Annu. Rev. Astron. Astrophys. 1998. 36: 599-654
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5.6. Testing Approximations for Nonlinear Gravitational Dynamics

Theoreticians devise approximations to nonlinear gravitational clustering with three purposes in mind: (a) to understand the nonlinear dynamics arising in simulations and thereby perhaps understand the Universe, (b) to replace expensive simulations with fast approximations, and (c) to relate the present-day distribution of galaxies directly to the initial conditions for structure formation. Many approaches have been adopted; only a brief and incomplete synopsis of recent work is given here. An early review was given by Shandarin & Zel'dovich (1989). Sahni & Coles (1995) provided an excellent comprehensive pedagogical review.

The Zel'dovich (1970) approximation (Equation 6) gives an accurate description of motion for pressureless dark matter (and even baryons or HDM on scales larger than the Jeans or free-streaming lengths). However, it breaks down once trajectories intersect; particles never turn around to orbit in bound systems. Several approximations have been suggested to cure this defect. The first was to add an approximate viscosity term to the equation of motion to prevent trajectories from crossing (the adhesion approximation of Gurbatov et al 1989). A simpler method is to prefilter the linear density fluctuation field with a window of radius large enough so that sigma(R) = 1 before applying the Zel'dovich approximation (the truncated Zel'dovich approximation of Kofman et al 1992). In both methods, coherent motion is reduced on small scales in virialized regions, which is in agreement with the fully nonlinear evolution of the gravitational potential (Melott et al 1996).

Sathyaprakash et al (1995) compared these modified Zel'dovich approximations and other dynamical approximations against N-body simulations, finding that the truncated Zel'dovich approximation is favored because of its simplicity and accuracy. Besides being slower, the adhesion approximation does not conserve comoving momentum. Shandarin & Sathyaprakash (1996) presented a promising new fast approximation that conserves momentum by replacing the Burgers equation of the adhesion approximation with the Navier-Stokes equation of viscous fluid flow.

The Zel'dovich approximation may be regarded as the first-order perturbation theory for the trajectories of mass elements. Higher-order Lagrangian perturbation theory would include additional terms in a power series in D in Equation 6. The second-order perturbation theory has been applied and compared against simulations of hierarchical models by Melott et al (1995), Bouchet et al (1995), who concluded that it gives significant improvements over the Zel'dovich approximation, particularly when the initial density field is smoothed (truncated) at high wavenumbers. Karakatsanis et al (1997) have improved these methods further by artificially slowing down the growth of D(t) in the perturbation series to prevent the displacements from growing too rapidly.

The least-action principle provides an alternative formulation of gravitational dynamics that underlies several new approximations. Peebles (1989, 1994; see also Giavalisco et al 1993) introduced the least-action method as a way to trace galaxy orbits back in time, given the final positions and requiring that the peculiar velocities vanish initially. By requiring the final velocities to match observations, the mean mass-to-light ratios of the galaxies can be deduced and, from this, Omega (Shaya et al 1995). Although simulations have raised questions about the reliability of this estimate (Branchini & Carlberg 1994, Dunn & Laflamme 1995), the least-action principle offers a powerful approach to dynamical approximations. Its difficulty lies in being nonlocal: The motion of all mass elements must be considered simultaneously to minimize the action. Its advantage over other techniques lies in the ability to reconstruct the initial conditions. Two impressive implementations of this idea have been published recently. Susperregi & Binney (1994) worked with Eulerian density and velocity fields, whereas Croft & Gaztañaga (1997) used straight-line Lagrangian trajectories. Their methods agree reasonably well with N-body simulations and offer the hope that, with data from large redshift surveys, similar methods may allow accurate reconstruction of the initial density fluctuation field.

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