As mentioned previously, Hamilton's principle was first introduced to the study of the dynamics of galaxy groups by Peebles in his letter to the Astrophysical Journal in 1989, then referring to the method as the "least action principle". In the study of the formation of large-scale structures in the universe, he had come to the conclusion that Hamilton's principle lent itself as a most applicable method when considering the orbits of galaxies as a mixed boundary-value problem. This paper gave a short presentation of the method and some of its applications in cosmology, concentrating on the method's ability to predict the present velocities. Two systems of galaxies were treated: the eight galaxy system that the numerical methods were tested on in Section 5.3 of this thesis, and a system that was extended to include the two nearby groups Sculptor and Maffei, treated as point-particles. The predicted radial velocities of the former system was in good agreement with the observed values (see Table 5.2), the assumptions and approximations (treating the galaxies as point particles, rough mass model, limited number of trial functions, etc.) considered. These results were a strong argument in support of the method and for the assumptions that it was based on, especially since the method left the present velocities unconstrained. Peebles had managed to show that the method had an utilitarian value in the study of the dynamics of galaxies.

Peebles also applied different cosmological models, with km s Mpc and being the two central ones. (In the model, a cosmological constant was included.) He showed that if the LG can be considered an isolated system (the LG being represented by the eight galaxies in Table 5.1), would be preferred, while when including the two external groups, this dense model over-predicts the velocities of these groups. The Maffei and Sculptor groups would then have a large positive radial velocity due to the local mass density being less than the background. This is in conflict what is observed-- the LG being a general galaxy concentration. For the system of ten mass tracers, gave a better fit, but not as good as for the eight galaxy system. The main reason for the discrepancy in the larger system is the rough mass model and the bad quality of the distance estimates to the two external groups. Even though there was a certain offset between the observed and the predicted radial velocities, the method did indicate a small value for the density parameter with a non-negligible cosmological constant.

For most of the mass tracers in the two systems, the discrepancies in the radial velocities were acceptable on the basis of the assumptions and the limited number of galaxies included, except in the case of the NGC 6822 galaxy. This galaxy has been observed to have a small positive radial velocity, but the action solutions insisted on giving it a rather large negative value. A different numerical method came to the rescue of this problem: In the following year, 1990, Peebles gave a more extensive presentation of the AVP, including an application of the Newton-Raphson method in locating stationary values of the action. Prior to this, he had used the method of Steepest Descent, which, as we have seen in Subsection 4.3.1, is only able to locate minimum points. Using the Newton-Raphson method, Peebles was able to reveal saddle points which gave a small positive value for the radial velocity of the NGC 6822 galaxy. The galaxy seemed not to be falling towards the MW for the first time, but rather to have already passed by and now receding. Peebles thus showed that saddle points may very well be acceptable solutions to the action.

Additionally, in his paper from 1990, Peebles introduced the Bernoulli-shaped trial functions in (3.23) and applied the AVP in alternative cosmological models. On the latter, he came to the conclusion that the model with km s Mpc , , and a non-zero cosmological constant was preferable, thereby ruling out the Einstein-de Sitter universe. The former value of 50 for the Hubble parameter was found to give an unreasonable large age of the universe. Apart from being a more elaborate version of his paper from 1989, Peebles also introduced a new area of application for the AVP, namely the estimation of the primeval mass density fluctuations. Based on the system of ten mass tracers, he was able to give an estimate of the size of the mean density contrast that produced the concentration of galaxies within the LG, which was reasonably consistent with observed fluctuations in galaxy counts on large scales.

The AVP had clearly been shown by Peebles to be an interesting method, with a large array of applications--if correct. D-L were some of the first to take interest in the method, using the resultant orbits to determine the present angular momentum of the LG, estimating the direction of the spin axes of the MW and M31, and studying the tidal force acting on the LG and its galaxies. In this study, they extended Peebles' mass tracer system to include the M81 and NGC 5128 (Centaurus) group, and splitting the Maffei group into Maffei I, Maffei II, and IC 342, resulting in a system consisting of 13 mass tracers. By applying Peebles' preferred cosmology, km s Mpc and , they came to the conclusion that in the initial state of the Universe, the gravitational force on MW and M31 were dominated by galaxies outside the LG, and that the group therefore can not be considered an isolated system. They also observed that the addition of the external groups degrades the agreement that Peebles had between the observed and predicted radial velocities, indicating that a somewhat large value for would give better results. It should be noted that they adjusted the mass of the system by reproducing the observed line-of-sight velocity of M31, which resulted in a considerably larger mass of the system than used by Peebles.

Later that year, [Giavalisco et al.1993] released a paper where they tested a different
implementation of the AVP, making use of the Zel'dovich approximation. [The details will
be given in the next section. Only the results of their study will be discussed here.]
They showed that by conjoining these two methods, the convergence of the AVP was
significantly more rapid and the representation significantly more accurate. They were
also able to reduce the number of possible solutions when dealing with large systems. A
problem with the method is that the flow is laminar; i.e. no orbit-crossing, which is
only achieved when dealing with sparse samples where the small-scale non-linear relative
motion has been suppressed. This is not the case for the galaxies in the LG and LN.
Gia also outlined an extension to their implementation using the density
and velocity *fields* instead of the particle distribution. This
field-implementation was discussed further by S-B.

The same year, a presentation of the collaboration between Shaya, Peebles, and Tully was given at a conference on cosmic velocity fields SPT-93. A different implementation of the AVP was used [for details, see next section], allowing as many as 500 nearby groups of galaxies within 3000 km s in the calculations. The primary goal of their work was to obtain both position and velocity for the mass tracers as a function of time, ultimately giving an estimate of the density parameter. A more detailed presentation of their work was given two years later, SPT-95, in which a catalog of as many as 1138 mass tracers within 3000 km was studied. A major difference in the implementations of the AVP in these two presentations from previous works is that redshift is used as input to the calculations instead of distance. This gave a considerably improved quality of the input data to the AVP, redshift being a parameter that can be measured to a much higher accuracy than distance. The positions in the sky and the redshift were used to predict the present distances. They also introduced a way of predicting the mass of each galaxy by using lumnosities and mass-to-light ratio, M/L, as input to the model. All galaxies and groups were assumed to have the same M/L, and the Hubble and density parameter were adjusted to give a preferred age of the Universe. The results of these calculation were and Gyr (the Hubble age), assuming a space curvature. The value of the expansion time seemed uncomfortably short, and SPT-95 mentioned that an introduction of a cosmological constant making the Universe cosmologically flat, would give Gyr for .

Peebles published another paper on the subject in 1994 P-94, meant as a more
detailed account of the computations and the nature of the solutions. The original
implementation from his first two papers were considered, with some slight modifications:
First, the system of mass tracers was increased to include more galaxies and groups of
galaxies in the LN, resulting in a system of 14 mass tracers. Second, the solutions were
located solely by the use of the Newton-Raphson method, finding both saddle and minimum
points. Third, the orbits were constrained to redshift as well as to distance, applying
the method used by SPT-93,SPT-95, in addition to the Newton-Raphson method. And
finally, the mass of the system was not being determined by arriving at the observed
value of the radial velocity of M31. Instead it was set by giving the system a
preassigned mass-to-light ratio, *M*/*L*, the choice of value loosely determined by the
agreement between the observed and the model velocities for all the mass tracers in the
system.

In addition to serve as an elaborate presentation of the AVP and its different solutions for different cosmologies and M/L-ratios, P-94 also presented some updated results. For example, it was shown that ``the formation of the LG can have been heavily influenced by the interaction with neighboring mass concentrations'', the best fit of predicted to observed radial velocities is for a cosmologically flat model with Gyr and , and that the mass-to-light ratio is in the range of 100-200 solar units, giving little room for a cold intragroup mass component. Another important point made was that the prediction of redshift from orbits fixed to present distances is to be preferred over distances measured from redshift.

The AVP was by now starting to pick up some attention, and some were rather critical to both the assumptions the method were based on and its results. B-C tested the method against a CDM N-body simulation in an Einstein-de Sitter universe, and concluded that the AVP ``cannot rule out an value on the LG scale''. This differed from the results presented previously by for example P-89,P-90,P-94, which strongly indicated a small value for the density parameter . B-C used the N-body simulations somewhat differently from P-90 in the sense that the N-body created orbits for an arbitrary LG-like system of galaxies, onto which the AVP was applied. The implementation of the AVP was identical to P-89. B-C found that the AVP systematically underestimates the mass of the system compared to the N-body solutions in a CDM universe. They explained this discrepancy by the existence of extended overlapping halos which would not be detected by the AVP. On the basis of this, they introduced another restriction onto which systems the AVP was applicable: in addition to demanding the system to be dynamically young, the average relative separation between the galaxies had to be larger than Mpc. This excludes the LG, and they therefore concluded that the AVP applied to the LG cannot rule out an universe.

D-L95 did a similar study, also comparing the AVP orbits to a CDM N-body
simulation. They came to the same conclusion as B-C, that the AVP
underestimates in a CDM universe, but presented a different explanation. The
shape of the halos were found *not* to be the reason for the discrepancy between the
two methods, but rather particles in the CDM N-body not linked to any halos, referred to
as ``orphans''. The AVP does not take these particles into account, which is in conflict
with the assumption that light traces mass. D-L95 therefore reach the same
conclusion as B-C: the analysis of the LG using the AVP does not exclude
the Einstein-de Sitter universe.

SPT-95 answered to this criticism in their papers from SPT-95 SPT-95. They agreed that if there is a significant mass component that is more smoothly distributed than the mass tracers usually used in the AVP, the method will not take any notice of it. But, they claimed that this is in conflict with the evidence from the redshift of the galaxies in the LG and LN, both from the AVP and other studies of the dynamics of the LG [e.g.][]Zar. These works strongly indicates that solar units, and that the mass is concentrated within relative compact halos. ``There is no room for substantially more mass in halos extending to 1 Mpc'' SPT-95. The authors also pointed out that even if the galaxies have halos that extend beyond 1 Mpc, they will rarely overlap since the mass tracers used in the calculations are chosen on the grounds of having a crossing time much larger than the Hubble time. Although SPT-95 argued against a mass component distributed on scales of a few megaparsec, they were open to the possibility of a dark matter component with a broader coherence length, but deemed it unlikely. If such a mass component should exist on scales of, say, 10 Mpc, it need not have any effect on the dynamics on smaller scales.

P-95 devoted a large portion of his paper from P-95 to this dispute P-95, arriving at the same conclusions as SPT-95. He found the AVP solutions to be in good agreement with the results of Zar: ``if the radial velocities of the dwarf spheroidal galaxy Leo I and the Andromeda Nebula are the results of the gravitational assembly of the Local Group, then the mass-to-light ratio of the two dominant group members is solar units, and the mass of the Milky Way is concentrated within a radius of about 200 kpc.'' In this study, Peebles used a system consisting of twelve mass tracers in the LG and its immediate neighbourhood, including the dwarfs Fornax, Leo I, and Leo II. The effort was here concentrated on modeling the radial velocity of these dwarfs, finding solutions that were comparable to the results of Zar.

Apart from the physical aspect of this paper, P-95 also presented a new implementation of the AVP, using a parameterization of the orbits by their positions at discrete points in time. [More on this in the next section.] It is also worth noting that Peebles in this paper mentioned an optimizing method that shows similarities to the Secant method, but did not use it in the calculations.

P-96 released yet another paper on the subject of AVP in P-96
P-96. This time the effort was concentrated on the measurement of the primeval
mass fluctuation power spectrum *P*(*k*). The AVP was used to determine the initial
peculiar velocities of 18 galaxies in the LG and LN, which in turn were used to estimate
the power spectrum at high redshift. Peebles made use of the parameterization of the
orbits at discrete times, and applied a mass-to-light ratio of *M*/*L*=150 solar units.
Some of the main conclusions from this work were as follows: Firstly, Peebles once again
insisted that the galaxies trace mass due to action solutions giving redshifts of the
galaxies that are in reasonable agreement with observations. Secondly, the estimates of
the mass fluctuation power spectrum showed good consistency with other studies in the
case of low-density cosmological models, adding to to the evidence that galaxies trace
the nearby mass distribution.

Sch returned to the idea of using the observed redshifts as input to the AVP calculations, as discussed and rejected earlier by P-94. They gave a slightly different implementation of the AVP as we shall see in the next section. The resultant distances from the calculations were in good agreement with observations, at least for the dwarf galaxies in the LG. Based on these preliminary results, this implementation seems promising, but need to be tested on a larger and deeper systems of galaxies.

Mon Jul 5 02:59:28 MET DST 1999