Observable Events Curve - Is Double Dipping Essential?

In Observable Events Curve - Hang On!  Aren't I Double Dipping? (SPOILER - No, I am not), I addressed a self-evident concern that I have had to revisit. I wrote:

“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.”

Then I referred to this as “apparent double dipping”.

From there, I went on my merry way, until I ran into a conceptual roadblock.  That roadblock made me reconsider the content of Observable Events Curve - World Line, in which I point out that the OE curve could be thought of as the world line of a photon.

The precise nature of the roadblock is difficult to explain, but I will have a go.  Let’s start with the event horizon that I have subsequently described as the causal horizon, a point sufficiently distant from the observer that it expands away at the speed of light.  The underlying concept is that any photon emitted at the causal horizon will never interact with the observer (assuming no reduction to the rate of expansion).  It certainly seems obvious that anything recessing from us at the speed of light will never be observed.

Anything closer to us than the causal horizon can, eventually, be observed because its location is recessing at a subluminal rate.  All well and good so far.

Now recall that a FUGE universe is, by definition, granular.  In such a universe, the distance to the causal horizon expands by one unit of Planck length every unit of Planck time.  We can imagine it like this, for any given direction:

The red line at the bottom is our notional observer’s position. 

The green square is the notional additional “grain” of space (Planck length) for every additional “grain” of time (Planck time).

Say that soon in the process we have an occurrence (avoiding the term “event”) that generates a photon that has a velocity (speed and direction) towards the observer, at the farthest possible location.  The farthest possible location is moving away from the observer at the speed of light and the photon is travelling towards the observer at the speed of light (noting that photons can travel from one grain of space to an adjoining grain of space in one grain of time).

So, what will happen is that there will be an additional grain of space that appears, expanding the distance, and in the same grain of time in which that happens, the photon moves closer towards the observer by one grain of space:

This seems to mean that a photon that is generated at that distance would never reach the observer.  It is, of course, not that simple.  There is no reason why the additional grain of space should always appear right next to the observer.  Instead, it makes more sense that if an additional grain of space were to appear, it would do so at a random location across the distance.  And that could include locations on the other side of the separation between the photon and the observer, meaning that eventually the photon would be observed.

Showing this graphically is going to take more time than I can justify, but I can show it in a chart.  Look at where the distance is three grains, in that case, there are four possible locations for an additional grain:

We can use RANDBETWEEN(0,<max before expansion>) to select the location for the additional grain.  Say the universe radius is 100 grains at the time that the photon is generated at the 100^th grain.  This is what the chart looks like:

That’s a problem.  And it’s not a problem that is limited to a granular universe.  It’s simply that this curve would be smoother otherwise which is also what happens when the grains are relatively smaller.  Here is what it looks like with grains that are about 30 times smaller (orange curve) versus the non-granular option (green curve):

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If we “double dip” the problem goes away.  Say that we add two randomly located grains between the observer and the causal horizon and look at a photon on that causal horizon, we get this:

It doesn’t matter how often I recalculate, the chart remains the same.

The next step is to consider a situation similar to that of the OE curve.  An observer at some nominal time (t₀) observes a photon that was generated a distance x' away at time t ago.  We can set the time by selecting a causal horizon distance (in my example that is 1500 grains of time after expansion started) and the initial separation as an offset from that causal horizon distance (in my example that is 1/6^th, so 250 grains of space).  Two grains of space are added, with a random distribution across the distance to the causal horizon.  This is what we get:

This may look similar to the calculated OE curve, and indeed it is.  We can calculate it because we can work out what the value of t₀ is from the intercept with the vertical axis.  Overlaying the calculated OE curve, we get:

The correlation is remarkable.

For this reason, I must walk back my walking back of the claim that the OE curve represents the world line of the photon, or in other words, the path of the photon between spacetime between generation and observation.

That said, the apparent essentiality of the “double dipping” does need some more thought.  I am not convinced that it is real and suspect that it might be an artefact of the modelling, especially since the OE curve was arrived at using the assumption of a Hubble parameter that matches that of a FUGE universe.

I would certainly welcome someone’s ideas on this conundrum.