Vr = HoD
where Vr is the radial velocity away from the observer, Ho is a constant (called Hubble's constant), and D is the distance to the galaxy. This linear relation between Vr and D means the Universe is in a state of uniform expansion. The slope of this linear relation is the expansion rate or Ho.
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Figure 2.4 Schematic representation of
velocity-distance relation. Each of the 5 galaxies is separated
from another by a distance D and thus each galaxy
has a velocity with respect to one another of V .
No matter which galaxy your are located on, you recover the slope
of the dashed line. The case illustrated here is from the perspective
of an observer on Galaxy 1.
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There is always a good deal of confusion that students have when it comes
to understanding what expansion actually means. As we will discuss
later, our modern cosmological model is known as the Big Bang model
and that always conveys the wrong impression to students concerning
the driver of the expanding Universe. We can best demonstrate
what is occurring by considering the following thought experiment
(which could be a real one if you found some cooperative ants).
Next, get a balloon and inflate it about half way. Sprinkle some ants onto the surface of the balloon and continue to inflate it. Notice that while you are inflating the balloon the ants are walking randomly around its surface. Some will be walking towards each other and some away. There will be no well-defined correlation between the distance between any two ants and their relative motion. This again is not what we observe and indicates there is some controlling agent that is causing the relative motion between galaxies to depend on distance.
Finally, get some glue (super glue works best) and glue the ants to the balloon (this is actually quite hard to do unless you have a really good pair of tweezers). As you inflate the balloon you will notice that the ants are no longer moving. The ants are stationary but the separation between each ant is increasing as the surface of the balloon increases as a result of it being inflated. If you pick any one ant on the balloon, all the other ants appear to be moving away from that ant. It matters not which ant you pick. Hence, every ant believes they are in the center of the ant distribution when in fact you know that there is no center because the ants are distributed on a surface. So what happens when you let the air out of the balloon? The ants all approach each other until they are all together (more or less) in one big ant glue-ball.
While this analog may seem silly, in fact, it's quite appropriate to the real Universe. Gravity is the "glue" that fixes the galaxies to the surface of the Universe. In the case of the Universe, this surface is a four dimensional spacetime surface which is embedded in some five dimensional volume. As this is rather difficult to draw on a two dimensional piece of paper, we use the analogy of the two dimensional balloon surface embedded in a three-dimensional volume. The key is to adopt the ant perspective. The ant is a two dimensional creature that really doesn't know it's on a surface. As we just saw above, the surface of the balloon has no center and the collapse of the balloon means the surface area has decreased (in principle, to zero).
In the case of our Universe, as the surface expands, the separation
between all the galaxies increases and the observational manifestation
of this is galaxy redshifts. This uniform expansion of the Universe
makes a clear and important prediction. If galaxies are getting farther
apart from each other due to the expansion of the surface, then in
the past the galaxies were much closer to one another. Indeed, there must
have even been a time when all the galaxies (all the matter) in the
Universe were together in the same space at the same time. This means
that the early Universe was a very small dense place and
in a physical state well-removed from how it is observed to be today.
If the early Universe is governed by a state of rapid expansion and cooling then its very dynamic in nature. The physicist George Gamow explored this nature in a series of theoretical investigations in the 1940s. In so doing, Gamow postulated a very important condition that must have existed in the early Universe. At sufficiently high temperatures where the average energy of a photon would exceed the rest mass energy of a neutron, then it would not be possible for any atomic nuclei to exist. If one formed, it would be immediately photo-dissociated by one of these high energy photons. Hence, in this state, the Universe had to consist of a sea of elementary particles at very high temperature. As a result of this state, the current abundance of helium (the simplest element that contains neutrons) must be related to how much radiation was present. If too much radiation was present, helium might never have formed. Gamow used the estimated helium abundance of the Universe at that time to make a prediction about the level of radiation that now exists in the Universe. He predicted that the Universe should be filled with photons of characteristic wavelength a few millimeters. At the time this prediction was made, no technology existed to try and detect this background. This again illustrates the principle that advances in cosmological models often require having the right kind of detector available to test predictions.
V = HD 
c = HD ==> D =c/H ==> This is our causal horizon -
beyond this distance something would have to travel faster than the
speed of light in order to communicate with us. All observers
are surrounded by such a horizon.
Horizons are okay. Our assumption about homogeneity means that the stuff beyond the horizon is the same stuff we already know about. This assumption must be correct due to horizon overlaps and causality.
Back to the Ants glued to the balloon:
Example: I attach the balloon to a slow pump which increases the
radius of the balloon by one foot each day. This is the expansion
rate that I measure. I measure the balloon to have a radius of
8 feet. This means the expansion age of the balloon is 8 days.
V = Hd ==> 1/H = D/V
Distance/Velocity = Time
1/H = the expansion age of the Universe.
This is how long the
Universe has been expanding. What it was doing prior to the expansion
is anybody's guess.
At some earlier time, all the galaxies had to have been together
in the same space at the same time
Run time backwards and
realize that all the galaxies used to be together.
Key Points:
If you know the rate of inflation of the balloon (the expansion
rate of the surface) and the surface area of the balloon (which
is proportional to its radius) then you can determine how long it
has taken for the balloon to reach its present size.
So what do we know now:
the Universe was once
really small. It is important to realize that the galaxies are
stuck to the surface of the universe by gravity and its the surface
that expands. The galaxies themselves are not moving but travel
along with the surface as shown here .