Lohan said:
How can can the observable universe be larger (i.e. larger in diameter, right?) than the Hubble volume??
Imagine you're living in a universe where the expansion rate (i.e. the Hubble constant) is constant not only in space (i.e., same everywhere at a given moment) but also in time (i.e., same everywhere and at any time).
In such a universe, if you draw a Hubble sphere (##H_R##) around yourself (observer), and then imagine a signal being sent from just outside ##H_R##, then its approach velocity of ##c## will not be sufficient to ever get closer to the observer because the recession velocity at that distance is greater than ##c##.
For example, a photon emitted just 1 metre outside the ##H_R## will have some very, very small net velocity away from you, that will only ever grow in time.
Now, make the Hubble constant change it time - specifically make it go down in time.
Look at the photon that was emitted 1 m outside the ##H_R##. Now, providing sufficient time has passed, and the Hubble constant went down sufficiently (which is equivalent to growth of the ##H_R##), the photon will find itself in a region where recession velocity is smaller than ##c##, and as a result will have some net velocity
towards the observer. From this point on, the photon is bound to eventually reach you to be observed. In other words, you will see the object that once emitted the photon - even though that object was outside the ##H_R##!
This shows that you can observe objects located at distances higher than ##H_R##
during emission.
Note that while the photon emitted just outside the ##H_R## is almost frozen in place (only from the observer's vantage point, locally it always moves at the speed of light), as its own velocity of c towards the observer combined with the recession velocity at that distance net something close to 0, the body that emitted the photon does not have such proper velocity and is instead carried away with full recession velocity. Hence while the photon of emitted light may sometime reach the observer, the object that emitted that photon will have been by that time carried away by the expansion much farther than where it had been when the light was emitted. This distance at which the emitter is NOW - at the time of reception of the once-emitted photon, is called proper distance. It is not where the emitter was when the light left it, but where it is now when the light reached the observer.
Proper distance of the farthest visible object (the CMBR) is called the Particle Horizon - or the radius of the observable universe.
The above paragraph works also for objects that emitted light from the inside of ##H_R##. All you need for the proper distance to be larger than ##H_R## is for the recession to carry the emitting galaxy away fast enough to overtake the growing ##H_R## by the time its signal reaches the observer.
This shows that you can observe objects located at distances higher than ##H_R##
during reception.In all of the above, the particulars of how far an emission of light can be outside the ##H_R## to be eventually observable, depends on the exact way the Hubble constant goes down with time. The more rapid the reduction, the farther a photon can be emitted.
Since the way it changes in our universe looks like this:
the ##H_R## was growing much faster in the past, and is approaching 0 growth in the far future (i.e., at the limit of infinity), which means photons could have been emitted farther away from the ##H_R## in the past than today, and still be observable. In the far future no photons emitted outside the ##H_R## will be ever observable.
