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The process of galaxy formation involves nonlinear physics and a wide variety of physical processes. As such, it is impossible to treat in full detail using analytic techniques. There are two major approaches that have been developed to circumvent this problem. The first, numerical N-body simulation, attempts to directly and numerically solve the fully nonlinear equations governing the physical processes inherent to galaxy formation. The second, semi-analytic modeling, attempts to construct a coherent set of analytic approximations which describe these same physics. Each has its strengths and weaknesses, as will be discussed below. As a result of this and of our ignorance of how some key processes work, both approaches contain a little of each other (rather like yin yang). A different, more empirical approach, utilizing so-called "halo occupation distributions" has become widely used in the past ten years and will be discussed briefly below.

7.1. N-body/Hydro

The most accurate computational method for solving the physics of galaxy formation is via direct simulation, in which the fundamental equations of gravitation, hydrodynamics and perhaps radiative cooling and transfer are solved far a large number of points (arranged either on a grid or following the trajectories of the fluid flow). I will not attempt to review the numerical methods utilized in this approach in any detail here (recent treatments of this can be found in Bertschinger 1998, Agertz et al. 2007, Rosswog 2009), but will instead merely highlight some of the approaches used.

Collisionless dark matter is (relatively) simple to model in this way, since it responds only to the gravitational force. For the velocities and gravitational fields occurring during structure and galaxy formation non-relativistic Newtonian dynamics is more than adequate and so solving the evolution of some initial distribution of dark matter (usually a Gaussian random field of density perturbations consistent with the power spectrum of the CMB) reduces to summing large numbers of 1 / r2 forces between pairs of particles. In practice, clever numerical techniques (such as particle-mesh, tree algorithms etc.) are usually used to reduce this N2 problem into something more manageable (Kravtsov et al. 1997; Springel 2005). The largest pure dark matter simulations carried out to date contain around 10 billion particles (Springel et al. 2005b). Dark matter only simulations of this type (carried out primarily for the cold dark matter scenario, but see White et al. 1984, Klypin et al. 1993, Bode et al. 2001, Davé et al. 2001, Colín et al. 2002, Ahn and Shapiro 2005) have been highly successful in determining the large scale structure of the Universe, as embodied in the so-called "cosmic web". As a result, the spatial and velocity correlation properties of dark matter and dark matter halos (Davis et al. 1985; White et al. 1987a, b; Efstathiou et al. 1988; Eke et al. 1996; Jenkins et al. 1998; Padilla and Baugh 2002; Bahcall et al. 2004; Kravtsov et al. 2004; Reed et al. 2009), together with the density profiles (Navarro et al. 1997; Bullock et al. 2001b; Navarro et al. 2004; Merritt et al. 2005; Prada et al. 2006), angular momenta (Barnes and Efstathiou 1987; Efstathiou et al. 1988; Warren et al. 1992; Cole and Lacey 1996; Lemson and Kauffmann 1999; Bullock et al. 2001a; van den Bosch et al. 2002; Bett et al. 2007; Gao et al. 2008) and internal structure (Moore et al. 1999; Klypin et al. 1999; Kuhlen et al. 2008; Springel et al. 2008) of dark matter halos are known to very high accuracy.

Of course, to study galaxy formation dark matter alone is insufficient, and baryonic material must be added in to the mix. This makes the problem much more difficult since, at the very least, pressure forces must be computed and the internal energy of the baryonic fluid tracked. Particle-based methods (most prominently smoothed particle hydrodynamics; Springel 2005) have been successful in this area, as have Eulerian grid methods (Ricker et al. 2000; Fryxell et al. 2000; Plewa and Müller 2001; Quilis 2004). The addition of radiative cooling is relatively straightforward (at least while the gas remains optically thin to its own radiation) by simply tabulating the rate at which gas of given density and temperature radiates energy.

Going beyond this level of detail becomes extremely challenging. Numerous simulation codes are now able to include star formation and feedback from supernovae explosions, while some even attempt to follow the formation of supermassive black holes in galactic centers. It should be kept in mind though that for galaxy scale simulations the real physics of these processes is happening on scales well below the resolution of the simulation and so the treatment of the physics is often at the "subgrid" level, which essentially means that it is put in by hand using a semi-analytic approach (Thacker and Couchman 2000; Kay et al. 2002; Marri and White 2003; White 2004; Stinson et al. 2006; Cox et al. 2006; Scannapieco et al. 2006a; Tasker and Bryan 2006; Stinson 2007; Vecchia and Schaye 2008; Oppenheimer and Davé 2008; Okamoto et al. 2008b; Booth and Schaye 2009; Ciotti 2009).

Beyond this, problems such as the inclusion of radiative transfer or magnetic fields complicate the problem further by introducing new sets of equations to be solved and the requirement to follow additional fields. For example, in radiative transfer one must follow photons (or photon packets) which have a position, direction of travel and wavelength, and determine the absorption of these photons as they traverse the baryonic material of the simulation, while simultaneously accounting for the re-emission of absorbed photons at other wavelengths. Despite these complexities, progress has been made on these issues using a variety of ingenious numerical techniques (Abel et al. 1999; Ciardi et al. 2001; Gnedin and Abel 2001; Ricotti et al. 2002; Petkova and Springel 2009; Li et al. 2008; Aubert and Teyssier 2008; Collins et al. 2009; Dolag and Stasyszyn 2008; Finlator et al. 2009; Laursen et al. 2009). Providing Moore's Law continues to hold true, the ability of simulations to provide ever more accurate pictures of galaxy formation should hold strong. A brief survey of current cosmological hydrodynamical codes and their functionalities is given in Table 1.

Table 1. A survey of physical processes included in several of the major hydrodynamical codes. The primary reference is indicated next to the name of the code. Where implementations of major physical processes are described elsewhere the reference is given next to the entry in the relevant row.

Feature Gadget-3 1 Gasoline 2 HART 3 Enzo(Zeus) 4 Flash 5

Gravity Tree Tree AMR PM AMR 6 PM 7 Multi-grid
Hydrodynamics SPH SPH 8 AMR 6 AMR 6 AMR 6
--> Multiphase subgrid model 10 × N/A N/A N/A
Radiative Cooling 11
--> Metal dependent 12 × 13 14 11
--> Molecular chemistry 15 × 13,16 17 ×
Thermal Conduction 18 × × ×
Star formation 19 20 13 21 ×
--> SNe feedback 19 20 13 21 ×
--> Chemical enrichment 19 20 13 21 ×
Black hole formation 22 × × × 23
--> AGN feedback 22 × × × ×
Radiative transfer OTVET 24,25 × OTVET 24 26 27
Magnetic fields 28 × × 29 30
1 "GAlaxies with Dark matter and Gas intEracT" (Springel, 2005); 2 Wadsley et al. (2004); 3 Hydrodynamic Adaptive Refinement Tree (Kravtsov et al., 2002); 4O'Shea et al. (2004); 5 (Fryxell et al., 2000); 6 Adaptive Mesh Refinement; 7 Particle-mesh; 8 Smoothed Particle Hydrodynamics; 9 Applicable only to SPH codes - used correctly, AMR codes naturally resolve multiphase media; 10 Scannapieco et al. (2006a); 11 Banerjee et al. (2006); 12 Scannapieco et al. (2005); 13 Tassis et al. (2008); 14 Smith et al. (2009); 15 Yoshida et al. (2003); 16 Equilibrium only; 17 Turk (2009); 18 Jubelgas et al. (2004); 19 Scannapieco et al. (2005); 20 Governato et al. (2007); 21 Tasker and Bryan (2008); 22 Matteo et al. (2005); 23 Federrath et al. (2010); 24 Optically Thin Variable Eddington Tensor; 25 Petkova and Springel (2009); 26 Flux-limited diffusion approximation (Norman et al. 2009; see also Wise and Abel 2008b); 27 Rijkhorst et al. (2006); Peters et al. (2010); 28 Dolag and Stasyszyn (2008); 29 Collins et al. (2009; see also Wang and Abel 2009); 30 Robinson et al. (2004);

7.2. Semi-Analytic

The technique of "semi-analytic modeling" or, perhaps, "phenomenological galaxy formation modeling" takes the approach of treating the various physical processes associated with galaxy formation using approximate, analytic techniques. As with N-body/hydro simulations, the degree of approximation varies considerably with the complexity of the physics being treated, ranging from precision-calibrated estimates of dark matter merger rates to empirically motivated scaling functions with large parameter uncertainty (e.g. in the case of star formation and feedback - just as in N-body/hydro simulations).

The primary advantage of the semi-analytic approach is that it is computationally inexpensive compared to N-body/hydro simulations. This facilitates the construction of samples of galaxies orders of magnitude larger than possible with N-body techniques and for the rapid exploration of parameter space (Henriques et al. 2009) and model space (i.e. adding in new physics and assessing the effects). The primary disadvantage is that they involve a larger degree of approximation. The extent to which this actually matters has not yet been well assessed. Comparison studies of semi-analytic vs. N-body/hydro calculations have shown overall quite good agreement (at least on mass scales well above the resolution limit of the simulation) but have been limited to either simplified physics (e.g. hydrodynamics and cooling only; Benson et al. 2001b, Yoshida et al. 2002, Helly et al. 2003b]) or to simulations of individual galaxies (Stringer et al. 2010).

Some of the earliest attempts to construct a self-consistent semi-analytic model of galaxy formation began with White and Frenk (1991), Cole (1991) and Lacey and Silk (1991), drawing on earlier work by Rees and Ostriker (1977) and White and Rees (1978). Since then numerous studies (Kauffmann et al. 1993; Baugh et al. 1999b, 1998; Somerville and Primack 1999; Cole et al. 2000; Benson et al. 2002a; Hatton et al. 2003; Monaco et al. 2007) have extended and improved this original framework. A recent review of semi-analytic techniques is given by Baugh (2006). Semi-analytic models have been used to investigate many aspects of galaxy formation including:

The term "semi-analytic" model has become somewhat insufficient, since this name now encompasses such a diverse range of models that the name alone does not convey enough information. Semi-analytic models in the literature contain a widely disparate range of physical phenomena and implementations. Table 2 is an attempt to assess in detail the physics included in currently implemented semi-analytic models 32.

Table 2. A survey of physical processes included in major semi-analytic models of galaxy formation. In each case we indicate how this process is implemented and give references where relevant. In many cases a single model has implemented a given physical process at different levels of complexity/realism. In such cases, we list the most "advanced" implementation that the model is capable of.

Feature Durham 1 Munich 2 Santa-Cruz 3 Morgana 4 Galics 5
Merger Trees
→ Analytic Modified ePS 6 ePS 7 ePS Pinocchio 8 ×
→ N-body 9 ×
Halo Profiles Einasto 10 Isothermal NFW NFW Empirical 11
Cooling Model
→ Metal-dependent
Star Formation
→ SNe
→ AGN 12 13
→ Reionization 10 × 14 15
→ Substructure 16 N-body 17 N-body 17 DF 18 DF 18 N-body 17
→ Substructure-Substructure 19 20, 10 × 21, 22 × 21
→ Ram Pressure Stripping 23 24 × × 25
→ Tidal Stripping 10 ×
→ Harassment × × × × ×
→ Disk Stability 26
→ Dynamical Friction 27 28 × × × ×
→ Thickness 28 × × × ×
→ Adiabatic contraction × ×
Chemical Enrichment ✓ [delayed 10] ✓ [instant 29] ✓ [delayed 30] ✓ [instant] ✓ [delayed 31]
Dust Grasil 32 Screen 33 Slab 34 Grasil 32, 35 Slab 34
1 Cole et al. (2000); 2 Croton et al. (2006); 3 Somerville et al. (2008b); 4 Monaco et al. (2007); 5 Hatton et al. (2003); 6 Parkinson et al. (2008); 7 Kauffmann and White (1993); 8 Monaco et al. (2002); 9 Helly et al. (2003a); 10 Benson and Bower (2010); 11 A "dark matter" core is included in calculations of disk sizes with an empirically selected dark matter fraction; 12 Bower et al. (2006); 13 Cattaneo et al. (2006); 14 Macció et al. (2009); 15 Lanzoni et al. (2005); 16 How does the model track substructures within halos?; 17 Substructure orbits and merging times are determined from N-body simulations; 18 Dynamical Friction: substructure merging times are computed from analytic estimates of dynamical friction timescales; 19 Does the model allow merging between pairs of subhalos orbiting in the same host halo?; 20 Using hierarchically nested substructures; 21 Using random collisions of subhalos; 22 Somerville and Primack (1999); 23 Font et al. (2008); 24 Brüggen and Lucia (2008); 25 Lanzoni et al. (2005); 26 Somerville et al. (2008a); 27 Does the model include dynamical friction forces exerted by a galaxy disk on orbiting satellites?; 28 Benson et al. (2004); 29 Lucia et al. (2004); 30 Arrigoni et al. (2009); 31 Pipino et al. (2008); 32 Silva et al. (1998);

7.3. Halo Occupation Distributions

Given the complexity of galaxy formation it is sometimes desirable to take a more empirical approach to the problem. This can be advantageous both to relate observations to the assumed underlying physical structure of the Universe (e.g. Zheng et al. 2007, Tinker et al. 2010), and to make predictions based on extrapolations from current data and which have a cosmological underpinning. "Halo occupation distributions", first described by Neyman and Scott (1952) and used to study galaxy clustering (see, for example, Benson et al. 2000c, Peacock and Smith 2000) in their simplest form specify the probability of finding N galaxies of some prescribed type in a dark matter halo of mass M, P(N|M). Given this probability distribution, knowledge of the distribution of dark matter halo masses and their spatial distribution plus some assumptions about the locations of galaxies within halos, one can construct (statistically or within an N-body simulation) the distribution of galaxies. This allows, for example, the abundance and clustering properties of those galaxies to be inferred. A detailed discussion of the use of halo occupation distributions in studying galaxy clustering is given by Cooray and Sheth (2002).

This approach has recently been extended by incorporating simple prescriptions relating, for example, star formation rates or quasar activity to halo mass and redshift. Several authors have demonstrated that such empirical models, while simple, can fit a wide variety of galaxy data and can be used to gain insight into phenomena such as "downsizing" (Yang et al. 2003, Conroy and Wechsler 2009, Croton 2009). At a fundamental level, the success of simple, empirical models such as these suggests that, despite the complexity of galaxy formation physics, its outcome is relatively simple.

32 One component missing from Table 2 is the specific implementation of stellar population synthesis used by each code. We have chosen to not include this because all the models listed treat this calculation in precisely the same way (see Section 6.5.1), differing only in which compilation of stellar spectra they choose to employ. Since it is trivial to replace one compilation of stellar spectra with another we do not consider this a fundamental difference between the models. Nevertheless, the choice of which stellar spectra to use can have important consequences for the predicted properties of galaxies (Conroy and Gunn 2010) and so should not be overlooked. Back.

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