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Galaxy morphology and structure allows a new way to compare with cosmologically based galaxy formation models, as well as those which include extensive physics such as star formation, AGN feedback and supernova in more detailed hydrodynamical models. This review only briefly discusses this large topic and how it relates to galaxy structure. For a more detailed recent review on the theory of galaxy formation from a theoretical prospective see Silk & Mamon (2012).

Galaxy formation models were first developed to explain the structures of galaxies, namely the bulge/disk/halo trichotomy, and the ages of the stars in these components (Eggen et al. 1962). The default initial assumption in the first galaxy formation models was that galaxies formed like stars in a relatively rapid collapse. In the 1980s the first computer simulations of structure formation showed that a universe dominated by Cold Dark Matter (CDM) matched observations of galaxy clustering on large scales (Davis et al. 1985), and that within this framework galaxy assembly should be hierarchical (Blumenthal et al. 1984), yet this is a fundamental prediction which is just now starting to be tested with only a few papers comparing the observations to the theoretical predictions (e.g., Bertone & Conselice 2009; Jogee et al. 2009; Hopkins et al. 2010; Lotz et al. 2011).

The situation today is that there are many simulations that are used to predict properties of the galaxy population, and how it evolves through time. These models are largely successful when predicting basic properties of nearby galaxies, such as their luminosities, masses, colors and star formation rates, as well as scaling relationships of galaxies. However problems still exist in predicting the abundances of low and high mass galaxies (e.g., Conselice et al. 2007; Guo et al. 2011). Within galaxy formation models there are very famous problems such as the satellite and the CDM dark matter profile, but there are also significant issues when examining how the evolution of galaxies occurs, and trying to match this with the theory. Another major problem is that there are several large disk galaxies without significant bulges in the nearby universe that are not predicted in CDM (e.g., Kormendy et al. 2010).

One of the ways to further test these models is to investigate how well CDM models can reproduce the formation history of galaxies as seen through the merging process using the so-called semi-analytical method (e.g., Bower et al. 2006; Guo et al. 2011). We show this comparison with the measured merger fractions in Figure 12 at two different stellar mass ranges of M* > 1010 M and M* > 1011 M. Plotted as the thin solid black line towards the lower part of each diagram is the prediction for the major merger fraction for galaxies from the Millennium simulation (Bertone & Conselice 2009). Also shown on these figures as the dotted blue line is the same predictions for major mergers for Warm Dark Matter models (e.g., Menci et al. 2012), which do a better job than CDM in matching the observed data. However, CDM better matches if minor mergers are taken into account, although the comparison merger fraction is only for major mergers based on the methodology used (Conselice 2003a; Lotz et al. 2010a).

Other recent attempts to predict the merger history of observed galaxies include the abundance matching technique (e.g., Stewart et al. 2008) where observed galaxies are matched to halos in models through their comparative abundance levels. Hopkins et al. (2010) predict based on this abundance matching the merger rate and fraction for galaxies. The result of this is show in Figure 12 for galaxies between M* = 1010-11 M . While the merger fractions from Hopkins et al. (2010) are higher than those from the CDM models, they are still lower than the observations (see also Jogee et al. 2009 and Lotz et al. 2011 for further discussions). Similar results from Stewart et al. (2008) are also shown in Figure 12, who find results similar to Hopkins et al. (2010).

Finally, as a contract to these Maller et al. (2006) present cosmological hydrodynamical simulation results for similar mass galaxies of a few times 1010 M, and find the highest merger fraction predictions of any simulation result (Figure 12). This shows that the predictions for merger histories are not correct or consistent with each other, and that more simulation work should be focused on this critical aspect of the galaxy population. This is an area where future work is certainly needed.

There are several other types of simulations in which galaxy structure and morphology can be directly compared with observations of galaxies through cosmic time. Perhaps the most direct of these is to compare the properties and structural features of distant galaxies to hydrodynamical models of galaxy formation. Some of this work for galaxy mergers is discussed in Section 3.4. Early work in this area showed that the components of galaxies - namely bulges and disks were the result of accretion events (e.g., Steinmetz & Navarro 2002) and argued from their simulations that the Hubble type of a galaxy is not stable for long periods of cosmic time. Governato et al. (2007) show that disk galaxies can be simulated which have properties that match the morphological properties and kinematics of nearby disks, although this simulation is not in a cosmological context. Overall however it is very difficult to predict the formation of galaxy morphology in simulations, and in a real sense this will be the ultimate test of galaxy formation models in the future.

Also, as discussed in Section 4.6 one of the most commonly seen properties in high redshift star forming galaxies is that they often contain large clumps of star formation within their disks. A major question is how these clumps form, evolve, and how they may play a role in the formation of other galaxy components such as bulges and AGN. Bournaud et al. (2013) examined this problem computationally to determine how clumps with stellar masses of M* = 108-9 M evolve in gaseous disks. The major question is whether these clumps dissipate within 50 Myr or so, the dynamical time-scale of the clumps, or if they regenerate and survive. Bournaud et al. (2013) find that these clumps can last around 300 Myr through acquiring new gas from its disk, although some mass is lost through tidal effects. This is enough time to migrate towards the center of the galaxy which can fuel the AGN or merge to form a bulge. This shows that these clumps may provide a significant route for galaxies to form. Thus, we have evidence for both inside-out and outside-in formation occurring in the galaxy population. What remains to be seen is whether one of these mechanisms is dominant, and the relative role of both in forming galaxies.

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