Expansion on Uncertainty

In Is the Universe Getting More Massive?  (Flatness, not Fatness), I concluded that (if the universe is and has always been flat then) the mass of the universe is increasing at a rate of M / r = c²/2G.  I should have put deltas in there, ie:

ΔM / Δr = c²/2G

I went on to write:

This implies, to me, that if the universe is and has always been flat, then the mass of universe is increasing by one unit of Planck mass every two units of Planck time.  (Note that I reached the same conclusion in Is the Universe (in) a Black Hole? but I expressed it in terms of energy.)

In the linked article I wrote (where ꬱ is the age of the universe):

Of interest is the fact that, with the assumption that the universe is a black hole and that it is expanding at the speed of light, we can recall the equation for the Schwarzschild radius and get this result a little more easily:

r_s = ꬱ.c = 2GM/c²  =>  M = ꬱ.c³/2G

This last equation is for mass of the universe now but the implication is that for a given period of time Δt, ΔM = Δt.c³/2G, or

ΔM / Δt = c³/2G

From which we can conclude that Δr / Δt = c, but all this is saying is that the universe is expanding at c, which we already know.

We can go further though, using this relationship, noting that l_pl / t_pl = c and thinking of incremental changes (increments of Planck length and Planck time):

ΔM / Δr = c²/2G

=> ΔM.c . Δr = c³/2G . Δr² (multiplying through by c. Δr²)

=> ΔM.c . Δr = c³/2G . ℏG/c³ (noting that Δr² = l_pl²)

=> Δp . Δx = ℏ/2

This is the lower limit of Heisenberg’s Uncertainty principle (Δp . Δx ≥ ℏ/2).  Alternatively:

ΔM / Δr = c²/2G

=> ΔM.c² . Δr/c = c³/2G . Δr²

=> ΔE . Δt = ℏ/2

Which is the lower limit of an alternate expression of Heisenberg’s Uncertainty principle (ΔE . Δt ≥ ℏ/2). 

How then should this be interpreted?  The way I understand it is that if we consider the tiniest meaningful increment of time, by which I mean one Planck time, then we are being told by the Heisenberg Uncertainty Principle that the minimum change in energy must be half a Planck energy.  Now this might be the wrong way around, since the expansion and the flatness of the universe point to the lower limit of the Heisenberg Uncertainty Principle, so it could be that this principle is merely pointing to an emergent feature of “flat expansion”.  Or it could just be another big fat coincidence.

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Interestingly, if you find your way to the vacuum energy page at Wikipedia, you will find that there is an “unsolved problem in physics” note:

Why does the zero-point energy of the vacuum not cause a large cosmological constant? What cancels it out?

The cosmological constant is the energy density of space and in Is the Universe Getting More Massive?  (Flatness, not Fatness) I concluded that that the density of mass-energy of the universe which is not baryonic or dark matter is about 6x10^-10 J/m³, which is precisely what is measured.  In my model, this “unsolved problem in physics” is not a problem.

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I should point out that when I was looking for more information on “Planck atoms”, a term that I think was used in The Story of Loop Quantum Gravity - From the Big Bounce to Black Holes (as mentioned in Another Teeny Tiny Struggle), I chanced upon some documents by José Garrigues-Baixauli.  It was when I was perusing those that it struck me that my one Planck energy per two Planck time result was reminiscent of the Heisenberg Uncertainly Principle equation.  I am not in a position to agree with everything that José has written there, but I do notice some parallels in that he has arrived at a couple of similar ideas from a different direction.

This image is particularly evocative considering the contents of Spherical Layers, the image that followed and the rather opaque follow up in The Messiness of Layered Spheres (I promise that it made sense to me even before the clarifying edit that I have just performed, but I was inside my head at the time of writing so I had an advantage).