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

**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
(*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,
(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.