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## Dark Matter Structure

In this section we review the process of growth of dissipationless structure in the Universe. The physics behind this topic is basically the dynamics of a matter density field under the eﬀect of its own gravity, as we will focus on the physics of the collisionless, non-baryonic dark matter.

We specify first the cosmological model, we go on then to describe the formation of the population of dark matter halos in this context, to finally make a pass on the detailed structure and formation history of a single halo. This top-down approach to link these two scales will allow us to tackle the first important problems in galaxy formation.

### Cosmological Scales

Currently we have a concordance cosmological model, and its parameters estimated with unprecedented accuracy [SBD+07]. In that model the dom-inant matter component is the cold dark matter † nevertheless the universe seems to be under the commanding presence of a cosmological constant [Car04] which today is believed to drive the accelerated expansion of the Universe. In this theoretical Universe we assume that the initial fluctuations, that originated the seeds for the structure formation in the universe, are Gaus-sian. Once we have set up the initial conditions for the density fluctuations with the cosmological parameters at hand, we can try to make a guess about the abundance of dark matter structures. The first attempt to do so was done by Press and Schechter [PS74, Sch76], long before the cold dark matter model was widely accepted in the community. They assumed a Gaussian density field and smoothed it on diﬀerent physical scales. By varying the physical scale of the smoothing window they could construct a mass for that scale, then the abundance of halos above a given mass is calculated from the fraction of volumes that have a given smoothed density.

The mass function (i.e. the number density of structures with a deter-mined mass) predicted by this simple approximation agrees very well with † A kind of non-collisional, non-baryonic kind of matter that only interacts with itself through gravity. We do not have, as of 2007, a stablished picture of its physical nature the results of numerical simulations of a dark matter density field. The accuracy with which we can produce mass functions of halos using the Press-Schechter approach have been improved in the nineties by further refinements, most notably by Sheth and Tormen [SMT01].

#### Galactic Scales

From the cosmological context, we can start to describe the structure of sin-gle dark matter halos, which are equilibrium entities created in the developed stages of nonlinear stages of dark matter collapse. In a first approximation we can say that a dark matter halo is spherical with a density profile de-pending only on the radius. Using numerical simulations in the early 90s Navarro, Frenk and White found that dark matter halos follow a universal density profile (known as NFW profile) [NFW97]. For this density profile we do not have a clear explanation for its emergence. A much simpler pro-file, with a longer career in astrophysical contexts, providing a rather good approximation to the results in numerical simulations for the most of the volume of the halo, is the isothermal profile DM 4πRvir r2 ρ (r) = Mvir 1 , (2.1).

where M is the mass of the halo, R its radius, and the subscript vir refers to virialized (equilibrium) quantities. We can associate a temperature to the halo, which is naturally called virial temperature 2kB Rvir Tvir = µmp GMvir , (2.2).

where µ = 0.57 is the mean molecular weight for an ionized gas composed of 75% hydrogen and 25% Helium, mp is the proton mass, kB is the Boltzmann constant and G the gravitational constant.

Another important characteristic of a dark matter halo is its rotation that is quantified through the spin parameter 1/2 ~ GMvir5/2 λ = |E| |J| , (2.3).

where E is the total energy of the halo and J is its angular momentum Later, when higher resolution simulations where done, the community saw that dark matter halos were not featureless [MGQ+98, MQG+99, KKVP99, CAV00]. They host a good deal of substructure, which can be thought as smaller halos inside a larger halo. Typically, around 15 percent of the total mass of dark matter halo is found in the form of those substructures [GMG+00]. How was created this substructure? Understanding that process imploes the study of growth of a dark matter halo as function of time. In the same way genetic information is passed from progenitor to descendant, some information of the halo final structure should be found in the history of its progenitors.

**Halos and Merger Trees**

The genealogical tree of a dark matter halo, in the context of structure formation, is known as a merger tree. This structure is fundamental to the construction of galaxies inside hierarchical scenarios. Accordingly, there is a great variety of ways to build such a tree.

For an analytical approach to merger trees, the natural step is to sample the Press-Schechter halo distribution at diﬀerent times, in order to link the diﬀerent populations. Usually the implementation of this idea starts from the final halo and draws a conditional probability for the mass of their parents, continuing recursively until every halo has a parent. Naturally, there is also a condition on the minimal mass a parent can have, implying that the tree cannot be infinitely large with smaller and smaller halos on it. This is known as a Monte-Carlo approach to merger trees [ND08, ?, PCH07]. It has been criticized in the past because a merger tree built in that way, won’t naturally reproduce the halo mass function at diﬀerent times. It means that if we select all the halos at present, and take all its parents at diﬀerent times, the halo mass functions we build from the parents in the tree won’t match the original halo mass functions from which we drew them [BKH05]. More recent implementations seem to have fixed this problem using additional ad-hoc conditions [ND08, ?, PCH07].

The fundamental assumption behind all the algorithms implementes in a Monte-Carlo approach is that the formation history of the halo depends only on its mass. This assumption has been challenged based on the results of large high resolution dark matter simulations. It has been shown, for instance, that the halo environment has an impact on its formation history [GSW05]. This can be seen as a major caveat to using merger trees from Monte-Carlo simulations, but it remains to be shown that indeed the merger tree properties change significantly with the halo environment. Another approach to merger trees is numerical. They are based on the same principle as before, detecting populations at diﬀerent cosmological times to link them trough parenthood relationships. The diﬀerence is that we define the structures at diﬀerent timesteps from the snapshots of numer-ical simulations. Now, all the magic (and variety of applications) is in the algorithm used to identify structures. The most popular algorithms identify structures based on a percolation process, where particles in simulation are linked to each other if they are found to be closer than a given distance. The linking length, as it is called this distance, is expressed usually as a fraction of the mean distance between particles. Its value is selected to have objects (sets of particles) of a particular global overdensity. The numerical techniques based on this percolation algorithm have been also extended to account for this substructure in halos. Substructure identification algorithms are based on the detection of the highest density regions in the simulation and proper profiling of the density morphology in the neighborhood of these density peaks. [DEFW85, KGKK99, SWTK01]. Once the dark matter structure has been identified inside the simulation, diﬀerent halos (and subhalos if the algorithm permits) can be linked from one timestep to the next. This way of constructing merger trees is in prin-ciple more exact and correct than the Monte-Carlo approach, nevertheless the construction of the trees is obstructed by the fact that dark matter sim-ulations have a finite resolution, meaning that for a given halo at redshift zero, we can only describe its progenitors above a given minimum mass [HDN+03]. This fact is in contrast with the Monte-Carlo trees that have, in comparison, an infinite resolution.

**Star Formation**

We do not understand star formation. However, we have relatively detailed observational studies that can correlate diﬀerent environmental properties in a galaxy with this star formation process. These empirical scaling relations are at the heart of the semi-analytic models, which can implement these relationships as black boxes with a few eﬃciency parameters that can be chosen to reproduce observations.

**Table of contents :**

**1 INTRODUCTION **

1.2 The Theoretical Effort

1.3 The Computational Trap

1.4 The Semi-Analytic Hope

**2 GALAXY FORMATION **

2.1 Dark Matter Structure

2.1.1 Cosmological Scales

2.1.2 Galactic Scales

2.1.3 Halos and Merger Trees

2.1.4 Cooling

2.1.5 Star Formation

2.1.6 Feedback

2.1.7 Merging

2.2 Spectra

2.2.1 Stars

2.2.2 Gas

2.3 Observations

2.3.1 Low redshift

2.3.2 High redshift

2.3.3 Bimodality

2.4 Our model

2.4.1 Galics

2.4.2 Momaf

2.5 Our results

2.5.1 Geometry of Merger Trees

2.5.2 Predictability of SAMs

**3 THE INFRARED UNIVERSE **

3.1 Physical Processes

3.1.1 Heating

3.1.2 ExtinctioN

3.1.3 Reemission

3.2 Infrared Observations

3.2.1 1983

3.2.2 1985

3.2.3 1997

3.2.4 2003

3.2.5 2006

3.2.6 2008

3.3 Theoretical Predictions

3.3.1 Phenomenological models

3.3.2 Hierarchical models

3.4 Our results

**4 WEAK GRAVITATIONAL LENSING **

4.1 Theoretical Bases

4.1.1 One Lens, Point Source

4.1.2 One Lens, Extended Source

4.1.3 Cosmological context

4.2 Observations

4.2.1 Image from the shear

4.2.2 Shear from the image

4.2.3 Systematics

4.3 Numerical Simulations

4.3.1 Multiple Plane Theory

4.3.2 Ray tracing algorithm

4.3.3 The galaxies

4.4 Our Results

Appendix 1 Cosmology

Appendix 2 Codes from the projects

Appendix 3 Submitted Paper: Merger Trees

Appendix 4 Submitted Paper: Predictability

Appendix 5 Published Paper: Weak Lensing

**Bibliography**