Mathematics for Taking Another Look at the Universe

In Taking Another Look at the Universe, I blithely introduced the equation x'=(ct₀-x).x/ct₀ (where x=ct) without given any derivation or real explanation as to what the terms mean.  I made the equation very slightly neater than it had been but, in the process, I might have made it less comprehensible for people like myself who like to work from first principles.

The central term is, of course, x – the distance to an event.  The other term, t, is the time of that event – but, in reality, it is more like the difference between the timing of the event and now, so it could be thought of as Δt.  Similarly, x could be more accurately described by Δx – but for reasons that may become clear shortly I dropped the “Δ” from both.

The term t₀ is used such that the subscript aligns with H₀, the current value of the Hubble parameter, and so t₀ is the age of the universe while x₀=ct₀ is the current Hubble length (or the current radius of a FUGE universe).  By analogy, for an observed event at (x,t), x=ct.

Note that this follows from the notion that we can only observe an event if there has been sufficient time for the photons from that event to reach us.  However, while photons are travelling to us from the event, the space in between is also expanding.  Therefore, for any observed event, there are two components, a temporal one (due to how long ago it happened) and a spatial one (the distance from our location to where the event happened at the time).  The latter is what x' on the y-axis represents in this chart:

It may get a little complicated here.  I suspect this because I have already explained it incorrectly (possibly more than once) and at one point even got close to persuading myself that the equation was wrong.

Consider it this way, all maximally distant observed events (MDOEs) in the very distant past were, when they happened, not as far from where we are in space (than other events) because the universe had not expanded very much.  Note that the equation x'=(ct₀-x).x/ct₀ specifically considers MDOEs (with less distant observed events lying under the curve).

The most distant MDOE (in any given direction) would have occurred when the universe was half its current age.  For ease we can call this the ½t₀ event, or “½t₀e” (half-toe, which can be thought of as both a distance and a time). 

Since ½t₀e, all MDOEs have by necessity been less distant because there has been less elapsed time for photons from those events to reach us.  Before ½t₀e, all MDOEs were less distant because the universe was smaller. 

We can consider the universe as being divided into two eras, a pre-½t₀e era and post-½t₀e era, with events in both eras that occurred at locations that were equally distant from us (at the time they occurred), meaning that they notionally travelled the same distance in unexpanded space, but photons from the event in the pre-½t₀e era will have experienced more expansion during transit.

A marked-up version of the image above may help to illustrate this fact:

Let us take the most extreme example, the instanton event happened about 13.787 billion years ago.  There is effectively no distance to where that event happened, because the maximum expansion one could consider to have happened at that time is one unit of Planck length.  As a consequence, the entirety of the distance between us and where the location of that event is now is due to expansion.

The next most extreme example illustrated above is an MDOE almost 2 billion years later, by which time the universe had expanded to a radius of 2 billion light years.  Photons from that MDOE were not at the full extent of the universe at the time however but rather at 1.565 billion light years.  Note that the location of that event (following the light green line up to the left) is currently 12 billion light years away, indicating the amount of expansion that has been incurred between our location and the location of the event at that time place is 10.435 billion light years.  Therefore, the time taken for a photon to reach us is the1.565 billion years due to the original separation plus 10.435 billion years due to expansion, or 12 billion years, precisely what we would expect.

The upright light green section can be calculated using the following:

Note that sinϴ=x'/√(x'²+(ct₀-ct)²)=x/√(x²+(ct₀)²), so, noting that x=ct:

x'²/(x'²+(ct₀-x)²)=x²/(x²+(ct₀)²)

x'².(x²+(ct₀)²)=x².(x'²+(ct₀-x)²)

x'².(ct₀)²=x²(ct₀-x)²

x'.(ct₀)= (ct₀-x).x

x'=(ct₀-x).x/ct₀=(ct₀-ct).t/t₀

This should come as no surprise, since this is the equation that I charted.

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Let us consider the (clean) chart again:

The apparent distance to any event is the length of the curve.  If we divide the curve into a relatively large number of finite elements and approximate the curve by summing the length of all those finite elements, we arrive at 15,800 million (light) years.  As mentioned in Taking Another Look at the Universe, this is well within the range used by Lineweaver and Egan (but they got it from integrating a(t) over the age of the universe, if I understand it correctly).

Note also that I asked a question about dark energy in Taking Another Look at the Universe.  We can use these finite elements to take a look at the apparent Hubble parameter value at all points along this curve as the universe expands, and we get a curve that looks like this:

The “apparent” H is based on a set of calculations, using a change in the age of the universe by a millionth of 1% and the consequent change to the values of x'.

The shape of the “Apparent H” curve is of particular interest.  Consider it with respect to the discussion in The Problem(s) with the Standard Cosmological Model and the eras discussed at the Scale Factor page at Wikipedia.

There really are only two eras observable in the chart, from about 7 billion years ago to now (to the left) corresponding to the “dark-energy-dominated era” and the period before that (to the right) corresponding to the “matter-dominated era”.  The radiation-dominated era and the purported era of inflation (plus era that preceded it) are not distinguishable at the scale used.

The chart indicates a very similar situation as that posited with the introduction of dark energy, but without requiring any actual dark energy.  The most recent era appears to have acceleration.  The only times that the apparent Hubble value is equivalent to the inverted age of the universe are at the transition between the “dark-energy-dominated” and the “matter-dominated era and maximally long ago/(apparently) far away.

Note that a lack of dark energy is consistent with the mass of the universe being ~10⁵³kg (the mass one would expect in FUGE universe that is 13.787 billion years old).

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I do acknowledge that the “apparent” values of H in the recent past/near vicinity are very high.  This may be worthy of further investigation.