Having specified the initial conditions and the cosmological framework, one can compute how small perturbations in the density field evolve..., in an expanding universe dominated by non-relativistic matter, perturbations grow with time. This is easy to understand. A region whose initial density is slightly higher than the mean will attract its surroundings slightly more strongly than average. Consequently, over-dense regions pull matter towards them and become even more over-dense. On the other hand, under-dense regions become even more rarefied as matter flows away from them. This amplification of density perturbations is referred to as gravitational instability and plays an important role in modern theories of structure formation. In a static universe, the amplification is a run-away process, and the density contrast δρ/ρ grows exponentially with time. In an expanding universe, however, the cosmic expansion damps accretion flows, and the growth rate is usually a power law of time, δρ/ρ ∝ tα , with α > 0..., the exact rate at which the perturbations grow depends on the cosmological model.
At early times, when the perturbations are still in what we call the linear regime (
), the physical size of an over-dense region increases with time due to the overall expansion of the universe. Once the perturbation reaches over-density δρ/ρ ∼ 1, it breaks away from the expansion and starts to collapse. This moment of ‘turn-around’, when the physical size of the perturbation is at its maximum, signals the transition from the mildly nonlinear regime to the strongly nonlinear regime.
The outcome of the subsequent nonlinear, gravitational collapse depends on the matter content of the perturbation. If the perturbation consists of ordinary baryonic gas, the collapse creates strong shocks that raise the entropy of the material. If radiative cooling is inefficient, the system relaxes to hydrostatic equilibrium, with its self-gravity balanced by pressure gradients. If the perturbation consists of collisionless matter (e.g. cold dark matter), no shocks develop, but the system still relaxes to a quasi-equilibrium state with a more-or-less universal structure. This process is called violent relaxation and will be discussed in Chapter 5. Nonlinear, quasi-equilibrium dark matter objects are called dark matter halos. Their predicted structure has been thoroughly explored using numerical simulations, and they play a pivotal role in modern theories of galaxy formation. Chapter 7 therefore presents a detailed discussion of the structure and formation of dark matter halos. As we shall see, halo density profiles, shapes, spins and internal substructure 8 Introduction all depend very weakly on mass and on cosmology, but the abundance and characteristic density of halos depend sensitively on both of these.
In cosmologies with both dark matter and baryonic matter, such as the currently favored CDM models, each initial perturbation contains baryonic gas and collisionless dark matter in roughly their universal proportions. When an object collapses, the dark matter relaxes violently to form a dark matter halo, while the gas shocks to the virial temperature, 
and may settle into hydrostatic equilibrium in the potential well of the dark matter halo if cooling is slow.
