At early times, for example the CMB epoch about 400,000 years after the big bang, or on very large scales at later times, linear calculations starting from the ΛCDM fluctuation spectrum allow accurate predictions. But on scales where structure forms, the fluctuations have grown large enough that they are strongly nonlinear, and we must resort to simulations. The basic idea is that regions that start out with slightly higher than average density expand a little more slowly than average because of gravity, and regions that start out with slightly lower density expand a little faster. Nonlinear structure forms by the process known by the somewhat misleading name "gravitational collapse" – misleading because what really happens is that when positive fluctuations have grown sufficiently that they are about twice as dense as typical regions their size, they stop expanding while the surrounding universe keeps expanding around them. The result is that regions that collapse earlier are denser than those that collapse later; thus galaxy dark matter halos are denser than cluster halos. The visible galaxies form because the ordinary baryonic matter can radiate away its kinetic energy and fall toward the centers of the dark matter halos; when the ordinary matter becomes dense enough it forms stars. Thus visible galaxies are much smaller than their host dark matter halos, which in turn are much smaller than the large scale structure of the cosmic web, as shown in Fig. 3.
Figure 3. The stellar disk of a large spiral galaxy like the Milky Way is about 100,000 light years across, which is tiny compared with the dark matter halo of such a galaxy (from the Aquarius dark matter simulation Springel et al. (2008)), and even much smaller compared with the large-scale cosmic web (from the Bolshoi simulation Klypin et al. (2011)).
Astronomical observations represent snapshots of moments long ago when the light we now observe left distant astronomical objects. It is the role of astrophysical theory to produce movies – both metaphorical and actual – that link these snapshots together into a coherent physical picture. To predict cosmological large-scale structures, it has been sufficient to treat all the mass as dark matter in order to calculate the growth of structure and dark matter halo properties. But hydrodynamic simulations – i.e., including baryonic matter – are necessary to treat the formation and evolution of galaxies.
An old criticism of ΛCDM has been that the order of cosmogony is wrong: halos grow from small to large by accretion in a hierarchical formation theory like ΛCDM, but the oldest stellar populations are found in the most massive galaxies – suggesting that these massive galaxies form earliest, a phenomenon known as "downsizing" Cowie et al. (1996). The key to explaining the downsizing phenomenon is the realization that star formation is most efficient in dark matter halos with masses in the band between about 1010 and 1012 M⊙ (Fig. 1 bottom in Behroozi et al. (2013)). This goes back at least as far as the original Cold Dark Matter paper Blumenthal et al. (1984): see Fig. 4. A dark matter halo that has the total mass of a cluster of galaxies today will have crossed this star-forming mass band at an early epoch, and it will therefore contain galaxies whose stars formed early. These galaxies will be red and dead today. A less massive dark matter halo that is now entering the star-forming band today will just be forming significant numbers of stars, and it will be blue today. The details of the origin of the star-forming band are still being worked out. Back in 1984, we argued that cooling would be inefficient for masses greater than about 1012 M⊙ because the density would be too low, and inefficient for masses less than about 108 M⊙ because the gas would not be heated enough by falling into these small potential wells. Now we know that reionization, supernovae Dekel and Silk (1986), and other energy input additionally impedes star formation for halo masses below about 1010 M⊙, and feedback from active galactic nuclei (AGN) additionally impedes star formation for halo masses above about 1012 M⊙.
Figure 4. The Star-Forming Band on a diagram of baryon density nb versus the three-dimensional r.m.s. velocity dispersion V and virial temperature T for structures of various sizes in the universe, where T = µ V2 / 3k, µ is mean molecular weight (≈ 0.6 for ionized primordial H + He) and k is Boltzmann's constant. Below the No Metals and Solar Metals cooling curves, the cooling timescale is more rapid than the gravitational timescale. Dots are groups and clusters. Diagonal lines show the halo masses in units of M⊙. (This is Fig. 3 in Blumenthal et al., 1984, with the Star-Forming Band added.)
Early simulations of disk galaxy formation found that the stellar disks had much lower rotation velocities than observed galaxies Navarro and Steinmetz (2000). This problem seemed so serious that it became known as the "angular momentum catastrophe." A major cause of this was excessive cooling of the gas in small halos before they merged to form larger galaxies Maller and Dekel (2002). Simulations with better resolution and more physical treatment of feedback from star formation appear to resolve this problem. In particular, the Eris cosmological simulation Guedes et al. (2011) produced a very realistic spiral galaxy, as have many simulations since then. Somerville and Davé (2014) is an excellent recent review of progress in understanding galaxy formation. In the following I summarize some of the latest developments. There are now two leading approaches to simulating galaxies:
Low resolution, ∼ 1 kiloparsecs. The advantages of this approach are that it is possible to simulate many galaxies and study galaxy populations and their interactions with the circumgalactic and intergalactic media, but the disadvantages are that we learn relatively little about how galaxies themselves form and evolve at high redshifts. The prime examples of this approach now are the Illustris Vogelsberger et al. (2014b) and EAGLE Schaye et al. (2015) simulations. Like semi-analytic models of galaxy formation (reviewed in Benson (2010)), these projects adjusted the parameters governing star-formation and feedback processes in order to reproduce key properties of galaxies at the present epoch, redshift z = 0. The Illustris simulation in a volume (106.5 Mpc)3 forms ∼ 40,000 galaxies at the present epoch with a reasonable mix of elliptical and spiral galaxies that have realistic appearances Snyder et al. (2015), obey observed scaling relations, have the observed numbers of galaxies as a function of their luminosity, and were formed with the observed cosmic star formation rate Vogelsberger et al. (2014a). It forms massive compact galaxies by redshift z = 2 via central starbursts in major mergers of gas-rich galaxies or else by assembly at very early times Wellons et al. (2015). Remarkably, the Illustris simulation also predicts a population of damped Lyman α absorbers (DLAs, small-galaxy-size clouds of neutral hydrogen) that agrees with some of the key observational properties of DLAs Bird et al. (2014), Bird et al. (2015). The EAGLE simulation in volumes up to (100 Mpc)3 reproduces the observed galaxy mass function from 108 to 1011 M⊙ at a level of agreement close to that attained by semi-analytic models Schaye et al. (2015), and the observed atomic and molecular hydrogen content of galaxies out to z ∼ 3 Rahmati et al. (2015), Lagos et al. (2015).
High resolution, ∼ 10s of parsecs. The advantages are that it is possible to compare simulation outputs in detail with high-redshift galaxy images and spectra to discover the drivers of morphological changes as galaxies evolve, but the disadvantage is that such simulations are so expensive computationally that it is as yet impossible to run enough cases to create statistical samples. Leading examples of this approach are FIRE simulations led by Phil Hopkins (e.g., Hopkins et al. (2014)) and the ART simulation suite led by Avishai Dekel and myself (e.g., Zolotov et al. (2014)). We try to compensate for the small number of high-resolution simulations by using simulation outputs to tune semi-analytic models, which in turn use cosmological dark-matter-only simulations like Bolshoi to follow the evolution of ∼106 galaxies in their cosmological context (e.g., Porter et al. (2014b), Porter et al. (2014a), Brennan et al. (2015)).
The high-resolution FIRE simulations, based on the GIZMO smooth particle hydrodynamics code Hopkins (2014) with supernova and stellar feedback, including radiative feedback (RF) pressure from massive stars, treated with zero adjusted parameters, reproduce the observed relation between stellar and halo mass up to Mhalo ∼ 1012 M⊙ and the observed star formation rates Hopkins et al. (2014). FIRE simulations predict covering fractions of neutral hydrogen with column densities from 1017 cm−2 (Lyman limit systems, LLS) to > 1020.3 cm−2 (DLAs) in agreement with observations at redshifts z = 2-2.5 Faucher-Giguère et al. (2015); this success is a consequence of the simulated galactic winds. FIRE simulations also correctly predict the observed evolution of the decrease of metallicity with stellar mass Ma et al. (2015), and produce dwarf galaxies that appear to agree with observations Oñorbe et al. (2015) as we will discuss in more detail below.
The high-resolution simulation suite based on the ART adaptive mesh refinement (AMR) approach Kravtsov et al. (1997), Ceverino and Klypin (2009) incorporates at the sub-grid level many of the physical processes relevant for galaxy formation. Our initial group of 30 zoom-in simulations of galaxies in dark matter halos of mass (1−30) × 1012 M⊙ at redshift z = 1 were run at 35-70 pc maximum (physical) resolution Ceverino et al. (2012), Ceverino et al. (2015a). The second group of 35 simulations (VELA01 to VELA35) with 17.5 to 35 pc resolution of halos of mass (2−20) × 1011 M⊙ at redshift z = 1 have now been run three times with varying inclusion of radiative pressure feedback (none, UV, UV+IR), as described in Ceverino et al. (2014). RF pressure including the effects of stellar winds Hopkins et al. (2012), Hopkins et al. (2014) captures essential features of star formation in our simulations. In particular, RF begins to affect the star-forming region as soon as massive stars form, long before the first supernovae occur, and the amount of energy released in RF greatly exceeds that released by supernovae Ceverino et al. (2014), Trujillo-Gomez et al. (2015). In addition to radiation pressure, the local UV flux from young star clusters also affects the cooling and heating processes in star-forming regions through photoheating and photoionization. We use our Sunrise code Jonsson (2006), Jonsson et al. (2006), Jonsson et al. (2010), Jonsson and Primack (2010) to make realistic images and spectra of these simulated galaxies in many wavebands and at many times during their evolution, including the effects of stellar evolution and of dust scattering, absorption, and re-emission, to compare with the imaging and photometry from CANDELS 3 and other surveys – see Fig. 5 for examples including the effect of CANDELization (reducing the resolution and adding noise) to allow direct comparison with HST images.
Figure 5. Face-on images of Vela26 simulated galaxy with UV radiation pressure feedback, at four redshifts (a) z = 3.6 when it is diffuse and star forming (dSF); (b) z = 2.7 when it has become compact and star forming (cSF) with a red ex-situ clump; (c) z = 2.3 still cSF, now with in situ clumps apparent in the V-band image; (d) compact and quenched (cQ) during a minor merger, with tidal features visible in the V-band image. Top panels: three-color composite images at high resolution; bottom panels: CANDELized V and H band images. The observed V band images correspond to ultraviolet radiation from massive young stars in the galaxy rest frame, while the observed H band images show optical light from the entire stellar population including old stars. The CANDELS survey took advantage of the infrared capability of the Wide Field Camera 3, installed on the last service visit to HST in 2009.
In comparing our simulations with HST observations, especially those from the CANDELS and 3D-HST surveys, we are finding that the simulations can help us interpret a variety of observed phenomena that we now realize are important in galaxy evolution. One is the formation of compact galaxies. Analysis of CANDELS images suggested Barro et al. (2013), Barro et al. (2014a), Barro et al. (2014b) that diffuse star-forming galaxies become compact galaxies (blue nuggets) which subsequently quench (red nuggets). We see very similar behavior in our VELA simulations with UV radiative feedback (Zolotov et al. (2014), see Figure 2), and we have identified in our simulations several mechanisms that lead to compaction often followed by rapid quenching, including major gas-rich mergers, disk instabilities often triggered by minor mergers, and opposing gas flows into the central galaxy Danovich et al. (2015).
Another aspect of galaxy formation seen in HST observations is massive star-forming clumps (Guo et al. (2012), Wuyts et al. (2013) and references therein), which occur in a large fraction of star-forming galaxies at redshifts z = 1-3 Guo et al. (2015). In our simulations there are two types of clumps. Some are a stage of minor mergers – we call those ex situ clumps. A majority of the clumps originate in situ from violent disk instabilities (VDI) in gas-rich galaxies Ceverino et al. (2012), Moody et al. (2014), Mandelker et al. (2014). Some of these in situ clumps are associated with gas instabilities that help to create compact spheroids, and some form after the central spheroid and are associated with the formation of surrounding disks. We find that there is not a clear separation between these processes, since minor mergers often trigger disk instabilities in our simulations Zolotov et al. (2014).
Star-forming galaxies with stellar masses M∗ ≲ 3 × 109 M⊙ at z > 1 have recently been shown to have mostly elongated (prolate) stellar distributions van der Wel et al. (2014) rather than disks or spheroids, based on their observed axis ratio distribution. In our simulations this occurs because most dark matter halos are prolate especially at small radii Allgood et al. (2006), and the first stars form in these elongated inner halos; at lower redshifts, as the stars begin to dominate the dark matter, the galaxy centers become disky or spheroidal Ceverino et al. (2015b).
Both the FIRE and ART simulation groups and many others are participating in the Assembling Galaxies of Resolved Anatomy (AGORA) collaboration Kim et al. (2014) to run high-resolution simulations of the same initial conditions with halos of masses 1010, 1011, 1012, and 1013 M⊙ at z = 0 with as much as possible the same astrophysical assumptions. AGORA cosmological runs using different simulation codes will be systematically compared with each other using a common analysis toolkit and validated against observations to verify that the solutions are robust – i.e., that the astrophysical assumptions are responsible for any success, rather than artifacts of particular implementations. The goals of the AGORA project are, broadly speaking, to raise the realism and predictive power of galaxy simulations and the understanding of the feedback processes that regulate galaxy "metabolism."
It still remains to be seen whether the entire population of galaxies can be explained in the context of ΛCDM. A concern regarding disk galaxies is whether the formation of bulges by both galaxy mergers and secular evolution will prevent the formation of as many pure disk galaxies as we see in the nearby universe Kormendy and Fisher (2008). A concern regarding massive galaxies is whether theory can naturally account for the relatively large number of ultra-luminous infrared galaxies. The bright sub-millimeter galaxies were the greatest discrepancy between our semi-analytic model predictions compared with observations out to high redshift Somerville et al. (2012). This could possibly be explained by a top-heavy stellar initial mass function, or perhaps more plausibly by more realistic simulations including self-consistent treatment of dust Hayward et al. (2011), Hayward et al. (2013). Clearly, there is much still to be done, both observationally and theoretically. It is possible that all the potential discrepancies between ΛCDM and observations of relatively massive galaxies will be resolved by better understanding of the complex astrophysics of their formation and evolution. But small galaxies might provide more stringent tests of ΛCDM.
3 CANDELS, the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey, was the largest-ever Hubble Space Telescope survey, see http://candels.ucolick.org/ Back.