- Gravitational Instability In the standard model of cosmology, structures form from small initial perturbations in an otherwise homogeneous and isotropic universe. The idea that structures can form via gravitational instability in this way originates from Jeans (1902), who showed that the stability of a perturbation depends on the competition between gravity and pressure. Density perturbations grow only if they are larger (heavier) than a characteristic length (mass) scale [now referred to as the Jeans’ length (mass)] beyond which gravity is able to overcome the pressure gradients. The application of this Jeans criterion to an expanding background was worked out by, among others, Gamow & Teller (1939) and Lifshitz (1946), with the result that perturbation growth is power-law in time, rather than exponential as for a static background.
- Initial Perturbations Most of the early models of structure formation assumed the Universe to contain two energy components, ordinary baryonic matter and radiation (CMB photons and relativistic neutrinos). In the absence of any theory for the origin of perturbations, two distinct models were considered, usually referred to as adiabatic and isothermal initial conditions. In adiabatic initial conditions all matter and radiation fields are perturbed in the same way, so that the total density (or local curvature) varies, but the ratio of photons to baryons, for example, is spatially invariant. Isothermal initial conditions, on the other hand, correspond to initial perturbations in the ratio of components, but with no associated spatial variation in the total density or curvature.
In the adiabatic case, the perturbations can be considered as applying to a single fluid with a constant specific entropy as long as the radiation and matter remain tightly coupled. At such times, the Jeans’ mass is very large and small-scale perturbations execute acoustic oscillations driven by the pressure gradients associated with the density fluctuations. Silk (1968) showed that towards the end of recombination, as radiation decouples from matter, small-scale oscillations are damped by photon diffusion, a process now called Silk damping. Depending on the matter density and the expansion rate of the Universe, the characteristic scale of Silk damping falls in the range of 
.
After radiation/matter decoupling the Jeans’ mass drops precipitously to 
. and perturbations above this mass scale can start to grow, but there are no perturbations left on the scale of galaxies at this time. Consequently, galaxies must form ‘topdown’, via the collapse and fragmentation of perturbations larger than the damping scale, an idea championed by Zel’dovich and colleagues.
In the case of isothermal initial conditions, the spatial variation in the ratio of baryons to photons remains fixed before recombination because of the tight coupling between the two fluids. The pressure is spatially uniform, so that there is no acoustic oscillation, and perturbations are not influenced by Silk damping. If the initial perturbations include small-scale structure, this survives until after the recombination epoch, when baryon fluctuations are no longer supported by photon pressure and so can collapse. Structure can then form ‘bottom-up’ through hierarchical clustering. This scenario of structure formation was originally proposed by Peebles (1965).
By the beginning of the 1970s, the linear evolution of both adiabatic and isothermal perturbations had been worked out in great detail (e.g. Lifshitz, 1946; Silk, 1968; Peebles & Yu, 1970; Sato, 1971; S. Weinberg, 1971). At that time, it was generally accepted that observed structures must have formed from finite amplitude perturbations which were somehow part of the
initial conditions set up at the Big Bang. Harrison (1970) and Zel’dovich (1972) independently argued that only one scaling of the amplitude of initial fluctuations with their wavelength could be consistent with the formation of galaxies from fluctuations imposed at very early times. Their suggestion, now known as the Harrison–Zel’dovich initial fluctuation spectrum, has the property that structure on every scale has the same dimensionless amplitude, corresponding to fluctuations in the equivalent Newtonian gravitational potential, 
In the early 1980s, immediately after the inflationary scenario was proposed, a number of authors realized almost simultaneously that quantum fluctuations of the scalar field (called the inflaton) that drives inflation can generate density perturbations with a spectrum that is close to the Harrison–Zel’dovich form (Hawking, 1982; Guth & Pi, 1982; Starobinsky, 1982; Bardeen et al., 1983). In the simplest models, inflation also predicts that the perturbations are adiabatic and that the initial density field is Gaussian. When parameters take their natural values, however, these models generically predict fluctuation amplitudes that are much too large, of order unity. This apparent fine-tuning problem is still unresolved.
In 1992 anisotropy in the cosmic microwave background was detected convincingly for the first time by the Cosmic Background Explorer (COBE) (Smoot et al., 1992). These anisotropies provide an image of the structure present at the time of radiation/matter decoupling, ∼400,000 years after the Big Bang. The resolved structures are all of very low amplitude and so can be used to probe the properties of the initial density perturbations. In agreement with the inflationary paradigm, the COBE maps were consistent with Gaussian initial perturbations with the Harrison– Zel’dovich spectrum. The fluctuation amplitudes are comparable to those inferred by Harrison and Zel’dovich. The COBE results have since been confirmed and dramatically refined by subsequent observations, most notably by the Wilkinson Microwave Anisotropy Probe (WMAP) (Bennett et al., 2003; Hinshaw et al., 2007). The agreement with simple inflationary predictions remains excellent.
c. Nonlinear Evolution In order to connect the initial perturbations to the nonlinear structures we see today, one has to understand the outcome of nonlinear evolution. In 1970 Zel’dovich published an analytical approximation (now referred to as the Zel’dovich approximation) which describes the initial nonlinear collapse of a coherent perturbation of the cosmic density field. This model shows that the collapse generically occurs first along one direction, producing a sheet-like structure, often referred to as a ‘pancake’. Zel’dovich imagined further evolution to take place via fragmentation of such pancakes. At about the same time, Gunn & Gott (1972) developed a simple spherically symmetric model to describe the growth, turn-around (from the general expansion), collapse and virialization of a perturbation. In particular, they showed that dissipationless collapse results in a quasi-equilibrium system with a characteristic radius that is about half the radius at turn-around. Although the nonlinear collapse described by the Zel’dovich approximation is more realistic, since it does not assume any symmetry, the spherical collapse model of Gunn & Gott has the virtue that it links the initial perturbation directly to the final quasiequilibrium state. By applying this model to a Gaussian initial density field, Press & Schechter (1974) developed a very useful formalism (now referred to as Press–Schechter theory) that allows one to estimate the mass function of collapsed objects (i.e. their abundance as a function of mass) produced by hierarchical clustering.
Hoyle (1949) was the first to suggest that perturbations (and the associated protogalaxies) might gain angular momentum through the tidal torques from their neighbors. A linear perturbation analysis of this process was first carried out correctly and in full generality by Doroshkevich (1970), and was later tested with the help of numerical simulations (Peebles, 1971; Efstathiou & Jones, 1979). The study of Efstathiou and Jones showed that clumps formed through gravitational collapse in a cosmological context typically acquire about 15% of the angular momentum needed for full rotational support. Better simulations in more recent years have shown that the correct value is closer to 10%. In the case of ‘top-down’ models, it was suggested that objects could acquire angular momentum not only through gravitational torques as pancakes fragment, but also via oblique shocks generated by their collapse (Doroshkevich, 1973).
