Classical and Modern Galaxy Formation

Table of Contents

• 1 Bernard Jones (1976): "The Origin of Galaxies"
  • 1.1 Historical Overview
  • 1.2 Gravitational Instability Theory
  • 1.3 Cosmic Turbulence Theory
  • 1.4 Cosmic Angular Momentum
  • 1.5 Comparison with Observations
• 2 J. Richard Gott, III (1977): "Recent Theories of Galaxy Formation"
  • 2.1 Galaxy Observations
  • 2.2 Galaxy Formation from Small Density Perturbations
  • 2.3 Collapse Picture of Galaxy Formation
  • 2.4 Other Models
• 3 (Biased) Galaxy Formation in a CDM Universe
  • 3.1 Blumenthal et al. (1984): Galaxy Formation with CDM
  • 3.2 Dekel & Rees (1987): "Physical mechanisms for biased galaxy formation"
  • 3.3 Cole et al. (2000): "Hierarchical galaxy formation"
• 4 Cosmic Angular Momentum By Tidal Torques
  • 4.1 Efstathiou & Jones (1979): Numerical investigations
  • 4.2 Fall & Efstathiou (1980): Formation of discs
  • 4.3 Mo, Mao & White (1998): "The formation of galactic discs"
• 5 Formation of the First Objects - Analytics
  • 5.1 Eggen et al. (1962): Protogalactic collapse into the Milky Way
  • 5.2 Saslaw & Zipoy (1967): "Molecular Hydrogen in Pre-galactic Gas Clouds"
  • 5.3 Peebles & Dicke (1968): "Origin of the Globular Star Clusters"
  • 5.4 Yoneyama (1972): Fragmentation of a Contracting Hydrogen Cloud
  • 5.5 Rees & Ostriker (1977): Properties of Massive Gas Clouds
  • 5.6 White & Rees (1978): Galaxy formation within Heavy Halos
• 6 Formation of the First Objects - Modern Analytics & Simulations
  • 6.1 Loeb & Rasio (1994): Black Halo Formation
  • 6.2 Haiman, Thoul, & Loeb (1996): "Cosmological Formation of Low-mass Objects"
  • 6.3 Haiman, Rees, & Loeb (1996): H₂ Formation in Primordial Gas
  • 6.4 Tegmark et al. (1997): "How Small were the First Cosmological Objects?"
  • 6.5 Gnedin & Ostriker (1997): Reionization and Metal Enrichment
  • 6.6 Abel et al. (1998, 2000, 2002): Formation of the First Star
  • 6.7 Ricotti et al. (2002): The First Galaxies
• 7 Edmund Bertschinger (1998): "Simulations of Structure Formation in the Universe"
  • 7.1 Foundation for Modern Techniques
  • 7.2 Gravity Solvers
    • 7.2.1 Barnes-Hut Tree Algorithm
    • 7.2.2 Particle-Mesh Algorithm
    • 7.2.3 P³M and Adaptive P³M
    • 7.2.4 Multi-resolution Mesh Methods
  • 7.3 Hydrodynamics Solvers
    • 7.3.1 Smoothed Particle Hydrodynamics (SPH)
    • 7.3.2 Eulerian Grid Algorithms
    • 7.3.3 Adaptive Mesh Refinement (AMR)
  • 7.4 Additional Physics
  • 7.5 Applications

1 Bernard Jones (1976): "The Origin of Galaxies"

Reviews of Modern Physics, 48, 1

After the discovery of the cosmic microwave background, momentous advances were made in the theories of galaxy formation in the hot Big Bang scenario. Although cold dark matter and inflation were not incorporated into these theories, many of the ideas put forth are still highly relevant in modern theories of galaxy formation.

Jones selects an excellent quote from Lemaitre (1950).

"The purpose of any Cosmogonic Theory is to seek out ideally simple conditions which could have initiated the work and from which, by the play of recognized forces, that world, in all its complexity, may have resulted."

1.1 Historical Overview

Dating back to ancient times, philosophers have questioned the origins of structures in the universe. Each advance in theory has been in some way influenced by previous works.

Lord Rosse discovered spiral structures in nebulae over the period 1842-1850, which sparked further interest in the origin of our solar system and galaxy. Alexander (1852) interpreted these observations as evidence in support of Laplace's "nebulae hypothesis" that states the Sun formed from a gravitational collapse of a rotating, hot diffuse cloud.

Some of the earliest photographs of galaxies (Roberts 1899, Keeler 1900) stimulated a fury of papers shortly after their publication. Interestingly, all of these theories were motivated in some part by Laplace's hypothesis, which demonstrates the impact of his work in early theories of galaxy formation.

Hubble (1925a,b,c, 1926, 1929) established these nebulae were extragalactic and ones farther away were receding at a greater velocity than closer galaxies. Before Hubble's discovery, there were great debates about the distance scale to galactic nebulae. Curtis (1919) was the main leader in theories of their extragalactic nature, whereas Shapley (1919) thought otherwise.

Now theories of these nebulae had to be rescaled to kiloparsecs and the formation of many stars instead of stellar birthplaces that results in a handful of stars and associated planetary systems. Jeans (1918, 1928) suggested that the origin of such systems came from gravitational collapse from a uniform universe. Although Jeans had not considered an expanding universe in his later works, Leimaitre (1945) generalized the collapse problem to account of this and thought of the universe as an enormous primeval gas cloud. Instabilities within this universal nebula led to the condensation of individual gas clouds and galaxy formation. Additionally, Gamow and Teller (1939) applied Jeans' theory to the same problem and gave a very comprehensible description of the gravitational instabilities of an expanding universe. These works were the foundation of galaxy formation theories considering gravitational instabilities.

The theory of cosmic turbulence was a parallel effort to explain galaxy formation, in which overdensities formed from an initial turbulent medium. von Weizsacker in the 1940's was the main driving force to this theory when he applied it to the formation of the solar system (1944), internal dynamics of galaxies (Heisenberg & von Weizsacker 1948), and galaxy formation (1948). Around this time, Gamow switched his standing and started to support the turbulence theory as seen in a paper (Gamow 1952) that supports von Weizsacker's view (1951). There were some concerns expressed by Gamow (1951, 1952) and Bonnor (1956) about the necessity of ab initio turbulence. Afterwards, there was a mysterious break in publications on the subject from the early 1950's until its resurgence after the hot Big Bang theory was established in 1965, which was postulated by Gamow (1953) a decade earlier.

The first paper to appear in the cosmogony brought forth by the discovery of the CMB was Peebles (1965) describing the dampening of density perturbations by Thomson drag forces when radiation and matter were coupled. The growth of adiabatic perturbations in an expanding universe was first postulated by Lifschitz (1946), whose theories were expanded to include the radiative diffusion of acoustic modes by numerous groups.

1.2 Gravitational Instability Theory

Dating back to Jeans (1902), the gravitational collapse of a gas cloud has been of much interest over the last century. Jeans expanded on his theory in his 1918 and 1928 works. These provided the basis for its cosmological application, where density perturbations grow in an expanding universe. This has been the topic of much research.

There is exact solution to this cosmological problem if there are no special symmetries. Most models of galaxy formation, therefore, use linear perturbation theory to evolve systems from an almost uniform state to study the growth of structure.

The growth of structure can be understood by three scales. Firstly, the horizon length scale (R = ct) determines the amount of matter that is causally connected and is (1/6) π ρ_m (ct)³. Secondly, the Jeans length scale controls the mass of an object whose gravitational attraction overcomes pressure forces. Lastly at scales much less than the Jeans length, perturbations are subjected to dissipative forces, such as Thomson drag forces and viscosity, and act as acoustic modes.

Before recombination and at sufficiently early times, an adiabatic perturbation will be larger than the horizon scale and will grow accordingly. Here it is smaller than the Jeans scale and damped. It starts to oscillate and diminish. At recombination, the temperature drops to 3000 K in approximately 20% of the Hubble time and may become Jeans unstable, in which it grows in amplitude.

There are two mass scales that coincide with galactic and globular cluster masses. Scales less than ~3 x 10¹² M_⊙ are severely attenuated. Protogalactic and galaxy cluster gas clouds were thought to originate from the clouds that above this mass scale. After recombination, the Jeans mass of an isothermal perturbation is roughly the mass of a globular cluster, which Peebles & Dicke (1968) pointed out. After the initial starburst, most of the gas would be expelled, and the stars would become unbound, creating the halos in present day galaxies.

Interestingly in 1965, Peebles discussed the beginnings of hierarchical structure formation, where smaller building blocks merge together to form larger systems. This depends on the power spectrum of density fluctuations and could (at the time) very well explain the origin of galaxies and galaxy clusters.

Note: See the appendix for a review on hydrodynamics basics.

1.3 Cosmic Turbulence Theory

After the discovery of the CMB, Ozernoi and co-workers reintroduced the cosmic turbulence theory that was previously untouched since the early 1950's. This point of view proposes that structure formation arises from a primordial turbulent velocity field. At recombination, the sound speed dramatically reduces to a few km/s, and the previously subsonic eddies "freeze-out" and can produce large density perturbations.

Here the largest eddy length scale determines the dynamics of all smaller scales because of the properties of turbulence, i.e. Kolmogorov. The mass scale of the largest eddy is approximately 3 x 10¹⁵ M_⊙, comparable with galaxy clusters. Masses larger than this never become turbulent. One consequence of cosmic turbulence theories is the simultaneous condensation of a given mass scale, which happens around redshift 200.

This theory fell out of favor in the 1970's as more evidence supported the gravitational instability theory of galaxy formation. In hindsight, the highly successful theory of hierarchical structure ultimately put cosmic turbulence in its deathbed.

1.4 Cosmic Angular Momentum

A previously outstanding problem in structure formation was the acquisition of angular momentum resulting in the rotating galaxies we observe today. How did it arise from an expanding, almost uniform, universe so the conservation of angular momentum is not violated?

Kelvin's circulation theorem states that an irrotational system must remain irrotational in the absence of dissipative forces. During the collapse of protogalactic gas clouds, there are certainly dissipative forces and thus Kelvin's theorem is inapplicable. This is also true in relaxing stellar systems (violent relaxation) where angular momentum is conserved.

However there must be an origin of pre-existing angular momentum before galaxies formed. One might conclude that there was an original vorticity field in the universe, but Hoyle (1951) and Peebles (1969) independently suggested that tidal torques from neighboring conglomerations induced galactic spins. This does not violate angular momentum conservation as the two systems gain orbital energy in response. Both analytic and numerical work by Peebles (1969, 1971), Silk (1974), Jones (1975) showed that there is a deficiency of a factor of 5 in the predicted amount of angular momentum in the Galaxy when compared with the observation of Innanen (1966).

1.5 Comparison with Observations

Although these theories examine very early times when galaxies have not yet formed and concentrate on the formation of protogalactic clouds and cosmic structure, the ultimate goal of galaxy formation theories is to explain the origin and properties of all of the galaxies seen in the observable universe. It is thus necessary to connect and test theories with observations. There are five broad characteristics of galaxies that can tested against.

1. Masses of galaxies: galactic internal dynamics, interacting galaxies, dynamics of groups, aggregation of galaxy clusters, galaxy luminosity function
2. The angular momenta and binding energies of galaxies
3. The origin of different morphologies of galaxies
4. The clustering of galaxies
5. Explanation of young galaxies if not all galaxies were formed early in the universe, as our own Galaxy.

2 J. Richard Gott, III (1977): "Recent Theories of Galaxy Formation"

Annual Reviews of Astronomy and Astrophysics, 15, 235

Gott reviews the development of collapse models of galaxy formation. In these models, galaxies are created by the collapse of density fluctuations that are quantified by a power spectrum. He first discusses observations of galaxies and then describes theories that aim to explain them. Elliptical galaxies form by the collapse of the largest density fluctuations, which then dissipate its energy through violent relaxation. Other theories of hierarchical structure formation and the associated galaxy mergers are also explored.

In the conclusion, he states the new ideas that emerged in the decade preceding 1977: violent relaxation, dynamical friction, "heavy halos", cosmological infall, and angular momentum by tidal interactions.

2.1 Galaxy Observations

Hubble (1926) first recognized that morphologies of galaxies are split into elliptical, spiral, S0, and irregular types. He also realized that the surface brightness of elliptical galaxies are approximately well by a decreasing function, I(r) = I₀ / (1 + r/a)². Another good fit was provided by deVaucouleurs (1959) by a exponential law that is proportional to exp(r^1/4). King (1966) developed a dynamical model that considered a Maxwellian velocity distribution of the stars and a tidal cutoff radius. It provides an even better fit to observed surface brightnesses. Faber & Jackson (1976) connected all of the previous insights to find that the luminosity of elliptical galaxies are proportional to the velocity dispersion to the fourth power (now dubbed as the Faber-Jackson relation).

Spiral and S0 galaxies exhibit an exponentially decaying surface brightness in the disk with respect to radius, i.e. I(r) = I₀ exp(-r/r_d). These galaxies also have three stellar components: halo, spheroidal, and disk. Low-luminosity irregular galaxies complete the observed morphology types of galaxies.

Ostriker & Peebles (1973) found that a cold, rotating, self-gravitating disk is subject to violent bar-like instabilities. It forms a strong central concentration that is incompatible with observations; however, a dark halo component can stabilize the disk against this trait. Schechter (1976) showed that luminosity function of elliptical galaxies in rich cluster (which was later applied to all galaxies) is an exponential with a cutoff at high luminosities above L_* = 3.4 x 10¹⁰ M_⊙.

2.2 Galaxy Formation from Small Density Perturbations

The standard view of galaxy formation is that small density perturbations grow with time and eventually form galaxies and galaxy clusters. When fluctuations on these mass scales become causally connected, matter is coupled with radiation through Thomson drag forces. The Jeans mass before decoupling is roughly the mass contained within the horizon. However after recombination, the Jeans mass sharply drops to ~10⁵ M_⊙ because of the cooler temperatures. Density fluctuations must be on the order of 0.1% to form present-day galaxies. These can be calculated using linear perturbation theory, and its evolution becomes non-linear once the overdensity (δρ/ρ) is approximately unity.

Primordial fluctuations are favored to be adiabatic or isothermal. Adiabatic fluctuations retain a particular entropy and are like sound waves with equal photon and baryon fluctuations. They are damped below a mass scale ~10¹² M_⊙ because of Thomson drag forces, i.e. photon viscosity. Isothermal fluctuations are density fluctuations immersed in a photon bath and do not conserve entropy. They cannot grow in amplitude before recombination because they are coupled with radiation. After recombination, an overdensity of one corresponds to a mass 10⁵ M_⊙ in a Einstein-deSitter universe. Both types of fluctuations result in a smaller mass scales becoming non-linear earlier (δρ/ρ ∝ M^-1/3), but as stated before, mass scales below 10¹² M_⊙ are suppressed in adiabatic perturbations. General models can contain a superposition of both types.

The physical picture painted by these two types of fluctuations differ greater in nature. Adiabatic ones create a scenario of top-down structure formation, in which objects that are large as galaxy clusters form first then fragment to form galaxies through Jeans instabilities in overdensities in shocks (Doroshkevich et al. 1974). Isothermal fluctuations produces a bottom-up scenario of structure formation, where smaller objects form first and combine to form larger entities (Peebles 1974). An analysis of the covariance function of galaxies provides evidence for the latter scenario because there is no preferred mass scale. Conversely, one would expect a strong feature at ~10¹⁴ M_⊙ if structure had formed top-down. Furthermore, the three-point correlation function indicates that galaxies are hierarchically clustered. Press & Schechter (1974) formulated a framework that can calculate the abundances of matter condensations, similar to the later developed Schechter galaxy luminosity function. This formalism only depends on the initial power spectrum of density perturbations.

2.3 Collapse Picture of Galaxy Formation

In a Ω₀ = 1 cosmology, any overdensity is bound. It will expand with the Hubble flow, and it then decouples from the Hubble flow and starts to gravitationally collapse. The "turn-around" time is 0.5 T_c, where T_c = (π / 2^1/2) (R₀³ / GM₀)^1/2. Before the development of violent relaxation (Lynden-Bell 1967), the relaxation time of a stellar system was greater than a Hubble time. However violent relaxation allows the previously ordered system to virialize and transform into a system with a Maxwellian velocity distribution that successfully reproduces the isophotes of elliptical galaxies and isothermal cores of King models.

Eggen et al. (1962) and Sandage et al. (1970) used these ideas of protogalactic collapse models to explain the structures in the Milky Way. They attributed high eccentricities of the metal-poor halo stars to violent relaxation, which suggested that they formed before the collapse of the galactic cloud. After virialization, the remaining gas collapsed and formed a gaseous disk where stars formed subsequently.

2.4 Other Models

Toomre & Toomre (1972) showed that numerical simulations of colliding galaxies can reproduce observations of interacting galaxies, which constitutes 10 of the 4000 NGC objects. They also propose that there had been a total of ~500 collisions in the NGC sample throughout the history of the universe, which could explain the ~400 ellipticals in the catalog. Two difficulties seen at the time was the reproduction of metallicity gradients (Strom 1977) and color versus magnitude relations (Faber 1973) in ellipticals. However Gott (also see White & Rees 1978) foresaw a solution by combining the Toomre picture of merging galaxies, hierarchical structure formation of Peebles & Dicke (1968), and initial isothermal perturbations to explain the observed properties of galaxies.

3 (Biased) Galaxy Formation in a CDM Universe

In the early and mid 1980's, there was growing evidence that universe was dominated by cold dark matter. These following articles highlight the upcoming of CDM cosmology as the favored cosmological model, and how galaxy formation proceeds in this cosmogony.

3.1 Blumenthal et al. (1984): Galaxy Formation with CDM

"Formation of galaxies and large-scale structure with cold dark matter", Nature, 311, 517

Theories of hot and warm dark matter are refuted on the theoretical and observational grounds. HDM models predict that galaxy clusters are the first to form at z ~ 2 with smaller objects fragmenting from them. However, this conflicts with the ages of globular clusters, stellar ages in galaxies, and dwarf galaxies. WDM models predict that galaxy sized clouds are the first to collapse, but this still cannot explain the abundances of dwarf galaxies.

From simulations, Peebles (1982) showed that CDM models with hierarchical clustering can account for all of the morphologies and masses of galaxies and clusters. UV heating of the IGM from stellar sources inhibit gravatational collapse and star formation in dwarf galaxies, which accounts for their high mass-to-light ratios. The growth of structure in a CDM universe also matches the differences of density and temperature in different Hubble types (E, S0, Sa, Sb, Sc, Irr). They also discuss the formation of galaxy clusters from progenitor galaxies, and how the central cD galaxy probably originated from a high-σ density fluctuation. On an even larger scale, hierarchical assembly explains topology of the largest structures, the voids and superclusters. Furthermore, UV heating from early star formation suppress star and thus galaxy formation is suppressed in these voids.

They list the following successful predictions made by CDM cosmologies.

1. Observed mass range of galaxies
2. Dissipationless nature of galaxy collapse
3. Observed Faber-Jackson and Tully-Fisher relations
4. Galaxy-environment correlations
5. Differences in angular momentum of elliptical and spiral galaxies.
6. Large-scale clustering

3.2 Dekel & Rees (1987): "Physical mechanisms for biased galaxy formation"

Nature, 326, 455

Do galaxies and baryons accurately trace dark matter? There is bias in galaxy formation in several ways. There is the bias in galaxies in the different σ peaks. Higher σ peaks collapse earlier and form in a different environment than the lower σ peaks of the same mass. The cooling and dynamical timescales scale as ρ^-1 (for bremsstrahlung and recombination) and ρ^-1/2, respectively. Additionally, high redshift objects are subject to Compton cooling, which is proportional to (1+z)⁴. Negative feedback, especially in the IGM, affect earlier forming galaxies less since the UV background and nearby structures have had less time to build up. Relevant to the formation of Pop III stars, the highest (e.g. ~3.5σ) peaks are highly clustered, with twice as many 2.5σ peaks as a in a 1σ peak on the same mass scale. This creates biasing in most feedback process in these first structures, which could lead to the observed differences of the galaxy-environment correlation. From these biasing effects, it is evident that galaxies do not trace the DM mass distributions on a whole.

3.3 Cole et al. (2000): "Hierarchical galaxy formation"

MNRAS, 319, 168

Semi-analytic models of galaxy formation have yielded success in predicting (1) the local field galaxy luminosity function, (2) the slope and scatter of the Tully-Fisher relation, (3) the counts and redshift distributions of faint galaxies, (4) the formation of most stars at z < 1.5, (5) a sharp decline in elliptical galaxies at high redshift, and (6) strong clustering of Lyman break galaxies at z = 3. This method is an alternative to studying galaxy formation with explicit numerical simulations, where baryonic physics and star formation are calculated with simplistic models set in a dark matter merger tree, produced by extended Press-Schechter formalism (i.e. Monte-Carlo). Furthermore the properties of dark matter halos, e.g. density profiles and spins, are accurately modeled from results of numerical simulations. Some of their simplications are however justified as some processes, in particular star formation and feedback, are not well understood. Furthermore, most of the free parameters are tuned to match observations, which allows the application of this method to other, not so well-known populations of galaxies. These models have been broadly successful in reproducing a large range of galaxy properties.

4 Cosmic Angular Momentum By Tidal Torques

If the universe was initially isotropic and galaxies gravitationally collapsed, how did they acquire angular momentum to produce the universal rotation in spiral galaxies? Hoyle (1951) and Peebles (1969) independently conjectured that galactic spins originated from tidal torques from neighboring structures. The early works about this were discussed in Jones (1976; sec. 1 of this work).

4.1 Efstathiou & Jones (1979): Numerical investigations

"The rotation of galaxies: numerical investigations of the tidal torque theory", MNRAS, 186, 133

Peebles (1969) predicted that the average spin parameter λ ≈ 0.08, and here it is verified through numerical simulations of a cosmologically expanding system. During the clustering of systems, most of the angular momentum is acquired during early times. Linear perturbation theory predicts that angular momentum is ∝ a(t)^5/2 ∝ t^5/3. They conclude that galactic discs must have dissipated a large amount of energy during the collapse from a large initial radius. Rotational is dynamically unimportant in the dissipationless collapses, i.e. ellipticals, and tidal torques cannot account for the flattening of such systems.

4.2 Fall & Efstathiou (1980): Formation of discs

"Formation and rotation of disc galaxies with haloes", MNRAS, 193, 189

Motivated by the results of Eggen et al. (1962) and Peebles (1969), Efstathiou & Jones investigate disc formation in the collapse of a slowly rotating cloud within a heavy halo. The angular momentum of a halo can be represented by the spin parameter λ. Their disc models agree remarkably well with observations whose density exponentially decreases. The collapse mass of a halo and the need for a dark halo agrees with White & Rees (1978) in order to explain the origins of galactic rotation by tidal torques. They also find that parts of galactic discs are unstable to fragmentation at 10⁴ K.

4.3 Mo, Mao & White (1998): "The formation of galactic discs"

MNRAS, 295, 319

Properties of self-gravitating discs embedded in a dark matter density profile (Navarro, Frenk, & White 1996) are computed. When comparing the populations of discs to the Tully-Fisher relation, they only include stable discs (see Christodoulou et al. 1995) that results in an excellent match when they tune their model to the zero-point of the relation. They also compute the scale lengths and formation times of such discs. Many modern semi-analytic models use their formalism for disc formation in the early universe.

5 Formation of the First Objects - Analytics

Here I review papers that consider the condensation of the first bound objects in the universe. These works were motivated by the gravitational instability picture of galaxy formation and the hot Big Bang hypothesis (Gamov 1953), which was confirmed by the CMB observation in 1965 and expanded upon by Dicke et al. (1965).

5.1 Eggen et al. (1962): Protogalactic collapse into the Milky Way

"Evidence from the Motions of Old Stars that the Galaxy Collapsed" Astrophysical Journal, 136, 748

The velocities of 221 Galactic dwarf stars were analyzed, along with their colors, to determine orbital parameters, namely eccentricities and angular momenta. It was already well-establish that there are two populations of stars that were "low" and "high" velocity stars (Oort 1926, Williams 1948, Schwarzschild 1952). In a self-gravitating cloud, the angular momentum of each star is conserved. This allows the determination of the birthplace of present-day stars, in the sense of "rings" of equal angular momentum in a solid-body rotator. They find that older stars have highly eccentric orbits that form during the initial collapse of the protogalactic cloud. They conclude that after this initial collapse, rotational support leads to a gaseous disk, where stars form. During the collapse, collisions between gas clouds relieved energy, and they settled into circular orbits, in which later generations of stars formed, explaining the disparity between "low" and "high" velocity stars.

5.2 Saslaw & Zipoy (1967): "Molecular Hydrogen in Pre-galactic Gas Clouds"

Nature, 216, 976

They examine the important cooling processes, in particular H₂, in the initial collapse of protogalaxies. Before this paper, H₂ had not been considered an important species in early galaxy formation. They show that above a number density of 10⁴ cm^-3, H₂ cooling starts to dominate, which decouples the gas dynamics from an adiabatic evolutionary track. They show that high density regions can cool to 300 K and continue to collapse until they become optically thick, H₂ dissociates, and hydrogen atoms ionize.

5.3 Peebles & Dicke (1968): "Origin of the Globular Star Clusters"

Astrophysical Journal, 154, 891

Globular clusters exhibit similar properties although they exist in varying environments, e.g. orbiting dwarf galaxies, elliptical galaxies, and isolated from any galaxy. Peebles & Dicke suggest that these systems are the first bound objects with masses ~5 x 10⁵ M_⊙ in the universe. They show that the radiative feedback from at least 30 B0 stars could blow out all of the gas in the object, stifling star formation until the cloud can contract again after the stars had died. They calculate the properties of these objects from the initial density perturbation power spectrum. Then they continue their analysis by following the initial contraction of the cloud, motivated by H₂ formation in the gas-phase by electron capture (McDowell 1961) because this is the main coolant at low temperatures of ~1000 K and low metallicity. They find that H₂ cooling is indeed efficient enough to drive a free-fall collapse, in which only small fraction of the total gas mass will form central stars.

5.4 Yoneyama (1972): Fragmentation of a Contracting Hydrogen Cloud

"On the Fragmentation of a Contracting Hydrogen Cloud in an Expanding Universe", PASP, 24, 87

Previous works had shown that molecular hydrogen was an important part of early galaxy formation, but Yoneyama followed the evolutionary track of the cloud in its cooling stage until the central parts fragmented. They confirmed that H₂ formation was possible in these clouds through inspection of relevant timescales (Hubble, free-fall, recombination, H₂ formation, H₂ dissociation, and cooling). H₂ formation becomes important at number densities of 10⁴ cm^-3 (cf. Saslaw & Zipoy 1967), and they it gradually heats as it condenses. Afterwards the molecular hydrogen dissociates and the central cloud evolves adiabatically. During appropriate values for one of the first bound objects (M ~ 10⁶ M_⊙), they determine that the central clump will fragment into subclumps of mass 60 M_⊙. Besides the prediction of "superstars" of mass 10⁵ M_⊙ by Doroshkevich et al. (1967), I think this is the first prediction of massive primordial stars.

5.5 Rees & Ostriker (1977): Properties of Massive Gas Clouds

"Cooling, dynamics and fragmentation of massive gas clouds: clues to the masses and radii of galaxies and clusters", MNRAS, 179, 541

The properties of massive gas clouds before they condensed into galaxies and clusters are investigated in this paper. Baryons are shock-heated to the virial temperature as they fall into the halo. Afterwards, the dynamics are regulated by gravity and radiative cooling. A comparison of the Jeans mass and cooling times shows that there are three categories of bound objects in the universe. First, objects that cannot cool within a Hubble time cannot collapse and are associated with galaxy clusters (M > 10¹⁴ M_⊙), whose intracluster gas is virially heated to high temperatures where cooling is inefficient. Second, objects (M ~ 10¹² M_⊙) that can cool within a Hubble time but the cooling time is greater than dynamical time is in quasi-statically pressure support. It will radiate its internal energy away and condense shortly afterwards. The third case involves objects with masses between 10¹⁰ and 10¹² M_⊙ that will efficiently cool (i.e. t_cool < t_dyn) and collapse on a free-fall timescale. Here they assume that objects with virial temperatures above 10⁴ K are the only ones to cool. They find that the most massive galaxies form at relatively recent times (z < 10). They comment that H₂ is important in star formation in the early universe. They do not investigate it in this paper because their main focus is pre-galactic mass scales.

5.6 White & Rees (1978): Galaxy formation within Heavy Halos

"Core condensation in heavy halos: a two-stage theory for galaxy formation and clustering", MNRAS, 183, 341

For the first time, galaxy formation is set in the context of hierarchical structure formation of the "missing matter", and the baryons follow the dark matter dynamics, condense, and form galaxies. Matter organizes in a self-similar nature that grows with the scale factor. As objects merge into a larger entity, the substructure of the progenitors are erased during virialization on a timescale less than t_dyn. For relevant galactic mass scales, gas temperatures will be much larger than 10⁴ K and supports the efficient cooling and condensation into galaxies. These objects condense on a free-fall time since they require t_cool < t_dyn < t_H.

Galaxies forming by hierarchical assembly will likely have an imprint at present-day. Low-mass galaxies form first before Milky Way type galaxies have reached the turn-around radius. However during the assembly, the luminous cores of these low-mass galaxies may survive to the present-day and exist in identifiable stellar systems. Based on the same arguments as Rees & Ostriker (1977), they conclude that the massive mass of galaxies is 10¹³ M_⊙.

6 Formation of the First Objects - Modern Analytics & Simulations

6.1 Loeb & Rasio (1994): Black Halo Formation

"Collapse of Primordial Gas Clouds and the Formation of Quasar Black Holes", Astrophysical Journal, 432, 52

The highest redshift quasars are massive (10⁶ - 10¹⁰ M_⊙) and must assemble in the first billion years of the universe. How did they become so massive so quickly? What were their seed black holes? Loeb & Rasio studies the collapse of an isolated, rotating, high-redshift density perturbation with SPH simulations. The infalling gas must (1) overcome an angular momentum barrier, (2) compete with star formation, (3) form a single black hole instead of a cluster, and (4) be fueled adequately after its birth. They find the material collapses into a central mass resembling a galactic bulge with a thin gaseous disk inside. For the disk to be stable, they conclude that a black hole of mass >10⁶ M_⊙ must reside in the center.

6.2 Haiman, Thoul, & Loeb (1996): "Cosmological Formation of Low-mass Objects"

Astrophysical Journal, 464, 523

Here one-dimensional collapse models of the first objects with a non-equilbrium chemistry solver (maybe for the first time in a cosmological simulation) are computed. They find that the first halos to collapse have a mass of 10⁵ M_⊙. They explore the collapse's dependence on the halo mass and initial overdensity of the system. They conclude that H₂ cooling is not possible before the gas becomes virialized in halos with virial tempeatures above 100 K.

6.3 Haiman, Rees, & Loeb (1996): H₂ Formation in Primordial Gas

"H₂ Cooling of Primordial Gas Triggered by UV Irradiation", Astrophysical Journal, 467, 522

The impact of a irradiating UV background on H₂ formation is considered in the high-redshift universe. This has implications on the formation and condensation of the first bound objects. They find that H₂ formation rates are significantly enhanced (t_cool < t_dyn) with a UV background that creates excess electrons that provides a catalyst for H₂ cooling in densities > 1 cm^-3. They suggest that this effect could lead to the increased star formation rates seen at z = 2-4.

6.4 Tegmark et al. (1997): "How Small were the First Cosmological Objects?"

Astrophysical Journal, 474, 1

There exists enough residual electrons after recombination so that H₂ cooling is important in virialized objects of mass 10⁵ M_⊙. Here they follow the evolution of the density, temperature, and chemical species of various virializing objects, starting from initial density perturbations at recombination. High-σ peaks of this mass collapse at z ~ 30 and have an H₂ fraction of 10^-3 and number density of 10² cm^-3 before rapidly collapsing. If the halo cannot cool efficiently (i.e. t_cool > t_H), it will remain in pressure support for a Hubble time.

6.5 Gnedin & Ostriker (1997): Reionization and Metal Enrichment

"Reionization of the Universe and the Early Production of Metals", Astrophysical Journal, 486, 581

The first stars and galaxies provide the first ionizing radiation and metals to the universe that eventually forms all of the present-day structure. They raise some important questions about this topic.

1. When should the universe become transparent blueward of Lyα? How closely do reheating and reionization follow each other?
2. How are metal abundances spatially characterized?
3. How clumpy is the gas before reionization?
4. Will large-area radio telescopes be able to detect 21cm emission and absorption from reionization?
5. How important were particular cooling (atomic, molecular, dust) and heating (virialization, UV heating, supernovae) processes? Is self-shielding important?
6. What were the properties of high-redshift stellar groups? Are they observable?

To answer these questions, they ran a set of numerical moving-mesh cosmological hydrodynamics simulations that focus on the formation of the first galaxies. They include star formation, a radiation background, metal enrichment, and a local approximation to shielding. The IGM is slowly reheated to 10⁴ K, but reionization occurs very suddenly, almost like a phase transition. The IGM is also enriched to 0.005 Z_⊙ by redshift 4 with a large dispersion, whereas high density regions are more uniform with 1/30 Z_⊙.

6.6 Abel et al. (1998, 2000, 2002): Formation of the First Star

"First Structure Formation. I. Primordial Star-forming Regions in Hierarchical Models", ApJ (1998), 508, 518

"The Formation and Fragmentation of Primordial Molecular Clouds", ApJ (2000), 540, 39

"The Formation of the First Star in the Universe", Science (2002), 295, 93

In a series of papers, Abel et al. concentrated on the collapse of one of the first objects in the universe. They used cosmological AMR (at first a static AMR code, HERCULES) hydrodynamics simulations to follow the collapse to eventually dynamic ranges of 10¹² in length scale. First they verified the findings of previous analytical and one-dimensional calculations in three-dimensions that H₂ formation indeed plays an important role in the first objects. A cool, dense core forms in the center of dark matter halos with masses ~10⁶ M_⊙. To follow the collapse to even higher densities, three-body H₂ formation becomes important at 10¹⁰ cm^-3. They end the final calculation when the core becomes optically thick to cooling radiation at a density of 10¹² cm^-3. There is no fragmentation during the free-fall collapse, where a single 1 M_⊙ protostar forms from a fully molecular cloud. They predict that stellar radiation cannot halt the rapid accretion, and the final stellar mass will be large, 30-300 M_⊙.

6.7 Ricotti et al. (2002): The First Galaxies

"The Fate of the First Galaxies. II. Effects of Radiative Feedback", Astrophysical Journal, 575, 49

This paper concerns the effects of radiative feedback during the formation of the first galaxies. They address this with numerical simulations that expand on the ones in Gnedin & Ostriker (1997) with the large improvement of detailed radiative feedback from stellar sources. The ~75 Lyman-Werner bands (H₂ dissociating radiation) are computed in detail, along with radiation transport using OTVET (Optically Thin Variable Eddington Tensor; Gnedin & Abel 2001). They find that star formation is regulated in low-mass objects and is intrinsically "bursting". Galaxy formation can also be triggered (positive feedback) by nearby galaxies, creating additional H₂ in relic HII regions. Negative feedback occurs through Lyman-Werner radiation in low-mass objects, which also dominate metal production in the high-redshift universe.

7 Edmund Bertschinger (1998): "Simulations of Structure Formation in the Universe"

Annual Reviews of Astronomy and Astrophysics, 15, 235

Numerical cosmology simulations have benefited the community and matured over the last decade. Here Bertschinger reviews the modern development of cosmological numerical techniques, including their strengths, weaknesses and applications. They have improved our understanding of quasar absorption lines, galaxy clusters, and cosmological models.

7.1 Foundation for Modern Techniques

The first gravitational N-body simulation was done by Holmberg (1941) using the flux from light bulbs to emulate the gravitational field. After the advent of digital computers, the first N-body simulations that calculated the dynamics of up to 100 particles were performed in the early 1960's (von Hoerner 1960, 1963, Aarseth 1963). The pioneering work of Larson (1969) involved the first hydrodynamical calculations in an astrophysical context. Press and Schechter (1974) conducted the first true cosmological simulations to confirm their framework of predicting the abundance of collapsed objects. Cosmological simulations were rare in the 1970's when collapse problems ruled the scene. However, several advances in cosmology allowed for tremendous growth in activity in the 1980's.

1. Plausible dark matter models were developed (Cowsik & McClelland 1972, Lee & Weinberg 1977, Bond et al. 1980), e.g. cold dark matter, hot dark matter (massive neutrinos).
2. Cosmic inflation (Guth 1981) provided an explanation for the generation of primordial density fluctuations and a scale-invariant Harrison-Zel'dovich spectrum (Harrison 1970, Zel'dovich 1972).
3. The evolution of fluctuations from the early universe through recombination was established by numerical calculations (Peebles & Yu 1970).
4. Theories of Gaussian random fields were formulated and applied to the study of density fluctuations in the early universe. Then linear perturbation theory could evolve these conditions to a sufficiently low redshift before the structure evolves non-linearly.

CDM started to become the favored cosmological model in the 1980's (Peebles 1982, Blumenthal et al. 1984, Davis et al. 1985), and cosmological simulations could readily test this cosmogony and compare it with observations and expectations. In the following years, baryonic physics was added to the calculations, providing access to even richer astrophysical problems.

7.2 Gravity Solvers

Particle trajectories are calculated by solving Newton's laws in comoving coordinates (Peebles 1980; see the review for them). The time integration is usually calculated with a second-order accurate leapfrog method (Efstathiou et al. 1985), which is more than adequate for cosmological applications. In practice, it is beneficial to use the time variable s = ∫ a^-2 dt instead of proper time because the equations of motion simplify to d²x/ds² = ag, which allows the use of a sympletic integrator (phase space volume preserving) (Quinn et al. 1997). The calculation of gravitational forces by direct summation of all particle pairs is prohibitive for cosmological simulations. Ingenious methods have thus been formulated to overcome this dilemma, which I describe below.

7.2.1 Barnes-Hut Tree Algorithm

Appel (1985), Barnes & Hut (1986)

Recursively divides the volume into cells that are smaller than some solid angle (e.g. size / distance < 1). These cells dramatically speed up the calculation by using a low-order multipole expansion instead of direct summation. Its speed is O(N log N). This algorithm is fully spatially adaptive, but its shortcoming is the large memory footprint of ~25N words (Hernquist 1987). It is publicly available and has been parallelized by many groups.

7.2.2 Particle-Mesh Algorithm

2D: Doroshkevich et al. (1980), Melott (1983), Bouchet et al. (1985)

3D: Centrella & Melott (1983), Klypin & Shandarin (1983), Miller (1983), White et al. (1983), etc.

The density field is mapped (some techniques include interpolation and anti-aliasing) to a Cartesian grid so the Poisson equation can be solved with an Fast Fourier Transforms (FFT), which only requires O(N log N) operations. After the gravitational field has been calculated, the forces are interpolated back to the particles, which are then integrated in time. The advantage is speed and light memory consumption, but the force resolution is poor for particle separations under several grid cells.

7.2.3 P³M and Adaptive P³M

Plasma physics: Hockney et al. (1974)

Cosmology: Efstathiou & Eastwood (1981), Davis et al. (1985)

This method combines the advantages of the PM and direct summation methods (and in most cases, the tree algorithm). At long distances, gravitational forces are calculated with the PM method, but nearby the forces are calculated by direct summation. The particle-particle limits the speed in strong overdensities, but this can be overcome by using the tree algorithm at short distances (Xu 1995). A more elegant solution is accomplished by adaptively placed subgrids (adaptive P³M) with isolated boundary conditions, where the PM method is still used (Couchman 1991). In the finest subgrids, pair summation is only calculated when the separation is less than 2-3 cell widths, dramatically increasing the speed of the calculation.

7.2.4 Multi-resolution Mesh Methods

1. Adaptive Mesh Refinement -- Jessop et al. (1994), Kravtsov et al. (1997)
2. Particle Splitting -- Anninos et al. (1994), Dutta (1995)
3. Moving Mesh -- Gnedin (1995), Pen (1995)

7.3 Hydrodynamics Solvers

The first cosmological gas dynamics calculations were one-dimensional, investigating the collapse of sheet-like structures (Zel'dovich pancakes) from initial density perturbations (Doroshkevich et al. 1978). The difficulty of baryonic physics arises from the non-linearity of strong shocks. There are three main techniques to solve the cosmological hydrodynamics equations: smoothed particle hydrodynamics, Eulerian grid, and adaptive mesh refinement.

7.3.1 Smoothed Particle Hydrodynamics (SPH)

Lucy (1977), Gingold & Monaghan (1977)

Cosmology: Evrard (1988)

SPH is a Lagrangian method that represents fluid parcels with particles that have gas properties, such as density, temperature, and velocity. These quantities are calculated by kernel smoothing over usually 20-30 of the nearest neighboring particles. Then a modified set of fluid equations using the kernel are solved, thus updating the gas dynamics. SPH benefits from the advances in collisionless algorithms because of its particle nature. It is inherently spatially adaptive but suffers from poor resolution in underdense regions. However, the accuracy of SPH is hard to assess in the limit of the continuum limit of the fluid equations.

7.3.2 Eulerian Grid Algorithms

Cosmology: Ryu et al. (1990), Cen et al. (1990), Yuan et al. (1991)

Grid algorithms have been utilized to solve the fluid equations since Richtmeyer & Morton (1967), who also established that additional terms (e.g. heating/cooling, gravity, cosmological expansion) could be solved by operator splitting. In order to capture strong shocks, robust schemes must be used, which have been extensively tested in the computational fluid dynamics community (e.g. Sod 1985, Le Veque 1992). There are two major approaches for shock capturing: the total-variational diminishing method (TVD) and the piecewise-parabolic method (PPM). TVD uses an approximation to the Riemann solution, are second-order accurate outside of shocks, and can resolve shocks in two grid cells (Ryu et al. 1993). PPM is a third-order accurate Godunov solution for the Riemann problem, which is solved using a quadratic interpolation of the densities that minimizes post-shock oscillations (Collela & Woodward 1984, Woodward & Collela 1984).

7.3.3 Adaptive Mesh Refinement (AMR)

Cosmology: Anninos et al. (1994; static), Bryan & Norman (1997; dynamic)

Eulerian methods can capture shocks well but cannot follow high density regions well since it is constrained to a fixed grid. This can be bypassed, however, by using AMR, which dynamically places higher resolution grids in regions of interest (i.e. high densities, shocks, etc.) An alternative method is the moving mesh method (Gnedin 1995), which allows the grid to contract and expand depending on the fluid flow and dens

7.4 Additional Physics

The accurate treatment of gas dynamics involves not only gravity and hydrodynamics but also additional physics, i.e. heating and cooling, chemistry, radiation transport. Cen et al. (1990) was the first to include radiative cooling in the form of a cooling function that considered equilibrium between recombinations and collisional ionizations. Considering gas cooling allows the calculation to follow the condensation of galaxies in virialized objects. The assumption of equilibrium chemistry breaks down behind shocks and in dense cooling regions. Here one must solve a rate equation network for non-equilibrium chemistry (Cen 1992, Haehnelt et al. 1996). These methods have also been successful in following the formation of the first objects with non-equilibrium solvers that include molecular hydrogen cooling (Haiman et al. 1996, Abel et al. 1997, Anninos et al. 1997, Gnedin & Ostriker 1997).

Gas heating from supernovae and stellar radiation is included by a phenomenological method that represents stars or galaxies as collisionless particles that return thermal energy (and metals) to the hydrodynamical grid or particles.

Radiative transfer is an essential part of most astrophysical scenarios; however its solution is difficult since it is a function of seven variables: position, frequency, direction, and time. There have been several methods in solving radiative transfer, which include:

1. A spatially homogeneous and isotropic radiation field that allows to focus on the energy dependence (Cen 1992). Gnedin & Ostriker (1997) extended this method to include an approximation to local absorption.
2. Variable Eddington factors (the ratio of radiation stress to energy density) provides accurate solutions in the optically thin and thick regimes (Ducloux et al. 1992, Stone et al. 1992, Abel & Gnedin 2001).

7.5 Applications

The first true application of cosmological simulations were to test various cosmology models and parameters, i.e. CDM, HDM, Ω, H₀. They proved useful in refining the CDM model and ruling out the HDM model.

1. Galaxy clusters -- they are the most recently virialized objects and provide a good problem for these simulations, which can determine the relations between mass, luminosity, and velocity dispersions. Furthermore comparing the substructure, morphologies, and radial profiles of clusters found in simulations to observations can constrain cosmological parameters.
2. Modelling of gravitational lenses
3. Quasar absorption lines and Lyα forest lines -- cosmology simulations aided the understanding of the origin of these lines. They originate from well-defined clouds, filamentary and sheet-like structures, and velocity caustics. They can accurately account for the column density distribution of Lyα lines (Zhang et al. 1997).
4. Radial profiles of DM halos -- Navarro, Frenk, and White (1995, 1996, 1997) found that ρ(r) ∝ [r(r + r_s)]^-2 in all DM halos regardless of their formation redshift and mass, initial power spectrum, and cosmological parameters. In the cores of halos, the profile flattens to r^-1 that may be caused by isotropic orbits of matter.
5. Self-similar clustering -- Structure exhibits a self-similar form that evolves as [a(t)]^α, where α = 2/(3+n) and n is the spectral index of the density fluctuation power spectrum.

Author: John Wise

Date: 2007/04/08 07:48:40 PM