One of the most challenging and exciting problems of Astronomy is to understand how structure forms and evolves in the Universe. The observable Universe presents itself to the human eye structured with different classes of celestial bodies and matter distributed on different length scales. On small scales the Universe looks heterogeneous with planets, stars, and interstellar gas appearing spatially distributed on preferred directions of the sky. On progressively larger scales, the Universe appears progressively less heterogeneous, with galaxies, groups of galaxies and galaxy clusters being spatially distributed in a structured pattern, expressively called the cosmic web, without apparent preferred directions. Why is the Universe structured in such a way? What’s the origin and physical mechanism responsible for such structure?  

The first module of the course will be dedicated to an overview of the main ideas and ingredients that lead to our present understanding of how structure forms and evolves in the Universe. We will review the Standard Model of Cosmology, the main observational evidences that support our present understanding of the structure formation process as well as the main cosmological probes capable of providing most valuable information about structure evolution.

Presently, the most popular mechanism of structure formation is based on fundamental discoveries, on both theoretical and observational fronts, during the first half of the twentieth century. Structure is believed to have grown via gravitational instability from passive density perturbations produced in a primordial phase of the Universe. In 1902 Jeans was the first to demonstrate that small density perturbations can grow with time in a homogeneous and isotropic self-gravitating fluid. The theory of gravitational instability was applied to an expanding universe by Lifshitz in 1946, which presented the first general analysis on the evolution of inhomogeneities in Friedman-Lemaitre-Robertson-Walker background models using linear perturbation theory. As long as the perturbations are small they are indeed well described in linear theory. However the accuracy of the theory starts to degrade as perturbations grow larger and higher-order terms need to be considered at later times. Eventually the perturbation theory itself breaks down at the point when complex non-linear structures form. At this stage other methods need to be used to follow the evolution of structure.

In this module we will discuss the linear versus non-linear growth of cosmological structure. After an introduction of the main statistical quantities used to describe the density field, we will review the main aspects of the linear growth of perturbations. The module ends with a discussion about methods that can be used to follow the quasi-linear and non-linear evolutionary regimes of cosmological structure.

Over the past two decades, numerical N-body simulations have become the most powerful technique to investigate the evolution of cosmological structure.  Complex non-linear physics, acting over a wide range of scales, makes the structure formation process hard or impossible to describe without the use of numerical computation methods. These methods follow the growth of structure in time by integrating the equations of motion of billions of individual particles that evolve under the action of gravity and other physical forces involved.  Numerical simulations which only account for the effects of gravity are appropriate to describe the evolution of colissionaless systems such as the cold dark matter component of the Universe. These are usually referred to as N-body simulations. To describe the evolution of collisinal systems such as those including baryonic matter (gas), simulations need also to account for the dynamics of the gas and to include the appropriate gas physics for the range of mass and time scales probed by the simulations. These are known as hydrodynamic N-body simulations.

This module is dedicated to the non-linear evolution of the (collissionaless) dark matter component of the Universe. We will introduce the equations of motions for collisionalless systems and discuss the main numerical techniques used to follow evolution of structure. Emphasis will be given to the discussion of the main features, performance and physical approximations involved, rather than discussing the detailed numerical implementation of different techniques. We will end with an overview of some of the most important achievements obtained with N-body simulations and discuss the importance of the gas dynamics to obtain a more realistic description of the formation and evolution of cosmological structure.

A description of structure formation exclusively based on the behaviour of collissionaless matter under its self-gravity is obviously insufficient. Gas dynamics, shocks, non-gravitational heating and pressure gradients become progressively important as structures evolve. Hydrodynamic effects are clearly important to describe the evolution of the baryonic component of the Universe, and play a central role in determining how collapsing structures, such as galaxy groups and clusters, reach virial equilibrium.

Hydrodynamic N-body simulations are today’s most powerful technique to study the non-linear evolution of structure in the Universe. They allow a direct treatment of physical effects such as gas shocks, radiative dissipation of energy and non-gravitational heating which are very difficult or impossible to describe with other methods. In this module we will review the main features of gas dynamical simulations, describing different techniques, physical approximations and main results obtained. We will end with a discussion of the main successes and today’s most pressing challenges posed to hydrodynamic simulations to attain the best possible description of how structure forms and evolves in the Universe.