Observable Events Curve - Simplified

This is an attempt to describe the observable events (OE) curve concept as simply as I can manage, without the complications that are introduced when trying to address predictable objections.  If you can’t suspend judgement long enough to grasp the concept without raising objections, the complicated version is discussed in:

Taking Another Look at the Universe,

Mathematics for Taking Another Look at the Universe,

Concepts in Taking Another Look at the Universe,

Observable Events Curve - Fast and Slow Expansion,

Observable Events Curve - Addressing a Point of Potential Confusion,

Observable Events Curve - Hang On!  Aren't I Double Dipping? (SPOILER - No, I am not.),

Observable Events Curve - Shifting About Redshift (Including an Alternate FUGE-like Universe), and

A Very Fast Rocket and a Rather Distant Planet. 

It’s possible that your objection has already been anticipated and addressed.  Some other objections may be raised in response to the FUGE universe.  There are currently 17 additional articles that refer to FUGE (with that label) and many earlier articles that lead to the concept, so I won’t list them here.

If you have additional objections that I have not already addressed, please let me know.

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If a photon hits your eye and is thus “observed”, then it must have travelled from where it originated via some path through spacetime.  To simplify, we will consider its path through empty space such there is no deviation in that path due to gravitational lensing.

The photon could be thought of travelling in a straight line from origin to observer, like this:

A complicating factor is that the universe through which the photon is travelling is expanding.  This means that the photon not only needs to cover the distance between the origin and observer, but also the additional space that is introduced due to expansion during its travel time.  So it looks a little like this:

Currently, the universal expansion rate is given by the inverse of the age of the universe (æ).  For a FUGE universe, the expansion rate is always the inverse of the age of the universe.  This means the expansion rate starts off very fast (approaching infinity as æ→0) and decreases as æ increases.

In the second animation, I have shown a situation in which the origin and observer are being moved apart by expansion at a rate of 0.5c.  While it might seem that this means that the photon is approaching the observer at c-0.5c=0.5c, it’s not that simple.  Some keen-eyed readers will have noticed that the photon covers a greater distance in the second half of its travel than in the first.

Let’s call the distance from the observer to the photon x' and say they were initially 5 billion light years apart (in a universe that is æ=10 billion years old [at that time]).  Saying that the photon is observed t=0 (where t is an indication of how long ago something happened). Tabling the results, we have:

x' (Gyr)	t (Gyr)
0.00	0
0.95	1
1.80	2
2.55	3
3.20	4
3.75	5
4.20	6
4.55	7
4.80	8
4.95	9
5.00	10

From this we can deduce that the universe is æ=20 billion years old when the photon is observed.  Then we can chart this, with t as the horizontal axis and x' as the vertical axis:

This is what I call an observable events (OE) curve, because all of the points on that curve are spacetime coordinates (also known as “events”) from which a photon observed at (t,x)=(0,0) may have originated.  The equation associated with this curve is x'=(ct₀-x).x/ct₀ for any time t ago, where x=ct and t₀=æ (the age of the universe at the time of observation).  Noting that all the events are relative to the observer, at (t,x)=(0,0), here is a final animation which may help cement understanding of the concept:

If we plug the value t₀=æ=14 billion years into the formula (as a rough approximation of the age of the universe) and plot across the period 0 to 14 billion years, we get the observable events curve for our universe (if it is a FUGE universe):

In GIF form similar to the above, this becomes (noting that an observed photon cannot start at (t,x')=(14,0) precisely, otherwise the observer and the photon are already collocated, it is just at a time [t→0] when [x'→0]).  Note that in this version, we no longer have the origin, but rather the causal horizon relative to the observer, which can be thought of as a location that is receding away from the observer at c:

Note that this should not necessarily be considered to be representing the path of an individual photon since it implies that the photon is being dragged away in the earlier travel.  I'm still mulling over this aspect.

In summary:

• all photon which we could possibly observe must have had a path through spacetime on their way to be being observed,
• the universe is expanding, so some element of the travelling distance covered by any photon on its way to us before being observed must have been due to expansion, and
• the longer ago a photon started on its way to us, the greater proportion of its travel must have been due to expansion – approaching 100% as the departure time approaches now minus the age of the universe.