WILLIAM & DEBORAH HILLYARD
Astronomy & Cosmology

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The Observable Universe

So; now let us consider that in the context of the real Universe.  The expansion rate is called the Hubble Constant, which is currently believed to be approximately 71 (±2.5) kilometers per second per Megaparsec, or about 13.5 miles/sec/million light-years.  So an object that is one Mpc away from us is receding at 71 km/sec; but an object one Gpc away, 1,000 times further, is receding at 71,000 km/sec!  At 13.7 billion light years away, 4.2 Gpc, the recession speed approaches that of light, but over the 13.7 billion years the light has traveled, the Universe has expanded so that the object that emitted that light is now 46.5 billion light years away.  Any objects further away from each other can not be seen as the fabric of space is expanding faster than the speed of light.  This means that an object that emitted light 13.7 billion years ago is now 46.5 billion light-years away from us as the expansion rate in that part of space has been more than the speed of light.
Ants on a Balloon Analogy

Firstly, imagine that our three dimensional Universe is represented as the two dimensional surface of a balloon; not the interior, just the surface.  It starts off deflated.  There are two ants on the surface, Anne and Betty, and they are two cms apart.  Anne starts to walk towards Betty (who remains stationary) at two cms per second, while the balloon inflates.  We measure expansion on the balloon's surface as a velocity-per-unit-distance; let us say that it's half a cm per second per cm.  This means that as Anne is 2 cms away from Betty, Betty expands away at one cm per second.  So, after one second, Anne has moved two cms towards Betty, but Betty has expanded away from Anne's starting point by approximately one cm, so Anne is still about one cm away from Betty.  The expansion continues as Anne walks so it takes just over half a second more to reach Betty.  Anne now looks back at his starting point, that was two cms from Betty.  After just over one and a half seconds, it is now just over one and a half cms FURTHER away, so the start point is now just over three and a half cms away, and moving further all the time at an ever increasing velocity, now at over 1.75 cms per second!  If you prefer Imperial measurements, just substitute inches for cms!
As we discussed earlier, the Universe (all the planets, stars, dust, galaxies etc) is NOT expanding into a void of empty space; it is space itself that is expanding and moving everything in it.  Of course, galaxies, stars, planets and the rest have their own motions, known as "proper motion" as Earth orbits the sun, the sun orbits the Galactic center, and the Galaxy moves toward the Andromeda Galaxy etc.  All this is overlaid on the underlying expansion of the fabric of space.  As space expands, it pulls all the matter with it.  Thus, the further apart two objects are, the faster they appear to recede from each other.  Eventually, objects are so far apart that the expansion rate, viewed cosmologically, is more than the speed of light, and we can not see them at all.  This is not a problem because it is not MOTION faster than light.  The fabric of space is not matter so it can do as it pleases, including expanding faster than light-speed.  Matter just gets taken along for the ride, but does not conflict with the rules of relativity.  In fact, while the Universe is around 13.73 billion years old, our horizon is a radius of approximately 46.5 billion light years (14 Gpc).  This is approximately 2.7x1023 miles!  I have included a brief explanation of this apparent paradox using the:
The sphere of space around us, with a diameter of 93 Gpc, is referred to as the "Observable Universe".  Taking a different location in space would result in a different observable universe.  If that location were outside our observable universe, then the two observable universes would have no volume in common.  If cosmic inflation theory is correct, Alan Guth, who first proposed the theory, estimates that the diameter of the entire Universe would be at least 1023 times larger than the observable universe, and possibly to 1026 times larger. 
Where "z" is the cosmological redshift, "a(tobs)" is the size of the Universe today, and "a(temt)" is the size of the Universe when the light was emitted. 
The redshift (zp) due to the proper motions between relatively close objects is approximated to "zp = v / c",
where "v" is the relative velocity and "c" is the speed of light. 
The Hubble Space Telescope is able to see objects that are "actually" about 32 billion light years from us.  This is known as the Comoving Distance.  Over time, the comoving distance to any distant object increases as it moves away from us.  However; at the time the object emitted the light we see today, it was only one or two billion light years away.  This distance correlates to the Angular Diameter Distance.  Looking at our "Ants on a Balloon Analogy", above, the Angular Diameter Distance would be two inches, while the Comoving Distance is a little over three and a half inches. 
Another measurement often seen for distant objects in particular is the redshift.  Essentially, because light always travels at the speed of light, when two objects are moving away from each other, the light is "stretched", looses energy, and appears to be redder.  Conversely, an object moving towards us would be shifted towards the blue; a blueshift.  The red-shift has two components; that caused by the expansion of space, called the cosmological redshift, and that caused by the local proper motions of the objects.  The cosmological redshift (z) is derived from the scale factor of the Universe quite simply as:
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