GALAXY FORMATION AND EVOLUTION

Contents

1 Introduction 1
1.1 The Diversity of the Galaxy Population 2
1.2 Basic Elements of Galaxy Formation 5
1.2.1 The Standard Model of Cosmology 6
1.2.2 Initial Conditions 6
1.2.3 Gravitational Instability and Structure Formation 7
1.2.4 Gas Cooling 8
1.2.5 Star Formation 8
1.2.6 Feedback Processes 9
1.2.7 Mergers 10
1.2.8 Dynamical Evolution 12
1.2.9 Chemical Evolution 12
1.2.10 Stellar Population Synthesis 13
1.2.11 The Intergalactic Medium 13
1.3 Time Scales 14
1.4 A Brief History of Galaxy Formation 15
1.4.1 Galaxies as Extragalactic Objects 15
1.4.2 Cosmology 16
1.4.3 Structure Formation 18
1.4.4 The Emergence of the Cold Dark Matter Paradigm 20
1.4.5 Galaxy Formation 22
2 Observational Facts 25
2.1 Astronomical Observations 25
2.1.1 Fluxes and Magnitudes 26
2.1.2 Spectroscopy 29
2.1.3 Distance Measurements 32
2.2 Stars 34
2.3 Galaxies 37
2.3.1 The Classification of Galaxies 38
2.3.2 Elliptical Galaxies 41
2.3.3 Disk Galaxies
2.3.4 The Milky Way 55
2.3.5 Dwarf Galaxies 57
2.3.6 Nuclear Star Clusters 59
2.3.7 Starbursts 60
2.3.8 Active Galactic Nuclei 60
2.4 Statistical Properties of the Galaxy Population 61
2.4.1 Luminosity Function 62
2.4.2 Size Distribution 63
2.4.3 Color Distribution 64
2.4.4 The Mass–Metallicity Relation 65
2.4.5 Environment Dependence 65
2.5 Clusters and Groups of Galaxies 67
2.5.1 Clusters of Galaxies 67
2.5.2 Groups of Galaxies 71
2.6 Galaxies at High Redshifts 72
2.6.1 Galaxy Counts 73
2.6.2 Photometric Redshifts 75
2.6.3 Galaxy Redshift Surveys at z ∼ 1 75
2.6.4 Lyman-Break Galaxies 77
2.6.5 Lyα Emitters 78
2.6.6 Submillimeter Sources 78
2.6.7 Extremely Red Objects and Distant Red Galaxies 79
2.6.8 The Cosmic Star-Formation History 80
2.7 Large-Scale Structure 81
2.7.1 Two-Point Correlation Functions 82
2.7.2 Probing the Matter Field via Weak Lensing 84
2.8 The Intergalactic Medium 85
2.8.1 The Gunn–Peterson Test 85
2.8.2 Quasar Absorption Line Systems 86
2.9 The Cosmic Microwave Background 89
2.10 The Homogeneous and Isotropic Universe 92
2.10.1 The Determination of Cosmological Parameters 94
2.10.2 The Mass and Energy Content of the Universe 95
3 Cosmological Background 100
3.1 The Cosmological Principle and the Robertson–Walker Metric 102
3.1.1 The Cosmological Principle and its Consequences 102
3.1.2 Robertson–Walker Metric 104
3.1.3 Redshift 106
3.1.4 Peculiar Velocities 107
3.1.5 Thermodynamics and the Equation of State 108
3.1.6 Angular-Diameter and Luminosity Distances 110
3.2 Relativistic Cosmology 112
3.2.1 Friedmann Equation 113
3.2.2 The Densities at the Present Time
3.2.3 Explicit Solutions of the Friedmann Equation 115
3.2.4 Horizons 119
3.2.5 The Age of the Universe 119
3.2.6 Cosmological Distances and Volumes 121
3.3 The Production and Survival of Particles 124
3.3.1 The Chronology of the Hot Big Bang 125
3.3.2 Particles in Thermal Equilibrium 127
3.3.3 Entropy 129
3.3.4 Distribution Functions of Decoupled Particle Species 132
3.3.5 The Freeze-Out of Stable Particles 133
3.3.6 Decaying Particles 137
3.4 Primordial Nucleosynthesis 139
3.4.1 Initial Conditions 139
3.4.2 Nuclear Reactions 140
3.4.3 Model Predictions 142
3.4.4 Observational Results 144
3.5 Recombination and Decoupling 146
3.5.1 Recombination 146
3.5.2 Decoupling and the Origin of the CMB 148
3.5.3 Compton Scattering 150
3.5.4 Energy Thermalization 151
3.6 Inflation 152
3.6.1 The Problems of the Standard Model 152
3.6.2 The Concept of Inflation 154
3.6.3 Realization of Inflation 156
3.6.4 Models of Inflation 158
4 Cosmological Perturbations 162
4.1 Newtonian Theory of Small Perturbations 162
4.1.1 Ideal Fluid 162
4.1.2 Isentropic and Isocurvature Initial Conditions 166
4.1.3 Gravitational Instability 166
4.1.4 Collisionless Gas 168
4.1.5 Free-Streaming Damping 171
4.1.6 Specific Solutions 172
4.1.7 Higher-Order Perturbation Theory 176
4.1.8 The Zel’dovich Approximation 177
4.2 Relativistic Theory of Small Perturbations 178
4.2.1 Gauge Freedom 179
4.2.2 Classification of Perturbations 181
4.2.3 Specific Examples of Gauge Choices 183
4.2.4 Basic Equations 185
4.2.5 Coupling between Baryons and Radiation 189
4.2.6 Perturbation Evolution 191
4.3 Linear Transfer Functions 196
4.3.1 Adiabatic Baryon Models
4.3.2 Adiabatic Cold Dark Matter Models 200
4.3.3 Adiabatic Hot Dark Matter Models 201
4.3.4 Isocurvature Cold Dark Matter Models 202
4.4 Statistical Properties 202
4.4.1 General Discussion 202
4.4.2 Gaussian Random Fields 204
4.4.3 Simple Non-Gaussian Models 205
4.4.4 Linear Perturbation Spectrum 206
4.5 The Origin of Cosmological Perturbations 209
4.5.1 Perturbations from Inflation 209
4.5.2 Perturbations from Topological Defects 213
5 Gravitational Collapse and Collisionless Dynamics 215
5.1 Spherical Collapse Models 215
5.1.1 Spherical Collapse in a Λ = 0 Universe 215
5.1.2 Spherical Collapse in a Flat Universe with Λ > 0 218
5.1.3 Spherical Collapse with Shell Crossing 219
5.2 Similarity Solutions for Spherical Collapse 220
5.2.1 Models with Radial Orbits 220
5.2.2 Models Including Non-Radial Orbits 224
5.3 Collapse of Homogeneous Ellipsoids 226
5.4 Collisionless Dynamics 230
5.4.1 Time Scales for Collisions 230
5.4.2 Basic Dynamics 232
5.4.3 The Jeans Equations 233
5.4.4 The Virial Theorem 234
5.4.5 Orbit Theory 236
5.4.6 The Jeans Theorem 240
5.4.7 Spherical Equilibrium Models 240
5.4.8 Axisymmetric Equilibrium Models 244
5.4.9 Triaxial Equilibrium Models 247
5.5 Collisionless Relaxation 248
5.5.1 Phase Mixing 249
5.5.2 Chaotic Mixing 250
5.5.3 Violent Relaxation 251
5.5.4 Landau Damping 253
5.5.5 The End State of Relaxation 254
5.6 Gravitational Collapse of the Cosmic Density Field 257
5.6.1 Hierarchical Clustering 257
5.6.2 Results from Numerical Simulations 258
6 Probing the Cosmic Density Field 262
6.1 Large-Scale Mass Distribution 262
6.1.1 Correlation Functions 262
6.1.2 Particle Sampling and Bias 264
6.1.3 Mass Moments
6.2 Large-Scale Velocity Field 270
6.2.1 Bulk Motions and Velocity Correlation Functions 270
6.2.2 Mass Density Reconstruction from the Velocity Field 271
6.3 Clustering in Real Space and Redshift Space 273
6.3.1 Redshift Distortions 273
6.3.2 Real-Space Correlation Functions 276
6.4 Clustering Evolution 278
6.4.1 Dynamics of Statistics 278
6.4.2 Self-Similar Gravitational Clustering 280
6.4.3 Development of Non-Gaussian Features 282
6.5 Galaxy Clustering 283
6.5.1 Correlation Analyses 284
6.5.2 Power Spectrum Analysis 288
6.5.3 Angular Correlation Function and Power Spectrum 290
6.6 Gravitational Lensing 292
6.6.1 Basic Equations 292
6.6.2 Lensing by a Point Mass 295
6.6.3 Lensing by an Extended Object 297
6.6.4 Cosmic Shear 300
6.7 Fluctuations in the Cosmic Microwave Background 302
6.7.1 Observational Quantities 302
6.7.2 Theoretical Expectations of Temperature Anisotropy 304
6.7.3 Thomson Scattering and Polarization of the Microwave Background 311
6.7.4 Interaction between CMB Photons and Matter 314
6.7.5 Constraints on Cosmological Parameters 316
7 Formation and Structure of Dark Matter Halos 319
7.1 Density Peaks 321
7.1.1 Peak Number Density 321
7.1.2 Spatial Modulation of the Peak Number Density 323
7.1.3 Correlation Function 324
7.1.4 Shapes of Density Peaks 325
7.2 Halo Mass Function 326
7.2.1 Press–Schechter Formalism 327
7.2.2 Excursion Set Derivation of the Press–Schechter Formula 328
7.2.3 Spherical versus Ellipsoidal Dynamics 331
7.2.4 Tests of the Press–Schechter Formalism 333
7.2.5 Number Density of Galaxy Clusters 334
7.3 Progenitor Distributions and Merger Trees 336
7.3.1 Progenitors of Dark Matter Halos 336
7.3.2 Halo Merger Trees 336
7.3.3 Main Progenitor Histories 339
7.3.4 Halo Assembly and Formation Times 340
7.3.5 Halo Merger Rates 342
7.3.6 Halo Survival Times
7.4 Spatial Clustering and Bias 345
7.4.1 Linear Bias and Correlation Function 345
7.4.2 Assembly Bias 348
7.4.3 Nonlinear and Stochastic Bias 348
7.5 Internal Structure of Dark Matter Halos 351
7.5.1 Halo Density Profiles 351
7.5.2 Halo Shapes 354
7.5.3 Halo Substructure 355
7.5.4 Angular Momentum 358
7.6 The Halo Model of Dark Matter Clustering 362
8 Formation and Evolution of Gaseous Halos 366
8.1 Basic Fluid Dynamics and Radiative Processes 366
8.1.1 Basic Equations 366
8.1.2 Compton Cooling 367
8.1.3 Radiative Cooling 367
8.1.4 Photoionization Heating 369
8.2 Hydrostatic Equilibrium 371
8.2.1 Gas Density Profile 371
8.2.2 Convective Instability 373
8.2.3 Virial Theorem Applied to a Gaseous Halo 374
8.3 The Formation of Hot Gaseous Halos 376
8.3.1 Accretion Shocks 376
8.3.2 Self-Similar Collapse of Collisional Gas 379
8.3.3 The Impact of a Collisionless Component 383
8.3.4 More General Models of Spherical Collapse 384
8.4 Radiative Cooling in Gaseous Halos 385
8.4.1 Radiative Cooling Time Scales for Uniform
Clouds 385
8.4.2 Evolution of the Cooling Radius 387
8.4.3 Self-Similar Cooling Waves 388
8.4.4 Spherical Collapse with Cooling 390
8.5 Thermal and Hydrodynamical Instabilities of Cooling Gas 393
8.5.1 Thermal Instability 393
8.5.2 Hydrodynamical Instabilities 396
8.5.3 Heat Conduction 397
8.6 Evolution of Gaseous Halos with Energy Sources 398
8.6.1 Blast Waves 399
8.6.2 Winds and Wind-Driven Bubbles 404
8.6.3 Supernova Feedback and Galaxy Formation 406
8.7 Results from Numerical Simulations 408
8.7.1 Three-Dimensional Collapse without Radiative
Cooling 408
8.7.2 Three-Dimensional Collapse with Radiative
Cooling
8.8 Observational Tests 410
8.8.1 X-ray Clusters and Groups 410
8.8.2 Gaseous Halos around Elliptical Galaxies 414
8.8.3 Gaseous Halos around Spiral Galaxies 416
9 Star Formation in Galaxies 417
9.1 Giant Molecular Clouds: The Sites of Star Formation 418
9.1.1 Observed Properties 418
9.1.2 Dynamical State 419
9.2 The Formation of Giant Molecular Clouds 421
9.2.1 The Formation of Molecular Hydrogen 421
9.2.2 Cloud Formation 422
9.3 What Controls the Star-Formation Efficiency 425
9.3.1 Magnetic Fields 425
9.3.2 Supersonic Turbulence 426
9.3.3 Self-Regulation 428
9.4 The Formation of Individual Stars 429
9.4.1 The Formation of Low-Mass Stars 429
9.4.2 The Formation of Massive Stars 432
9.5 Empirical Star-Formation Laws 433
9.5.1 The Kennicutt–Schmidt Law 434
9.5.2 Local Star-Formation Laws 436
9.5.3 Star-Formation Thresholds 438
9.6 The Initial Mass Function 440
9.6.1 Observational Constraints 441
9.6.2 Theoretical Models 443
9.7 The Formation of Population III Stars 446
10 Stellar Populations and Chemical Evolution 449
10.1 The Basic Concepts of Stellar Evolution 449
10.1.1 Basic Equations of Stellar Structure 450
10.1.2 Stellar Evolution 453
10.1.3 Equation of State, Opacity, and Energy Production 453
10.1.4 Scaling Relations 460
10.1.5 Main-Sequence Lifetimes 462
10.2 Stellar Evolutionary Tracks 463
10.2.1 Pre-Main-Sequence Evolution 463
10.2.2 Post-Main-Sequence Evolution 464
10.2.3 Supernova Progenitors and Rates 468
10.3 Stellar Population Synthesis 470
10.3.1 Stellar Spectra 470
10.3.2 Spectral Synthesis 471
10.3.3 Passive Evolution 472
10.3.4 Spectral Features 474
10.3.5 Age–Metallicity Degeneracy
10.3.6 K and E Corrections 475
10.3.7 Emission and Absorption by the Interstellar Medium 476
10.3.8 Star-Formation Diagnostics 482
10.3.9 Estimating Stellar Masses and Star-Formation Histories of Galaxies 484
10.4 Chemical Evolution of Galaxies 486
10.4.1 Stellar Chemical Production 486
10.4.2 The Closed-Box Model 488
10.4.3 Models with Inflow and Outflow 490
10.4.4 Abundance Ratios 491
10.5 Stellar Energetic Feedback 492
10.5.1 Mass-Loaded Kinetic Energy from Stars 492
10.5.2 Gas Dynamics Including Stellar Feedback 493
11 Disk Galaxies 495
11.1 Mass Components and Angular Momentum 495
11.1.1 Disk Models 496
11.1.2 Rotation Curves 498
11.1.3 Adiabatic Contraction 501
11.1.4 Disk Angular Momentum 502
11.1.5 Orbits in Disk Galaxies 503
11.2 The Formation of Disk Galaxies 505
11.2.1 General Discussion 505
11.2.2 Non-Self-Gravitating Disks in Isothermal Spheres 505
11.2.3 Self-Gravitating Disks in Halos with Realistic Profiles 507
11.2.4 Including a Bulge Component 509
11.2.5 Disk Assembly 509
11.2.6 Numerical Simulations of Disk Formation 511
11.3 The Origin of Disk Galaxy Scaling Relations 512
11.4 The Origin of Exponential Disks 515
11.4.1 Disks from Relic Angular Momentum Distribution 515
11.4.2 Viscous Disks 517
11.4.3 The Vertical Structure of Disk Galaxies 518
11.5 Disk Instabilities 521
11.5.1 Basic Equations 521
11.5.2 Local Instability 523
11.5.3 Global Instability 525
11.5.4 Secular Evolution 528
11.6 The Formation of Spiral Arms 531
11.7 Stellar Population Properties 534
11.7.1 Global Trends 535
11.7.2 Color Gradients 537
11.8 Chemical Evolution of Disk Galaxies 538
11.8.1 The Solar Neighborhood 538
11.8.2 Global Relations
12 Galaxy Interactions and Transformations 544
12.1 High-Speed Encounters 545
12.2 Tidal Stripping 548
12.2.1 Tidal Radius 548
12.2.2 Tidal Streams and Tails 549
12.3 Dynamical Friction 553
12.3.1 Orbital Decay 556
12.3.2 The Validity of Chandrasekhar’s Formula 559
12.4 Galaxy Merging 561
12.4.1 Criterion for Mergers 561
12.4.2 Merger Demographics 563
12.4.3 The Connection between Mergers, Starbursts and AGN 564
12.4.4 Minor Mergers and Disk Heating 565
12.5 Transformation of Galaxies in Clusters 568
12.5.1 Galaxy Harassment 569
12.5.2 Galactic Cannibalism 570
12.5.3 Ram-Pressure Stripping 571
12.5.4 Strangulation 572
13 Elliptical Galaxies 574
13.1 Structure and Dynamics 574
13.1.1 Observables 575
13.1.2 Photometric Properties 576
13.1.3 Kinematic Properties 577
13.1.4 Dynamical Modeling 579
13.1.5 Evidence for Dark Halos 581
13.1.6 Evidence for Supermassive Black Holes 582
13.1.7 Shapes 584
13.2 The Formation of Elliptical Galaxies 587
13.2.1 The Monolithic Collapse Scenario 588
13.2.2 The Merger Scenario 590
13.2.3 Hierarchical Merging and the Elliptical Population 593
13.3 Observational Tests and Constraints 594
13.3.1 Evolution of the Number Density of Ellipticals 594
13.3.2 The Sizes of Elliptical Galaxies 595
13.3.3 Phase-Space Density Constraints 598
13.3.4 The Specific Frequency of Globular Clusters 599
13.3.5 Merging Signatures 600
13.3.6 Merger Rates 601
13.4 The Fundamental Plane of Elliptical Galaxies 602
13.4.1 The Fundamental Plane in the Merger Scenario 604
13.4.2 Projections and Rotations of the Fundamental Plane 604
13.5 Stellar Population Properties 606
13.5.1 Archaeological Records 606
13.5.2 Evolutionary Probes
13.5.3 Color and Metallicity Gradients 610
13.5.4 Implications for the Formation of Elliptical Galaxies 610
13.6 Bulges, Dwarf Ellipticals and Dwarf Spheroidals 613
13.6.1 The Formation of Galactic Bulges 614
13.6.2 The Formation of Dwarf Ellipticals 616
14 Active Galaxies 618
14.1 The Population of Active Galactic Nuclei 619
14.2 The Supermassive Black Hole Paradigm 623
14.2.1 The Central Engine 623
14.2.2 Accretion Disks 624
14.2.3 Continuum Emission 626
14.2.4 Emission Lines 631
14.2.5 Jets, Superluminal Motion and Beaming 633
14.2.6 Emission-Line Regions and Obscuring Torus 637
14.2.7 The Idea of Unification 638
14.2.8 Observational Tests for Supermassive Black Holes 639
14.3 The Formation and Evolution of AGN 640
14.3.1 The Growth of Supermassive Black Holes and the Fueling of AGN 640
14.3.2 AGN Demographics 644
14.3.3 Outstanding Questions 647
14.4 AGN and Galaxy Formation 648
14.4.1 Radiative Feedback 649
14.4.2 Mechanical Feedback 650
15 Statistical Properties of the Galaxy Population 652
15.1 Preamble 652
15.2 Galaxy Luminosities and Stellar Masses 654
15.2.1 Galaxy Luminosity Functions 654
15.2.2 Galaxy Counts 658
15.2.3 Extragalactic Background Light 660
15.3 Linking Halo Mass to Galaxy Luminosity 663
15.3.1 Simple Considerations 663
15.3.2 The Luminosity Function of Central Galaxies 665
15.3.3 The Luminosity Function of Satellite Galaxies 666
15.3.4 Satellite Fractions 668
15.3.5 Discussion 669
15.4 Linking Halo Mass to Star-Formation History 670
15.4.1 The Color Distribution of Galaxies 670
15.4.2 Origin of the Cosmic Star-Formation History 673
15.5 Environmental Dependence 674
15.5.1 Effects within Dark Matter Halos 675
15.5.2 Effects on Large Scales 677
15.6 Spatial Clustering and Galaxy Bias 679
15.6.1 Application to High-Redshift Galaxies
15.7 Putting it All Together 684
15.7.1 Semi-Analytical Models 684
15.7.2 Hydrodynamical Simulations 686
16 The Intergalactic Medium 689
16.1 The Ionization State of the Intergalactic Medium 690
16.1.1 Physical Conditions after Recombination 690
16.1.2 The Mean Optical Depth of the IGM 690
16.1.3 The Gunn–Peterson Test 692
16.1.4 Constraints from the Cosmic Microwave Background 694
16.2 Ionizing Sources 695
16.2.1 Photoionization versus Collisional Ionization 695
16.2.2 Emissivity from Quasars and Young Galaxies 697
16.2.3 Attenuation by Intervening Absorbers 699
16.2.4 Observational Constraints on the UV Background 701
16.3 The Evolution of the Intergalactic Medium 702
16.3.1 Thermal Evolution 702
16.3.2 Ionization Evolution 704
16.3.3 The Epoch of Re-ionization 705
16.3.4 Probing Re-ionization with 21-cm Emission and Absorption 707
16.4 General Properties of Absorption Lines 709
16.4.1 Distribution Function 709
16.4.2 Thermal Broadening 710
16.4.3 Natural Broadening and Voigt Profiles 711
16.4.4 Equivalent Width and Column Density 712
16.4.5 Common QSO Absorption Line Systems 714
16.4.6 Photoionization Models 714
16.5 The Lyman α Forest 714
16.5.1 Redshift Evolution 715
16.5.2 Column Density Distribution 716
16.5.3 Doppler Parameter 717
16.5.4 Sizes of Absorbers 718
16.5.5 Metallicity 719
16.5.6 Clustering 720
16.5.7 Lyman α Forests at Low Redshift 721
16.5.8 The Helium Lyman α Forest 722
16.6 Models of the Lyman α Forest 723
16.6.1 Early Models 723
16.6.2 Lyman α Forest in Hierarchical Models 724
16.6.3 Lyman α Forest in Hydrodynamical Simulations 731
16.7 Lyman-Limit Systems 732
16.8 Damped Lyman α Systems 733
16.8.1 Column Density Distribution 734
16.8.2 Redshift Evolution 734
16.8.3 Metallicities 736
16.8.4 Kinematics
16.9 Metal Absorption Line Systems 738
16.9.1 MgII Systems 739
16.9.2 CIV and OVI Systems 740
A Basics of General Relativity 741
A1.1 Space-time Geometry 741
A1.2 The Equivalence Principle 743
A1.3 Geodesic Equations 744
A1.4 Energy–Momentum Tensor 746
A1.5 Newtonian Limit 747
A1.6 Einstein’s Field Equation 747
B Gas and Radiative Processes 748
B1.1 Ideal Gas 748
B1.2 Basic Equations 749
B1.3 Radiative Processes 751
B1.3.1 Einstein Coefficients and Milne Relation 752
B1.3.2 Photoionization and Photo-excitation 755
B1.3.3 Recombination 756
B1.3.4 Collisional Ionization and Collisional Excitation 757
B1.3.5 Bremsstrahlung 758
B1.3.6 Compton Scattering 759
B1.4 Radiative Cooling 760
C Numerical Simulations 764
C1.1 N-Body Simulations 764
C1.1.1 Force Calculations 766
C1.1.2 Issues Related to Numerical Accuracy 767
C1.1.3 Boundary Conditions 769
C1.1.4 Initial Conditions 769
C1.2 Hydrodynamical Simulations 770
C1.2.1 Smoothed-Particle Hydrodynamics (SPH) 770
C1.2.2 Grid-Based Algorithms 772
D Frequently Used Abbreviations 775
E Useful Numbers 776
References 777
Index

Preface
The vast ocean of space is full of starry islands called galaxies. These objects, extraordinarily
beautiful and diverse in their own right, not only are the localities within which stars form
and evolve, but also act as the lighthouses that allow us to explore our Universe over cosmological
scales. Understanding the majesty and variety of galaxies in a cosmological context is
therefore an important, yet daunting task. Particularly mind-boggling is the fact that, in the current
paradigm, galaxies only represent the tip of the iceberg in a Universe dominated by some
unknown ‘dark matter’ and an even more elusive form of ‘dark energy’.
How do galaxies come into existence in this dark Universe, and how do they evolve? What is
the relation of galaxies to the dark components? What shapes the properties of different galaxies?
How are different properties of galaxies correlated with each other and what physics underlies
these correlations? How do stars form and evolve in different galaxies? The quest for the answers
to these questions, among others, constitutes an important part of modern cosmology, the study of
the structure and evolution of the Universe as a whole, and drives the active and rapidly evolving
research field of extragalactic astronomy and astrophysics.
The aim of this book is to provide a self-contained description of the physical processes and
the astronomical observations which underlie our present understanding of the formation and
evolution of galaxies in a Universe dominated by dark matter and dark energy. Any book on
this subject must take into account that this is a rapidly developing field; there is a danger that
material may rapidly become outdated. We hope that this can be avoided if the book is appropriately
structured. Our premises are the following. In the first place, although observational data
are continually updated, forcing revision of the theoretical models used to interpret them, the
general principles involved in building such models do not change as rapidly. It is these principles,
rather than the details of specific observations or models, that are the main focus of this
book. Secondly, galaxies are complex systems, and the study of their formation and evolution is
an applied and synthetic science. The interest of the subject is precisely that there are so many
unsolved problems, and that the study of these problems requires techniques from many branches
of physics and astrophysics – the formation of stars, the origin and dispersal of the elements, the
link between galaxies and their central black holes, the nature of dark matter and dark energy,
the origin and evolution of cosmic structure, and the size and age of our Universe. A firm grasp
of the basic principles and the main outstanding issues across this full breadth of topics is needed
by anyone preparing to carry out her/his own research, and this we hope to provide.
These considerations dictated both our selection of material and our style of presentation.
Throughout the book, we emphasize the principles and the important issues rather than the details
of observational results and theoretical models. In particular, special attention is paid to bringing
out the physical connections between different parts of the problem, so that the reader will not
lose the big picture while working on details. To this end, we start in each chapter with an
introduction describing the material to be presented and its position in the overall scenario. In a
field as broad as galaxy formation and evolution, it is clearly impossible to include all relevant
material. The selection of the material presented in this book is therefore unavoidably biased by
our prejudice, taste, and limited knowledge of the literature, and we apologize to anyone whose
important work is not properly covered.
This book can be divided into several parts according to the material contained. Chapter 1 is
an introduction, which sketches our current ideas about galaxies and their formation processes.
Chapter 2 is an overview of the observational facts related to galaxy formation and evolution.
Chapter 3 describes the cosmological framework within which galaxy formation and evolution
must be studied. Chapters 4–8 contain material about the nature and evolution of the cosmological
density field, both in collisionless dark matter and in collisional gas. Chapters 9 and 10
deal with topics related to star formation and stellar evolution in galaxies. Chapters 11–15 are
concerned with the structure, formation, and evolution of individual galaxies and with the statistical
properties of the galaxy population, and Chapter 16 gives an overview of the intergalactic
medium. In addition, we provide appendixes to describe the general concepts of general relativity
(Appendix A), basic hydrodynamic and radiative processes (Appendix B), and some commonly
used techniques of N-body and hydrodynamical simulations (Appendix C).
The different parts are largely self-contained, and can be used separately for courses or seminars
on specific topics. Chapters 1 and 2 are particularly geared towards novices to the field of
extragalactic astronomy. Chapter 3, combined with parts of Chapters 4 and 5, could make up a
course on cosmology, while a more advanced course on structure formation might be constructed
around the material presented in Chapters 4–8. Chapter 2 and Chapters 11–15 contain material
suited for a course on galaxy formation. Chapters 9, 10 and 16 contain special topics related to
the formation and evolution of galaxies, and could be combined with Chapters 11–15 to form an
extended course on galaxy formation and evolution. They could also be used independently for
short courses on star formation and stellar evolution (Chapters 9 and 10), and on the intergalactic
medium (Chapter 16).
Throughout the book, we have adopted a number of abbreviations that are commonly used
by galaxy-formation practitioners. In order to avoid confusion, these abbreviations are listed in
Appendix D along with their definitions. Some important physical constants and units are listed
in Appendix E.
References are provided at the end of the book. Although long, the reference list is by no
means complete, and we apologize once more to anyone whose relevant papers are overlooked.
The number of references citing our own work clearly overrates our own contribution to the
field. This is again a consequence of our limited knowledge of the existing literature, which
is expanding at such a dramatic pace that it is impossible to cite all the relevant papers. The
references given are mainly intended to serve as a starting point for readers interested in a more
detailed literature study. We hope, by looking for the papers cited by our listed references, one
can find relevant papers published in the past, and by looking for the papers citing the listed
references, one can find relevant papers published later. Nowadays this is relatively easy to do
with the use of the search engines provided by The SAO/NASA Astrophysics Data System1 and
the arXiv e-print server.2
We would not have been able to write this book without the help of many people. We benefitted
greatly from discussions with and comments by many of our colleagues, including E. Bell,
A. Berlind, G. B¨orner, A. Coil, J. Dalcanton, A. Dekel, M. H¨ahnelt, M. Heyer, W. Hu, Y. Jing,
N. Katz, R. Larson, M. Longair, M. Mac Low, C.-P. Ma, S. Mao, E. Neistein, A. Pasquali,
J. Peacock, M. Rees, H.-W. Rix, J. Sellwood, E. Sheldon, R. Sheth, R. Somerville, V. Springel,
R. Sunyaev, A. van derWel, R.Wechsler, M.Weinberg, and X. Yang.We are also deeply indebted
to our many students and collaborators who made it possible for us to continue to publish
scientific papers while working on the book, and who gave us many new ideas and insights,
some of which are presented in this book.

May 2009
Houjun Mo
Frank van den Bosch
Simon White

1
Introduction
This book is concerned with the physical processes related to the formation and evolution of
galaxies. Simply put, a galaxy is a dynamically bound system that consists of many stars. A
typical bright galaxy, such as our own Milky Way, contains a few times 10^10 stars and has a
diameter (∼ 20kpc) that is several hundred times smaller than the mean separation between
bright galaxies. Since most of the visible stars in the Universe belong to a galaxy, the number
density of stars within a galaxy is about 10^7 times higher than the mean number density of
stars in the Universe as a whole. In this sense, galaxies are well-defined, astronomical identities.
They are also extraordinarily beautiful and diverse objects whose nature, structure and origin
have intrigued astronomers ever since the first galaxy images were taken in the mid-nineteenth
century.
The goal of this book is to show how physical principles can be used to understand the formation
and evolution of galaxies. Viewed as a physical process, galaxy formation and evolution
involve two different aspects: (i) initial and boundary conditions; and (ii) physical processes
which drive evolution. Thus, in very broad terms, our study will consist of the following parts:
• Cosmology: Since we are dealing with events on cosmological time and length scales, we
need to understand the space-time structure on large scales. One can think of the cosmological
framework as the stage on which galaxy formation and evolution take place.
• Initial conditions: These were set by physical processes in the early Universe which are
beyond our direct view, and which took place under conditions far different from those we
can reproduce in Earth-bound laboratories.
• Physical processes: As we will show in this book, the basic physics required to study galaxy
formation and evolution includes general relativity, hydrodynamics, dynamics of collisionless
systems, plasma physics, thermodynamics, electrodynamics, atomic, nuclear and particle
physics, and the theory of radiation processes.
In a sense, galaxy formation and evolution can therefore be thought of as an application of (relatively)
well-known physics with cosmological initial and boundary conditions. As in many other
branches of applied physics, the phenomena to be studied are diverse and interact in many different
ways. Furthermore, the physical processes involved in galaxy formation cover some 23 orders
of magnitude in physical size, from the scale of the Universe itself down to the scale of individual
stars, and about four orders of magnitude in time scales, from the age of the Universe to that of
the lifetime of individual, massive stars. Put together, it makes the formation and evolution of
galaxies a subject of great complexity.
From an empirical point of view, the study of galaxy formation and evolution is very different
from most other areas of experimental physics. This is due mainly to the fact that even the
shortest time scales involved are much longer than that of a human being. Consequently, we
cannot witness the actual evolution of individual galaxies. However, because the speed of light
is finite, looking at galaxies at larger distances from us is equivalent to looking at galaxies when
the Universe was younger. Therefore, we may hope to infer how galaxies form and evolve by
comparing their properties, in a statistical sense, at different epochs. In addition, at each epoch
we can try to identify regularities and correspondences among the galaxy population. Although
galaxies span a wide range in masses, sizes, and morphologies, to the extent that no two galaxies
are alike, the structural parameters of galaxies also obey various scaling relations, some of which
are remarkably tight. These relations must hold important information regarding the physical
processes that underlie them, and any successful theory of galaxy formation has to be able to
explain their origin.
Galaxies are not only interesting in their own right, they also play a pivotal role in our study
of the structure and evolution of the Universe. They are bright, long-lived and abundant, and so
can be observed in large numbers over cosmological distances and time scales. This makes them
unique tracers of the evolution of the Universe as a whole, and detailed studies of their large
scale distribution can provide important constraints on cosmological parameters. In this book we
therefore also describe the large scale distribution of galaxies, and discuss how it can be used to
test cosmological models.
In Chapter 2 we start by describing the observational properties of stars, galaxies and the large
scale structure of the Universe as a whole. Chapters 3 through 10 describe the various physical
ingredients needed for a self-consistent model of galaxy formation, ranging from the cosmological
framework to the formation and evolution of individual stars. Finally, in Chapters 11–16 we
combine these physical ingredients to examine how galaxies form and evolve in a cosmological
context, using the observational data as constraints.
The purpose of this introductory chapter is to sketch our current ideas about galaxies and
their formation process, without going into any detail. After a brief overview of some observed
properties of galaxies, we list the various physical processes that play a role in galaxy formation
and outline how they are connected. We also give a brief historical overview of how our current
views of galaxy formation have been shaped.
1.1 The Diversity of the Galaxy Population
Galaxies are a diverse class of objects. This means that a large number of parameters is required
in order to characterize any given galaxy. One of the main goals of any theory of galaxy formation
is to explain the full probability distribution function of all these parameters. In particular, as we
will see in Chapter 2, many of these parameters are correlated with each other, a fact which any
successful theory of galaxy formation should also be able to reproduce.
Here we list briefly the most salient parameters that characterize a galaxy. This overview is
necessarily brief and certainly not complete. However, it serves to stress the diversity of the
galaxy population, and to highlight some of the most important observational aspects that galaxy
formation theories need to address. A more thorough description of the observational properties
of galaxies is given in Chapter 2.
(a) Morphology One of the most noticeable properties of the galaxy population is the existence
of two basic galaxy types: spirals and ellipticals. Elliptical galaxies are mildly flattened, ellipsoidal
systems that are mainly supported by the random motions of their stars. Spiral galaxies, on
the other hand, have highly flattened disks that are mainly supported by rotation. Consequently,
they are also often referred to as disk galaxies. The name ‘spiral’ comes from the fact that the gas
and stars in the disk often reveal a clear spiral pattern. Finally, for historical reasons, ellipticals
and spirals are also called early- and late-type galaxies, respectively.
Most galaxies, however, are neither a perfect ellipsoid nor a perfect disk, but rather a combination
of both. When the disk is the dominant component, its ellipsoidal component is generally
called the bulge. In the opposite case, of a large ellipsoidal system with a small disk, one typically
talks about a disky elliptical. One of the earliest classification schemes for galaxies, which is still
heavily used, is the Hubble sequence. Roughly speaking, the Hubble sequence is a sequence
in the admixture of the disk and ellipsoidal components in a galaxy, which ranges from earlytype
ellipticals that are pure ellipsoids to late-type spirals that are pure disks. As we will see
in Chapter 2, the important aspect of the Hubble sequence is that many intrinsic properties of
galaxies, such as luminosity, color, and gas content, change systematically along this sequence.
In addition, disks and ellipsoids most likely have very different formation mechanisms. Therefore,
the morphology of a galaxy, or its location along the Hubble sequence, is directly related to
its formation history.
For completeness, we stress that not all galaxies fall in this spiral vs. elliptical classification.
The faintest galaxies, called dwarf galaxies, typically do not fall on the Hubble sequence. Dwarf
galaxies with significant amounts of gas and ongoing star formation typically have a very irregular
structure, and are consequently called (dwarf) irregulars. Dwarf galaxies without gas and
young stars are often very diffuse, and are called dwarf spheroidals. In addition to these dwarf
galaxies, there is also a class of brighter galaxies whose morphology neither resembles a disk nor
a smooth ellipsoid. These are called peculiar galaxies and include, among others, galaxies with
double or multiple subcomponents linked by filamentary structure and highly distorted galaxies
with extended tails. As we will see, they are usually associated with recent mergers or tidal
interactions. Although peculiar galaxies only constitute a small fraction of the entire galaxy population,
their existence conveys important information about how galaxies may have changed
their morphologies during their evolutionary history.
(b) Luminosity and Stellar Mass Galaxies span a wide range in luminosity. The brightest
galaxies have luminosities of ∼10^12 L, where L indicates the luminosity of the Sun. The exact
lower limit of the luminosity distribution is less well defined, and is subject to regular changes,
as fainter and fainter galaxies are constantly being discovered. In 2007 the faintest galaxy known
was a newly discovered dwarf spheroidal Willman I, with a total luminosity somewhat below
1000L.
Obviously, the total luminosity of a galaxy is related to its total number of stars, and thus to its
total stellar mass. However, the relation between luminosity and stellar mass reveals a significant
amount of scatter, because different galaxies have different stellar populations. As we will see in
Chapter 10, galaxies with a younger stellar population have a higher luminosity per unit stellar
mass than galaxies with an older stellar population.
An important statistic of the galaxy population is its luminosity probability distribution function,
also known as the luminosity function. As we will see in Chapter 2, there are many more
faint galaxies than bright galaxies, so that the faint ones clearly dominate the number density.
However, in terms of the contribution to the total luminosity density, neither the faintest nor the
brightest galaxies dominate. Instead, it is the galaxies with a characteristic luminosity similar
to that of our Milky Way that contribute most to the total luminosity density in the present-day
Universe. This indicates that there is a characteristic scale in galaxy formation, which is accentuated
by the fact that most galaxies that are brighter than this characteristic scale are ellipticals,
while those that are fainter are mainly spirals (at the very faint end dwarf irregulars and dwarf
spheroidals dominate). Understanding the physical origin of this characteristic scale has turned
out to be one of the most challenging problems in contemporary galaxy formation modeling.
(c) Size and Su***ce Brightness As we will see in Chapter 2, galaxies do not have well-defined
boundaries. Consequently, several different definitions for the size of a galaxy can be found in
the literature. One measure often used is the radius enclosing a certain fraction (e.g. half) of the
total luminosity. In general, as one might expect, brighter galaxies are bigger. However, even for
a fixed luminosity, there is a considerable scatter in sizes, or in su***ce brightness, defined as the
luminosity per unit area.
The size of a galaxy has an important physical meaning. In disk galaxies, which are rotation
supported, the sizes are a measure of their specific angular momenta (see Chapter 11). In the
case of elliptical galaxies, which are supported by random motions, the sizes are a measure
of the amount of dissipation during their formation (see Chapter 13). Therefore, the observed
distribution of galaxy sizes is an important constraint for galaxy formation models.
(d) Gas Mass Fraction Another useful parameter to describe galaxies is their cold gas mass
fraction, defined as fgas = Mcold/[Mcold +M], with Mcold and M the masses of cold gas and
stars, respectively. This ratio expresses the efficiency with which cold gas has been turned into
stars. Typically, the gas mass fractions of ellipticals are negligibly small, while those of disk
galaxies increase systematically with decreasing su***ce brightness. Indeed, the lowest su***ce
brightness disk galaxies can have gas mass fractions in excess of 90 percent, in contrast to our
Milky Way which has fgas ∼ 0.1.
(e) Color Galaxies also come in different colors. The color of a galaxy reflects the ratio of
its luminosity in two photometric passbands. A galaxy is said to be red if its luminosity in the
redder passband is relatively high compared to that in the bluer passband. Ellipticals and dwarf
spheroidals generally have redder colors than spirals and dwarf irregulars. As we will see in
Chapter 10, the color of a galaxy is related to the characteristic age and metallicity of its stellar
population. In general, redder galaxies are either older or more metal rich (or both). Therefore, the
color of a galaxy holds important information regarding its stellar population. However, extinction
by dust, either in the galaxy itself, or along the line-of-sight between the source and the
observer, also tends to make a galaxy appear red. As we will see, separating age, metallicity and
dust effects is one of the most daunting tasks in observational astronomy.
(f) Environment As we will see in §§2.5–2.7, galaxies are not randomly distributed throughout
space, but show a variety of structures. Some galaxies are located in high-density clusters containing
several hundreds of galaxies, some in smaller groups containing a few to tens of galaxies,
while yet others are distributed in low-density filamentary or sheet-like structures. Many of these
structures are gravitationally bound, and may have played an important role in the formation
and evolution of the galaxies. This is evident from the fact that elliptical galaxies seem to prefer
cluster environments, whereas spiral galaxies are mainly found in relative isolation (sometimes
called the field). As briefly discussed in §1.2.8 below, it is believed that this morphology–density
relation reflects enhanced dynamical interaction in denser environments, although we still lack a
detailed understanding of its origin.
(g) Nuclear Activity For the majority of galaxies, the observed light is consistent with what
we expect from a collection of stars and gas. However, a small fraction of all galaxies, called
active galaxies, show an additional non-stellar component in their spectral energy distribution.
As we will see in Chapter 14, this emission originates from a small region in the centers of these
galaxies, called the active galactic nucleus (AGN), and is associated with matter accretion onto a
supermassive black hole. According to the relative importance of such non-stellar emission, one
can separate active galaxies from normal (or non-active) galaxies.
(h) Redshift Because of the expansion of the Universe, an object that is farther away will have a
larger receding velocity, and thus a larger redshift. Since the light from high-redshift galaxies was
emitted when the Universe was younger, we can study galaxy evolution by observing the galaxy
population at different redshifts. In fact, in a statistical sense the high-redshift galaxies are the
progenitors of present-day galaxies, and any changes in the number density or intrinsic properties
of galaxies with redshift give us a direct window on the formation and evolution of the galaxy
population. With modern, large telescopes we can now observe galaxies out to redshifts beyond
six, *** it possible for us to probe the galaxy population back to a time when the Universe
was only about 10 percent of its current age.

1.2 Basic Elements of Galaxy Formation
Before diving into details, it is useful to have an overview of the basic theoretical framework
within which our current ideas about galaxy formation and evolution have been developed. In
this section we give a brief overview of the various physical processes that play a role during
the formation and evolution of galaxies. The goal is to provide the reader with a picture
of the relationships among the various aspects of galaxy formation to be addressed in greater
detail in the chapters to come. To guide the reader, Fig. 1.1 shows a flow chart of galaxy formation,
which illustrates how the various processes to be discussed below are intertwined. It
is important to stress, though, that this particular flow chart reflects our current, undoubtedly
incomplete view of galaxy formation. Future improvements in our understanding of galaxy formation
and evolution may add new links to the flow chart, or may render some of the links shown
obsolete.
Fig. 1.1. A logic flow chart for galaxy formation. In the standard scenario, the initial and boundary conditions
for galaxy formation are set by the cosmological framework. The paths leading to the formation of
various galaxies are shown along with the relevant physical processes. Note, however, that processes do
not separate as neatly as this figure suggests. For example, cold gas may not have the time to settle into a
gaseous disk before a major merger takes place.
1.2.1 The Standard Model of Cosmology
Since galaxies are observed over cosmological length and time scales, the description of their
formation and evolution must involve cosmology, the study of the properties of space-time on
large scales. Modern cosmology is based upon the cosmological principle, the hypothesis that
the Universe is spatially homogeneous and isotropic, and Einstein’s theory of general relativity,
according to which the structure of space-time is determined by the mass distribution in the
Universe. As we will see in Chapter 3, these two assumptions together lead to a cosmology (the
standard model) that is completely specified by the curvature of the Universe, K, and the scale
factor, a(t), describing the change of the length scale of the Universe with time. One of the basic
tasks in cosmology is to determine the value of K and the form of a(t) (hence the space-time
geometry of the Universe on large scales), and to show how observables are related to physical
quantities in such a universe.
Modern cosmology not only specifies the large-scale geometry of the Universe, but also has
the potential to predict its thermal history and matter content. Because the Universe is expanding
and filled with microwave photons at the present time, it must have been smaller, denser and
hotter at earlier times. The hot and dense medium in the early Universe provides conditions
under which various reactions among elementary particles, nuclei and atoms occur. Therefore,
the application of particle, nuclear and atomic physics to the thermal history of the Universe in
principle allows us to predict the abundances of all species of elementary particles, nuclei and
atoms at different epochs. Clearly, this is an important part of the problem to be addressed in this
book, because the formation of galaxies depends crucially on the matter/energy content of the
Universe.
In currently popular cosmologies we usually consider a universe consisting of three main components.
In addition to the ‘baryonic’ matter, the protons, neutrons and electrons1 that make up
the visible Universe, astronomers have found various indications for the presence of dark matter
and dark energy (see Chapter 2 for a detailed discussion of the observational evidence). Although
the nature of both dark matter and dark energy is still unknown, we believe that they are responsible
for more than 95 percent of the energy density of the Universe. Different cosmological
models differ mainly in (i) the relative contributions of baryonic matter, dark matter, and dark
energy, and (ii) the nature of dark matter and dark energy. At the time of writing, the most popular
model is the so-called ΛCDM model, a flat universe in which ∼ 75 percent of the energy
density is due to a cosmological constant, ∼ 21 percent is due to ‘cold’ dark matter (CDM),
and the remaining 4 percent is due to the baryonic matter out of which stars and galaxies are
made. Chapter 3 gives a detailed description of these various components, and describes how
they influence the expansion history of the Universe.
1.2.2 Initial Conditions
If the cosmological principle held perfectly and the distribution of matter in the Universe were
perfectly uniform and isotropic, there would be no structure formation. In order to explain the
presence of structure, in particular galaxies, we clearly need some deviations from perfect uniformity.
Unfortunately, the standard cosmology does not in itself provide us with an explanation
for the origin of these perturbations. We have to go beyond it to search for an answer.
A classical, general relativistic description of cosmology is expected to break down at very
early times when the Universe is so dense that quantum effects are expected to be important. As
we will see in §3.6, the standard cosmology has a number of conceptual problems when applied
to the early Universe, and the solutions to these problems require an extension of the standard
cosmology to incorporate quantum processes. One generic consequence of such an extension
is the generation of density perturbations by quantum fluctuations at early times. It is believed
that these perturbations are responsible for the formation of the structures observed in today’s
Universe.
As we will see in §3.6, one particularly successful extension of the standard cosmology is the
inflationary theory, in which the Universe is assumed to have gone through a phase of rapid,
exponential expansion (called inflation) driven by the vacuum energy of one or more quantum
fields. In many, but not all, inflationary models, quantum fluctuations in this vacuum energy can
produce density perturbations with properties consistent with the observed large scale structure.
Inflation thus offers a promising explanation for the physical origin of the initial perturbations.
Unfortunately, our understanding of the very early Universe is still far from complete, and we are
currently unable to predict the initial conditions for structure formation entirely from first principles.
Consequently, even this part of galaxy formation theory is still partly phenomenological:
typically initial conditions are specified by a set of parameters that are constrained by observational
data, such as the pattern of fluctuations in the microwave background or the present-day
abundance of galaxy clusters.
1.2.3 Gravitational Instability and Structure Formation
Having specified the initial conditions and the cosmological framework, one can compute how
small perturbations in the density field evolve. As we will see in Chapter 4, in an expanding
universe dominated by non-relativistic matter, perturbations grow with time. This is easy
to understand. A region whose initial density is slightly higher than the mean will attract its
surroundings slightly more strongly than average. Consequently, over-dense regions pull matter
towards them and become even more over-dense. On the other hand, under-dense regions become
even more rarefied as matter flows away from them. This amplification of density perturbations is
referred to as gravitational instability and plays an important role in modern theories of structure
formation. In a static universe, the amplification is a run-away process, and the density contrast
δρ/ρ grows exponentially with time. In an expanding universe, however, the cosmic expansion
damps accretion flows, and the growth rate is usually a power law of time, δρ/ρ ∝ tα , with
α > 0. As we will see in Chapter 4, the exact rate at which the perturbations grow depends on
the cosmological model.
At early times, when the perturbations are still in what we call the linear regime (δρ/ρ 1),
the physical size of an over-dense region increases with time due to the overall expansion of
the universe. Once the perturbation reaches over-density δρ/ρ ∼ 1, it breaks away from the
expansion and starts to collapse. This moment of ‘turn-around’, when the physical size of
the perturbation is at its maximum, signals the transition from the mildly nonlinear regime to
the strongly nonlinear regime.
The outcome of the subsequent nonlinear, gravitational collapse depends on the matter content
of the perturbation. If the perturbation consists of ordinary baryonic gas, the collapse creates
strong shocks that raise the entropy of the material. If radiative cooling is inefficient, the system
relaxes to hydrostatic equilibrium, with its self-gravity balanced by pressure gradients. If the
perturbation consists of collisionless matter (e.g. cold dark matter), no shocks develop, but the
system still relaxes to a quasi-equilibrium state with a more-or-less universal structure. This process
is called violent relaxation and will be discussed in Chapter 5. Nonlinear, quasi-equilibrium
dark matter objects are called dark matter halos. Their predicted structure has been thoroughly
explored using numerical simulations, and they play a pivotal role in modern theories of galaxy
formation. Chapter 7 therefore presents a detailed discussion of the structure and formation of
dark matter halos. As we shall see, halo density profiles, shapes, spins and internal substructure
all depend very weakly on mass and on cosmology, but the abundance and characteristic density
of halos depend sensitively on both of these.
In cosmologies with both dark matter and baryonic matter, such as the currently favored CDM
models, each initial perturbation contains baryonic gas and collisionless dark matter in roughly
their universal proportions. When an object collapses, the dark matter relaxes violently to form a
dark matter halo, while the gas shocks to the virial temperature, Tvir (see §8.2.3 for a definition)
and may settle into hydrostatic equilibrium in the potential well of the dark matter halo if cooling
is slow.
1.2.4 Gas Cooling
Cooling is a crucial ingredient of galaxy formation. Depending on temperature and density,
a variety of cooling processes can affect gas. In massive halos, where the virial temperature
Tvir∼>
107 K, gas is fully collisionally ionized and cools mainly through bremsstrahlung emission
from free electrons. In the temperature range 104K < Tvir < 106 K, a number of excitation and
de-excitation mechanisms can play a role. Electrons can recombine with ions, emitting a photon,
or atoms (neutral or partially ionized) can be excited by a collision with another particle,
thereafter decaying radiatively to the ground state. Since different atomic species have different
excitation energies, the cooling rates depend strongly on the chemical composition of the gas.
In halos with Tvir < 104 K, gas is predicted to be almost completely neutral. This strongly suppresses
the cooling processes mentioned above. However, if heavy elements and/or molecules are
present, cooling is still possible through the collisional excitation/de-excitation of fine and hyperfine
structure lines (for heavy elements) or rotational and/or vibrational lines (for molecules).
Finally, at high redshifts (z∼>
6), inverse Compton scattering of cosmic microwave background
photons by electrons in hot halo gas can also be an effective cooling channel. Chapter 8 will
discuss these cooling processes in more detail.
Except for inverse Compton scattering, all these cooling mechanisms involve two particles.
Consequently, cooling is generally more effective in higher density regions. After nonlinear gravitational
collapse, the shocked gas in virialized halos may be dense enough for cooling to be
effective. If cooling times are short, the gas never comes to hydrostatic equilibrium, but rather
accretes directly onto the central protogalaxy. Even if cooling is slow enough for a hydrostatic
atmosphere to develop, it may still cause the denser inner regions of the atmosphere to lose pressure
support and to flow onto the central object. The net effect of cooling is thus that the baryonic
material segregates from the dark matter, and accumulates as dense, cold gas in a protogalaxy at
the center of the dark matter halo.
As we will see in Chapter 7, dark matter halos, as well as the baryonic material associated
with them, typically have a small amount of angular momentum. If this angular momentum is
conserved during cooling, the gas will spin up as it flows inwards, settling in a cold disk in
centrifugal equilibrium at the center of the halo. This is the standard paradigm for the formation
of disk galaxies, which we will discuss in detail in Chapter 11.
1.2.5 Star Formation
As the gas in a dark matter halo cools and flows inwards, its self-gravity will eventually dominate
over the gravity of the dark matter. Thereafter it collapses under its own gravity, and in the
presence of effective cooling, this collapse becomes catastrophic. Collapse increases the density
and temperature of the gas, which generally reduces the cooling time more rapidly than it reduces
the collapse time. During such runaway collapse the gas cloud may fragment into small, highdensity
cores that may eventually form stars (see Chapter 9), thus giving rise to a visible galaxy.
Unfortunately, many details of these processes are still unclear. In particular, we are still unable
to predict the mass fraction of, and the time scale for, a self-gravitating cloud to be transformed
into stars. Another important and yet poorly understood issue is concerned with the mass distribution
with which stars are formed, i.e. the initial mass function (IMF). As we will see in
Chapter 10, the evolution of a star, in particular its luminosity as function of time and its eventual
fate, is largely determined by its mass at birth. Predictions of observable quantities for model
galaxies thus require not only the birth rate of stars as a function of time, but also their IMF.
In principle, it should be possible to derive the IMF from first principles, but the theory of star
formation has not yet matured to this level. At present one has to assume an IMF ad hoc and
check its validity by comparing model predictions to observations.
Based on observations, we will often distinguish two modes of star formation: quiescent star
formation in rotationally supported gas disks, and starbursts. The latter are characterized by
much higher star-formation rates, and are typically confined to relatively small regions (often
the nucleus) of galaxies. Starbursts require the accumulation of large amounts of gas in a small
volume, and appear to be triggered by strong dynamical interactions or instabilities. These processes
will be discussed in more detail in §1.2.8 below and in Chapter 12. At the moment,
there are still many open questions related to these different modes of star formation. What
fraction of stars formed in the quiescent mode? Do both modes produce stellar populations
with the same IMF? How does the relative importance of starbursts scale with time? As we
will see, these and related questions play an important role in contemporary models of galaxy
formation.
1.2.6 Feedback Processes
When astronomers began to develop the first dynamical models for galaxy formation in a CDM
dominated universe, it immediately became clear that most baryonic material is predicted to
cool and form stars. This is because in these ‘hierarchical’ structure formation models, small
dense halos form at high redshift and cooling within them is predicted to be very efficient. This
disagrees badly with observations, which show that only a relatively small fraction of all baryons
are in cold gas or stars (see Chapter 2). Apparently, some physical process must either prevent
the gas from cooling, or reheat it after it has become cold.
Even the very first models suggested that the solution to this problem might lie in feedback
from supernovae, a class of exploding stars that can produce enormous amounts of energy (see
§10.5). The radiation and the blast waves from these supernovae may heat (or reheat) surrounding
gas, blowing it out of the galaxy in what is called a galactic wind. These processes are described
in more detail in §§8.6 and 10.5.
Another important feedback source for galaxy formation is provided by active galactic nuclei
(AGN), the active accretion phase of supermassive black holes (***BH) lurking at the centers of
almost all massive galaxies (see Chapter 14). This process releases vast amounts of energy – this
is why AGN are bright and can be seen out to large distances, which can be tapped by surrounding
gas. Although only a relatively small fraction of present-day galaxies contain an AGN, observations
indicate that virtually all massive spheroids contain a nuclear ***BH (see Chapter 2).
Therefore, it is believed that virtually all galaxies with a significant spheroidal component have
gone through one or more AGN phases during their life.
Although it has become clear over the years that feedback processes play an important role
in galaxy formation, we are still far from understanding which processes dominate, and when
and how exactly they operate. Furthermore, to make accurate predictions for their effects, one
also needs to know how often they occur. For supernovae this requires a prior understanding of
the star-formation rates and the IMF. For AGN it requires understanding how, when and where
supermassive black holes form, and how they accrete mass.
Fig. 1.2. A flow chart of the evolution of an individual galaxy. The galaxy is represented by the dashed box
which contains hot gas, cold gas, stars and a supermassive black hole (***BH). Gas cooling converts hot gas
into cold gas, star formation converts cold gas into stars, and dying stars inject energy, metals and gas into
the gas components. In addition, the ***BH can accrete gas (both hot and cold) as well as stars, producing
AGN activity which can release vast amounts of energy which affect primarily the gaseous components of
the galaxy. Note that in general the box will not be closed: gas can be added to the system through accretion
from the intergalactic medium and can escape the galaxy through outflows driven by feedback from the
stars and/or the ***BH. Finally, a galaxy may merge or interact with another galaxy, causing a significant
boost or suppression of all these processes.
It should be clear from the above discussion that galaxy formation is a subject of great complexity,
involving many strongly intertwined processes. This is illustrated in Fig. 1.2, which
shows the relations between the four main baryonic components of a galaxy: hot gas, cold gas,
stars, and a supermassive black hole. Cooling, star formation, AGN accretion, and feedback
processes can all shift baryons from one of these components to another, thereby altering the
efficiency of all the processes. For example, increased cooling of hot gas will produce more
cold gas. This in turn will increases the star-formation rate, hence the supernova rate. The additional
energy injection from supernovae can reheat cold gas, thereby suppressing further star
formation (negative feedback). On the other hand, supernova blast waves may also compress the
surrounding cold gas, so as to boost the star-formation rate (positive feedback). Understanding
these various feedback loops is one of the most important and intractable issues in contemporary
models for the formation and evolution of galaxies.
1.2.7 Mergers
So far we have considered what happens to a single, isolated system of dark matter, gas and
stars. However, galaxies and dark matter halos are not isolated. For example, as illustrated in
Fig. 1.2, systems can accrete new material (both dark and baryonic matter) from the intergalactic
medium, and can lose material through outflows driven by feedback from stars and/or AGN. In
addition, two (or more) systems may merge to form a new system with very different properties
from its progenitors. In the currently popular CDM cosmologies, the initial density fluctuations
Fig. 1.3. A schematic merger tree, illustrating the merger history of a dark matter halo. It shows, at three
different epochs, the progenitor halos that at time t4 have merged to form a single halo. The size of each
circle represents the mass of the halo. Merger histories of dark matter halos play an important role in
hierarchical theories of galaxy formation.
have larger amplitudes on smaller scales. Consequently, dark matter halos grow hierarchically, in
the sense that larger halos are formed by the coalescence (merging) of smaller progenitors. Such
a formation process is usually called a hierarchical or ‘bottom-up’ scenario.
The formation history of a dark matter halo can be described by a ‘merger tree’ that traces
all its progenitors, as illustrated in Fig. 1.3. Such merger trees play an important role in modern
galaxy formation theory. Note, however, that illustrations such as Fig. 1.3 can be misleading. In
CDM models part of the growth of a massive halo is due to merging with a large number of much
smaller halos, and to a good approximation, such mergers can be thought of as smooth accretion.
When two similar mass dark matter halos merge, violent relaxation rapidly transforms the orbital
energy of the progenitors into the internal binding energy of the quasi-equilibrium remnant. Any
hot gas associated with the progenitors is shock-heated during the merger and settles back into
hydrostatic equilibrium in the new halo. If the progenitor halos contained central galaxies, the
galaxies also merge as part of the violent relaxation process, producing a new central galaxy in
the final system. Such a merger may be accompanied by strong star formation or AGN activity if
the merging galaxies contained significant amounts of cold gas. If two merging halos have very
different mass, the dynamical processes are less violent. The smaller system orbits within the
main halo for an extended period of time during which two processes compete to determine its
eventual fate. Dynamical friction transfers energy from its orbit to the main halo, causing it to
spiral inwards, while tidal effects remove mass from its outer regions and may eventually dissolve
it completely (see Chapter 12). Dynamical friction is more effective for more massive satellites,
but if the mass ratio of the initial halos is large enough, the smaller object (and any galaxy
associated with it) can maintain its identity for a long time. This is the process for the build-up of
clusters of galaxies: a cluster may be considered as a massive dark matter halo hosting a relatively
massive galaxy near its center and many satellites that have not yet dissolved or merged with the
central galaxy.
As we will see in Chapters 12 and 13, numerical simulations show that the merger of two
galaxies of roughly equal mass produces an object reminiscent of an elliptical galaxy, and the
result is largely independent of whether the progenitors are spirals or ellipticals. Indeed, current
hierarchical models of galaxy formation assume that most, if not all, elliptical galaxies are merger
remnants. If gas cools onto this merger remnant with significant angular momentum, a new disk
may form, producing a disk–bulge system like that in an early-type spiral galaxy.
It should be obvious from the above discussion that mergers play a crucial role in galaxy
formation. Detailed descriptions of halo mergers and galaxy mergers are presented in Chapter 7
and Chapter 12, respectively.

1.2.8 Dynamical Evolution
When satellite galaxies orbit within dark matter halos, they experience tidal forces due to the
central galaxy, due to other satellite galaxies, and due to the potential of the halo itself. These
tidal interactions can remove dark matter, gas and stars from the galaxy, a process called tidal
stripping (see §12.2), and may also perturb its structure. In addition, if the halo contains a hot
gas component, any gas associated with the satellite galaxy will experience a drag force due to
the relative motion of the two fluids. If the drag force exceeds the restoring force due to the
satellite’s own gravity, its gas will be ablated, a process called ram-pressure stripping. These
dynamical processes are thought to play an important role in driving galaxy evolution within
clusters and groups of galaxies. In particular, they are thought to be partially responsible for the
observed environmental dependence of galaxy morphology (see Chapter 15).
Internal dynamical effects can also reshape galaxies. For example, a galaxy may form in a
configuration which becomes unstable at some later time. Large scale instabilities may then
redistribute mass and angular momentum within the galaxy, thereby changing its morphology. A
well-known and important example is the bar-instability within disk galaxies. As we shall see in
§11.5, a thin disk with too high a su***ce density is susceptible to a non-axisymmetric instability,
which produces a bar-like structure similar to that seen in barred spiral galaxies. These bars
may then buckle out of the disk to produce a central ellipsoidal component, a so-called ‘pseudobulge’.
Instabilities may also be triggered in otherwise stable galaxies by interactions. Thus, an
important question is whether the sizes and morphologies of galaxies were set at formation, or are
the result of later dynamical process (‘secular evolution’, as it is termed). Bulges are particularly
interesting in this context. They may be a remnant of the first stage of galaxy formation, or, as
mentioned in §1.2.7, may reflect an early merger which has grown a new disk, or may result from
buckling of a bar. It is likely that all these processes are important for at least some bulges.
1.2.9 Chemical Evolution
In astronomy, all chemical elements heavier than helium are collectively termed ‘metals’. The
mass fraction of a baryonic component (e.g. hot gas, cold gas, stars) in metals is then referred to
as its metallicity. As we will see in §3.4, the nuclear reactions during the first three minutes of the
Universe (the epoch of primordial nucleosynthesis) produced primarily hydrogen (∼ 75%) and
helium (∼ 25%), with a very small admixture of metals dominated by lithium. All other metals
in the Universe were formed at later times as a consequence of nuclear reactions in stars. When
stars expel mass in stellar winds, or in supernova explosions, they enrich the interstellar medium
(I***) with newly synthesized metals.
Evolution of the chemical composition of the gas and stars in galaxies is important for several
reasons. First of all, the luminosity and color of a stellar population depend not only on its age
and IMF, but also on the metallicity of the stars (see Chapter 10). Secondly, the cooling efficiency
of gas depends strongly on its metallicity, in the sense that more metal-enriched gas cools faster
(see §8.1). Thirdly, small particles of heavy elements known as dust grains, which are mixed with
the interstellar gas in galaxies, can absorb significant amounts of the starlight and reradiate it in
infrared wavelengths. Depending on the amount of the dust in the I***, which scales roughly
linearly with its metallicity (see §10.3.7), this interstellar extinction can significantly reduce the
brightness of a galaxy.
As we will see in Chapter 10, the mass and detailed chemical composition of the material
ejected by a stellar population as it evolves depend both on the IMF and on its initial metallicity.
In principle, observations of the metallicity and abundance ratios of a galaxy can therefore be
used to constrain its star-formation history and IMF. In practice, however, the interpretation of
the observations is complicated by the fact that galaxies can accrete new material of different
metallicity, that feedback processes can blow out gas, perhaps preferentially metals, and that
mergers can mix the chemical compositions of different systems.
1.2.10 Stellar Population Synthesis
The light we receive from a given galaxy is emitted by a large number of stars that may have
different masses, ages, and metallicities. In order to interpret the observed spectral energy distribution,
we need to predict how each of these stars contributes to the total spectrum. Unlike
many of the ingredients in galaxy formation, the theory of stellar evolution, to be discussed in
Chapter 10, is reasonably well understood. This allows us to compute not only the evolution of
the luminosity, color and spectrum of a star of given initial mass and chemical composition, but
also the rates at which it ejects mass, energy and metals into the interstellar medium. If we know
the star-formation history (i.e. the star-formation rate as a function of time) and IMF of a galaxy,
we can then synthesize its spectrum at any given time by adding together the spectra of all the
stars, after evolving each to the time under consideration. In addition, this also yields the rates
at which mass, energy and metals are ejected into the interstellar medium, providing important
ingredients for modeling the chemical evolution of galaxies.
Most of the energy of a stellar population is emitted in the optical, or, if the stellar population
is very young (∼<
10Myr), in the ultraviolet (see §10.3). However, if the galaxy contains a lot of
dust, a significant fraction of this optical and UV light may get absorbed and re-emitted in the
infrared. Unfortunately, predicting the final emergent spectrum is extremely complicated. Not
only does it depend on the amount of the radiation absorbed, it also depends strongly on the
properties of the dust, such as its geometry, its chemical composition, and (the distribution of)
the sizes of the dust grains (see §10.3.7).
Finally, to complete the spectral energy distribution emitted by a galaxy, we also need to
add the contribution from a possible AGN. Chapter 14 discusses various emission mechanisms
associated with accreting ***BHs. Unfortunately, as we will see, we are still far from being able
to predict the detailed spectra for AGN.
1.2.11 The Intergalactic Medium
The intergalactic medium (IGM) is the baryonic material lying between galaxies. This is and
has always been the dominant baryonic component of the Universe and it is the material from
which galaxies form. Detailed studies of the IGM can therefore give insight into the properties
of the pregalactic matter before it condensed into galaxies. As illustrated in Fig. 1.2, galaxies
do not evolve as closed boxes, but can affect the properties of the IGM through exchanges of
mass, energy and heavy elements. The study of the IGM is thus an integral part of understanding
how galaxies form and evolve. As we will see in Chapter 16, the properties of the IGM can be
probed most effectively through the absorption it produces in the spectra of distant quasars (a
certain class of active galaxies; see Chapter 14). Since quasars are now observed out to redshifts
beyond 6, their absorption line spectra can be used to study the properties of the IGM back to a
time when the Universe was only a few percent of its present age.

1.3 Time Scales
As discussed above, and as illustrated in Fig. 1.1, the formation of an individual galaxy in the
standard, hierarchical formation scenario involves the following processes: the collapse and virialization
of dark matter halos, the cooling and condensation of gas within the halo, and the
conversion of cold gas into stars and a central supermassive black hole. Evolving stars and AGN
eject energy, mass and heavy elements into the interstellar medium, thereby determining its structure
and chemical composition and perhaps driving winds into the intergalactic medium. Finally,
galaxies can merge and interact, reshaping their morphology and triggering further starbursts and
AGN activity. In general, the properties of galaxies are determined by the competition among all
these processes, and a simple way to characterize the relative importance of these processes is to
use the time scales associated with them. Here we give a brief summary of the most important
time scales in this context.
• Hubble time: This is an estimate of the time scale on which the Universe as a whole evolves.
It is defined as the inverse of the Hubble constant (see §3.2), which specifies the current cosmic
expansion rate. It would be equal to the time since the Big Bang if the Universe had always
expanded at its current rate. Roughly speaking, this is the time scale on which substantial
evolution of the galaxy population is expected.
• Dynamical time: This is the time required to orbit across an equilibrium dynamical system.
For a system with mass M and radius R, we define it as tdyn =
（3π/16Gρ）^1/2, where
ρ = 3M/4πR^3. This is related to the free-fall time, defined as the time required for a uniform,
pressure-free sphere to collapse to a point, as tff = tdyn/
√2.
• Cooling time: This time scale is the ratio between the thermal energy content and the energy
loss rate (through radiative or conductive cooling) for a gas component.
• Star-formation time: This time scale is the ratio of the cold gas content of a galaxy to its
star-formation rate. It is thus an indication of how long it would take for the galaxy to run out
of gas if the fuel for star formation is not replenished.
• Chemical enrichment time: This is a measure for the time scale on which the gas is enriched
in heavy elements. This enrichment time is generally different for different elements, depending
on the lifetimes of the stars responsible for the bulk of the production of each element (see
§10.1).
• Merging time: This is the typical time that a halo or galaxy must wait before experiencing a
merger with an object of similar mass, and is directly related to the major merger frequency.
• Dynamical friction time: This is the time scale on which a satellite object in a large halo
loses its orbital energy and spirals to the center. As we will see in §12.3, this time scale is
proportional to Msat/Mmain, where Msat is the mass of the satellite object and Mmain is that of
the main halo. Thus, more massive galaxies will merge with the central galaxy in a halo more
quickly than smaller ones.
These time scales can provide guidelines for incorporating the underlying physical processes
in models of galaxy formation and evolution, as we describe in later chapters. In particular,
comparing time scales can give useful insights. As an illustration, consider the following
examples:
• Processes whose time scale is longer than the Hubble time can usually be ignored. For example,
satellite galaxies with mass less than a few percent of their parent halo normally have
dynamical friction times exceeding the Hubble time (see §12.3). Consequently, their orbits do
not decay significantly. This explains why clusters of galaxies have so many ‘satellite’ galaxies
– the main halos are so much more massive than a typical galaxy that dynamical friction is
ineffective.
• If the cooling time is longer than the dynamical time, hot gas will typically be in hydrostatic
equilibrium. In the opposite case, however, the gas cools rapidly, losing pressure support,
and collapsing to the halo center on a free-fall time without establishing any hydrostatic
equilibrium.
• If the star formation time is comparable to the dynamical time, gas will turn into stars during
its initial collapse, a situation which may lead to the formation of something resembling an
elliptical galaxy. On the other hand, if the star formation time is much longer than the cooling
and dynamical times, the gas will settle into a centrifugally supported disk before forming
stars, thus producing a disk galaxy (see §1.4.5).
• If the relevant chemical evolution time is longer than the star-formation time, little metal
enrichment will occur during star formation and all stars will end up with the same, initial
metallicity. In the opposite case, the star-forming gas is continuously enriched, so that stars
formed at different times will have different metallicities and abundance patterns (see §10.4).
So far we have avoided one obvious question, namely, what is the time scale for galaxy formation
itself? Unfortunately, there is no single useful definition for such a time scale. Galaxy
formation is a process, not an event, and as we have seen, this process is an amalgam of many
different elements, each with its own time scale. If, for example, we are concerned with its stellar
population, we might define the formation time of a galaxy as the epoch when a fixed fraction
(e.g. 1% or 50%) of its stars had formed. If, on the other hand, we are concerned with its structure,
we might want to define the galaxy’s formation time as the epoch when a fixed fraction
(e.g. 50% or 90%) of its mass was first assembled into a single object. These two ‘formation’
times can differ greatly for a given galaxy, and even their ordering can change from one galaxy
to another. Thus it is important to be precise about definition when talking about the formation
times of galaxies.
1.4 A Brief History of Galaxy Formation
The picture of galaxy formation sketched above is largely based on the hierarchical cold dark
matter model for structure formation, which has been the standard paradigm since the beginning
of the 1980s. In the following, we give an historical overview of the development of ideas and
concepts about galaxy formation up to the present time. This is not intended as a complete historical
account, but rather as a summary for young researchers of how our current ideas about
galaxy formation were developed. Readers interested in a more extensive historical review can
find some relevant material in the book The Cosmic Century: A History of Astrophysics and
Cosmology by Malcolm Longair (2006).
1.4.1 Galaxies as Extragalactic Objects
By the end of the nineteenth century, astronomers had discovered a large number of astronomical
objects that differ from stars in that they are fuzzy rather than point-like. These objects were
collectively referred to as ‘nebulae’. During the period 1771 to 1784 the French astronomer
Charles Messier cataloged more than 100 of these objects in order to avoid confusing them
with the comets he was searching for. Today the Messier numbers are still used to designate a
number of bright galaxies. For example, the Andromeda Galaxy is also known as M31, because
it is the 31st nebula in Messier’s catalog. A more systematic search for nebulae was carried
out by the Herschels, and in 1864 John Herschel published his General Catalogue of Galaxies
which contains 5079 nebular objects. In 1888, Dreyer published an expanded version as his New
General Catalogue of Nebulae and Clusters of Stars. Together with its two supplementary Index
Catalogues, Dreyer’s catalogue contained about 15,000 objects. Today, NGC and IC numbers
are still widely used to refer to galaxies.
For many years after their discovery, the nature of the nebular objects was controversial. There
were two competing ideas: one assumed that all nebulae are objects within our Milky Way,
the other that some might be extragalactic objects, individual ‘island universes’ like the Milky
Way. In 1920 the National Academy of Sciences inWashington invited two leading astronomers,
Harlow Shapley and Heber Curtis, to debate this issue, an event which has passed into astronomical
folklore as ‘The Great Debate’. The controversy remained unresolved until 1925, when
Edwin Hubble used distances estimated from Cepheid variables to demonstrate conclusively
that some nebulae are extragalactic, individual galaxies comparable to our Milky Way in size
and luminosity. Hubble’s discovery marked the beginning of extragalactic astronomy. During
the 1930s, high-quality photographic images of galaxies enabled him to classify galaxies into a
broad sequence according to their morphology. Today Hubble’s sequence is still widely adopted
to classify galaxies.
Since Hubble’s time, astronomers have made tremendous progress in systematically searching
the skies for galaxies. At present deep CCD imaging and high-quality spectroscopy are available
for about a million galaxies.
1.4.2 Cosmology
Only four years after his discovery that galaxies truly are extragalactic, Hubble made his second
fundamental breakthrough: he showed that the recession velocities of galaxies are linearly related
to their distances (Hubble, 1929; see also Hubble & Humason 1931), thus demonstrating that
our Universe is expanding. This is undoubtedly the greatest single discovery in the history of
cosmology. It revolutionized our picture of the Universe we live in.
The construction of mathematical models for the Universe actually started somewhat earlier.
As soon as Albert Einstein completed his theory of general relativity in 1916, it was realized that
this theory allowed, for the first time, the construction of self-consistent models for the Universe
as a whole. Einstein himself was among the first to explore such solutions of his field equations.
To his dismay, he found that all solutions require the Universe either to expand or to contract, in
contrast with his belief at that time that the Universe should be static. In order to obtain a static
solution, he introduced a cosmological constant into his field equations. This additional constant
of gravity can oppose the standard gravitational attraction and so make possible a static (though
unstable) solution. In 1922 Alexander Friedmann published two papers exploring both static and
expanding solutions. These models are today known as Friedmann models, although this work
drew little attention until Georges Lemaitre independently rediscovered the same solutions in
1927.
An expanding universe is a natural consequence of general relativity, so it is not surprising
that Einstein considered his introduction of a cosmological constant as ‘the biggest blunder of
my life’ once he learned of Hubble’s discovery. History has many ironies, however. As we will
see later, the cosmological constant is now back with us. In 1998 two teams independently used
the distance–redshift relation of Type Ia supernovae to show that the expansion of the Universe is
accelerating at the present time. Within general relativity this requires an additional mass/energy
component with properties very similar to those of Einstein’s cosmological constant. Rather than
just counterbalancing the attractive effects of ‘normal’ gravity, the cosmological constant today
overwhelms them to drive an ever more rapid expansion.
Since the Universe is expanding, it must have been denser and perhaps also hotter at earlier
times. In the late 1940s this prompted George Gamow to suggest that the chemical elements may
have been created by thermonuclear reactions in the early Universe, a process known as primordial
nucleosynthesis. Gamow’s model was not considered a success, because it was unable to
explain the existence of elements heavier than lithium due to the lack of stable elements with
atomic mass numbers 5 and 8. We now know that this was not a failure at all; all heavier elements
are a result of nucleosynthesis within stars, as first shown convincingly by Fred Hoyle
and collaborators in the 1950s. For Gamow’s model to be correct, the Universe would have to
be hot as well as dense at early times, and Gamow realized that the residual heat should still
be visible in today’s Universe as a background of thermal radiation with a temperature of a few
degrees kelvin, thus with a peak at microwave wavelengths. This was a remarkable prediction
of the cosmic microwave background radiation (CMB), which was finally discovered in 1965.
The thermal history suggested by Gamow, in which the Universe expands from a dense and hot
initial state, was derisively referred to as the Hot Big Bang by Fred Hoyle, who preferred an
unchanging steady state cosmology. Hoyle’s cosmological theory was wrong, but his name for
the correct model has stuck.
The Hot Big Bang model developed gradually during the 1950s and 1960s. By 1964, it had
been noticed that the abundance of helium by mass is everywhere about one third that of hydrogen,
a result which is difficult to explain by nucleosynthesis in stars. In 1964, Hoyle and Tayler
published calculations that demonstrated how the observed helium abundance could emerge
from the Hot Big Bang. Three years later, Wagoner et al. (1967) made detailed calculations
of a complete network of nuclear reactions, confirming the earlier result and suggesting that
the abundances of other light isotopes, such as helium-3, deuterium and lithium, could also be
explained by primordial nucleosynthesis. This success provided strong support for the Hot Big
Bang. The 1965 discovery of the cosmic microwave background showed it to be isotropic and
to have a temperature (2.7K) exactly in the range expected in the Hot Big Bang model (Penzias
& Wilson, 1965; Dicke et al., 1965). This firmly established the Hot Big Bang as the standard
model of cosmology, a status which it has kept up to the present day. Although there have been
changes over the years, these have affected only the exact matter/energy content of the model
and the exact values of its characteristic parameters.
Despite its success, during the 1960s and 1970s it was realized that the standard cosmology
had several serious shortcomings. Its structure implies that the different parts of the Universe
we see today were never in causal contact at early times (e.g. Misner, 1968). How then can
these regions have contrived to be so similar, as required by the isotropy of the CMB? A second
shortcoming is connected with the spatial flatness of the Universe (e.g. Dicke & Peebles, 1979).
It was known by the 1960s that the matter density in the Universe is not very different from the
critical density for closure, i.e. the density for which the spatial geometry of the Universe is flat.
However, in the standard model any tiny deviation from flatness in the early Universe is amplified
enormously by later evolution. Thus, extreme fine tuning of the initial curvature is required to
explain why so little curvature is observed today. A closely related formulation is to ask how our
Universe has managed to survive and to evolve for billions of years, when the time scales of all
physical processes in its earliest phases were measured in tiny fractions of a nanosecond. The
standard cosmology provides no explanations for these puzzles.
A conceptual breakthrough came in 1981 when Alan Guth proposed that the Universe may
have gone through an early period of exponential expansion (inflation) driven by the vacuum
energy of some quantum field. His original model had some problems and was revised in 1982
by Linde and by Albrecht & Steinhardt. In this scenario, the different parts of the Universe
we see today were indeed in causal contact before inflation took place, thereby allowing physical
processes to establish homogeneity and isotropy. Inflation also solves the flatness/time-scale
problem, because the Universe expanded so much during inflation that its curvature radius grew
to be much larger than the presently observable Universe. Thus, a generic prediction of the
inflation scenario is that today’s Universe should appear flat.
1.4.3 Structure Formation
(a) Gravitational Instability In the standard model of cosmology, structures form from small
initial perturbations in an otherwise homogeneous and isotropic universe. The idea that structures
can form via gravitational instability in this way originates from Jeans (1902), who showed that
the stability of a perturbation depends on the competition between gravity and pressure. Density
perturbations grow only if they are larger (heavier) than a characteristic length (mass) scale [now
referred to as the Jeans’ length (mass)] beyond which gravity is able to overcome the pressure
gradients. The application of this Jeans criterion to an expanding background was worked out
by, among others, Gamow & Teller (1939) and Lif***z (1946), with the result that perturbation
growth is power-law in time, rather than exponential as for a static background.
(b) Initial Perturbations Most of the early models of structure formation assumed the Universe
to contain two energy components, ordinary baryonic matter and radiation (CMB photons
and relativistic neutrinos). In the absence of any theory for the origin of perturbations, two distinct
models were considered, usually referred to as adiabatic and isothermal initial conditions.
In adiabatic initial conditions all matter and radiation fields are perturbed in the same way, so
that the total density (or local curvature) varies, but the ratio of photons to baryons, for example,
is spatially invariant. Isothermal initial conditions, on the other hand, correspond to initial perturbations
in the ratio of components, but with no associated spatial variation in the total density
or curvature.2
In the adiabatic case, the perturbations can be considered as applying to a single fluid with
a constant specific entropy as long as the radiation and matter remain tightly coupled. At such
times, the Jeans’ mass is very large and small-scale perturbations execute acoustic oscillations
driven by the pressure gradients associated with the density fluctuations. Silk (1968) showed that
towards the end of recombination, as radiation decouples from matter, small-scale oscillations
are damped by photon diffusion, a process now called Silk damping. Depending on the matter
density and the expansion rate of the Universe, the characteristic scale of Silk damping falls
in the range of 1012–1014M. After radiation/matter decoupling the Jeans’ mass drops precipitously
to  106M and perturbations above this mass scale can start to grow,3 but there are no
perturbations left on the scale of galaxies at this time. Consequently, galaxies must form ‘topdown’,
via the collapse and fragmentation of perturbations larger than the damping scale, an idea
championed by Zel’dovich and colleagues.
In the case of isothermal initial conditions, the spatial variation in the ratio of baryons to
photons remains fixed before recombination because of the tight coupling between the two fluids.
The pressure is spatially uniform, so that there is no acoustic oscillation, and perturbations are
not influenced by Silk damping. If the initial perturbations include small-scale structure, this
survives until after the recombination epoch, when baryon fluctuations are no longer supported
by photon pressure and so can collapse. Structure can then form ‘bottom-up’ through hierarchical
clustering. This scenario of structure formation was originally proposed by Peebles (1965).
By the beginning of the 1970s, the linear evolution of both adiabatic and isothermal perturbations
had been worked out in great detail (e.g. Lif***z, 1946; Silk, 1968; Peebles & Yu, 1970;
Sato, 1971; S. Weinberg, 1971). At that time, it was generally accepted that observed structures
must have formed from finite amplitude perturbations which were somehow part of the

Vlasov equation
From Wikipedia, the free encyclopedia


[ltr]The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range (for example, Coulomb) interaction. The equation was first suggested for description of plasma by Anatoly Vlasov in 1938^[1] (see also ^[2]) and later discussed by him in detail in a monograph.^[3]

[/ltr]

• 1Difficulties of the standard kinetic approach
  
• 2The Vlasov–Maxwell system of equations (gaussian units)
  
• 3The Vlasov–Poisson equation
  
• 4Moment equations
  
  
  
  • 4.1Continuity equation
    
  • 4.2Momentum equation
    
  
  
  
• 5The frozen-in approximation
  
• 6See also
  
• 7References
  

[size][ltr]

Difficulties of the standard kinetic approach[edit]

First, Vlasov argues that the standard kinetic approach based on the Boltzmann equation has difficulties when applied to a description of the plasma with long-range Coulomb interaction. He mentions the following problems arising when applying the kinetic theory based on pair collisions to plasma dynamics:
[/ltr][/size][list="margin-top: 0.3em; margin-right: 0px; margin-left: 3.2em; padding-right: 0px; padding-left: 0px; list-style-image: none;"]
[*]Theory of pair collisions disagrees with the discovery by Rayleigh, Irving Langmuir and Lewi Tonks of natural vibrations in electron plasma.
[*]Theory of pair collisions is formally not applicable to Coulomb interaction due to the divergence of the kinetic terms.
[*]Theory of pair collisions cannot explain experiments by Harrison Merrill and Harold Webb on anomalous electron scattering in gaseous plasma.^[4]
[/list]
[size][ltr]
Vlasov suggests that these difficulties originate from the long-range character of Coulomb interaction. He starts with the collisionless Boltzmann equation(sometimes called the Vlasov equation, anachronistically in this context), in generalized coordinates:

explicitly a PDE:

and adapted it to the case of a plasma, leading to the systems of equations shown below.^[5]

The Vlasov–Maxwell system of equations (gaussian units)[edit]

Instead of collision-based kinetic description for interaction of charged particles in plasma, Vlasov utilizes a self-consistent collective field created by the charged plasma particles. Such a description uses distribution functions  and  for electrons and (positive) plasma ions. The distribution function  for species α describes the number of particles of the species α having approximately the momentum  near the position  at time t. Instead of the Boltzmann equation, the following system of equations was proposed for description of charged components of plasma (electrons and positive ions):

Here e is the electron charge, c is the speed of light, m_i is the mass of the ion,  and  represent collective self-consistent electromagnetic field created in the point  at time moment t by all plasma particles. The essential difference of this system of equations from equations for particles in an external electromagnetic field is that the self-consistent electromagnetic field depends in a complex way on the distribution functions of electrons and ions  and .

The Vlasov–Poisson equation[edit]

The Vlasov–Poisson equations are an approximation of the Vlasov–Maxwell equations in the nonrelativistic zero-magnetic field limit:

and Poisson's equation for self-consistent electric field:

Here q_α is the particle's electric charge, m_α is the particle's mass,  is the self-consistent electric field,  the self-consistent electric potential and ρ is the electric charge density.
Vlasov–Poisson equations are used to describe various phenomena in plasma, in particular Landau damping and the distributions in a double layer plasma, where they are necessarily strongly non-Maxwellian, and therefore inaccessible to fluid models.

Moment equations[edit]

In fluid descriptions of plasmas (see plasma modeling and magnetohydrodynamics (MHD)) one does not consider the velocity distribution. This is achieved by replacing  with plasma moments such as number density n, flow velocity u and pressure p.^[6] They are named plasma moments because the n-th moment of  can be found by integrating  over velocity. These variables are only functions of position and time, which means that some information is lost. In multifluid theory, the different particle species are treated as different fluids with different pressures, densities and flow velocities. The equations governing the plasma moments are called the moment or fluid equations.
Below the two most used moment equations are presented (in SI units). Deriving the moment equations from the Vlasov equation requires no assumptions about the distribution function.

Continuity equation[edit]

The continuity equation describes how the density changes with time. It can be found by integration of the Vlasov equation over the entire velocity space.

After some calculations, one ends up with

The number density n, and the momentum density nu, are zeroth and first order moments:

Momentum equation[edit]

The rate of change of momentum of a particle is given by the Lorentz equation:

By using this equation and the Vlasov Equation, the momentum equation for each fluid becomes
,
where p is the pressure tensor. The material derivative is

The pressure tensor is defined as the particle mass times the covariance matrix of the velocity:

The frozen-in approximation[edit]

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As for ideal MHD, the plasma can be considered as tied to the magnetic field lines when certain conditions are fulfilled. One often says that the magnetic field lines are frozen into the plasma. The frozen-in conditions can be derived from Vlasov equation.
We introduce the scales T, L and V for time, distance and speed respectively. They represent magnitudes of the different parameters which give large changes in . By large we mean that

We then write

Vlasov equation can now be written

So far no approximations have been done. To be able to proceed we set , where  is the gyro frequency and R is the gyroradius. By dividing by ω_g, we get

If  and , the two first terms will be much less than  since  and  due to the definitions of T, L and Vabove. Since the last term is of the order of , we can neglect the two first terms and write

This equation can be decomposed into a field aligned and a perpendicular part:

The next step is to write , where

It will soon be clear why this is done. With this substitution, we get

If the parallel electric field is small,

This equation means that the distribution is gyrotropic.^[7] The mean velocity of a gyrotropic distribution is zero. Hence,  is identical with the mean velocity, u, and we have

To summarize, the gyro period and the gyro radius must be much smaller than the typical times and lengths which give large changes in the distribution function. The gyro radius is often estimated by replacing V with the thermal velocity or the Alfvén velocity. In the latter case R is often called the inertial length. The frozen-in conditions must be evaluated for each particle species separately. Because electrons have much smaller gyro period and gyro radius than ions, the frozen-in conditions will more often be satisfied.

See also[edit]

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• A. A. Vlasov, Many-Particle Theory and Its Application to Plasma, Gordon and Breach, 1961.
  
• List of plasma (physics) articles
  

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References[edit]

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[*]Jump up^ A. A. Vlasov (1938). "On Vibration Properties of Electron Gas". J. Exp. Theor. Phys. (in Russian) 8 (3): 291.
[*]Jump up^ A. A. Vlasov (1968). "The Vibrational Properties of an Electron Gas". Soviet Physics Uspekhi 10 (6): 721. Bibcode:1968SvPhU..10..721V.doi:10.1070/PU1968v010n06ABEH003709.
[*]Jump up^ A. A. Vlasov (1945). Theory of Vibrational Properties of an Electron Gas and Its Applications.
[*]Jump up^ H. J. Merrill & H. W. Webb (1939). "Electron Scattering and Plasma Oscillations". Physical Review 55 (12): 1191. Bibcode:1939PhRv...55.1191M.doi:10.1103/PhysRev.55.1191.
[*]Jump up^ "Vlasov equation?", M. Hénon, Astronomy and Astrophysics 114, #1 (October 1982), pp. 211-212, Bibcode: 1982A&A...114..211H
[*]Jump up^ W. Baumjohann and R. A. Treumann, Basic Space Plasma Physics, Imperial College Press, 1997
[*]Jump up^ P. C. Clemmow and J. Dougherty, Electrodynamics of Particles and Plasmas, Addison-Wesley series in advanced physics, Addison-Wesley Publishing Company, 1969
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Categories: 
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• Statistical mechanics
  
• Non-equilibrium thermodynamics
  
• Plasma physics
  
• Equations