Annu. Rev. Astron. Astrophys. 1998. 36: 599-654
Copyright © 1998 by . All rights reserved

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During the past twenty years, numerical simulations of cosmic structure formation have become a powerful theoretical tool to accompany, interpret, and sometimes to lead cosmological observations. Simulations bridge the gap that often exists between basic theory and observation. They have found many uses, including testing and calibrating methods used to measure cosmological parameters, providing insight into nonlinear gravitational clustering and hydrodynamic turbulence, helping to explain the nature of systems such as quasistellar object (QSO) absorption lines, and highlighting shortcomings in the current physical modeling of galaxy formation. However, their main use has been and continues to be testing the viability of cosmological models of structure formation, such as the cold dark matter (CDM) model and its variants.

Structure formation models attempt to reduce cosmology to an initial value problem. Given the initial conditions - a background cosmological model with specified composition of matter, radiation, and exotic fields, such as a cosmological constant, and primordial fluctuations in the matter, radiation, and spacetime geometry - the goal is to compute using well-known laws of physics the evolution of structure from the Big Bang to the present day. At any useful level of abstraction, the Universe is an exceedingly complex system, hence analytic theory has only a limited (but nonetheless valuable) place in cosmic structure formation.

The current generation of cosmological simulations has antecedents that date back several decades. The first gravitational N-body simulation of interacting galaxies was performed using an analog optical computer (Holmberg 1941): Gravity was represented by the flux from 37 lightbulbs, with photocells and galvanometers used to measure and display the inverse square law force. The first astronomical N-body computations using digital computers were made in the early 1960s (von Hoerner 1960, 1963, Aarseth 1963). These early simulations were limited to at most about 100 particles. Gas dynamical simulations of galaxy formation began with the pioneering spherically symmetric calculations of Larson (1969). Increasingly large simulations of cluster collapse and evolution were performed throughout the 1970s (e.g. Peebles 1970, White 1976).

The first truly cosmological simulations of structure formation were the N-body integrations of Press & Schechter (1974) in their influential paper on the mass distribution of bound clumps formed by hierarchical clustering. At almost the same time, smaller simulations of cosmological clustering were performed by Haggerty & Janin (1974). This work was followed by numerous studies of the evolution of the two-point correlation function, a measure of galaxy clustering (Miyoshi & Kihara 1975, Groth & Peebles 1976, Fall 1978, Aarseth et al 1979, Efstathiou 1979, Gott et al 1979). Early work on galaxy formation was reviewed by Gott (1977).

The early 1980s saw several important developments leading to an explosion of activity in simulations of cosmic structure formation:

  1. Plausible physical models for dark matter had been proposed (Cowsik & McClelland 1972, Lee & Weinberg 1977, Bond et al 1980), including massive neutrinos (also known as hot dark matter or HDM) and cold dark matter (CDM; see Trimble 1987 for a review of dark matter).
  2. Cosmic inflation (Guth 1981) was shown to produce naturally the scale-invariant Harrison-Zel'dovich spectrum (Harrison 1970, Zel'dovich 1972) of primordial fluctuations in matter and radiation (Guth & Pi 1985 and references therein). In Omega = 1 (critical density) models, the initial conditions were thus reduced to specification of one number (the fluctuation amplitude) plus the composition of matter and radiation.
  3. Accurate numerical computations were made of the evolution of density fluctuations from their generation in the early Universe through recombination to the onset of nonlinear evolution (Peebles & Yu 1970, Bond & Szalay 1983).
  4. The theory of Gaussian random fields was developed and applied to the statistics of primordial density fields (Doroshkevich 1970, Bardeen et al 1986). Methods were developed to simulate Gaussian random fields with arbitrary power spectra (Efstathiou et al 1985, Peacock & Heavens 1985), using the Zel'dovich (1970) approximation to produce only the growing mode (Doroshkevich et al 1980, Dekel 1982).
  5. Grid-based N-body algorithms were applied to cosmology, enabling dark matter simulations with more than 105 particles to be performed (Section 2.1 below).

With all these developments occurring within a few years, there was great optimism among many working in this area that cosmologists were on the verge of understanding the formation of large-scale structure in the Universe. The CDM model became the paradigm of this new understanding (Peebles 1982, Blumenthal et al 1984, Davis et al 1985). As interest in structure formation grew during the 1980s, increasingly sophisticated tests were made of the CDM model. Problems began to appear, with the model seeming to show too little clustering on large (~ 50 h-1 Mpc, h = H0 / 100 km s-1 Mpc-1) scales compared with the real Universe when normalized to produce the correct amplitude on galaxy and cluster scales (see Ostriker 1993 for a review). The long sought after measurement of anisotropy in the cosmic microwave background radiation (Smoot et al 1992) highlighted and recast this problem: The CDM model has excessive power on small scales when normalized to produce the measured microwave background anisotropy (Efstathiou et al 1992). Although the optimism of the early 1980s waned, it was replaced by an appreciation that structure formation is a richer problem that needs the incorporation of much more physics into cosmological simulations, especially of gas dynamics for the ordinary ("baryonic") matter that is all we can see directly.

With the demise of the simplest detailed model of structure formation, attention has turned to variants that retain many of the attractive features of the CDM model while attempting to repair its deficiencies. These include replacing some of the CDM with light massive neutrinos (i.e. with HDM) or a cosmological constant; tilting the primordial spectrum; or including spatial curvature. Another class of models has the fluctuations seeded from topological defects like cosmic strings or global textures (Brandenberger 1994, Vilenkin & Shellard 1994) instead of quantum fluctuations produced during inflation. Texture models now appear to be inconsistent with the measured anisotropy of the cosmic microwave background (Pen et al 1997).

The last decade has seen an impressive growth not only in the size of cosmological simulations - hydrodynamic grids of 5123 with 16 million or more particles tracing the dark matter are now almost common - but also in the sophistication of the physics. Although baryons are thought to contribute anywhere from about 3-30% of the total mass in the Universe (depending on the uncertain value of Omega, the mean density of nonrelativistic matter in units of the critical density), they are responsible for 100% of the light we see, and they dominate the mass of the bulges and disks of galaxies. Cosmological gas dynamics has now come into maturity with a variety of algorithms being applied and compared with each other and with observations (e.g. Kang et al 1994).

This article reviews the techniques and results of cosmological structure formation simulations since the early 1980s. For the purposes of this article, cosmological simulations begin, by definition, with small-amplitude stochastic fluctuations in an expanding universe generated at high redshift. No attempt is made to review simulations of galaxies, galaxy groups, or clusters treated in isolation without such initial conditions. Only a limited discussion space is given to simulations of topological defects, computations of primary microwave background anisotropy (see White et al 1994, Bond 1996 for pedagogical reviews), analytic and semianalytic models of galaxy formation (White 1996), and simulations based on approximate quasilinear dynamics. While many simulations have been devoted to large-scale structure, that subject has been reviewed recently elsewhere (Dekel 1994, Strauss & Willick 1995, Efstathiou 1996) and no attempt is made at a comprehensive summary here. Sections 2 and 3, summarizing simulation and analysis methods, present technical matter of interest primarily to experts. Others may wish to skip directly to Section 4.

As this article was prepared, a bibliography of approximately 900 refereed articles relevant to cosmological simulations was compiled. This bibliography is available on the World Wide Web in the Supplemental Materials Section of the Annual Reviews site (

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