Observable Events Curve - Not Quite a Drunkard's Walk

I retracted this for a while, before reinstating it.  There might be more some edits required.

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In Observable Events Curve - Is Double Dipping Essential? I showed the output of a relatively simple Excel sheet which models how a photon moves towards an observer in a universe with granular expansion.

I then got to thinking about how I could use the underlying principles to trace back the possible path(s) of an observed photon, at a given time, with different characteristics of the universe in the past.

So I wrote a script in VBA to generate a table containing sufficient data to create the chart that represents one possible path of the photon, which I could refresh as often as I liked to see what happens.

In brief, I wanted two randomly generated locations between the observer and the causal horizon (where the causal horizon is the limit beyond which a photon travelling in the direction of the observer would never reach that observer where the causal horizon is the limit [at the time] at which an object [at that distance] would recede from the observer at the speed of light).  I assumed that the reverse path of the photon would be one “grain” of additional space each “grain” of time, minus any “grains” of space that are inserted between the photon and the observer.  This might sound odd, but note that, if ran forward, the same logic would lead to a path of the photon that reduces by one “grain” of space per “grain” of time, and any expansion “grains” between the photon and the observer would increase the separation between them.

The effect is a little like a random walk (or drunkard’s walk) with the photon going through the repeated process of taking one step away and then a random number of steps (zero, one or two) back towards the observer’s location.  The likelihood of each number is determined by the separation between the observer and the photon as compared to the separation between the observer and the causal horizon at the time.  As the ratio of separations approaches zero, zero becomes increasingly more likely and as the ratio approaches one, two becomes more likely.

The script produces a table which I then charted to create a cloud of random locations at which “grains” of space are located at a given time (plotted just as points, not the lines between the points, with a brown point and a blue point for each given time), the path of the photon through spacetime (red), the OE Curve for a FUGE universe (green) and the causal horizon (grey).  For a FUGE universe that is 14 billion years old, it looks like this:

If I modify the inputs to make it a Standard Model universe – meaning that the expansion rate is initially H(t)=2/3t and then such that there is accelerated expansion post the beginning of a Dark-Energy-Dominated Era 4 billion years ago – I get this:

 

Note the problem: a photon that is observed today cannot have originated at recombination, 380,000 years after the Big Bang.  This implies that, in the Standard Model, the most ancient photon that we could possibly observe would have originated from an event about 12.27 billion years ago.

Note also that for the past 2.25 billion years, the path of the photon would have been very similar to that of a photon in a FUGE universe (within a couple of percent).

Of course, I may have made an outrageous assumption here that has resulted in me being completely incorrect, but it doesn’t seem so.  This is all very close to first principles.  Note also that I can refresh repeatedly, getting different clouds of random points and the curve does not budge.  Also, the number of points is just a display thing.  Every time, the script processes 14000 time divisions, of which I have plotted 1000 because it gets a bit busy and takes much longer when I use more.

You might wonder, what happens if I make that number larger, say 1400000?

It takes about 100 times longer to process, sure, and you get these:

 

and

 

It might be difficult to see in these images, unless you look very closely, but the effect of more time divisions is to smooth out the curves and (in the case of the FUGE universe curve) make them more closely align with the equation.  With the Standard Model universe curve, the point at which there is an intersection with the causal horizon is still approximately 12.27 billion years ago and the overall difference between the FUGE and Standard Model curves is unchanged throughout.

If anyone wants to get a copy of the macro-enabled Excel spreadsheet in order to play around with it and confirm (or refute) what I have said above, just let me know in the comments.