As seen in the previous chapter, one of the central problems in cosmology is to understand the generation of structures in the Universe; i.e. evolution of cosmic density and velocity fields. There are epochs in the history of the Universe that one understands better than others. For example, the evolution of small perturbations ( ) is well understood, while models describing the growth of perturbations greater than unity are numerous (see Subsection 2.2.2). As of today, there are no model that fully recreates the gravitational formation of structures in the non-linear domain--so is also the case for the AVP. In this section, a detailed account of the approximations that AVP are based on will be given.

One of the most common and most simplifying approximations used is the Newtonian approximation. Einstein's theory of relativity is general and can be applied to most of the universe, but since its mathematics are not feasible (except for very simple forms of the metric), one usually tries to use the Newtonian approximation whenever possible. In the study of evolution of density perturbations, the choice of this approximation is well justified if one limits oneself to perturbations much smaller than the Hubble radius. This will be the case in this study, and hence, the Newtonian approximation applies.

Two propositions concerning the cosmic density and velocity fields are widely accepted:

**(1)**- The evolution of cosmic density and velocity fields on scales significantly larger than that of individual galaxies has been dominated by gravity.
**(2)**- The cosmic density and velocity fields started from fluctuations of very small
amplitude at early times.

These propositions leave strong constraints on the cosmic density and velocity fields, and are part of the standard model. Proposition (1) states that other forces than gravity on galactic scales can be ignored, including pressure.

Proposition (2) is of special interest here. It states that the mass that ended up in for example a galaxy, had practically no peculiar velocity in the early Universe. This makes, as first pointed out by P-89, cosmology into a two-point boundary-value problem; proposition (2) provides us with one of the boundary values and observations of the current positions of galaxies provide us with the other. More on this later in this section.

In the realm restricted by the approximations above, the linear theory can easily be
applied. The problem is, as commonly known, that the relative density of galaxies can
reach values of a few, even when smoothed on a scale 10 Mpc. This is well into
the non-linear regime. Two main approaches have been proposed for the calculations of
gravitational instability in the non-linear regime, namely the Zel'dovich approximation
(see Subsection 2.2.2) and Peebles' AVP [used a combination of the
two]Gia,S-B. The AVP is a variant of the classical Hamilton's principle from
mechanics, where the point particles have been replaced by galaxies, and ranges from
scale factor *a*=0 until today, . The system of galaxies is described by a scalar
potential created by gravity alone, and the correct path for these particles is where
the action is stationary, subject to the fixed end-points. More on Hamilton's principle
in Section 3.2.

From this short description of the AVP, one see that there are a few additional approximations to the ones mentioned above that needs to be justified before it can be applied to the study of evolution of galaxies. These are: galaxies must be treated as point masses, galaxies trace mass, and the end-points have to be fixed. Let us start with the latter, which is the most accepted one.

Hamilton's principle normally demands that the endpoints of the action are fixed; i.e. the positions at the end-points have to be known. As mentioned above, P-89 suggested that instead of having the position constrained at both end-points, one could have the velocity constrained at one of them, according to proposition (2). We then have a situation where the initial position and the final velocity are left unconstrained, while the initial velocity and the final position are fixed. This is an example of ``mixed'' boundary conditions. More on this in Section 3.3.

Treating the galaxies as point masses is to some extent a quite controversial approximation. One argument against it is that the galaxies can not have been distinct objects at early epochs of the universe, due to the mass distribution being almost homogeneous. P-90 stated that it still is a useful approximation, because at high redshifts the point-particles represent the center-of-mass motion of the matter out of which the galaxies is going to be assembled. This argument also holds for galaxy mergers at low redshifts; since we are only looking at the center-of-mass movement of the material now in a single galaxy--there is no reason why this could not have been two galaxies that have merged. The same argument goes for treating groups of galaxies as one mass tracer (this is the case for the groups in the LN).

Another argument against the point-mass picture is that galaxies are know to have quite
extensive halos (see Section 2.1), and that if two galaxies have had a
close passage, their halos may have overlapped. This is avoided in the calculations by
introducing a cutoff length *c* into the inverse square force law (see Subsection
5.1.1). However, the cutoff can often be ignored, due the systems usually being
so sparse that close passages between particles are very rare.

The question of galaxies being good tracers of mass has also been a subject of dispute. P-90 argued for the assumption both on the ground of observations and theoretical predictions. B-C and D-L95 compared the AVP solutions to a CDM N-body simulation in an Einstein-de Sitter universe, and showed that in that model there is a significant mass component that is more smoothly distributed than the mass tracers in the AVP, and therefore not measured by the motion of these mass tracers. The computation in the AVP will therefore underestimate the mean mass density, . SPT-95 and P-95 argued that this is contrary to the evidence from the redshifts of the galaxies in and near the LG, redshifts reached by AVP and other methods. More on on this dispute in Section 6.1.

The most powerful justification for the use of the approximations applied in the AVP is simply the success of the method; even though it leaves the final velocities of the galaxies unconstrained, it manages to reproduce the observed radial velocities to a surprising accuracy.

Mon Jul 5 02:59:28 MET DST 1999