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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

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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.

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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:

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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.

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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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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

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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.

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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

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

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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

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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.

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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.

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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

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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.

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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:

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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

δ(x) =

δ

(2π)3

ke−ikxd3k,

(5)

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 ≡ (δM/M)2R 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

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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

index-24_1.png

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),

index-26_1.png

index-26_2.png

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.