[...] Some areas of the universe, the majority to be sure, have a current density below the critical density,
(#1)
of their local area, and they have a tendency to expand. Other areas have a density above the critical density and they will want to collapse. In the past the critical density was no doubt different and the actual density was different, but I'm sure there still would have been a density inhomogeneity where some areas were above the critical density of the time and others below.
#1: ...where lambda is zero and H is the recession velocity in the area under consideration
[...]
The formula,
, works not only for finding the critical density of the visible universe as a whole, but any particular part of it as well. If you draw a sphere around an area of interest and you want to know if that area will collapse or expand in the future then mark a point in the center of the sphere. The velocity between the point and the edge is H. Using that to find pc and comparing it to the real measured density, p, will say if the area will eventually collapse or expand.
The important point, i think, is that inhomogeneities existed in the past. Some areas were above their local pc and some areas below. The areas of segregated mass that we see now are a result of that.
[...]
Ignoring the cosmological constant (just as an approximation) if the distance you are talking about is D, the measured density of the area you are talking about is p, the measured velocity of the "far out-into-deep-space" point as compared to the starting point is V, and the gravitational constant is G then the area will expand if,
and will contract if,
You can get this by solving the escape velocity of an expanding sphere.