Understanding Galaxy Formation and

Evolution

Vladimir Avila-Reese1

Instituto de Astronom´ıa, Universidad Nacional Autónoma de México, A.P. 70-264,

04510, México,D.F. avila@astroscu.unam.mx

The old dream of integrating into one the study of micro and macrocosmos

is now a reality. Cosmology, astrophysics, and particle physics intersect in a

scenario (but still not a theory) of cosmic structure formation and evolution

called Λ Cold Dark Matter (ΛCDM) model. This scenario emerged mainly to

explain the origin of galaxies. In these lecture notes, I first present a review

of the main galaxy properties, highlighting the questions that any theory of

galaxy formation should explain. Then, the cosmological framework and the

main aspects of primordial perturbation generation and evolution are ped-

agogically detached. Next, I focus on the “dark side” of galaxy formation,

presenting a review on ΛCDM halo assembling and properties, and on the

main candidates for non–baryonic dark matter. It is shown how the nature of

elemental particles can influence on the features of galaxies and their systems.

Finally, the complex processes of baryon dissipation inside the non–linearly

evolving CDM halos, formation of disks and spheroids, and transformation

of gas into stars are briefly described, remarking on the possibility of a few

driving factors and parameters able to explain the main body of galaxy prop-

erties. A summary and a discussion of some of the issues and open problems

of the ΛCDM paradigm are given in the final part of these notes.

arXiv:astro-ph/0605212v1 9 May 2006

1 Introduction

Our vision of the cosmic world and in particular of the whole Universe has

been changing dramatically in the last century. As we will see, galaxies were

repeatedly the main protagonist in the scene of these changes. It is about

80 years since E. Hubble established the nature of galaxies as gigantic self-

bound stellar systems and used their kinematics to show that the Universe as

a whole is expanding uniformly at the present time. Galaxies, as the building

blocks of the Universe, are also tracers of its large–scale structure and of its

evolution in the last 13 Gyrs or more. By looking inside galaxies we find

that they are the arena where stars form, evolve and collapse in constant

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Vladimir Avila-Reese

interaction with the interstellar medium (ISM), a complex mix of gas and

plasma, dust, radiation, cosmic rays, and magnetics fields. The center of a

significant fraction of galaxies harbor supermassive black holes. When these

“monsters” are fed with infalling material, the accretion disks around them

release, mainly through powerful plasma jets, the largest amounts of energy

known in astronomical objects. This phenomenon of Active Galactic Nuclei

(AGN) was much more frequent in the past than in the present, being the

high–redshift quasars (QSO’s) the most powerful incarnation of the AGN

phenomenon. But the most astonishing surprise of galaxies comes from the

fact that luminous matter (stars, gas, AGN’s, etc.) is only a tiny fraction

(∼ 1 − 5%) of all the mass measured in galaxies and the giant halos around

them. What this dark component of galaxies is made of? This is one of the

most acute enigmas of modern science.

Thus, exploring and understanding galaxies is of paramount interest to cos-

mology, high–energy and particle physics, gravitation theories, and, of course,

astronomy and astrophysics. As astronomical objects, among other questions,

we would like to know how do they take shape and evolve, what is the origin of

their diversity and scaling laws, why they cluster in space as observed, follow-

ing a sponge–like structure, what is the dark component that predominates

in their masses. By answering to these questions we would able also to use

galaxies as a true link between the observed universe and the properties of the

early universe, and as physical laboratories for testing fundamental theories.

The content of these notes is as follows. In §2 a review on main galaxy

properties and correlations is given. By following an analogy with biology,

the taxonomical, anatomical, ecological and genetical study of galaxies is pre-

sented. The observational inference of dark matter existence, and the baryon

budget in galaxies and in the Universe is highlighted. Section 3 is dedicated

to a pedagogical presentation of the basis of cosmic structure formation the-

ory in the context of the Λ Cold Dark Matter (ΛCDM) paradigm. The main

questions to be answered are: why CDM is invoked to explain the formation of

galaxies? How is explained the origin of the seeds of present–day cosmic struc-

tures? How these seeds evolve?. In §4 an updated review of the main results on

properties and evolution of CDM halos is given, with emphasis on the aspects

that influence the propertied of the galaxies expected to be formed inside the

halos. A short discussion on dark matter candidates is also presented (§§4.2).

The main ingredients of disk and spheroid galaxy formation are reviewed and

discussed in §5. An attempt to highlight the main drivers of the Hubble and

color sequences of galaxies is given in §§5.3. Finally, some selected issues and

open problems in the field are resumed and discussed in §6.

2 Galaxy properties and correlations

During several decades galaxies were considered basically as self–gravitating

stellar systems so that the study of their physics was a domain of Galactic

Understanding Galaxy Formation and Evolution

3

Dynamics. Galaxies in the local Universe are indeed mainly conglomerates of

hundreds of millions to trillions of stars supported against gravity either by

rotation or by random motions. In the former case, the system has the shape

of a flattened disk, where most of the material is on circular orbits at radii that are the minimal ones allowed by the specific angular momentum of the material. Besides, disks are dynamically fragile systems, unstable to perturbations.

Thus, the mass distribution along the disks is the result of the specific angular

momentum distribution of the material from which the disks form, and of the

posterior dynamical (internal and external) processes. In the latter case, the

shape of the galactic system is a concentrated spheroid/ellipsoid, with mostly

(disordered) radial orbits. The spheroid is dynamically hot, stable to pertur-

bations. Are the properties of the stellar populations in the disk and spheroid

systems different?

Stellar populations

Already in the 40’s, W. Baade discovered that according to the ages, metal-

licities, kinematics and spatial distribution of the stars in our Galaxy, they

separate in two groups: 1) Population I stars, which populate the plane of the

disk; their ages do not go beyond 10 Gyr –a fraction of them in fact are young

( < 106 yr) luminous O,B stars mostly in the spiral arms, and their metallicites

∼

are close to the solar one, Z ≈ 2%; 2) Population II stars, which are located

in the spheroidal component of the Galaxy (stellar halo and partially in the

bulge), where velocity dispersion (random motion) is higher than rotation

velocity (ordered motion); they are old stars (> 10 Gyr) with very low metal-

licities, on the average lower by two orders of magnitude than Population I

stars. In between Pop’s I and II there are several stellar subsystems. 1.

Stellar populations are true fossils of the galaxy assembling process. The

differences between them evidence differences in the formation and evolution

of the galaxy components. The Pop II stars, being old, of low metallicity, and

dominated by random motions (dynamically hot), had to form early in the

assembling history of galaxies and through violent processes. In the meantime,

the large range of ages of Pop I stars, but on average younger than the Pop

II stars, indicates a slow star formation process that continues even today

in the disk plane. Thus, the common wisdom says that spheroids form early

in a violent collapse (monolithic or major merger), while disks assemble by

continuous infall of gas rich in angular momentum, keeping a self–regulated

SF process.

1 Astronomers suspect also the existence of non–observable Population III of pris-

tine stars with zero metallicities, formed in the first molecular clouds ∼ 4 108

yrs (z ∼ 20) after the Big Bang. These stars are thought to be very massive,

so that in scaletimes of 1Myr they exploded, injected a big amount of energy to

the primordial gas and started to reionize it through expanding cosmological HII

regions (see e.g., [20, 27] for recent reviews on the subject).

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Vladimir Avila-Reese

Interstellar Medium (ISM)

Galaxies are not only conglomerates of stars. The study of galaxies is incom-

plete if it does not take into account the ISM, which for late–type galaxies

accounts for more mass than that of stars. Besides, it is expected that in

the deep past, galaxies were gas–dominated and with the passing of time

the cold gas was being transformed into stars. The ISM is a turbulent, non–

isothermal, multi–phase flow. Most of the gas mass is contained in neutral

instable HI clouds (102 < T < 104K) and in dense, cold molecular clouds

(T < 102K), where stars form. Most of the volume of the ISM is occuppied by

diffuse (n ≈ 0.1cm−3), warm–hot (T ≈ 104 − 105K) turbulent gas that con-

fines clouds by pressure. The complex structure of the ISM is related to (i)

its peculiar thermodynamical properties (in particular the heating and cool-

ing processes), (ii) its hydrodynamical and magnetic properties which imply

development of turbulence, and (iii) the different energy input sources. The

star formation unities (molecular clouds) appear to form during large–scale

compression of the diffuse ISM driven by supernovae (SN), magnetorotational

instability, or disk gravitational instability (e.g., [7]). At the same time, the energy input by stars influences the hydrodynamical conditions of the ISM: the

star formation results self–regulated by a delicate energy (turbulent) balance.

Galaxies are true “ecosystems” where stars form, evolve and collapse in

constant interaction with the complex ISM. Following a pedagogical analogy

with biological sciences, we may say that the study of galaxies proceeded

through taxonomical, anatomical, ecological and genetical approaches.

2.1 Taxonomy

As it happens in any science, as soon as galaxies were discovered, the next step

was to attempt to classify these news objects. This endeavor was taken on by

E. Hubble. The showiest characteristics of galaxies are the bright shapes pro-

duced by their stars, in particular those most luminous. Hubble noticed that

by their external look (morphology), galaxies can be divided into three prin-

cipal types: Ellipticals (E, from round to flattened elliptical shapes), Spirals

(S, characterized by spiral arms emanating from their central regions where

an spheroidal structure called bulge is present), and Irregulars (Irr, clumpy

without any defined shape). In fact, the last two classes of galaxies are disk–

dominated, rotating structures. Spirals are subdivided into Sa, Sb, Sc types

according to the size of the bulge in relation to the disk, the openness of the

winding of the spiral arms, and the degree of resolution of the arms into stars

(in between the arms there are also stars but less luminous than in the arms).

Roughly 40% of S galaxies present an extended rectangular structure (called

bar) further from the bulge; these are the barred Spirals (SB), where the bar

is evidence of disk gravitational instability.

From the physical point of view, the most remarkable aspect of the mor-

phological Hubble sequence is the ratio of spheroid (bulge) to total luminosity.

Understanding Galaxy Formation and Evolution

5

This ratio decreases from 1 for the Es, to ∼ 0.5 for the so–called lenticulars

(S0), to ∼ 0.5 − 0.1 for the Ss, to almost 0 for the Irrs. What is the origin of

this sequence? Is it given by nature or nurture? Can the morphological types

change from one to another and how frequently they do it? It is interesting

enough that roughly half of the stars at present are in galaxy spheroids (Es

and the bulges of S0s and Ss), while the other half is in disks (e.g., [11]), where some fraction of stars is still forming.

2.2 Anatomy

The morphological classification of galaxies is based on their external aspect

and it implies somewhat subjective criteria. Besides, the “showy” features

that characterize this classification may change with the color band: in blue

bands, which trace young luminous stellar populations, the arms, bar and

other features may look different to what it is seen in infrared bands, which

trace less massive, older stellar populations. We would like to explore deeper

the internal physical properties of galaxies and see whether these properties

correlate along the Hubble sequence. Fortunately, this seems to be the case in

general so that, in spite of the complexity of galaxies, some clear and sequential

trends in their properties encourage us to think about regularity and the

possibility to find driving parameters and factors beyond this complexity.

Figure 1 below resumes the main trends of the “anatomical” properties of galaxies along the Hubble sequence.

The advent of extremely large galaxy surveys made possible massive and

uniform determinations of global galaxy properties. Among others, the Sloan

Digital Sky Survey (SDSS2) and the Two–degree Field Galaxy Redshift Sur-

vey (2dFGRS3) currently provide uniform data already for around 105 galaxies

in limited volumes. The numbers will continue growing in the coming years.

The results from these surveys confirmed the well known trends shown in

Fig. 1; moreover, it allowed to determine the distributions of different properties. Most of these properties present a bimodal distribution with two main

sequences: the red, passive galaxies and the blue, active galaxies, with a frac-

tion of intermediate types (see for recent results [68, 6, 114, 34, 127] and more references therein). The most distinct segregation in two peaks is for

the specific star formation rate ( ˙

Ms/Ms); there is a narrow and high peak

of passive galaxies, and a broad and low peak of star forming galaxies. The

two sequences are also segregated in the luminosity function: the faint end is

dominated by the blue, active sequence, while the bright end is dominated by

the red, passive sequence. It seems that the transition from one sequence to

the other happens at the galaxy stellar mass of ∼ 3 × 1010M⊙.

2 www.sdss.org/sdss.html

3 www.aao.gov.au/2df/

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Vladimir Avila-Reese

Fig. 1. Main trends of physical properties of galaxies along the Hubble morpholog-

ical sequence. The latter is basically a sequence of change of the spheroid–to–disk

ratio. Spheroids are supported against gravity by velocity dispersion, while disks by

rotation.

The hidden component

Under the assumption of Newtonian gravity, the observed dynamics of galax-

ies points out to the presence of enormous amounts of mass not seen as stars

or gas. Assuming that disks are in centrifugal equilibrium and that the orbits

are circular (both are reasonable assumptions for non–central regions), the

measured rotation curves are good tracers of the total (dynamical) mass dis-

tribution (Fig. 2). The mass distribution associated with the luminous galaxy (stars+gas) can be inferred directly from the surface brightness (density) profiles. For an exponential disk of scalelength Rd (=3 kpc for our Galaxy), the

rotation curve beyond the optical radius (Ropt ≈ 3.2Rd) decreases as in the

Keplerian case. The observed HI rotation curves at radii around and beyond

Ropt are far from the Keplerian fall–off, implying the existence of hidden mass

called dark matter (DM) [99, 18]. The fraction of DM increases with radius.

It is important to remark the following observational facts:

Understanding Galaxy Formation and Evolution

7

Fig. 2. Under the assumption of circular orbits, the observed rotation curve of disk

galaxies traces the dynamical (total) mass distribution. The outer rotation curve of

a nearly exponential disk decreases as in the Keplerian case. The observed rotation

curves are nearly flat, suggesting the existence of massive dark halos.

• the outer rotation curves are not universally flat as it is as-

sumed in hundreds of papers. Following, Salucci & Gentile [101], let us define the average value of the rotation curve logarithmic slope,

▽ ≡ (dlogV/dlogR) between two and three Rd. A flat curve means

▽ = 0; for an exponential disk without DM, ▽ = −0.27 at 3Rs. Ob-

servations show a large range of values for the slope: −0.2 ≤ ▽ ≤ 1

• the rotation curve shape (▽) correlates with the luminosity and

surface brightness of galaxies [95, 123, 132]: it increases according the galaxy is fainter and of lower surface brightness

• at the optical radius Ropt, the DM–to–baryon ratio varies from

≈ 1 to 7 for luminous high–surface brightness to faint low–surface

brightness galaxies, respectively

• the roughly smooth shape of the rotation curves implies a fine

coupling between disk and DM halo mass distributions [24]

The HI rotation curves extend typically to 2 − 5Ropt. The dynamics at

larger radii can be traced with satellite galaxies if the satellite statistics allows

for that. More recently, the technique of (statistical) weak lensing around

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Vladimir Avila-Reese

galaxies began to emerge as the most direct way to trace the masses of galaxy

halos. The results show that a typical L∗ galaxy (early or late) with a stellar

mass of Ms ≈ 6 × 1010M⊙ is surrounded by a halo of ≈ 2 × 1012M⊙ ([80] and more references therein). The extension of the halo is typically ≈ 200−250kpc.

These numbers are very close to the determinations for our own Galaxy.

The picture has been confirmed definitively: luminous galaxies are just

the top of the iceberg (Fig. 3). The baryonic mass of (normal) galaxies is only

∼ 3 − 5% of the DM mass in the halo! This fraction could be even lower for

dwarf galaxies (because of feedback) and for very luminous galaxies (because

the gas cooling time > Hubble time). On the other hand, the universal baryon–

to–DM fraction (ΩB/ΩDM ≈ 0.04/0.022, see below) is fB,Un ≈ 18%. Thus,

galaxies are not only dominated by DM, but are much more so than the

average in the Universe! This begs the next question: if the majority of baryons

is not in galaxies, where it is? Recent observations, based on highly ionized

absorption lines towards low redshfit luminous AGNs, seem to have found a

fraction of the missing baryons in the interfilamentary warm–hot intergalactic

medium at T < 105 − 107 K [89].

∼

Fig. 3. Galaxies are just the top of the iceberg. They are surrounded by enormous

DM halos extending 10–20 times their sizes, where baryon matter is only less than

5% of the total mass. Moreover, galaxies are much more DM–dominated than the

average content of the Universe. The corresponding typical baryon–to–DM mass

ratios are given in the inset.

Understanding Galaxy Formation and Evolution

9

Global baryon inventory: The different probes of baryon abundance in the

Universe (primordial nucleosynthesis of light elements, the ratios of odd and

even CMBR acoustic peaks heights, absorption lines in the Lyα forest) have

been converging in the last years towards the same value of the baryon density:

Ωb ≈ 0.042 ± 0.005. In Table 1 below, the densities (Ω′s) of different baryon

components at low redshfits and at z > 2 are given (from [48] and [89]).

Table 1. Abundances of the different baryon components (h = 0.7)

Component

Contribution to Ω

Low redshifts

Galaxies: stars

0.0027 ± 0.0005

Galaxies: HI

(4.2 ± 0.7)×10−4

Galaxies: H2

(1.6 ± 0.6)×10−4

Galaxies: others

(≈ 2.0)×10−4

Intracluster gas

0.0018 ± 0.0007

IGM: (cold-warm)

0.013 ± 0.0023

IGM: (warm-hot)

≈ 0.016

z > 2

Lyα forest clouds

> 0.035

The present–day abundance of baryons in virialized objects (normal stars,

gas, white dwarfs, black holes, etc. in galaxies, and hot gas in clusters) is

therefore ΩB ≈ 0.0037, which accounts for ≈ 9% of all the baryons at low

redshifts. The gas in not virialized structures in the Intergalactic Medium

(cold-warm Lyα/β gas clouds and the warm–hot phase) accounts for ≈ 73%

of all baryons. Instead, at z > 2 more than 88% of the universal baryonic

fraction is in the Lyα forest composed of cold HI clouds. The baryonic budget’s

outstanding questions: Why only ≈ 9% of baryons are in virialized structures at the present epoch?

2.3 Ecology

The properties of galaxies vary systematically as a function of environment.

The environment can be relatively local (measured through the number of

nearest neighborhoods) or of large scale (measured through counting in de-

fined volumes around the galaxy). The morphological type of galaxies is earlier

in the locally denser regions (morphology–density relation),the fraction of el-

lipticals being maximal in cluster cores [40] and enhanced in rich [96] and poor groups. The extension of the morphology–density relation to low local–density

environment (cluster outskirts, low mass groups, field) has been a matter of

debate. From an analysis of SDSS data, [54] have found that (i) in the sparsest regions both relations flatten out, (ii) in the intermediate density regions (e.g.,

cluster outskirts) the intermediate–type galaxy (mostly S0s) fraction increases

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Vladimir Avila-Reese

towards denser regions whereas the late–type galaxy fraction decreases, and

(iii) in the densest regions intermediate–type fraction decreases radically and

early–type fraction increases. In a similar way, a study based on 2dFGRS

data of the luminosity functions in clusters and voids shows that the popu-

lation of faint late–type galaxies dominates in the latter, while, in contrast,

very bright early–late galaxies are relatively overabundant in the former [34].

This and other studies suggest that the origin of the morphology–density (or

morphology-radius) relation could be a combination of (i) initial (cosmologi-

cal) conditions and (ii) of external mechanisms (ram-pressure and tidal stripping, thermal evaporation of the disk gas, strangulation, galaxy harassment,

truncated star formation, etc.) that operate mostly in dense environments,

where precisely the relation steepens significantly.

The morphology–environment relation evolves. It systematically flattens

with z in the sense that the grow of the early-type (E+S0) galaxy fraction with

density becomes less rapid ([97] and more references therein) the main change being in the high–density population fraction. Postman et al. conclude that

the observed flattening of the relation up to z ∼ 1 is due mainly to a deficit

of S0 galaxies and an excess of Sp+Irr galaxies relative to the local galaxy

population; the E fraction-density relation does not appear to evolve over the

range 0 < z < 1.3! Observational studies show that other properties besides

morphology vary with environment. The galaxy properties most sensitive to

environment are the integral color and specific star formation rate (e.g. [68,

114, 127]. The dependences of both properties on environment extend typically to lower densities than the dependence for morphology. These properties are

tightly related to the galaxy star formation history, which in turn depends on

internal formation/evolution processes related directly to initial cosmological

conditions as well as to external astrophysical mechanisms able to inhibit or

induce star formation activity.

2.4 Genetics

Galaxies definitively evolve. We can reconstruct the past of a given galaxy by

matching the observational properties of its stellar populations and ISM with

(parametric) spectro–photo–chemical models (inductive approach). These are

well–established models specialized in following the spectral, photometrical

and chemical evolution of stellar populations formed with different gas in-

fall rates and star formation laws (e.g. [16] and the references therein). The inductive approach allowed to determine that spiral galaxies as our Galaxy

can not be explained with closed–box models (a single burst of star forma-

tion); continuous infall of low–metallicity gas is required to reproduce the local

and global colors, metal abundances, star formation rates, and gas fractions.

On the other hand, the properties of massive ellipticals (specially their high

α-elements/Fe ratios) are well explained by a single early fast burst of star

formation and subsequent passive evolution.

Understanding Galaxy Formation and Evolution

11

A different approach to the genetical study of galaxies emerged after cos-

mology provided a reliable theoretical background. Within such a background

it is possible to “handle” galaxies as physical objects that evolve according

to the initial and boundary conditions given by cosmology. The deductive

construction of galaxies can be confronted with observations corresponding to

different stages of the proto-galaxy and galaxy evolution. The breakthrough

for the deductive approach was the success of the inflationary theory and the

consistency of the so–called Cold Dark Matter (CDM) scenario with parti-

cle physics and observational cosmology. The main goal of these notes is to

describe the ingredients, predictions, and tests of this scenario.

Galaxy evolution in action

The dramatic development of observational astronomy in the last 15 years

or so opened a new window for the study of galaxy genesis: the follow up of

galaxy/protogalaxy populations and their environment at different redshifts.

The Deep and Ultra Deep Fields of the Hubble Spatial Telescope and other

facilities allowed to discover new populations of galaxies at high redshifts,

as well as to measure the evolution of global (per unit of comoving volume)

quantities associated with galaxies: the cosmic star formation rate density

(SFRD), the cosmic density of neutral gas, the cosmic density of metals, etc.

Overall, these global quantities change significantly with z, in particular the

SFRD as traced by the UV–luminosity at rest of galaxies [79]: since z ∼ 1.5−2

to the present it decreased by a factor close to ten (the Universe is literally

lightening off), and for higher redshifts the SFRD remains roughly constant or

slightly decreases ([51, 61] and the references therein). There exists indications that the SFRD at redshifts 2–4 could be approximately two times higher if

considering Far Infrared/submmilimetric sources (SCUBA galaxies), where

intense bursts of star formation take place in a dust–obscured phase.

Concerning populations of individual galaxies, the Deep Fields evidence

a significant increase in the fraction of blue galaxies at z ∼ 1 for the blue

sequence that at these epochs look more distorted and with higher SFRs than

their local counterparts. Instead, the changes observed in the red sequence

are small; it seems that most red elliptical galaxies were in place long ago.

At higher redshifts (z > 2), galaxy objects with high SFRs become more and

∼

more common. The most abundant populations are:

Lyman Break Galaxies (LBG) , selected via the Lyman break at 912˚

A in the

rest–frame. These are star–bursting galaxies (SFRs of 10 − 1000M⊙/yr) with

stellar masses of 109 − 1011M⊙ and moderately clustered.

Sub-millimeter (SCUBA) Galaxies, detected with sub–millimeter bolometer

arrays. These are strongly star–bursting galaxies (SFRs of ∼ 1000M⊙/yr)

obscured by dust; they are strongly clustered and seem to be merging galaxies,

probably precursors of ellipticals.

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Vladimir Avila-Reese

Lyman α emitters (LAEs), selected in narrow–band studies centered in the Lyman α line at rest at z > 3; strong emission Lyman α lines evidence phases

of rapid star formation or strong gas cooling. LAEs could be young (disk?)

galaxies in the early phases of rapid star formation or even before, when the

gas in the halo was cooling and infalling to form the gaseous disk.

Quasars (QSOs), easily discovered by their powerful energetics; they are as-

sociated to intense activity in the nuclei of galaxies that apparently will end

as spheroids; QSOs are strongly clustered and are observed up to z ≈ 6.5.

There are many other populations of galaxies and protogalaxies at high

redshifts (Luminous Red Galaxies, Damped Lyα disks, Radiogalaxies, etc.).

A major challenge now is to put together all the pieces of the high–redshift

puzzle to come up with a coherent picture of galaxy formation and evolution.

3 Cosmic structure formation

In the previous section we have learn that galaxy formation and evolution

are definitively related to cosmological conditions. Cosmology provides the

theoretical framework for the initial and boundary conditions of the cosmic

structure formation models. At the same time, the confrontation of model

predictions with astronomical observations became the most powerful testbed

for cosmology. As a result of this fruitful convergence between cosmology

and astronomy, there emerged the current paradigmatic scenario of cosmic

structure formation and evolution of the Universe called Λ Cold Dark Mat-

ter (ΛCDM). The ΛCDM scenario integrates nicely: (1) cosmological theories

(Big Bang and Inflation), (2) physical models (standard and extensions of the

particle physics models), (3) astrophysical models (gravitational cosmic struc-

ture growth, hierarchical clustering, gastrophysics), and (4) phenomenology

(CMBR anisotropies, non-baryonic DM, repulsive dark energy, flat geometry,

galaxy properties).

Nowadays, cosmology passed from being the Cinderella of astronomy to

be one of the highest precision sciences. Let us consider only the Inflation/Big

Bang cosmological models with the F-R-W metric and adiabatic perturba-

tions. The number of parameters that characterize these models is high,

around 15 to be more precise. No single cosmological probe constrain all of

these parameters. By using multiple data sets and probes it is possible to

constrain with precision several of these parameters, many of which correlate

among them (degeneracy). The main cosmological probes used for precision

cosmology are the CMBR anisotropies, the type–Ia SNe and long Gamma–

Ray Bursts, the Lyα power spectrum, the large–scale power spectrum from

galaxy surveys, the cluster of galaxies dynamics and abundances, the peculiar

velocity surveys, the weak and strong lensing, the baryonic acoustic oscillation

in the large–scale galaxy distribution. There is a model that is systematically

consistent with most of these probes and one of the goals in the last years has

Understanding Galaxy Formation and Evolution

13

been to improve the error bars of the parameters for this ’concordance’ model.

The geometry in the concordance model is flat with an energy composition

dominated in ∼ 2/3 by the cosmological constant Λ (generically called Dark

Energy), responsible for the current accelerated expansion of the Universe.

The other ∼ 1/3 is matter, but ∼ 85% of this 1/3 is in form of non–baryonic

DM. Table 2 presents the central values of different parameters of the ΛCDM

cosmology from combined model fittings to the recent 3–year W M AP CMBR

and several other cosmological probes [109] (see the WMAP website).

Table 2. Constraints to the parameters of the ΛCDM model

Parameter

Constraint

Total density

Ω = 1

Dark Energy density

ΩΛ = 0.74

Dark Matter density

ΩDM = 0.216

Baryon Matter dens.

ΩB = 0.044

Hubble constant

h = 0.71

Age

13.8 Gyr

Power spectrum norm.

σ8 = 0.75

Power spectrum index ns(0.002) = 0.94

In the following, I will describe some of the ingredients of the ΛCDM sce-

nario, emphasizing that most of these ingredients are well established aspects

that any alternative scenario to ΛCDM should be able to explain.

3.1 Origin of fluctuations

The Big Bang4 is now a mature theory, based on well established observational

pieces of evidence. However, the Big Bang theory has limitations. One of

them is namely the origin of fluctuations that should give rise to the highly

inhomogeneous structure observed today in the Universe, at scales of less

than ∼ 200Mpc. The smaller the scales, the more clustered is the matter.

For example, the densities inside the central regions of galaxies, within the

galaxies, cluster of galaxies, and superclusters are about 1011, 106, 103 and

few times the average density of the Universe, respectively.

The General Relativity equations that describe the Universe dynamics in

the Big Bang theory are for an homogeneous and isotropic fluid (Cosmologi-

cal Principle); inhomogeneities are not taken into account in this theory “by

definition”. Instead, the concept of fluctuations is inherent to the Inflation-

ary theory introduced in the early 80’s by A. Guth and A. Linde namely to

4 It is well known that the name of ’Big Bang’ is not appropriate for this theory. The

key physical conditions required for an explosion are temperature and pressure

gradients. These conditions contradict the Cosmological Principle of homogeneity

and isotropy on which is based the ’Big Bang’ theory.

14

Vladimir Avila-Reese

overcome the Big Bang limitations. According to this theory, at the energies

of Grand Unification ( > 1014GeV or T > 1027K!), the matter was in the state

∼

∼

known in quantum field theory as vacuum. Vacuum is characterized by quan-

tum fluctuations –temporary changes in the amount of energy in a point in

space, arising from Heisenberg uncertainty principle. For a small time interval

∆t, a virtual particle–antiparticle pair of energy ∆E is created (in the GU

theory, the field particles are supposed to be the X- and Y-bossons), but then

the pair disappears so that there is no violation of energy conservation. Time

and energy are related by ∆E∆t ≈ h . The vacuum quantum fluctuations

2π

are proposed to be the seeds of present–day structures in the Universe.

How is that quantum fluctuations become density inhomogeneities? Dur-

ing the inflationary period, the expansion is described approximately by the

de Sitter cosmology, a ∝ eHt, H ≡ ˙a/a is the Hubble parameter and it is con-

stant in this cosmology. Therefore, the proper length of any fluctuation grows

as λp ∝ eHt. On the other hand, the proper radius of the horizon for de Sitter

metric is equal to c/H =const, so that initially causally connected (quan-

tum) fluctuations become suddenly supra–horizon (classical) perturbations to

the spacetime metric. After inflation, the Hubble radius grows proportional

to ct, and at some time a given curvature perturbation cross again the hori-

zon (becomes causally connected, λp < LH). It becomes now a true density

perturbation. The interesting aspect of the perturbation ’trip’ outside the

horizon is that its amplitude remains roughly constant, so that if the ampli-

tude of the fluctuations at the time of exiting the horizon during inflation is

constant (scale invariant), then their amplitude at the time of entering the

horizon should be also scale invariant. In fact, the computation of classical per-

turbations generated by a quantum field during inflation demonstrates that

the amplitude of the scalar fluctuations at the time of crossing the horizon is

nearly constant, δφH ∝const. This can be understood on dimensional grounds:

due to the Heisenberg principle δφ/δt ∝ const, where δt ∝ H−1. Therefore,

δφH ∝ H, but H is roughly constant during inflation, so that δφH ∝const.

3.2 Gravitational evolution of fluctuations

The ΛCDM scenario assumes the gravitational instability paradigm: the cos-

mic structures in the Universe were formed as a consequence of the growth of

primordial tiny fluctuations (for example seeded in the inflationary epochs)

by gravitational instability in an expanding frame. The fluctuation or pertur-

bation is characterized by its density contrast,

δρ

ρ − ρ

δ ≡

=

,

(1)

ρ

ρ

where ρ is the average density of the Universe and ρ is the perturbation den-

sity. At early epochs, δ << 1 for perturbation of all scales, otherwise the

homogeneity condition in the Big Bang theory is not anymore obeyed. When

δ << 1, the perturbation is in the linear regime and its physical size grows

Understanding Galaxy Formation and Evolution

15

with the expansion proportional to a(t). The perturbation analysis in the lin-

ear approximation shows whether a given perturbation is stable (δ ∼ const or

even → 0) or unstable (δ grows). In the latter case, when δ → 1, the linear

approximation is not anymore valid, and the perturbation “separates” from

the expansion, collapses, and becomes a self–gravitating structure. The grav-

itational evolution in the non–linear regime is complex for realistic cases and is studied with numerical N–body simulations. Next, a pedagogical review of

the linear evolution of perturbations is presented. More detailed explanations

on this subject can be found in the books [72, 94, 90, 30, 77, 92].

Relevant times and scales.

The important times in the problem of linear gravitational evolution of per-

turbations are: (a) the epoch when inflation finished (tinf ≈ 10−34s, at this

time the primordial fluctuation field is established); (b) the epoch of matter–

radiation equality teq (corresponding to æ ≈ 1/3.9 × 104(Ω0h2), before teq the

dynamics of the universe is dominated by radiation density, after teq dominates

matter density); (c) the epoch of recombination trec, when radiation decouples

from baryonic matter (corresponding to arec = 1/1080, or trec ≈ 3.8 × 105yr

for the concordance cosmology).

Scales: first of all, we need to characterize the size of the perturbation. In

the linear regime, its physical size expands with the Universe: λp = a(t)λ0,

where λ0 is the comoving size, by convention fixed (extrapolated) to the

present epoch, a(t0) = 1. In a given (early) epoch, the size of the pertur-

bation can be larger than the so–called Hubble radius, the typical radius

over which physical processes operate coherently (there is causal connection):

LH ≡ (a/ ˙a)−1 = H−1 = n−1ct. For the radiation or matter dominated cases,

a(t) ∝ tn, with n = 1/2 and n = 2/3, respectively, that is n < 1. Therefore,

LH grows faster than λp and at a given “crossing” time tcross, λp < LH. Thus,

the perturbation is supra–horizon sized at epochs t < tcross and sub–horizon

sized at t > tcross. Notice that if n > 1, then at some time the perturbation

“exits” the Hubble radius. This is what happens in the inflationary epoch,

when a(t) ∝ et: causally–connected fluctuations of any size are are suddenly

“taken out” outside the Hubble radius becoming causally disconnected.

For convenience, in some cases it is better to use masses instead of sizes.

Since in the linear regime δ << 1 (ρ ≈ ρ), then M ≈ ρM (a)ℓ3, where ℓ is the

size of a given region of the Universe with average matter density ρM . The

mass of the perturbation, Mp, is invariant.

Supra–horizon sized perturbations.

In this case, causal, microphysical processes are not possible, so that it does

not matter what perturbations are made of (baryons, radiation, dark mat-

ter, etc.); they are in general just perturbations to the metric. To study the

gravitational growth of metric perturbations, a General Relativistic analysis

is necessary. A major issue in carrying out this program is that the metric

16

Vladimir Avila-Reese

perturbation is not a gauge invariant quantity. See e.g., [72] for an outline of how E. Lifshitz resolved brilliantly this difficult problem in 1946. The result is

quite simple and it shows that the amplitude of metric perturbations outside

the horizon grows kinematically at different rates, depending on the dominant

component in the expansion dynamics. For the critical cosmological model

(at early epochs all models approach this case), the growing modes of metric

perturbations according to what dominates the background are:

δm,+ ∝ a(t) ∝ t2/3, .................matter

(2)

δm,+ ∝ a(t)2 ∝ t, .................radiation

δm,+ ∝ a(t)−2 ∝ e−2Ht, ..Λ (deSitter)

(3)

Sub–horizon sized perturbations.

Once perturbations are causally connected, microphysical processes are switched

on (pressure, viscosity, radiative transport, etc.) and the gravitational evolu-

tion of the perturbation depends on what it is made of. Now, we deal with

true density perturbations. For them applies the classical perturbation anal-

ysis for a fluid, originally introduced by J. Jeans in 1902, in the context of

the problem of star formation in the ISM. But unlike in the ISM, in the cos-

mological context the fluid is expanding. What can prevent the perturbation

amplitude from growing gravitationally? The answer is pressure support. If

the fluid pressure gradient can re–adjust itself in a timescale tpress smaller

than the gravitational collapse timescale, tgrav, then pressure prevents the

gravitational growth of δ. Thus, the condition for gravitational instability is:

1

λ

t

p

grav ≈

< t

,

(4)

(Gρ)1/2

press ≈ v

where ρ is the density of the component that is most gravitationally domi-

nant in the Universe, and v is the sound speed (collisional fluid) or velocity

dispersion (collisionless fluid) of the perturbed component. In other words,

if the perturbation scale is larger than a critical scale λJ ∼ v(Gρ)−1/2, then

pressure loses, gravity wins.

The perturbation analysis applied to the hydrodynamical equations of a

fluid at rest shows that δ grows exponentially with time for perturbations

obeying the Jeans instability criterion λp > λJ , where the exact value of λJ

is v(π/Gρ)1/2. If λp < λJ , then the perturbations are described by stable

gravito–acustic oscillations. The situation is conceptually similar for pertur-

bations in an expanding cosmological fluid, but the growth of δ in the unstable

regime is algebraical instead of exponential. Thus, the cosmic structure forma-

tion process is relatively slow. Indeed, the typical epochs of galaxy and cluster

of galaxies formation are at redshifts z ∼ 1 − 5 (ages of ∼ 1.2 − 6 Gyrs) and

z < 1 (ages larger than 6 Gyrs), respectively.

Understanding Galaxy Formation and Evolution

17

Baryonic matter. The Jeans instability analysis for a relativistic (plasma)

fluid of baryons ideally coupled to radiation and expanding at the rate H =

˙a/a shows that there is an instability critical scale λJ = v(3π/8Gρ)1/2, where

√

the sound speed for adiabatic perturbations is v = p/ρ = c/ 3; the latter

equality is due to pressure radiation. At the epoch when radiation dominates,

ρ = ρr ∝ a−4 and then λJ ∝ a2 ∝ ct. It is not surprising that at this epoch

λJ approximates the Hubble scale LH ∝ ct (it is in fact ∼ 3 times larger).

Thus, perturbations that might collapse gravitationally are in fact outside

the horizon, and those that already entered the horizon, have scales smaller

than λJ : they are stable gravito–acoustic oscillations. When matter dominates,

ρ = ρM ∝ a−3, and a ∝ t2/3. Therefore, λJ ∝ a ∝ t2/3 < L

∼

H , but still radiation

is coupled to baryons, so that radiation pressure is dominant and λJ remains

large. However, when radiation decouples from baryons at trec, the pressure

support drops dramatically by a factor of Pr/Pb ∝ nrT/nbT ≈ 108! Now, the

Jeans analysis for a gas mix of H and He at temperature Trec ≈ 4000 K shows

that baryonic clouds with masses > 106M

∼

⊙ can collapse gravitationally, i.e. all

masses of cosmological interest. But this is literally too “ideal” to be true.

The problem is that as the Universe expands, radiation cools (Tr = T0a−1)

and the photon–baryon fluid becomes less and less perfect: the mean free path

for scattering of photons by electrons (which at the same time are coupled

electrostatically to the protons) increases. Therefore, photons can diffuse out

of the bigger and bigger density perturbations as the photon mean free path

increases. If perturbations are in the gravito–acoustic oscillatory regime, then

the oscillations are damped out and the perturbations disappear. The “iron-

ing out” of perturbations continues until the epoch of recombination. In a

pioneering work, J. Silk [104] carried out a perturbation analysis of a relativistic cosmological fluid taking into account radiative transfer in the diffusion

approximation. He showed that all photon–baryon perturbations of masses

smaller than MS are “ironed out” until trec by the (Silk) damping process.

The first crisis in galaxy formation theory emerged: calculations showed that

MS is of the order of 1013 − 1014M⊙h−1! If somebody (god, inflation, ...)

seeded primordial fluctuations in the Universe, by Silk damping all galaxy–

sized perturbation are “ironed out”. 5

Non–baryonic matter. The gravito–acoustic oscillations and their damping by

photon diffusion refer to baryons. What happens for a fluid of non–baryonic

DM? After all, astronomers, since Zwicky in the 1930s, find routinely pieces

5 In the 1970s Y. Zel’dovich and collaborators worked out a scenario of galaxy for-

mation starting from very large perturbations, those that were not affected by

Silk damping. In this elegant scenario, the large–scale perturbations, considered

in a first approximation as ellipsoids, collapse most rapidly along their shortest

axis, forming flattened structures (“pancakes”), which then fragment into galax-

ies by gravitational or thermal instabilities. In this ’top-down’ scenario, to obtain

galaxies in place at z ∼ 1, the amplitude of the large perturbations at recombina-

tion should be ≥ 3 × 10−3. Observations of the CMBR anisotropies showed that

the amplitudes are 1–2 order of magnitudes smaller than those required.

18

Vladimir Avila-Reese

Fig. 4. Free–streaming damping kills perturbations of sizes roughly smaller than

the horizon length if they are made of relativistic particles. The epoch tn.r. when

thermal–coupled particles become non–relativistic is inverse proportional to the

square of the particle mass mX . Typical particle masses of CDM, WDM and HDM

are given together with the corresponding horizon (filtering) masses.

of evidence for the presence of large amounts of DM in the Universe. As

DM is assumed to be collisionless and not interacting electromagnetically,

then the radiative or thermal pressure supports are not important for linear

DM perturbations. However, DM perturbations can be damped out by free

streaming if the particles are relativistic: the geodesic motion of the particles at the speed of light will iron out any perturbation smaller than a scale close to

the particle horizon radius, because the particles can freely propagate from an

overdense region to an underdense region. Once the particles cool and become

non relativistic, free streaming is not anymore important. A particle of mass

mX and temperature TX becomes non relativistic when kBTX ∼ mXc2. Since

TX ∝ a−1, and a ∝ t1/2 when radiation dominates, one then finds that the

epoch when a thermal–relic particle becomes non relativistic is tnr ∝ m−2.

X

Understanding Galaxy Formation and Evolution

19

The more massive the DM particle, the earlier it becomes non relativistic,

and the smaller are therefore the perturbations damped out by free streaming

(those smaller than ∼ ct; see Fig. 4). According to the epoch when a given thermal DM particle species becomes non relativistic, DM is called Cold Dark

Matter (CDM, very early), Warm Dark Matter (WDM, early) and Hot Dark

Matter (HDM, late)6.

The only non–baryonic particles confirmed experimentally are (light) neu-

trinos (HDM). For neutrinos of masses ∼ 1 − 10eV, free streaming attains to

iron out perturbations of scales as large as massive clusters and superclus-

ters of galaxies (see Fig. 4). Thus, HDM suffers the same problem of baryonic matter concerning galaxy formation7. At the other extreme is CDM, in

which case survive free streaming practically all scales of cosmological inter-

est. This makes CDM appealing to galaxy formation theory. In the minimal

CDM model, it is assumed that perturbations of all scales survive, and that

CDM particles are collisionless (they do not self–interact). Thus, if CDM

dominates, then the first step in galaxy formation study is reduced to the

calculation of the linear and non–linear gravitational evolution of collisionless

CDM perturbations. Galaxies are expected to form in the centers of collapsed

CDM structures, called halos, from the baryonic gas, first trapped in the

gravitational potential of these halos, and second, cooled by radiative (and

turbulence) processes (see §5).

The CDM perturbations are free of any physical damping processes and

in principle their amplitudes may grow by gravitational instability. However,

when radiation dominates, the perturbation growth is stagnated by expansion.

The gravitational instability timescale for sub–horizon linear CDM perturba-

tions is tgrav ∼ (GρDM )−2, while the expansion (Hubble) timescale is given by

texp ∼ (Gρ)−2. When radiation dominates, ρ ≈ ρr and ρr >> ρM . Therefore

texp << tgrav, that is, expansion is faster than the gravitational shrinking.

Fig. 5 resumes the evolution of primordial perturbations. Instead of spatial scales, in Fig. 5 are shown masses, which are invariant for the perturbations. We highlight the following conclusions from this plot: (1) Photon–

baryon perturbations of masses < MS are washed out (δB → 0) as long

as baryon matter is coupled to radiation. (2) The amplitude of CDM per-

turbations that enter the horizon before teq is “freezed-out” (δDM ∝const)

as long as radiation dominates; these are perturbations of masses smaller

than MH,eq ≈ 1013(ΩM h2)−2M⊙, namely galaxy scales. (3) The baryons are

trapped gravitationally by CDM perturbations, and within a factor of two

in z, baryon perturbations attain amplitudes half that of δDM . For WDM

6 The reference to “early” and “late” is given by the epoch and the correspond-

ing radiation temperature when the largest galaxy–sized perturbations (M ∼

1013M⊙) enter the horizon: agal ∼ æ ≈ 1/3.9 × 104(Ω0h2) and Tr ∼ 1KeV.

7 Neutrinos exist and have masses larger than 0.05 eV according to determinations

based on solar neutrino oscillations. Therefore, neutrinos contribute to the matter

density in the Universe. Cosmological observations set a limit: Ωνh2 < 0.0076,

otherwise too much structure is erased.

20

Vladimir Avila-Reese

or HDM perturbations, the free–streaming damping introduces a mass scale

Mfs ≈ MH,n.r. in Fig. 5, below which δ → 0; Mfs increases as the DM mass particle decreases (Fig. 4).

Fig. 5. Different evolutive regimes of perturbations δ. The suffixes “B” and “DM”

are for baryon–photon and DM perturbations, respectively. The evolution of the

horizon, Jeans and Silk masses (MH, MJ , and MS) are showed. Mf1 and Mf2 are

the masses of two perturbations. See text for explanations.

The processed power spectrum of perturbations. The exact solution to the

problem of linear evolution of cosmological perturbations is much more com-

plex than the conceptual aspects described above. Starting from a primordial

fluctuation field, the perturbation analysis should be applied to a cosmolog-

ical mix of baryons, radiation, neutrinos, and other non–baryonic dark mat-

ter components (e.g., CDM), at sub– and supra–horizon scales (the fluid as-

sumption is relaxed). Then, coupled relativistic hydrodynamic and Boltzmann

equations in a general relativity context have to be solved taking into account

radiative and dissipative processes. The outcome of these complex calculations

is the full description of the processed fluctuation field at the recombination

epoch (when fluctuations at almost all scales are still in the linear regime).

The goal is double and of crucial relevance in cosmology and astrophysics:

Understanding Galaxy Formation and Evolution

21

1) to predict the physical and statistical properties of CMBR anisotropies,

which can be then compared with observations, and 2) to provide the initial

conditions for calculating the non–linear regime of cosmic structure formation

and evolution. Fortunately, there are now several public friendly-to-use codes

that numerically solve the cosmological linear perturbation equations (e.g.,

CMBFast and CAMB 8).

The description of the density fluctuation field is statistical. As any random

field, it is convenient to study perturbations in the Fourier space. The Fourier

expansion of δ(x) is:

V

Z

δ(x) =

δ

(2π)3

ke−ikxd3k,

(5)

Z

δk = V −1

δ(x)eikxd3x

(6)

The Fourier modes δk evolve independently while the perturbations are in

the linear regime, so that the perturbation analysis can be applied to this

quantity. For a Gaussian random field, any statistical quantity of interest can

be specified in terms of the power spectrum P (k) ≡ |δk|2, which measures

the amplitude of the fluctuations at a given scale k9. Thus, from the linear

perturbation analysis we may follow the evolution of P (k). A more intuitive

quantity than P (k) is the mass variance σ2M ≡ h(δM/M)2Ri of the fluctuation

field. The physical meaning of σM is that of an rms density contrast on a

given sphere of radius R associated to the mass M = ρVW (R), where W (R)

is a window (smoothing) function. The mass variance is related to P (k). By

assuming a power law power spectrum, P (k) ∝ kn, it is easy to show that

σM ∝ R−(3+n) ∝ M−(3+n)/3 = M−2α

(7)

3 + n

α =

,

6

for 4 < n < −3 using a Gaussian window function. The question is: How the

scaling law of perturbations, σM , evolves starting from an initial (σM )i?

In the early 1970s, Harrison and Zel’dovich independently asked them-

selves about the functionality of σM (or the density contrast) at the time

adiabatic perturbations cross the horizon, that is, if (σM )H ∝ MαH , then

what is the value of αH? These authors concluded that −0.1 ≤ αH ≤ 0.2, i.e.

8 http://www.cmbfast.org and http://camb.info/

9 The phases of the Fourier modes in the Gaussian case are uncorrelated. Gaus-

sianity is the simplest assumption for the primordial fluctuation field statistics

and it seems to be consistent with some variants of inflation. However, there are

other variants that predict non–Gaussian fluctuations (for a recent review on this

subject see e.g. [8]), and the observational determination of the primordial fluctuation statistics is currently an active field of investigation. The properties of

cosmic structures depend on the assumption about the primordial statistics, not

only at large scales but also at galaxy scales; see for a review and new results [4].

22

Vladimir Avila-Reese

αH ≈ 0 (nH ≈ −3). If αH >> 0 (nH >> −3), then σM → ∞ for M → 0; this

means that for a given small mass scale M , the mass density of the perturba-

tion at the time of becoming causally connected can correspond to the one of a

(primordial) black hole. Hawking evaporation of black holes put a constraint

on M

<

BH,prim

1015g, which corresponds to α

∼

H ≤ 0.2, otherwise the γ–ray

background radiation would be more intense than that observed. If αH << 0

(nH << −3), then larger scales would be denser than the small ones, contrary

to what is observed. The scale–invariant Harrison–Zel’dovich power spectrum,

PH(k) ∝ k−3 , is for perturbations at the time of entering the horizon. How

should the primordial power spectrum Pi(k) = Akni or (σM )i = BM−αi (de-

fined at some fixed initial time) be to produce such scale invariance? Since ti

until the horizon crossing time tcross(M ) for a given perturbation of mass M ,

σM (t) evolves as a(t)2 (supra–horizon regime in the radiation era). At tcross,

the horizon mass MH is equal by definition to M . We have seen that MH ∝ a3

(radiation dominion), so that across ∝ M1/3 = M1/3. Therefore,

H

σM (tcross) ∝ (σM )i(across/ai)2 ∝ M−αiM2/3,

(8)

i.e. αH = 2/3 − αi or nH = ni − 4. A similar result is obtained if the pertur-

bation enters the horizon during the matter dominion era. From this analysis

one concludes that for the perturbations to be scale invariant at horizon cross-

ing (αH = 0 or nH = −3), the primordial (initial) power spectrum should be

Pi(k) = Ak1 or (σM )i ∝ M−2/3 ∝ λ−2

0

(i.e. ni = 1 and α = 2/3; A is a nor-

malization constant). Does inflation predict such power spectrum? We have

seen that, according to the quantum field theory and assuming that H =const

during inflation, the fluctuation amplitude is scale invariant at the time to exit

the horizon, δH ∼const. On the other hand, we have seen that supra–horizon

curvature perturbations during a de Sitter period evolve as δ ∝ a−2 (eq. 4).

Therefore, at the end of inflation we have that δinf = δH(λ0)(ainf /aH)−2. The

proper size of the fluctuation when crossing the horizon is λp = aHλ0 ≈ H−1;

therefore, aH ≈ 1/(λ0H). Replacing now this expression in the equation for

δinf we get that:

δinf ≈ δH(λ0)(ainf λ0H)−2 ∝ λ−2

0

∝ M−2/3,

(9)

if δH ∼const. Thus, inflation predicts αi nearly equal to 2/3 (ni ≈ 1)! Recent

results from the analysis of CMBR anisotropies by the WMAP satellite [109]

seem to show that ni is slightly smaller than 1 or that ni changes with the

scale (running power–spectrum index). This is in more agreement with several

inflationary models, where H actually slightly vary with time introducing

some scale dependence in δH.

The perturbation analysis, whose bases were presented in §3.2 and resumed

in Fig. 5, show us that σM grows (kinematically) while perturbations are in the supra–horizon regime. Once perturbations enter the horizon (first the smaller

ones), if they are made of CDM, then the gravitational growth is “freezed

out” whilst radiation dominates (stangexpantion). As shown schematically

Understanding Galaxy Formation and Evolution

23

Fig. 6. Linear evolution of the perturbation mass variance σM . The perturbation

amplitude in the supra–horizon regime grow kinematically. DM perturbations (solid

curve) that cross the horizon during the radiation dominion, freeze–out their grow

due to stangexpantion, producing a flattening in the scaling law σM for all scales

smaller than the corresponding to the horizon at the equality epoch (galaxy scales).

Baryon–photon perturbations smaller than the Silk mass MS are damped out (dot-

ted curve) and those larger than MS but smaller than the horizon mass at recom-

bination are oscillating (Baryonic Acoustic Oscillation, BAO).

in Fig. 6, this “flattens” the variance σM at scales smaller than MH,eq; in fact, σM ∝ ln(M) at these scales, corresponding to galaxies! After teq the

CDM variance (or power spectrum) grows at the same rate at all scales. If

perturbations are made out of baryons, then for scales smaller than MS, the

gravito–acoustic oscillations are damped out, while for scales close to the

Hubble radius at recombination, these oscillations are present. The “final”

processed mass variance or power spectrum is defined at the recombination

epoch. For example, the power spectrum is expressed as:

Prec(k) = Akni × (D(trec)/D(ti))2 × T 2(k),

(10)

where the first term is the initial power spectrum Pi(k); the second one is

how much the fluctuation amplitude has grown in the linear regime (D(t) is

the so–called linear growth factor), and the third one is a transfer function

that encapsulates the different damping and freezing out processes able to

deform the initial power spectrum shape. At large scales, T 2(k) = 1, i.e. the

primordial shape is conserved (see Fig. 6).

Besides the mass power spectrum, it is possible to calculate the angu-

lar power spectrum of temperature fluctuation in the CMBR. This spectrum

consists basically of 2 ranges divided by a critical angular scales: the an-

gle θh corresponding to the horizon scale at the epoch of recombination

24

Vladimir Avila-Reese

((LH)rec ≈ 200(Ωh2)−1/2 Mpc, comoving). For scales grander than θh the

spectrum is featureless and corresponds to the scale–invariant supra–horizon

Sachs-Wolfe fluctuations. For scales smaller than θh, the sub–horizon fluctu-

ations are dominated by the Doppler scattering (produced by the gravito–

acoustic oscillations) with a series of decreasing in amplitude peaks; the po-

sition (angle) of the first Doppler peak depends strongly on Ω, i.e. on the

geometry of the Universe. In the last 15 years, high–technology experiments

as COBE, Boomerang, WMAP provided valuable information (in particular the latter one) on CMBR anisotropies. The results of this exciting branch of

astronomy (called sometimes anisotronomy) were of paramount importance

for astronomy and cosmology (see for a review [62] and the W. Hu website10).

Just to highlight some of the key results of CMBR studies, let us mention

the next ones: 1) detailed predictions of the ΛCDM scenario concerning the

linear evolution of perturbations were accurately proved, 2) several cosmolog-

ical parameters as the geometry of the Universe, the baryonic fraction ΩB,

and the index of the primordial power spectrum, were determined with high

precision (see the actualized, recently delivered results from the 3 year analy-

sis of WMAP in [109]), 3) by studying the polarization maps of the CMBR it was possible to infer the epoch when the Universe started to be significantly

reionized by the formation of first stars, 4) the amplitude (normalization) of

the primordial fluctuation power spectrum was accurately measured. The lat-

ter is crucial for further calculating the non–linear regime of cosmic structure

formation. I should emphasize that while the shape of the power spectrum is

predicted and well understood within the context of the ΛCDM model, the

situation is fuzzy concerning the power spectrum normalization. We have a

phenomenological value for A but not a theoretical prediction.

4 The dark side of galaxy formation and evolution

A great triumph of the ΛCDM scenario was the overall consistency found

between predicted and observed CMBR anisotropies generated at the recom-

bination epoch. In this scenario, the gravitational evolution of CDM pertur-

bations is the driver of cosmic structure formation. At scales much larger than

galaxies, (i) mass density perturbations are still in the (quasi)linear regime,

following the scaling law of primordial fluctuations, and (ii) the dissipative

physics of baryons does not affect significantly the matter distribution. Thus,

the large–scale structure (LSS) of the Universe is determined basically by

DM perturbations yet in their (quasi)linear regime. At smaller scales, non–

linearity strongly affects the primordial scaling law and, moreover, the dissi-

pative physics of baryons “distorts” the original DM distribution, particularly

inside galaxy–sized DM halos. However, DM in any case provides the original

“mold” where gas dynamics processes take place.

10 http://background.uchicago.edu/∼ whu/physics/physics.html

Understanding Galaxy Formation and Evolution

25

The ΛCDM scenario describes successfully the observed LSS of the Uni-

verse (for reviews see e.g., [49, 58], and for some recent observational results see e.g. [115, 102, 109]). The observed filamentary structure can be explained as a natural consequence of the CDM gravitational instability occurring pref-erentially in the shortest axis of 3D and 2D protostructures (the Zel’dovich

panckakes). The clustering of matter in space, traced mainly by galaxies, is

also well explained by the clustering properties of CDM. At scales r much

larger than typical galaxy sizes, the galaxy 2-point correlation function ξgal(r)

(a measure of the average clustering strength on spheres of radius r) agrees

rather well with ξCDM (r). Current large statistical galaxy surveys as SDSS

and 2dFGRS, allow now to measure the redshift–space 2-point correlation

function at large scales with unprecedented accuracy, to the point that weak

“bumps” associated with the baryon acoustic oscillations at the recombina-

tion epoch begin to be detected [41]. At small scales ( < 3Mpch−1), ξ

∼

gal(r)

departs from the predicted pure ξCDM (r) due to the emergence of two pro-

cesses: (i) the strong non–linear evolution that small scales underwent, and

(ii) the complexity of the baryon processes related to galaxy formation. The

difference between ξgal(r) and ξCDM (r) is parametrized through one “igno-

rance” parameter, b, called bias, ξgal(r) = bξCDM (r). Below, some basic ideas

and results related to the former processes will be described. The baryonic

process will be sketched in the next Section.

4.1 Nonlinear clustering evolution

The scaling law of the processed ΛCDM perturbations, is such that σM at

galaxy–halo scales decreases slightly with mass (logarithmically) and for larger

scales, decreases as a power law (see Fig. 6). Because the perturbations of higher amplitudes collapse first, the first structures to form in the ΛCDM

scenario are typically the smallest ones. Larger structures assemble from the

smaller ones in a process called hierarchical clustering or bottom–up mass

assembling. It is interesting to note that the concept of hierarchical clustering

was introduced several years before the CDM paradigm emerged. Two seminal

papers settled the basis for the current theory of galaxy formation: Press &

Schechter 1974 [98] and White & Rees 1979 [131]. In the latter it was proposed that “the smaller–scale virialized [dark] systems merge into an amorphous

whole when they are incorporated in a larger bound cluster. Residual gas

in the resulting potential wells cools and acquires sufficient concentration to

self–gravitate, forming luminous galaxies up to a limiting size”.

The Press & Schechter (P-S) formalism was developed to calculate the

mass function (per unit of comoving volume) of halos at a given epoch,

n(M, z). The starting point is a Gaussian density field filtered (smoothed)

at different scales corresponding to different masses, the mass variance σM

being the characterization of this filtering process. A collapsed halo is iden-

tified when the evolving density contrast of the region of mass M , δM (z),

26

Vladimir Avila-Reese

attains a critical value, δc, given by the spherical top–hat collapse model11.

This way, the Gaussian probability distribution for δM is used to calculate

the mass distribution of objects collapsed at the epoch z. The P-S formalism

assumes implicitly that the only objects to be counted as collapsed halos at a

given epoch are those with δM (z) = δc. For a mass variance decreasing with

mass, as is the case for CDM models, this implies a “hierarchical” evolution

of n(M, z): as z decreases, less massive collapsed objects disappear in favor

of more massive ones (see Fig. 8). The original P-S formalism had an error of 2 in the sense that integrating n(M, z) half of the mass is lost. The authors

multiplied n(M, z) by 2, argumenting that the objects duplicate their masses

by accretion from the sub–dense regions. The problem of the factor of 2 in the

P-S analysis was partially solved using an excursion set statistical approach

[17, 73].

To get an idea of the typical formation epochs of CDM halos, the spheri-

cal collapse model can be used. According to this model, the density contrast

of given overdense region, δ, grows with z proportional to the growing fac-

tor, D(z), until it reaches a critical value, δc, after which the perturbation is

supposed to collapse and virialize12. at redshift zcol (for example see [90]): δ(zcol) ≡ δ0D(zcol) = δc,0.

(11)

The convention is to fix all the quantities to their linearly extrapolated values

at the present epoch (indicated by the subscript “0”) in such a way that D(z =

0) ≡ D0 = 1. Within this convention, for an Einstein–de Sitter cosmology,

δc,0 = 1.686, while for the ΛCDM cosmology, δc,0 = 1.686Ω0.0055, and the

M,0

growing factor is given by

g(z)

D(z) =

,

(12)

g(z0)(1 + z)

11 The spherical top–hat model refers to the exact calculation of the collapse of

a uniform spherical density perturbation in an otherwise uniform Universe; the

dynamics of such a region is the same of a closed Universe. The solution of the

equations of motion shows that the perturbation at the beginning expands as the

background Universe (proportional to a), then it reaches a maximum expansion

(size) in a time tmax, and since that moment the perturbation separates of the

expanding background, collapsing in a time tcol = 2tmax.

12 The mathematical solution gives that the spherical perturbed region collapses

into a point (a black hole) after reaching its maximum expansion. However, real

perturbations are lumpy and the particle orbits are not perfectly radial. In this

situation, during the collapse the structure comes to a dynamical equilibrium un-

der the influence of large scale gravitational potential gradients, a process named

by the oxymoron “violent relaxation” (see e.g. [14]); this is a typical collective phenomenon. The end result is a system that satisfies the virial theorem: for

a self–gravitating system this means that the internal kinetic energy is half the

(negative) gravitational potential energy. Gravity is supported by the velocity dis-

persion of particles or lumps. The collapse factor is roughly 1/2, i.e. the typical

virial radius Rv of the collapsed structure is ≈ 0.5 the radius of the perturbation

at its maximum expansion.

Understanding Galaxy Formation and Evolution

27

where a good approximation for g(z) is [23]:

5

−1

4

Ω

Ω

g(z) ≃

Ω

M

Λ

7

1 +

1 +

,

(13)

2

M − ΩΛ +

2

70

and where ΩM = ΩM,0(1 + z)3/E2(z), ΩΛ = ΩΛ/E2(z), with E2(z) = ΩΛ +

ΩM,0 (1 + z)3. For the Einstein–de Sitter model, D(z) = (1 + z). We need

now to connect the top–hat sphere results to a perturbation of mass M . The

processed perturbation field, fixed at the present epoch, is characterized by the

mass variance σM and we may assume that δ0 = νσM , where δ0 is δ linearly

extrapolated to z = 0, and ν is the peak height. For average perturbations, ν =

1, while for rare, high–density perturbations, from which the first structures

arose, ν >> 1. By introducing δ0 = νσM into eq. (11) one may infer zcol for a given mass. Fig. 7 shows the typical zcol of 1σ, 2σ, and 3σ halos. The collapse of galaxy–sized 1σ halos occurs within a relatively small range of

redshifts. This is a direct consequence of the “flattening” suffered by σM

during radiation–dominated era due to stangexpansion (see §3.2). Therefore,

in a ΛCDM Universe it is not expected to observe a significant population of

galaxies at z > 5.

∼

Fig. 7. Collapse redshifts of spherical top–hat 1σ, 2σ and 3σ perturbations in a

ΛCDM cosmology with σ8 = 0.9. Note that galaxy–sized (M ∼ 108 − 1013M⊙)

1σ halos collapse in a redshift range, from z ∼ 3.5 to z = 0, respectively; the

corresponding ages are from ∼ 1.9 to 13.8 Gyr, respectively.

The problem of cosmological gravitational clustering is very complex due

to non–linearity, lack of symmetry and large dynamical range. Analytical

and semi–analytical approaches provide illuminating results but numerical

N–body simulations are necessary to tackle all the aspects of this problem. In

the last 20 years, the “industry” of numerical simulations had an impressive

development. The first cosmological simulations in the middle 80s used a few

104 particles (e.g., [36]). The currently largest simulation (called the Mille-

28

Vladimir Avila-Reese

nium simulation [111]) uses ∼ 1010 particles! A main effort is done to reach larger and larger dynamic ranges in order to simulate encompassing volumes

large enough to contain representative populations of all kinds of halos (low

mass and massive ones, in low– and high–density environments, high–peak

rare halos), yet resolving the inner structure of individual halos.

Halo mass function

The CDM halo mass function (comoving number density of halos of different

masses at a given epoch z, n(M, z)) obtained in the N–body simulations is

consistent with the P-S function in general, which is amazing given the ap-

proximate character of the P-S analysis. However, in more detail, the results

of large N–body simulations are better fitted by modified P-S analytical func-

tions, as the one derived in [103] and showed in Fig. 8. Using the Millennium simulation, the halo mass function has been accurately measured in the range

that is well sampled by this run (z ≤ 12, M ≥ 1.7 × 1010M⊙h−1). The mass

function is described by a power law at low masses and an exponential cut–off

at larger masses. The “cut-off”, most typical mass, increases with time and

is related to the hierarchical evolution of the 1σ halos shown in Fig. 7. The halo mass function is the starting point for modeling the luminosity function of galaxies. From Fig. 8 we see that the evolution of the abundances of massive halos is much more pronounced than the evolution of less massive

halos. This is why observational studies of abundance of massive galaxies or

cluster of galaxies at high redshifts provide a sharp test to theories of cosmic

structure formation. The abundance of massive rare halos at high redshifts

are for example a strong function of the fluctuation field primordial statistics

(Gaussianity or non-Gaussianity).

Subhalos. An important result of N–body simulations is the existence of sub-

halos, i.e. halos inside the virial radius of larger halos, which survived as

self–bound entities the gravitational collapse of the higher level of the hier-

archy. Of course, subhalos suffer strong mass loss due to tidal stripping, but

this is probably not relevant for the luminous galaxies formed in the inner-

most regions of (sub)halos. This is why in the case of subhalos, the maximum

circular velocity Vm (attained at radii much smaller than the virial radius) is

used instead of the virial mass. The Vm distribution of subhalos inside cluster–

sized and galaxy–sized halos is similar [83]. This distribution agrees with the distribution of galaxies seen in clusters, but for galaxy–sized halos the number

of subhalos overwhelms by 1–2 orders of magnitude the observed number of

satellite galaxies around galaxies like Milky Way and Andromeda [70, 83].

Fig. 9 (right side) shows the subhalo cumulative Vm−distribution for a CDM Milky Way–like halo compared to the observed satellite Vm−distribution.

In this Fig. are also shown the Vm−distributions obtained for the same Milky–

Way halo but using the power spectrum of three WDM models with particle

masses mX ≈ 0.6, 1, and 1.7 KeV. The smaller mX, the larger is the free–

streaming (filtering) scale, Rf , and the more substructure is washed out (see

Understanding Galaxy Formation and Evolution

29

Fig. 8. Evolution of the comoving number density of collapsed halos (P–S mass

function) according to the ellipsoidal modification by [103]. Note that the “cut–off”

mass grows with time. Most of the mass fraction in collapsed halos at a given epoch

are contained in halos with masses around the “cut–off” mass.

§3.2). In the left side of Fig. 9 is shown the DM distribution inside the Milky–

Way halo simulated by using a CDM power spectrum (top) and a WDM

power spectrum with mX ≈ 1KeV (sterile neutrino, bottom). For a student it

should be exciting to see with her(his) own eyes this tight connection between

micro– and macro–cosmos: the mass of the elemental particle determines the

structure and substructure properties of galaxy halos!

Halo density profiles

High–resolution N–body simulations [87] and semi–analytical techniques (e.g.,

[3]) allowed to answer the following questions: How is the inner mass distribution in CDM halos? Does this distribution depend on mass? How universal

is it? The two–parameter density profile established in [87] (the Navarro-Frenk-White, NFW profile) departs from a single power law, and it was

proposed to be universal and not depending on mass. In fact the slope

β(r) ≡ −dlogρ(r)/dlogr of the NFW profile changes from −1 in the cen-

ter to −3 in the periphery. The two parameters, a normalization factor, ρs

and a shape factor, rs, were found to be related in a such a way that the profile

depends only on one shape parameter that could be expressed as the concen-

tration, cNF W ≡ rs/Rv. The more massive the halo, the less concentrated on

the average. For the ΛCDM model, c ≈ 20−5 for M ∼ 2×108−2×1015M⊙h−1,

respectively [42]. However, for a given M , the scatter of cNF W is large (≈ 30 − 40%), and it is related to the halo formation history [3, 21, 125] (see

30

Vladimir Avila-Reese

Fig. 9. Dark matter distribution in a sphere of 400Mpch−1 of a simulated Galaxy–

sized halo with CDM (a) and WDM (mX = 1KeV, b). The substructure in the

latter case is significantly erased. Right panel shows the cumulative maximum Vc

distribution for both cases (open crosses and squares, respectively) as well as for an

average of observations of satellite galaxies in our Galaxy and in Andromeda (dotted

error bars). Adapted from [31] .

below). A significant fraction of halos depart from the NFW profile. These

are typically not relaxed or disturbed by companions or external tidal forces.

Is there a “cusp” crisis? More recently, it was found that the inner density

profile of halos can be steeper than β = −1 (e.g. [84]). However, it was shown that in the limit of resolution, β never is as steep a −1.5 [88]. The inner structure of CDM halos can be tested in principle with observations of (i) the

inner rotation curves of DM dominated galaxies (Irr dwarf and LSB galaxies;

the inner velocity dispersion of dSph galaxies is also being used as a test

), and (ii) strong gravitational lensing and hot gas distribution in the inner

regions of clusters of galaxies. Observations suggest that the DM distribution

in dwarf and LSB galaxies has a roughly constant density core, in contrast

to the cuspy cores of CDM halos (the literature on this subject is extensive;

see for recent results [37, 50, 107, 128] and more references therein). If the observational studies confirm that halos have constant–density cores, then

either astrophysical mechanisms able to expand the halo cores should work

efficiently or the ΛCDM scenario should be modified. In the latter case, one of

Understanding Galaxy Formation and Evolution

31

the possibilities is to introduce weakly self–interacting DM particles. For small

cross sections, the interaction is effective only in the more dense inner regions

of galaxies, where heat inflow may expand the core. However, the gravo–

thermal catastrophe can also be triggered. In [32] it was shown that in order to avoid the gravo–thermal instability and to produce shallow cores with densities

approximately constant for all masses, as suggested by observations, the DM

cross section per unit of particle mass should be σDM /mX = 0.5 − 1.0v−1

100

cm2/gr, where v100 is the relative velocity of the colliding particles in unities

of 100 km/s; v100 is close to the halo maximum circular velocity, Vm.

The DM mass distribution was inferred from the rotation curves of dwarf

and LSB galaxies under the assumptions of circular motion, halo spherical

symmetry, the lack of asymmetrical drift, etc. In recent studies it was discussed

that these assumptions work typically in the sense of lowering the observed

inner rotation velocity [59, 100, 118]. For example, in [118] it is demonstrated that non-circular motions (due to a bar) combined with gas pressure support

and projection effects systematically underestimate by up to 50% the rotation

velocity of cold gas in the central 1 kpc region of their simulated dwarf galaxies,

creating the illusion of a constant density core.

Mass–velocity relation. In a very simplistic analysis, it is easy to find that

M ∝ V 3

c

if the average halo density ρh does not depend on mass. On one

hand, Vc ∝ (GM/R)1/2, and on the other hand, ρh ∝ M/R3, so that Vc ∝

M 1/3ρ1/6. Therefore, for ρ

h

h =const, M ∝ V 3

c . We have seen in §3.2 that

the CDM perturbations at galaxy scales have similar amplitudes (actually

σM ∝ lnM) due to the stangexpansion effect in the radiation–dominated era.

This implies that galaxy–sized perturbations collapse within a small range

of epochs attaining more or less similar average densities. The CDM halos

actually have a mass distribution that translates into a circular velocity profile

Vc(r). The maximum of this profile, Vm, is typically the circular velocity that

characterizes a given halo of virial mass M . Numerical and semi–numerical

results show that (ΛCDM model):

V

3.2

M ≈ 5.2 × 104

m

M

kms−1

⊙h−1,

(14)

Assuming that the disk infrared luminosity LIR ∝ M, and that the disk

maximum rotation velocity Vrot,m ∝ Vm, one obtains that LIR ∝ V 3.2

rot,m,

amazingly similar to the observed infrared Tully–Fisher relation [116], one of the most robust and intriguingly correlations in the galaxy world! I conclude

that this relation is a clear imprint of the CDM power spectrum of fluctuations.

Mass assembling histories

One of the key concepts of the hierarchical clustering scenario is that cos-

mic structures form by a process of continuous mass aggregation, opposite to

the monolithic collapse scenario. The mass assembly of CDM halos is charac-

terized by the mass aggregation history (MAH), which can alternate smooth

32

Vladimir Avila-Reese

mass accretion with violent major mergers. The MAH can be calculated by using semi–analytical approaches based on extensions of the P-S formalism.

The main idea lies in the estimate of the conditional probability that given a

collapsed region of mass M0 at z0, a region of mass M1 embedded within the

volume containing M0, had collapsed at an earlier epoch z1. This probability

is calculated based on the excursion set formalism starting from a Gaussian

density field characterized by an evolving mass variance σM [17, 73]. By using the conditional probability and random trials at each temporal step, the

“backward” MAHs corresponding to a fixed mass M0 (defined for instance at

z = 0) can be traced. The MAHs of isolated halos by definition decrease to-

ward the past, following different tracks (Fig. 10), sometimes with abrupt big jumps that can be identified as major mergers in the halo assembly history.

Fig. 10. Upper panels (a). A score of random halo MAHs for a present–day virial mass of 3.5 × 1011M⊙ and the corresponding circular velocity profiles of the virialized halos. Lower panels (b). The average MAH and two extreme deviations from

104 random MAHs for the same mass as in (a), and the corresponding halo cir-

cular velocity profiles. The MAHs are diverse for a given mass and the Vc (mass)

distribution of the halos depend on the MAH. Adapted from [45] .

Understanding Galaxy Formation and Evolution

33

To characterize typical behaviors of the halo MAHs, one may calculate the

average MAH for a given virial mass M0, for a given “population” of halos

selected by its environment, etc. In the left panels of Fig. 10 are shown 20

individual MAHs randomly selected from 104 trials for M0 = 3.5 × 1011M⊙ in

a ΛCDM cosmology [45]. In the bottom panel are plotted the average MAH

from these 104 trials as well as two extreme deviations from the average. The

average MAHs depend on mass: more massive halos have a more extended

average MAH, i.e. they aggregate a given fraction of M0 latter than less mas-

sive halos. It is a convention to define the typical halo formation redshift, zf ,

when half of the current halo mass M0 has been aggregated. For instance, for

the ΛCDM cosmology the average MAHs show that zf ≈ 2.2, 1.2 and 0.7 for

M0 = 1010M⊙, 1012M⊙ and 1014M⊙, respectively. A more physical definition

of halo formation time is when the halo maximum circular velocity Vm attains

its maximum value. After this epoch, the mass can continue growing, but the

inner gravitational potential of the system is already set.

Right panels of Fig. 10 show the present–day halo circular velocity profiles, Vc(r), corresponding to the MAHs plotted in the left panels. The average

Vc(r) is well described by the NFW profile. There is a direct relation between

the MAH and the halo structure as described by Vc(r) or the concentration

parameter. The later the MAH, the more extended is Vc(r) and the less con-

centrated is the halo [3, 125]. Using high–resolution simulations some authors have shown that the halo MAH presents two regimes: an early phase of fast

mass aggregation (mainly by major mergers) and a late phase of slow aggre-

gation (mainly by smooth mass accretion) [133, 75]. The potential well of a present–day halo is set mainly at the end of the fast, major–merging driven,

growth phase.

From the MAHs we may infer: (i) the mass aggregation rate evolution of

halos (halo mass aggregated per unit of time at different z′s), and (ii) the ma-

jor merging rates of halos (number of major mergers per unit of time per halo

at different z′s). These quantities should be closely related to the star forma-

tion rates of the galaxies formed within the halos as well as to the merging of

luminous galaxies and pair galaxy statistics. By using the ΛCDM model, sev-

eral studies showed that most of the mass of the present–day halos has been

aggregated by accretion rather than major mergers (e.g., [85]). Major merging was more frequent in the past [55], and it is important for understanding the formation of massive galaxy spheroids and the phenomena related to this

process like QSOs, supermassive black hole growth, obscured star formation

bursts, etc. Both the mass aggregation rate and major merging rate histories

depend strongly on environment: the denser the environment, the higher is

the merging rate in the past. However, in the dense environments (group and

clusters) form typically structures more massive than in the less dense regions

(field and voids). Once a large structure virializes, the smaller, galaxy–sized

halos become subhalos with high velocity dispersions: the mass growth of the

subhalos is truncated, or even reversed due to tidal stripping, and the merging

probability strongly decreases. Halo assembling (and therefore, galaxy assem-

34

Vladimir Avila-Reese

bling) definitively depends on environment. Overall, by integrating the MAHs

of the whole galaxy–sized ΛCDM halo population in a given volume, the gen-

eral result is that the peak in halo assembling activity was at z ≈ 1 − 2. After

these redshifts, the global mass aggregation rate strongly decreases (e.g., [121].

To illustrate the driving role of DM processes in galaxy evolution, I men-

tion briefly here two concrete examples:

1). Distributions of present–day specific mass aggregation rate, ( ˙

M /M )0 ,

and halo lookback formation time, T1/2 . For a ΛCDM model, these distributions are bimodal, in particular the former. We have found that roughly

40% of halos (masses larger than ≈ 1011M⊙h−1) have ( ˙

M /M )0 ≤ 0; they

are basically subhalos. The remaining 60% present a broad distribution of

( ˙

M /M )0 > 0 peaked at ≈ 0.04Gyr−1. Moreover, this bimodality strongly

changes with large–scale environment: the denser is the environment the,

higher is the fraction of halos with ( ˙

M /M )0 ≤ 0. It is interesting enough

that similar fractions and dependences on environment are found for the spe-

cific star formation rates of galaxies in large statistical surveys (§§2.3); the

situation is similar when confronting the distributions of T1/2 and observed

colors. Therefore, it seems that the the main driver of the observed bimodal-

ities in z = 0 specific star formation rate and color of galaxies is the nature of the CDM halo mass aggregation process. Astrophysical processes of course

are important but the main body of the bimodalities can be explained just at

the level of DM processes.

2. Major merging rates. The observational inference of galaxy major merg-

ing rates is not an easy task. The two commonly used methods are based on

the statistics of galaxy pairs (pre–mergers) and in the morphological distor-

tions of ellipticals (post–mergers). The results show that the merging rate

increases as (1 + z)x, with x ∼ 0 − 4. The predicted major merging rates in

the ΛCDM scenario agree roughly with those inferred from statistics of galaxy

pairs. From the fraction of normal galaxies in close companions (with sepa-

rations less than 50 kpch−1) inferred from observations at z = 0 and z = 0.3

[91], and assuming an average merging time of ∼ 1 Gyr for these separations, we estimate that the major merging rate at the present epoch is ∼ 0.01 Gyr−1

for halos in the range of 0.1 − 2.0 1012M⊙, while at z = 0.3 the rate increased

to ∼ 0.018 Gyr−1. These values are only slightly lower than predictions for

the ΛCDM model.

Angular momentum

The origin of the angular momentum (AM) is a key ingredient in theories of

galaxy formation. Two mechanisms of AM acquirement were proposed for the

CDM halos (e.g., [93, 22, 78]): 1. tidal torques of the surrounding shear field when the perturbation is still in the linear regime, and 2. transfer of orbital AM

to internal AM in major and minor mergers of collapsed halos. The angular

momentum of DM halos is parametrized in terms of the dimensionless spin

√

parameter λ ≡ J E/(GM5/2, where J is the modulus of the total angular

Understanding Galaxy Formation and Evolution

35

momentum and E is the total (kinetic plus potential). It is easy to show that λ

can be interpreted as the level of rotational support of a gravitational system,

λ = ω/ωsup, where ω is the angular velocity of the system and ωsup is the

angular velocity needed for the system to be rotationally supported against

gravity (see [90]).

For disk and elliptical galaxies, λ ∼ 0.4−0.8 and ∼ 0.01−0.05, respectively.

Cosmological N–body simulations showed that the CDM halo spin parameter

is log–normal distributed, with a median value λ ≈ 0.04 and a standard de-

viation σλ ≈ 0.5; this distribution is almost independent from cosmology. A

related quantity, but more straightforward to compute is λ′ ≡

J

√

[22],

2MVv Rv

where Rv is the virial radius and Vv the circular velocity at this radius. Recent

simulations show that (λ′, σλ′ ) ≈ (0.035, 0.6), though some variations with en-

vironment and mass are measured [5]. The evolution of the spin parameter depends on the AM acquirement mechanism. In general, a significant systematical change of λ with time is not expected, but relatively strong changes are

measured in short time steps, mainly after merging of halos, when λ increases.

How is the internal AM distribution in CDM halos? Bullock et al. [22]

found that in most of cases this distribution can be described by a simple

(universal) two–parameter function that departs significantly from the solid–

body rotation distribution. In addition, the spatial distribution of AM in

CDM halos tends to be cylindrical, being well aligned for 80% of the halos, and

misaligned at different levels for the rest. The mass distribution of the galaxies

formed within CDM halos, under the assumption of specific AM conservation,

is established by λ, the halo AM distribution, and its alignment.

4.2 Non–baryonic dark matter candidates

The non–baryonic DM required in cosmology to explain observations and cos-

mic structure formation should be in form of elemental or scalar field particles

or early formed quark nuggets. Modifications to fundamental physical theories

(modified Newtonian Dynamics, extra–dimensions, etc.) are also plausible if

DM is not discovered.

There are several docens of predicted elemental particles as DM candi-

dates. The list is reduced if we focus only on well–motivated exotic particles

from the point of view of particle physics theory alone (see for a recent review

[53]). The most popular particles beyond the standard model are the supersymmetric (SUSY) particles in supersymmetric extensions of the Standard

Model of particle physics. Supersymmetry is a new symmetry of space–time

introduced in the process of unifying the fundamental forces of nature (includ-

ing gravity). An excellent CDM candidate is the lightest stable SUSY particle

under the requirement that superpartners are only produced or destroyed in

pairs (called R-parity conservation). This particle called neutralino is weakly interacting and massive (WIMP). Other SUSY particles are the gravitino and

the sneutrino; they are of WDM type. The predicted masses for neutralino

range from ∼ 30 to 5000 GeV. The cosmological density of neutralino (and of

36

Vladimir Avila-Reese

other thermal WIMPs) is naturally as required when their interaction cross

section is of the order of a weak cross section. The latter gives the possibility

to detect neutralinos in laboratory.

The possible discovery of WIMPs relies on two main techniques:

(i) Direct detections. The WIMP interactions with nuclei (elastic scattering)

in ultra–low–background terrestrial targets may deposit a tiny amount of en-

ergy (< 50 keV) in the target material; this kinetic energy of the recoiling

nucleus is converted partly into scintillation light or ionization energy and

partly into thermal energy. Dozens of experiments worldwide -of cryogenic

or scintillator type, placed in mines or underground laboratories, attempt to

measure these energies. Predicted event rates for neutralinos range from 10−6

to 10 events per kilogram detector material and day. The nuclear recoil spec-

trum is featureless, but depends on the WIMP and target nucleus mass. To

convincingly detect a WIMP signal, a specific signature from the galactic halo

particles is important. The Earth’s motion through the galaxy induces both a

seasonal variation of the total event rate and a forward–backward asymmetry

in a directional signal. The detection of structures in the dark velocity space,

as those predicted to be produced by the Sagittarius stream, is also an specific

signature from the Galactic halo; directional detectors are needed to measure

this kind of signatures.

The DAMA collaboration reported a possible detection of WIMP particles

obeying the seasonal variation; the most probable value of the WIMP mass

was ∼ 60 GeV. However, the interpretation of the detected signal as WIMP

particles is controversial. The sensitivity of current experiments (e.g., CDMS

and EDEL-WEISS) limit already the WIMP–proton spin–independent cross

sections to values < 2 10−42 − 10−40cm−2 for the range of masses ∼ 50 − 104

∼

GeV, respectively; for smaller masses, the cross–section sensitivities are larger,

and WIMP signals were not detected. Future experiments will be able to test

the regions in the cross-section–WIMP mass diagram, where most of models

make certain predictions.

(ii) Indirect detections. We can search for WIMPS by looking for the prod-

ucts of their annihilation. The flux of annihilation products is proportional

to the square of the WIMP density, thus regions of interest are those where

the WIMP concentration is relatively high. There are three types of searches

according to the place where WIMP annihilation occur: (i) in the Sun or the

Earth, which gives rise to a signal in high-energy neutrinos; (ii) in the galactic

halo, or in the halo of external galaxies, which generates γ−rays and other

cosmic rays such as positrons and antiprotons; (iii) around black holes, spe-

cially around the black hole at the Galactic Center. The predicted radiation

fluxes depend on the particle physics model used to predict the WIMP candi-

date and on astrophysical quantities such as the dark matter halo structure,

the presence of sub–structure, and the galactic cosmic ray diffusion model.

Most of WIMPS were in thermal equilibrium in the early Universe (thermal

relics). Particles which were produced by a non-thermal mechanism and that

Understanding Galaxy Formation and Evolution

37

never had the chance of reaching thermal equilibrium are called non-thermal

relics (e.g., axions, solitons produced in phase transitions, WIMPZILLAs pro-

duced gravitationally at the end of inflation). From the side of WDM, the most

popular candidate are the ∼ 1KeV sterile neutrinos. A sterile neutrino is a

fermion that has no standard model interactions other than a coupling to the

standard neutrinos through their mass generation mechanism. Cosmological

probes, mainly the power spectrum of Lyα forest at high redshifts, constrain

the mass of the sterile neutrino to values larger than ∼ 2KeV.

5 The bright side of galaxy formation and evolution

The ΛCDM scenario of cosmic structure formation has been well tested for

perturbations that are still in the linear or quasilinear phase of evolution.

These tests are based, among other cosmological probes, on accurate mea-

surements of:

• the CMBR temperature fluctuations at large and small angular scales

• the large–scale mass power spectrum as traced by the spatial distribu-

tion of galaxies and cluster of galaxies, by the Lyα forest clouds, by maps of

gravitational weak and strong lensing, etc.

• the peculiar large–scale motions of galaxies13.

• the statistics of strong gravitational lensing (multiple–lensed arcs).

Although these cosmological probes are based on observations of lumi-

nous (baryonic) objects, the physics of baryons plays a minor or indirect role

in the properties of the linear mass perturbations. The situation is different

at small (galaxy) scales, where perturbations went into the non–linear regime

and the dissipative physics of baryons becomes relevant. The interplay of DM

and baryonic processes is crucial for understanding galaxy formation and evo-

lution. The progress in this field was mostly heuristic; the ΛCDM scenario

provides the initial and boundary conditions for modeling galaxy evolution,

but the complex physics of the baryonic processes, in the absence of funda-

mental theories, requires a model adjustment through confrontation with the

observations.

Following, I will outline some key concepts, ingredients, and results of the

galaxy evolution study based on the ΛCDM scenario. Some of the pioneer pa-

pers in this field are those of Gunn [57], White & Reese [131], Fall & Efstathiou

[43], Blumental et al. [15], Davis et al. [36], Katz & Gunn [65], White & Frenk

[130], Kauffmann et al. [66]. For useful lecture notes and recent reviews see e.g., Longair [76, 77], White [129], Steinmetz [113], Firmani & Avila-Reese

[46].

13 Recall that linear theory relates the peculiar velocity, that is the velocity deviation from the Hubble flow, to the density contrast. It is said that the cosmological

velocity field is potential; any primordial rotational motion able to give rise to a density perturbation decays as the Universe expands due to angular momentum

conservation.

38

Vladimir Avila-Reese

The main methods of studying galaxy formation and evolution in the

ΛCDM context are:

• Semi-analytical Models (e.g., [130, 66, 28, 9, 108, 29, 12, 10]), where the halo mass assembling histories are calculated with the extended Press–

Schechter formalism and galaxies are seeded within the halos by means of

phenomenological recipes. This method is very useful for producing whole

populations of galaxies at a given epoch and predicting statistical properties

as the luminosity function and the morphological mix.

• Semi-numerical Models (e.g, [45, 2, 119, 16]), where the internal physics of the galaxies, including those of the halos, are modeled numerically but

under simplifying assumptions; the initial and boundary conditions are taken

from the ΛCDM scenario by using the extended Press–Schechter formalism

and halo AM distributions from simulations. This method is useful to predict

the local properties of galaxies and correlations among the global properties,

as well as to follow the overall evolution of individual galaxies.

• Numerical N–body+hydrodyamical simulations (e.g., [65, 26, 64, 86, 112,

126, 1, 110, 56]), where the DM and baryonic processes are followed in cosmological simulations. This is the most advanced and complete approach to

galaxy evolution. However, current limitations in the computational capabili-

ties and the lack of fundamental theories for several of the physical processes

involved, do not allow yet to exploit optimally this method. A great advance

is being made currently with an hybrid approach: in the high–resolution cos-

mological N–body simulations of only DM, galaxies are grafted by using the

semi–analytical models (e.g., [67, 60, 38, 13, 111, 63]).

5.1 Disks

The formation of galaxy disks deep inside the CDM halos is a generic process

in the ΛCDM scenario. Let us outline the (simplified) steps of disk galaxy

formation in this scenario:

1. DM halo growth. The “mold” for disk formation is provided by the mass

and AM distributions of the virialized halo, which grows hierarchically. A

description of these aspects were presented in the previous Section.

2. Gas cooling and infall, and the maximum mass of galaxies. It is common

to assume that the gas in a halo is shock–heated during collapse to the virial

temperature [131]. The gas then cools radiatively and falls in a free–fall time to the center. The cooling function Λ(n, Tk; Z) depends on the gas density,

temperature, and composition14. Since the seminal work by White & Frenk

(1990) [130], the rate infall of gas available to form the galaxy is assumed to 14 The main cooling processes for the intrahalo gas are collisional excitation and

ionization, recombination, and bremsstrahlung. The former is the most efficient

for kinetic temperatures Tk ≈ 104 −105K and for neutral hydrogen and single ion-

ized helium; for a meta–enriched gas, cooling is efficient at temperatures between

105 − 107K. At higher temperatures, where the gas is completely ionized, the

Understanding Galaxy Formation and Evolution

39

be driven either by the free–fall time, tff , if tff > tcool or by the cooling time

tcool if tff < tcool. The former case applies to halos of masses smaller than

approximately 5×1011M⊙, whilst the latter applies to more massive halos. The

cooling flow from the quasistatic hot atmosphere is the process that basically

limits the baryonic mass of galaxies [105], and therefore the bright end of the galaxy luminosity function; for the outer, dilute hot gas in large halos, tcool

becomes larger than the Hubble time. However, detailed calculations show

that even so, in massive halos too much gas cools, and the bright end of the

predicted luminosity function results with a decrease slower than the observed

one [12]. Below we will see some solutions proposed to this problem.

More recently it was shown that the cooling of gas trapped in filaments

during the halo collapse may be so rapid that the gas flows along the filaments

to the center, thus avoiding shock heating [69]. However, this process is efficient only for halos less massive than 2.5 × 1011M⊙, which in any case (even if

shock–heating happens), cool their gas very rapidly [19]. Thus, for modeling the formation of disks, and for masses smaller than ∼ 5 × 1011M⊙, we may

assume that gas infalls in a dynamical time since the halo has virialized, or in

two dynamical times since the protostructure was at its maximum expansion.

3. Disk formation, the origin of exponentially, and rotation curves. The gas,

originally distributed in mass and AM as the DM, cools and collapses until

it reaches centrifugal balance in a disk. Therefore, assuming detailed AM

conservation, the radial mass distribution of the disk can be calculated by

equating its specific AM to the AM of its final circular orbit in centrifugal

equilibrium. The typical collapse factor of the gas within a DM halo is ∼

10 − 1515, depending on the initial halo spin parameter λ; the higher the λ,

the more extended (lower surface density) is the resulting disk. The surface

density profile of the disks formed within CDM halos is nearly exponential,

which provides an explanation to the long–standing question of why galaxy

disks are exponential. This is a direct consequence of the AM distribution

acquired by the halos by tidal torques and mergers. In more detail, however,

the profiles are more concentrated in the center and with a slight excess in the

periphery than the exponential law [45, 22]. The cusp in the central disk could give rise to either a photometrical bulge [120] or to a real kinematical bulge due to disk gravitational instability enhanced by the higher central surface density

[2] (bulge secular formation). In a few cases (high–λ, low–concentrated halos), purely exponential disks can be formed.

Baryons are a small mass fraction in the CDM halos, however, the disk

formed in the center is very dense (recall the high collapse factors), so that

dominant cooling process is bremsstrahlung. At temperatures lower than 104K

(small halos) and in absence of metals, the main cooling process is by H2 and

HD molecule line emission.

15 It is interesting to note that in the absence of a massive halo around galaxies, the

collapse factor would be larger by ∼ M/Md ≈ 20, where M and Md are the total

halo and disk masses, respectively [90].

40

Vladimir Avila-Reese

the contribution of the baryonic disk to the inner gravitational potential is

important or even dominant. The formed disk will drag gravitationally DM,

producing an inner halo contraction that is important to calculate for obtain-

ing the rotation curve decomposition. The method commonly used to calculate

it is based on the approximation of radial adiabatic invariance, where spher-

ical symmetry and circular orbits are assumed (e.g., [47, 82]). However, the orbits in CDM halos obtained in N–body simulations are elliptical rather than

circular; by generalizing the adiabatic invariance to elliptical orbits, the halo

contraction becomes less efficient [132, 52].

The rotation curve decomposition of disks within contracted ΛCDM halos

are in general consistent with observations [82, 45, 132] (nearly–flat total rotation curves; maximum disk for high–surface brightness disks; submaximum

disk for the LSB disks; in more detail, the outer rotation curve shape depends

on surface density, going from decreasing to increasing at the disk radius for

higher to lower densities, respectively). However, there are important non–

solved issues. For example, from a large sample of observed rotation curves,

Persic et al. [95] inferred that the rotation curve shapes are described by an

“universal” profile that (i) depends on the galaxy luminosity and (ii) implies

a halo profile different from the CDM (NFW) profile. Other studies confirm

only partially these claims [123, 132, 25]. Statistical studies of rotation curves are very important for testing the ΛCDM scenario.

In general, the structure and dynamics of disks formed within ΛCDM halos

under the assumption of detailed AM conservation seem to be consistent with

observations. An important result to remark is the successful prediction of the

infrared Tully–Fisher relation and its scatter16. The core problem mentioned

in §4.2 is the most serious potential difficulty. Other potential difficulties are:

(i) the predicted disk size (surface brightness) distribution implies a P (λ)

distribution narrower than that corresponding to ΛCDM halos by almost a

factor of two [74]; (ii) the internal AM distribution inferred from observations of dwarf galaxies seems not to be in agreement with the ΛCDM halo AM

distribution [122]; (iii) the inference of the halo profile from the statistical study of rotation curve shapes seems not to be agreement with CMD halos.

In N–body+hydrodynamical simulations of disk galaxy formation there was

common another difficulty called the ’angular momentum catastrophe’: the

simulated disks ended too much concentrated, apparently due to AM trans-

ference of baryons to DM during the gas collapse. The formation of highly

concentrated disks also affects the shape of the rotation curve (strongly de-

creasing), as well as the zero–point of the Tully–Fisher relation. Recent nu-

16 In §4.1 we have shown that the basis of the Tully–Fisher relation is the CDM halo

M − Vm relation. From the pure halo to the disk+halo system there are several

intermediate processes that could distort the original M − Vm relation. However,

it was shown that the way in which the CDM halo couples with the disk and the

way galaxies transform their gas into stars “conspire” to keep the relation. Due

to this conspiring, the Tully–Fisher relation is robust to variations in the baryon

fraction fB (or mass–to–luminosity ratios) and in the spin parameter λ [45].

Understanding Galaxy Formation and Evolution

41

merical simulations are showing that the ’angular momentum catastrophe’,

rather than a physical problem, is a problem related to the resolution of the

simulations and the correct inclusion of feedback effects.

4. Star formation and feedback. We are coming to the less understood and

most complicated aspects of the models of galaxy evolution, which deserve

separate notes. The star formation (SF) process is studied at two levels (each

one by two separated communities!): (i) the small–scale physics, related to the

complex processes by which the cold gas inside molecular clouds fragments and

collapses into stars, and (ii) the large–scale physics, related to the disk global

instabilities that give rise to the largest unities of SF, the molecular clouds.

The SF physics incorporated to galaxy evolution models is still oversimpli-

fied, phenomenological and refers to the latter item. The large-scale SF cycle

in normal galaxies is believed to be self–regulated by a balance between the

energy injection due to SF (mainly SNe) and dissipation (radiative or turbu-

lent). Two main approaches have been used to describe the SF self–regulation

in models of galaxy evolution: (a) the halo cooling-feedback approach [130]), (b) the disk turbulent ISM approach [44, 124].

According to the former, the cool gas is reheated by the “galaxy” SF feed-

back and driven back to the intrahalo medium until it again cools radiatively

and collapses into the galaxy. This approach has been used in semi–analytical

models of galaxy formation where the internal structure and hydrodynamics

of the disks are not treated in detail. The reheating rate is assumed to depend

on the halo circular velocity Vc: ˙

Mrh ∝ ˙

Ms/V α

c , where

˙

Ms is the SF rate

(SFR) and α ≥ 2. Thus, the galaxy SFR, gas fraction and luminosity depend

on Vc. In these models, the disk ISM is virtually ignored and the SN–energy

injection is assumed to be as efficient as to reheat the cold gas up to the virial

temperature of the halo. A drawback of the model is that it predicts hot X-ray

halos around disk galaxies much more luminous than those observed.

Approach (b) is more appropriate for models where the internal processes

of the disk are considered. In this approach, the SF at a given radius r is

assumed to be triggered by disk gravitational instabilities (Toomre criterion)

and self–regulated by a balance between energy injection (mainly by SNe)

and dissipation in the turbulent ISM in the direction perpendicular to the

disk plane:

v

Q

g (r)κ(r)

g(r) ≡

< Q

πGΣ

crit

(15)

g (r)

Σg(r)v2

γ

˙

g (r)

SN ǫSN Σ∗(r) + ˙

ΣE,accr(r) =

,

(16)

2td(r)

where vg and Σg are the gas velocity dispersion and surface density, κ is

the epicyclic frequency, Qcrit is a critical value for instability, γSN and ǫSN

are the kinetic energy injection efficiency of the SN into the gas and the

SN energy generated per gram of gas transformed into stars, respectively,

˙

Σ∗ is the surface SFR, and ˙

ΣE,accr is the kinetic energy input due to mass

42

Vladimir Avila-Reese

accretion rate (or eventually any other energy source as AGN feedback). The

key parameter in the self–regulating process is the dissipation time td. The

disk ISM is a turbulent, non-isothermal, multi-temperature flow. Turbulent

dissipation in the ISM is typically efficient (td ∼ 107−108yr) in such a way that

self–regulation happens at the characteristic vertical scales of the disk. Thus,

there is not too much room for strong feedback with the gas at heights larger

than the vertical scaleheigth of normal present–day disks: self–regulation is

at the level of the disk, but not at the level of the gas corona around. With

this approach the predicted SFR is proportional to Σn

g (Schmidt law), with

n ≈ 1.4 − 2 varying along the disk, in good agreement with observational

inferences. The typical SF timescales are not longer than 3 − 4Gyr. Therefore,

to keep active SFRs in the disks, gas infall is necessary, a condition perfectly

fulfilled in the ΛCDM scenario.

Given the SFR radius by radius and time by time, and assuming an IMF,

the corresponding luminosities in different color bands can be calculated with

stellar population synthesis models. The final result is then an evolving inside–

out luminous disk with defined global and local colors.

5. Secular evolution

The “quiet” evolution of galaxy disks as described above can be disturbed by

minor mergers (satellite accretion) and interactions with close galaxy com-

panions. However, as several studies have shown, the disk may suffer even

intrinsic instabilities which lead to secular changes in its structure, dynam-

ics, and SFR. The main effects of secular evolution, i.e. dynamical processes

that act in a timescale longer than the disk dynamical time, are the vertical

thickening and “heating” of the disk, the formation of bars, which are efficient

mechanisms of radial AM and mass redistribution, and the possible formation

of (pseudo)bulges (see for recent reviews [71, 33]). Models of disk galaxy evolution should include these processes, which also can affect disk properties,

for example increasing the disk scale radii [117].

5.2 Spheroids

As mentioned in §2, the simple appearance, the dominant old stellar popu-

lations, the α–elements enhancement, and the dynamically hot structure of

spheroids suggest that they were formed by an early (z > 4) single violent event

∼

with a strong burst of star formation, followed by passive evolution of their

stellar population ( monolithic mechanism). Nevertheless, both observations

and theory point out to a more complex situation. There are two ways to de-

fine the formation epoch of a spheroid: when most of its stars formed or when

the stellar spheroid acquired its dynamical properties in violent or secular

processes. For the monolithic collapse mechanism both epochs coincide.

In the context of the ΛCDM scenario, spheroids are expected to be formed

basically as the result of major mergers of disks. However,

Understanding Galaxy Formation and Evolution

43

• if the major mergers occur at high redshifts, when the disks are

mostly gaseous, then the situation is close to the monolithic collapse;

• if the major mergers occur at low redshifts, when the galaxies

have already transformed a large fraction of their gas into stars, then

the spheroids assemble by the “classical” dissipationless collision.

Besides, stellar disks may develop spheroids in their centers (bulges) by

secular evolution mechanisms, both intrinsic or enhanced by minor mergers

and interactions; this channel of spheroid formation should work for late–

type galaxies and it is supported by a large body of observations [71]. But the picture is even more complex in the hierarchical cosmogony as galaxy

morphology may be continuously changing, depending on the MAH (smooth

accretion and violent mergers) and environment. An spheroid formed early

should continue accreting gas so that a new, younger disk grows around.

A naive expectation in the context of the ΛCDM scenario is that massive

elliptical galaxies should be assembled mainly by late major mergers of the

smaller galaxies in the hierarchy. It is also expected that the disks in galaxies

with small bulge–to–disk ratios should be on average redder than those in

galaxies with large bulge–to–disk ratios, contrary to observations.

Although it is currently subject of debate, a more elaborate picture of

spheroid formation is emerging now in the context of the ΛCDM hierarchical

scenario (see [106, 46, 39] and the references therein). The basic ideas are that massive ellipticals formed early (z > 3) and in a short timescale by the

∼

merging of gas–rich disks in rare high–peak, clustered regions of the Universe.

The complex physics of the merging implies (i) an ultraluminous burst of SF

obscured by dust (cool ULIRG phase) and the establishment of a spheroidal

structure, (ii) gas collapse to the center, a situation that favors the growth of

the preexisting massive black hole(s) through an Eddington or even super–

Eddington regime (warm ULIRG phase), (iii) the switch on of the AGN activ-

ity associated to the supermassive black hole when reaching a critical mass,

reverting then the gas inflow to gas outflow (QSO phase), (iv) the switch off

of the AGN activity leaving a giant stellar spheroid with a supermassive black

hole in the center and a hot gas corona around (passive elliptical evolution).

In principle, the hot corona may cool by cooling flows and increase the mass

of the galaxy, likely renewing a disk around the spheroid. However, it seems

that recurrent AGN phases (less energetic than the initial QSO phase) are

possible during the life of the spheroid. Therefore, the energy injected from

AGN in the form of radio jets (feedback) can be responsible for avoiding the

cooling flow. This way is solved the problem of disk formation around the

elliptical, as well as the problem of the extended bright end in the luminosity

function. It is also important to note that as soon as the halo hosting the

elliptical becomes a subhalo of the group or cluster, the MAH is truncated

(§4). According to the model just described, massive elliptical galaxies were

in place at high redshifts, while less massive galaxies (collapsing from more

common density peaks) assembled later. This model was called downsizing or

44

Vladimir Avila-Reese

anti-hierarchical. In spite of the name, it fits perfectly within the hierarchical

ΛCDM scenario.

5.3 Drivers of the Hubble sequence

• Disks are generic objects formed by gas dissipation and collapse inside the

growing CDM halos. Three (cosmological) initial and boundary conditions

related to the halos define the main properties of disks in isolated halos:

1. The virial mass, which determines extensive properties

2. The spin parameter λ, which determines mainly the disk surface

brightness (SB; it gives rise to the sequence from high SB to low SB

disks) and strongly influences the rotation curve shape and the bulge–

to–disk ratio (within the secular scenario). λ also plays some role in

the SFR history.

3. The MAH, which drives the gas infall rate and, therefore, the

disk SFR and color; the MAH determines also the halo concentration,

and its scatter is reflected in the scatter of the Tully–Fisher relation.

The two latter determine the intensive properties of disks, suggesting a

biparametrical sequence in SB and color. There is a fourth important param-

eter, the galaxy baryon fraction fB, which influences the disk SB and rotation

curve shape. We have seen that fB in galaxies is 3–5 times lower than the uni-

versal ΩB/ΩDM fraction. This parameter is related probably to astrophysical

processes as gas dissipation and feedback.

• The clustering of CDM halos follows an spatial distribution with very

different large–scale environments. In low–density environments, halos live

mostly isolated, favoring the formation of disks, whose properties are driven

by the factors mentioned above. However, as we move to higher–density envi-

ronments, halos form from more and more clustered high–peak perturbations

that assemble early by violent major mergers: this is the necessary condition

to form massive ellipticals. At some time, the larger scale in the hierarchy col-

lapses and the halo becomes a subhalo: the mass aggregation is then truncated

and the probability of merging decreases dramatically. Elliptical galaxies are

settled and continue evolving passively. Thus, the environment of CDM halos

is another important driver of the Hubble sequence, able to establish the main

body of the observed blue–red and early–type morphology sequences and their

dependences on density.

• Although the initial, boundary and environmental conditions provided

by the ΛCDM scenario are drivers of several of the main properties and cor-

relations of galaxies, astrophysical processes should also play an important

role. The driving astrophysical processes are global SF and feedback. They

should come in two modes that drive the disk and elliptical sequences: (i)

the quiescent disk mode, where disk instabilities trigger SF and local (nega-

tive) feedback self–regulates the SFR, and (ii) the bursting mode of violent

mergers of gaseous galaxies, where local shocks and gravothermal catastrophe

trigger SF, and presumably a positive feedback increases its efficiency. Other

Understanding Galaxy Formation and Evolution

45

important astrophysical drivers of galaxy properties are: (i) the SN–induced

wind–driven outflows, which are important to shape the properties of dwarf

galaxies (M < 1010M

< 80km/s), (ii) the AGN–induced hydrodynamical

∼

⊙, Vm ∼

outflows, which are important to prevent cooling flows in massive ellipticals,

(iii) several processes typical of high–density environments such as ram pres-

sure, harassment, strangulation, etc., presumably important to shape some

properties of galaxies in clusters.

6 Issues and outlook

Our understanding of galaxy formation and evolution is in its infancy. So far,

only the first steps were given in the direction of consolidating a theory in

this field. The process is apparently so complex and non–linear that several

specialists do not expect the emergence of a theory in the sense that a few

driving parameters and factors might explain the main body of observations.

Instead, the most popular trend now is to attain some description of galaxy

evolution by simulating it in expensive computational runs. I believe that

simulations are a valuable tool to extend a bridge between reality and the

distorted (biased) information given by observations. However, the search of

basic theories for explaining galaxy formation and evolution should not be

replaced by the only effort of simulating in detail what in fact we want to get.

The power of science lies in its predictive capability. Besides, if galaxy theory

becomes predictive, then its potential to test fundamental and cosmological

theories will be enormous.

Along this notes, potential difficulties or unsolved problems of the ΛCDM

scenario were discussed. Now I summarize and complement them:

Physics

• What is non–baryonic DM? From the structure formation side, the preferred

(and necessary!) type is CDM, though WDM with filtering masses below ∼

109M⊙ is also acceptable. So far none of the well–motivated cold or warm non–

baryonic particles have been detected in Earth experiments. The situation is

even worth for proposals not based on elemental particles as DM from extra–

dimensions.

• What is Dark Energy? Dark Energy does not play apparently a signif-

icant role in the internal evolution of perturbations but it crucially defines

the cosmic timescale and expansion rate, which are important for the grow-

ing factor of perturbations. The simplest interpretation of Dark Energy is the

homogeneous and inert cosmological constant Λ, with equation of state pa-

rameter w = −1 and ρΛ =const. The combinations of different cosmological

probes tend to favor the flat-geometry Λ models with (ΩM , ΩΛ)≈(0.26, 0.74).

However, the cosmological constant explanation of Dark Energy faces serious

theoretical problems. Several alternatives to Λ were proposed to ameliorate

46

Vladimir Avila-Reese

partially these problems (e.g. quintaessence, k–essence, Chaplygin gas, etc.).

Also have been proposed unifying schemes of DM and Dark Energy through

scalar fields (e.g, [81]).

Cosmology

• Inflation provides a natural mechanism for the generation of primordial

fluctuations. The nearly scale–invariance of the primordial power spectrum is

well predicted by several inflation models, but its amplitude, rather than being

predicted, is empirically inferred from observations of CMBR anisotropies.

Another aspect of primordial fluctuations not well understood is related to

their statistics, i.e., whether they are Gaussian–distributed or not. And this

is crucial for cosmic structure formation.

• Indirect pieces of evidence are consistent with the main predictions of

inflation regarding primordial fluctuations. However, more direct tests of this

theory are highly desirable. Hopefully, CMBR anisotropy observations will

allow for some more direct tests (e.g., effects from primordial gravitational

waves).

Astrophysics

• Issues at small scales. The excess of substructure (satellite galaxies) can

be apparently solved by inhibition of galaxy formation in small halos due to

UV–radiation produced by reionization and due to feedback, rather than to

modifications to the scenario (e.g., the introduction of WDM). Observational

inferences of the inner volume and phase–space densities of dwarf satellite

galaxies are crucial to explore this question. The direct detection (with gravi-

tational lensing) of the numerous subhalo (dark galaxy) population predicted

by CDM for the Galaxy halo is a decisive test on the problem of substruc-

ture. The CDM prediction of cuspy halos is a more involved problem when

confronting it with observational inferences. If the disagreement persists, then

either the ΛCDM scenario will need a modification (e.g., introduction of self–

interaction or annihilation), or astrophysical processes involving gas baryon

physics should be in action. However, there are still unsolved issues at the

intermediate level: for example, the central halo density profile of galaxies is

inferred from observations of inner rotation curves under several assumptions

that could be incorrect. An interesting technique to overcome this problem

is being currently developed: to simulate as realistically as possible a given

galaxy, “observe” its rotation curve and then compare with that of the real

galaxy (see §§4.1).

• The early formation of massive red elliptical galaxies can be accommo-

dated in the hierarchical ΛCDM scenario (§§5.2) if spheroids are produced by

the major merger of gaseous disks, and if the cold gas is transformed rapidly

into stars during the merger in a dynamical time or so. Both conditions should

be demonstrated, in particular the latter. A kind of positive feedback seems

Understanding Galaxy Formation and Evolution

47

to be necessary for such an efficient star formation rate (ISM shocks produced

by the jets generated in the vicinity of supermassive black holes?).

• Once the elliptical has formed early, the next difficulty is how to avoid

further (disk) growth around it. The problem can be partially solved by con-

sidering that ellipticals form typically in dense, clustered environments, and

at some time they become substructures of larger virialized groups or clusters,

truncating any possible accretion to the halo/galaxy. However, (i) galaxy ha-

los, even in clusters, are filled with a reservoir of gas, and (ii) there are some

ellipticals in the field. Therefore, negative feedback mechanisms are needed to

stop gas cooling and accretion. AGN–triggered radio jets have been proposed

as a possible mechanism, but further investigation is necessary.

• The merging mechanism of bulge formation within the hierarchical model

implies roughly bluer (later formed) disks as the bulge–to–disk ratio is larger,

contrary to the observed trend. The secular scenario could solve this problem

but it is not still clear whether bars disolve or not in favor of pseudobulges. It

is not clear also if the secular scenario could predict the central supermassive

black hole mass–velocity dispersion relation.

• We lack a fundamental theory of star formation. So far, simple models,

or even just phenomenological recipes, have been used in galaxy formation

studies. The two proposed modes of star formation (the quiescent, inefficient,

disk self–regulated regime, and the violent efficient star–bursting regime in

mergers) are oversimplifications of a much more complex problem with more

physical mechanisms (shocks, turbulence, etc.). Closely related to star forma-

tion is the problem of feedback. The feedback mechanisms are different in the

ISM of disks, in the gaseous medium of merging galaxies with a powerful en-

ergy source (the AGN) other than stars, and in the diluted and hot intrahalo

medium around galaxies.

• We have seen in §§2.2 that at the present epoch only ≈ 9% of baryons

are within virialized structures. Where are the remaining 91% of the baryons?

The fraction of particles in halos measured in ΛCDM N–body cosmologi-

cal simulations is ∼ 50%. This sounds good but still we have to explain,

within the ΛCDM scenario, the ∼ 40% of missing baryons. The question is

were these baryons never trapped by collapsed halos or were they trapped but

later expelled due to galaxy feedback. Large–scale N–body+hdydrodynamical

simulations have shown that the gravitational collapse of filaments may heat

the gas and keep a big fraction of baryons outside the collapsed halos [35].

Nevertheless, feedback mechanisms, especially at high redshifts, are also pre-

dicted to be strong enough as to expel enriched gas back to the Intergalactic

Medium. The problem is open.

The field has plenty of open and exciting problems. The ΛCDM scenario

has survived many observational tests but it still faces the difficulties typical

of a theory constructed phenomenologically and heuristically. Even if in the

future it is demonstrated that CDM does not exist (which is little probable),

the ΛCDM scenario would serve as an excellent “fitting” model to reality,

which would strongly help researchers in developing new theories.

48

Vladimir Avila-Reese

Acknowledgments.- I am in debt with Dr. I. Alcántara-Ayala and R. Nú˜

nez-

López for their help in the preparation of the figures. I am also grateful to J.

Benda for grammar corrections, and to the Editors for their infinite patience.

References

1. Abadi, M. G. et al. 2003, ApJ, 591, 499

2. Avila-Reese, V., & Firmani, C. 2000, RevMexAA, 36, 23

3. Avila-Reese, V., Firmani, C. & Hernández, X. 1998, ApJ, 505, 37

4. Avila-Reese, V., et al. 2003, ApJ, 598, 36

5. Avila-Reese, V. et al. 2005, ApJ, 634, 51

6. Balogh, M. L. et al. 2004, ApJ, 615, L101

7. Ballesteros–Paredes, J. et al. 2006, in “Protostars and Planets V”, in press

(astro-ph/0603357)

8. Bartolo, N. et al. 2004, Phys.Rep., 402, 103

9. Baugh, C.M., Cole, S., & Frenk, C.S. 1996, MNRAS, 283, 136

10. Baugh, C.M. et al. 2005, MNRAS, 356, 1191

11. Bell E. F. et al. 2003, ApJSS, 149, 289

12. Benson, A.J. et al. 2002, ApJ, 599, 38

13. Berlind, A. A. et al. 2005, ApJ, 629, 625

14. Binney, J. & Tremaine S. 1987, Galactic Dynamics, Princeton Univ. Press, Princeton

15. Blumenthal, G. R. et al. 1984, Nature, 311, 517

16. Boissier, S. & Prantzos, N. 2001, MNRAS, 325, 321

17. Bond, J. R. et al. 1991, ApJ, 379, 440

18. Bosma, A. 1981, AJ, 86, 1791

19. Bower, R.G. et al. 2005, preprint, astro-ph/0511338

20. Bromm, V., & Larson, R. B. 2004, ARA&A, 42, 79

21. Bullock, J. S. et al. 2001, MNRAS, 321, 559

22. Bullock, J.S. et al. 2001, ApJ, 555, 240

23. Carroll S.M., Press W.H., Turner E.L., 1992, ARA&A, 30, 499

24. Casertano, S., & van Gorkom, J.H. 1991, AJ, 101, 1231

25. Catinella, B., Giovanelli, R., & Haynes, M. P. 2005, astro-ph/0512051

26. Cen, R., & Ostriker, J. 1992, ApJ, 393, 22

27. Ciardi, B. & Ferrara, A. 2005, Space Science Reviews, 116, 625

28. Cole, S. et al. 1994, MNRAS, 271, 781

29. Cole, S. et al. 2000, MNRAS, 319, 168

30. Coles, P., & Lucchin, F. 1995, Cosmology. The origin and evolution of cosmic structure, Chichester: Wiley, —c1995,

31. Col´ın, P., Avila-Reese, V., & Valenzuela, O. 2000, ApJ, 542, 622

32. Col´ın, P., Avila-Reese, V., Valenzuela, O., & Firmani, C. 2002, ApJ, 581, 777

33. Combes, F., preprint, astro-ph/0506265

34. Croton, D. J. et al. 2005, MNRAS, 356, 1155

35. Davé, R. et al. 2001, ApJ, 552, 473

36. Davis, M. et al. 1985, ApJ, 292, 371

37. de Blok, W. J. G. 2005, ApJ, 634, 227

38. De Lucia, G., Kauffmann, G., & White, S. D. M. 2004b, MNRAS, 349, 1101

39. De Lucia, G. et al. 2006, MNRAS, 366, 499

Understanding Galaxy Formation and Evolution

49

40. Dressler, A. 1980, ApJ, 236, 351

41. Eisenstein D. J. et al., 2005, ApJ, 633, 560

42. Eke, V.R., Navarro, J.F. & Steinmetz, M., 2001, ApJ, 554, 114

43. Fall, S.M. & Efstathiou, G. (1980), MNRAS, 193, 189

44. Firmani, C., & Tutukov, A.V. 1994, A&A, 288, 713

45. Firmani, C., & Avila-Reese, V. 2000, MNRAS, 315, 457

46. Firmani, C., & Avila-Reese, V. 20003, RevMexAA (SC), v. 17, 106

47. Flores R.A. et al. 1993, ApJ, 412, 443

48. Fukugita, M., & Peebles, P. J. E. 2004, ApJ, 616, 643

49. Frenk, C.S. 2002, preprint, astro-ph/0208219

50. Gentile, G. et al. 2004, MNRAS, 351, 903

51. Giavalisco, M. et al. 2004, ApJ, 600, L103

52. Gnedin, O. Y. et al. 2004, ApJ, 616, 16

53. Gondolo, P. 2004, preprint, astro-ph/0403064

54. Goto, T. et al. 2003, MNRAS, 346, 601

55. Gottlöber, S., Klypin, A., & Kravtsov, A.V. 2001, ApJ, 546, 223

56. Governato, F., et al. 2004, ApJ, 607, 688

57. Gunn, J. E. 1977, ApJ, 218, 592

58. Guzzo, L. 2002, in Modern Cosmology, eds. S. Bonometto, V. Gorini & U.

Moschella, Bristol, UK, 344

59. Hayashi, E., et al. 2004, MNRAS, 355, 794

60. Helly, J. C. et al. 2003, MNRAS, 338, 903

61. Hopkins, A.M. & Beacom, J. F. 2006, preprint, astro-ph:0601463

62. Hu, W., & Dodelson, S. 2002, ARA&A, 40, 171

63. Kang, X. et al. 2005, ApJ, 631, 21

64. Katz, N. 1992, ApJ, 391, 502

65. Katz, N., & Gunn, J. E. 1991, ApJ, 377, 365

66. Kauffmann, G., White, S.D.M., & Guiderdoni, B. 1993, MNRAS, 264, 201

67. Kauffmann, G. et al. 1999, MNRAS, 303, 188

68. Kauffmann, G. et al. 2004, MNRAS, 353, 713

69. Kereš, D., Katz, N., Weinberg, D. H., & Davé, R. 2005, MNRAS, 363, 2

70. Klypin, A.A. et al. 1999, ApJ, 522, 82

71. Kormendy, J., & Kennicutt, R. C. 2004, ARA&A, 42, 603

72. Kolb, E.W. & Turner, M. S. 1990, The Early Universe, Redwood City,

California-Addison-Wesley Publishing Company

73. Lacey, C.G., & Cole, S. 1993, MNRAS, 262, 627

74. de Jong, R. S., & Lacey, C. 2000, ApJ, 545, 781

75. Lin, Y., Mo, H.J. & van den Bosch F.C., 2005, astro-ph/0510372

76. Longair, M. S. 1989, LNP Vol. 333: Evolution of Galaxies: Astronomical Ob-

servations, 333, 1

77. Longair, M. S. 1998, Galaxy formation. Springer-Verlag (Germany)

78. Maller, A.H., Dekel, A., & Somerville, R. 2002, MNRAS, 329, 423

79. Madau, P., Pozzetti, L., & Dickinson, M. 1998, ApJ, 498, 106

80. Mandelbaum, R. et al. 2005, preprint, astro-ph/0511164

81. Matos, T., & Ure˜

na-López, L. 2001, Phys.Rev.D, 63, 063506

82. Mo, H.J., Mao, S., White, S.D.M. 1998, MNRAS, 295,319

83. Moore, B. et al. 1999, ApJ, 524, L19

84. Moore, B. et al. 1998, ApJ, 499, L5

85. Murali, C. et al. 2002, ApJ, 571, 1

50

Vladimir Avila-Reese

86. Navarro, J. F., & White, S. D. M. 1993, MNRAS, 265, 271

87. Navarro, J.F., Frenk, C.S., & White, S.D.M. 1997, ApJ, 490, 493

88. Navarro, J. F., et al. 2004, MNRAS, 349, 1039

89. Nicastro, F., et al. 2005, Nature, 433, 495

90. Padmanabhan, T. 1993, Cosmic structure formation in the universe, Cambridge

Univ. Press

91. Patton, D.R. et al. 2002, ApJ, 565, 208

92. Peacock, J. A. 1999, Cosmological Physics, Cambridge University Press

93. Peebles, P.J.E. 1969, ApJ, 155, 393

94. Peebles, P.J.E. 1993, Principles of Physical Cosmology, Princeton: Princeton University Press

95. Persic, M., Salucci, P. & Stel, F. 1996, MNRAS, 281, 27

96. Postman, M. & Geller, M. J. 1984, ApJ, 281, 95

97. Postman, M., et al. 2005, ApJ, 623, 721

98. Press, W. H., & Schechter, P. 1974, ApJ, 187, 425

99. Rubin, V.C., Thonnard, N., & Ford, W.K. 1980, ApJ, 238, 471

100. Rhee, G. et al. 2004, ApJ, 617, 1059

101. Salucci, P., & Gentile, G. 2005, preprint, astro-ph/0510716

102. Seljak U. et al., 2005, Phys. Rev. D, 71, 103515

103. Sheth, R. K., Mo, H. J., & Tormen, G. 2001, MNRAS, 323, 1

104. Silk, J. 1968, ApJ, 151, 459

105. Silk, J. 1977, ApJ, 211, 638

106. Silk, J., & Rees, M. J. 1998, A&A, 331, L1

107. Simon, J. et al. 2005, ApJ, 621, 757

108. Somerville, R.S., & Primack, J.R. 1999, MNRAS, 310, 1087

109. Spergel, D. N. e t al. 2006, prepint, astro-ph/0603449

110. Springel, V., & Hernquist, L. 2003, MNRAS, 339, 289

111. Springel, V., et al. 2005, Nature, 435, 629

112. Steinmetz, M., & Müller, E. 1994, A&A, 281, L97

113. Steinmetz, M. 1996, in Dark Matter in the Universe, Eds. S. Bonometto,

J.R. Primack, and A. Provenzale Oxford, GB: IOS Press, 1996, p.479.

114. Tanaka, M., et al. 2004, AJ, 128, 2677

115. Tegmark M. et al., 2004, Phys.Rev. D, 69, 103501

116. Tully, R.B., & Pierce, M.J. 2000, ApJ, 533, 744

117. Valenzuela, O.& Klypin, A. 2003, MNRAS, 345, 406

118. Valenzuela, O. et al. 2005, preprint, astro-ph/0509644

119. van den Bosch, F. C. 2000, ApJ, 530, 177

120. van den Bosch, F. C. 2001, MNRAS, 327, 1334

121. van den Bosch, F. C. 2002, MNRAS, 331, 98

122. van den Bosch 2001, F. C., Burkert, A., & Swaters, R. A. MNRAS, 326, 1205

123. Verheijen, M.A.W. 1997, PhD. Thesis, Groningen University

124. Wang, B., & Silk, J. 1994, ApJ, 427, 759

125. Wechsler, R.H., et al. 2002, ApJ, 568, 52

126. Weil, M., Eke, V. R., & Efstathiou, G. P. 1998, MNRAS, 300, 773

127. Weinmann, S. M. et al. 2006, MNRAS, 366, 2

128. Weldrake, D. T. F., de Blok, W. J. G., & Walter, F. 2003, MNRAS, 340, 12

129. White,

S.D.M.

1996,

Cosmology

and

Large

Scale

Structure,

349,

astro-ph/9410043

130. White, S.D.M., & Frenk, C.S. 1991, ApJ, 379, 52

131. White, S.D.M. & Rees, M. J. 1978, MNRAS, 183, 341

132. Zavala, J. et al. 2003, A&A, 412, 633

133. Zhao, D. H. et al. 2003, ApJ, 597, L9

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Table of Contents

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