Elliptical galaxies are thought to be formed by merging events in the hierarchical picture. Semi-analytic model (SAM) is a representative way to study this scenario.

In fact, the best way to study lambda CDM model is hydrodynamic N-body simulation. However, this is chalenging work in the sense of computation, and thus, it is very difficult to investigate the physics of cosmological scale.

To circumvent this difficulty, SAM adopts simplified physical processes. At first, one need to construct dark matter merger tree, and inject baryonic physics into the tree.

The most popular way to construct DM merger tree is to use Monte Carlo realization based on the extended Press-Schechter formalism (EPS). Also possible way is to adopt dark matter clustering history from dissipationless N-body simulation. Once the tree is obtained, baryonic quantities such as mass or metallicity are calculated. In early universe, each dark matter halo is assumed to contain a central galaxy. In order to distinguish the physical properties, galaxies are assumed to have bulge, disk, and halo (subhalo) component. Also only three types of matter are allowed : star, cold gas, hot gas.

Using these components, one can obtain the magnitude, mean metallicity, mass and so forth by combining with evolutionary population synthesis and chemical enrichment model.

Since SAM simplifies baryonic physic using free parameter, they should be normalized by observed quantity. Nevertheless, it provides useful framework to study galaxy formation. Furthermore, it also could let us know which physics have room to be improved.