**3.4. Local Group Dynamics**

The mixed-boundary problem of a cosmological gravitating system
is solved for the trajectories of galaxies in the Local Group.
It is done using the least-action principle, under the assumption
that the mass is concentrated around the galaxies (*i.e.*, strict
no biasing).

*New Developments:*
Simulations are being used to estimate the biases in the method.

*Pro:*
This method is making use of accurate local measurements.

*Con:*
The assumption that the mass is all in point-like galaxies
during the whole evolution of the Local Group is likely to be wrong,
especially if _{m} is
high. It causes an overestimate of the forces
and thus an underestimate of
_{m}.
This is a stronger assumption than assuming no biasing in the statistical
sense, *b* = 1.
A related problem is the neglect of possible merging
that has taken place among the
initial subsystems in the Local Group.
The solution to the mixed-boundary problem may be non-unique.
The method treats the Local Group as an isolated system and
neglects possible tidal effects from external material.

*Current Results:*
Peebles [30] obtains
~ 0.15, under the assumption that
galaxies strictly trace mass.
However, it has been shown using N-body simulations
[31] that
the assumption that the mass is all in galaxies causes an underestimate
of _{m} by a factor 4 -
5 for any true _{m} in
the range 0.2 - 1.0. Shaya *et al. *
[32]
obtain a similarly low value using a similar
analysis inside a sphere of radius
~ 30 *h*^{-1}*Mpc*.