy haveexperienced a (major) merger during their lifetime. Note, however, that the fractiondrops to about percent if we concentrate on major mergers that have taken place lessthan Myr ago, which are the mergers that are likely to be causally linked to highstar formation rates. For the micron band, the results are very similar; about per cent of high-redshift galaxies with S > mJy have experienced a (major)merger. If we again concentrate only on recent mergers in which the major mergingevent took place less than Myr ago, the merger fraction drops to ∼ per cent.Thus the model used in Paper IV makes an interesting prediction: a significant fraction(half or more) of IR-luminous galaxies at high redshift ( z > ) have not experienceda recent merger. This implies that the high gas accretion rates and efficient feeding viacold flows predicted by cosmological simulations at high redshift can fuel a significantfraction of the galaxies detected by Herschel . Interestingly, this appears consistent withpreliminary observational results of Sturm et al. (2010) who concluded that their twogalaxies add support to recent results which indicate that with an increased gas reservoirstar forming galaxies at high redshifts can achieve ultra-high luminosities without beingmajor mergers. However, as the samples are still small, more observational evidence isrequired for more robust conclusions. A major merger is here defined as a galaxy-galaxy merger in which the mass ratio of the merging pairis > . CHAPTER 4. GALAXY EVOLUTION In Paper III isolated field elliptical galaxies (IfEs) were studied in detail by using cosmo-logical N -body simulations. Elliptical galaxies are usually found to be red and located indense environments such as cores of groups and clusters of galaxies, unlike IfEs. Hence,the obvious question Paper III sought answer to was how do elliptical galaxies form inunderdense regions?Paper III shows that the formation of isolated field elliptical galaxies is not linked tofossil groups, albeit this has been suggested before, as they reside in significantly lighterdark matter haloes. This also renders the argument, also suggested in literature, thatthe majority of IfEs are progenitors of collapsed groups, impossible. Instead, simula-tions show that three different yet typical formation scenarios can lead to an IfE all inagreement with current observations. It was also noted in Paper III that IfEs have highbaryon to dark matter fractions, making IfEs therefore good candidates for studies ofdark matter poor haloes and their host galaxies.The most significant result of Paper III is a prediction of a previously unobservedpopulation of faint and blue isolated field elliptical galaxies. This population comprises ∼ per cent of all IfEs, thus, a significant number of IfEs identified from simulationsseem to be missing in observations (although, see Kannappan et al., 2009, for possiblecandidates). Deeper observations of the surroundings of faint and blue elliptical galaxiesare required if we are to identify this population. Moreover, these galaxies show signifi-cant star formation and can therefore be important for the study of evolution of ellipticalgalaxies as a whole. Thus, an effort to identify them in the future should be made using,for example, large redshift surveys or dedicated observations. With the Herschel space observatory, we can finally probe the infrared light of faintgalaxies all the way from 70 to 500 microns. Unfortunately, at such long wavelengthscontamination and crowding becomes often a limiting factor rather than the depth of theobservations. This is especially true at higher redshifts. Interpretation of IR observationscan therefore be complicated and less than robust. To help to overcome such complica-tions a theoretical study of high-redshift (2 ≤ z < infrared luminous galaxies wasundertaken in Paper IV. The simulated galaxies of Paper IV were generated using a semi-analytical galaxy formation model and they were used to study the evolution of galaxiesin the Herschel IR bands and to make detailed predictions for the physical propertiesand evolution of luminous IR galaxies in hope that the study could help to shed somelight on galaxy evolution. Specifically, the goal of Paper IV was to present quantitative .6. SUMMARIES OF PAPERS III AND IV predictions for the relationship between the observed SPIRE 250 micron flux and phys-ical quantities such as halo mass, stellar mass, cold gas mass, star formation rate, andtotal infrared luminosity, at different redshifts. The goal was also to quantify the corre-lation between SPIRE 250 micron flux and the probability that a galaxy has experienceda recent major or minor merger.Results of Paper IV imply, for example, that SPIRE detectable galaxies are on aver-age significantly larger than IR faint galaxies with comparable stellar masses. The resultsalso show that in case of luminous IR galaxies the largest stellar disks are not always as-sociated with the galaxies of the highest stellar masses. If so, how has all the stellar massformed then? Not surprisingly Paper IV shows that at high redshift (2 ≤ z < the av-erage star formation rate of luminous IR galaxies is relatively high ( ∼ M (cid:12) yr − ).For high-redshift galaxies the highest SFRs found from the simulation are as high as afew thousand M (cid:12) yr − , leading to the question what fuels such a rapid star formation.Paper IV tries to answer this question, and shows that about per cent of the sim-ulated high-redshift IR bright galaxies have experienced a recent merger. Almost per cent of these mergers have been major events that are assumed to trigger starburst,emphasising the importance of major mergers for IR brightness. Furthermore, PaperIV finds a fairly strong trend between the 250 micron flux and the probability that agalaxy has had a recent merger, indicating that brighter galaxies are more likely to bemerger driven. 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