Observable Events Curve - Hang On! Aren't I Double Dipping?

It could be argued that, in Observable Events Curve - Addressing a Point of Potential Confusion, I double dipped.

Note the argument involving a hypothetical racetrack along which a photon was approaching us (as observers).  I said that the location from which a photon would be travelling would recess away at a rate determined by its proper distance.  I effectively said that the photon would advance towards us at a rate given by c-v, where v is the speed of recession.  So, over an increment of time Δt, the photon would advance by (c-v).Δt.  During that increment of time, however, the entire racetrack will have expanded by a factor of (1+H(æ).Δt), where æ is the age of the universe (at that time, t).  Note that v=x'(t).H(æ), where x'(t) is the proper distance at t (and that the value of H(æ) will vary over the period Δt).

It could be argued that I used the expansion twice, first to retard the advancement of the photon and then to stretch the racetrack, pushing a sufficiently distant photon even further away than it started.

Well, yes, I did.  The question is whether this is valid.

I think it is. 

What this apparent double dipping is doing is accounting for the fact that the entirely of the metaphorical racetrack is expanding, but we have metaphorically put a pin in the observer’s location.  For a sufficiently long period ago, the proper distance to the location of the photon is small but the rate of expansion is great.

Speaking in terms of comoving distance, the photon is always advancing towards the observer.  For example, consider the cleaned-up chart in Observable Events Curve - Addressing a Point of Potential Confusion:

The purple dotted line illustrates the comoving distance to the location of the observed photon that it had at t=10,000 million years.  Note that the comoving distance is greater than any proper distance after the event and less than any proper distance before the event.  In that sense, the photon is always advancing on the observer, which is precisely what we would expect.  If we normalise the chart to consider distances from a comoving distance perspective, rather than the proper distance perspective above, we get this:

As could be expected, the photon actually advances towards the observer at a constant rate.  Which indicates that I was not, in fact, double dipping.

This graph also offer a different perspective on the underlying mathematics.  The events on the blue line are given by t.(c-v). Note the similarity of this equation in form to x'=x.(ct₀-x)/ct₀=ct.(ct₀-ct)/ct₀=t.(c-ct/t₀).

The implication, when understanding the nature of a FUGE universe, is that (in comoving terms) v=H₀.ct=ct/t₀.

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See a later post to see that "double dipping" may in fact be unavoidable.