Observable Events Curve - Fast and Slow Expansion

The fundamental concept of an OE curve applies irrespective of what sort of expanding universe we are considering – so it applies to both a FUGE universe and one described by the Standard Cosmological Model.  The transit distance of all photons that are observed will include both the initial proper distance (the separation between the emitting phenomenon and the observer’s location, at the time of the emitting phenomenon, according to the observer) and a component due to expansion that occurred during the time of transit (although this second component will be so tiny as to be negligible for small proper distances, for example a photon from Proxima Centauri will have experienced ~0.04 light seconds of expansion while transiting the 4.24 light years distance).

If expansion is consistently rapid (H(t)>1/t), then there will be phenomena that will never be observed from a specific observation point.  If, on the other hand, the universe expands slowly (H(t)<1/t), then all past events will be observable from any given point eventually (barring something catastrophic in the meantime like proton decay).  In a modified FUGE universe, where only the value of H(t) is different, this would mean that we could only see a subset of events that occurred in the past.

Imagine that a phenomenon in the past emits a photon that later interacts with an observer that is stationary with respect to the phenomenon (or rather their coordinates are “comoving”).  That event is at a proper distance x' at time of emission.  During transit to the observer, space will expand by a certain amount, call that Δx.  The time between emission and observation is t, so the observer considers the phenomenon to have occurred a period t ago, at an apparent distance of x=ct.  This gives us one definition for Δx: since x=x'+Δx, then Δx=x-x', and thus also x'=x-Δx.  The other definition for Δx is based on an expansion factor.  Note that we are using t to refer to time elapsed since events in the past occurred.  Because of this usage, the age of the universe is given by t₀-t.  Therefore, the equation for the Hubble parameter in a FUGE universe would be given by H(t₀-t)=1/(t₀-t).  And, for an observer at t=0, H(t₀-t)=H₀=1/t₀.

Assuming constant expansion since the instanton, Δx=H(t₀-t).x.t, so:

x'=x-H(t₀-t).x.t=x-H(t₀-t).x.ct/c=x.(1-H(t₀-t).x/c)

Which, currently, where t₀-t=t₀ would be, given that H₀=1/t₀:

x'=x.(1-H₀.x/c)=(1-x/ct₀).x=(ct₀-x).x/ct₀

Therefore, for a rapidly expanding universe (H(t)=5/(4t)):

and this for a slowly expanding universe (H(t)=4/(5t)):

In the first chart, the events above the red line are those which will never be observed (at the selected observation point).  The second is more difficult to interpret, at least above and to the right of the grey line illustrating the expansion of the universe.  There are events along the curve that are “outside” the universe and it’s unclear what would happen there, perhaps the observer would see the back of the head of an ancient version of itself?  Or perhaps the universe would just appear to end at about 0.76t₀ ago (and this would be the case for everyone in the universe, irrespective of their location)?