Is the Universe Getting More Massive? (Flatness, not Fatness)

First and foremost, I should point out that it is not my intention to fat shame the universe.  By “more massive” I just mean “having more mass”.  It’s undeniable that the universe is big, but we like it that way.

Second, I've been accused of crack-pottery which hurt more than I had expected, but I'd recently learned that one of our dogs might be dying of cancer so I was a bit more fragile than normal.  A quick search on "Schwarzschild radius / Hubble mysticism" did indeed bring up some worrisome results.

Let me assure you that I am not going anywhere close to suggesting cosmic consciousness, that crystals work because of (insert quantum woo here) or anything like that.  I'm just noticing some coincidences (which might be easily explained by someone who is both sufficiently patient and well-informed) and wondering if there might be a simple set of rules in the background from which the beautiful complexity of the universe emerges.  I'm not denying that complexity at all, but 1) I don't really want to be distracted by it and 2) I currently have neither the time nor the education to immerse myself in it anyway.

I'm happy with the notion that I might be wrong, but so far people who said with great confidence (and perhaps good cause) that I am wrong have been pointing at the wrong things, namely interpretations rather than measurements.  If a measurement or observation simply won't conform with the model, I'll have to throw it away but I am going to foolish/arrogant enough to hang onto it while there is no empirical evidence against it because to me at least, and if only to me, it all makes sense.

Given the above as a caveat, if any reader has a reason why what I am suggesting simply cannot happen, please do me the favour of explaining exactly why.

Okay, onto the main event ... fattening up the universe.

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What I want to address here an issue that I have raised before, for example in My Universal (and Expanding) Struggles, where I finished off with:

The second struggle is associated with the second model, and was mentioned in Is the Universe (in) a Black Hole?, namely that the argument leads to a need for the mass in the universe to increase.  In my defence though, this is a problem that also apparently exists with the standard model.  Again, this will require more thought.

Well, I’ve given it more thought.

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Take two facts from the universe, as we observe today (in cosmological terms).  First, the Hubble parameter at this point in time, H_o, is such that H_o = 1/ꬱ is a very good approximation, where ꬱ is the age of the universe.  Second, the density of the universe is very close to the critical density, which is the density that the universe would have if it had zero curvature, meaning that it is “flat” (no, not “fat”, "flat").

We have two options here, either it’s a big fat coincidence that the Hubble parameter at this point in time is equal to the inverse of the age of the universe and that the universe looks to be absolutely flat, or … it’s not a coincidence – in other words the Hubble parameter is the inverse of the age of the universe and the universe is flat.  I went with the no coincidence option and see where we would get.

We can note that the critical density is equal to the density of within the event horizon of a black hole with a Schwarzschild radius of the Hubble length (speed of light divided by the Hubble parameter), if the universe is flat:

r = c/H => H = c/r

r = 2GM/c² => M = r.c²/2G

V = 4πr³/3

ρ = M/V = 3c²/8πGr² = 3H²/8πG = ρ_c

As noted, this introduces a new issue specifically with relation to the mass M.  In retrospect, there was something staring me in the face in the second line of equations above and I just didn’t see it.

We have to go back a bit, to before my time, to get some context.  The term Big Bang was coined by Fred Hoyle and was meant to be pejorative.  He didn’t like the idea that the universe was non-existent one moment and then suddenly exploded into existence.  Hoyle, together with Gold and Bondi, developed the Steady State model in which the density of the universe remains unchanged while the universe expands, via the continuous creation of matter.  A problem with this model however is that it posits a universe that is eternal into the past as well as into the future.  This model has some issues and basically no-one holds it high regard so the Big Bang model prevailed.

With the Big Bang model (at least initially), there was no “continuous creation”, the matter that exists in the universe now was here in the beginning (note that I prefer to call this mass-energy, but this might be little more than a stylistic thing).  This is not to say that the mass of the universe should have remained constant because stars are busily turning some mass into energy via nuclear fission.  The mass of the universe should, therefore, be decreasing (but not the mass-energy).  For the purposes of the argument, I’m going to ignore this decreasing mass and consider all mass-energy as mass.

(Note that this is all prior to dark matter and dark energy.)

If the mass of the universe were constant, then the argument goes that if you wind the close back, we eventually arrive at a singularity in which all the mass in the universe is squeezed into basically no space at all.  This implies that there is a positive curve of universal density from today back to the big bang - when density was not only maximum but effectively infinite (although this could just be an indication that mathematics has broken down at that point).

There is a relationship between the mass of a Schwarzschild black hole and density, meditated by its event horizon.  The more mass such a black hole it has, the less dense it is but when you have a given mass, then it has a given event horizon and for the density of our universe, that mass gives us an event horizon which 13.8 billion light years, which is 1) the Hubble length at this time and 2) the distance that light can travel in 13.8 billion years, which just happens to be the age of the universe.  This could mean one of two things, either the universe just happens to be at a point in its development at which it is entirely flat … or, the mass in the universe is not constant.

The former option is another big fat coincidence, so we’d be swapping one big fat coincidence for another big fat coincidence if it were the case.  However, great minds than mine tell me “that a spatially flat universe remains spatially flat forever, so this isn’t telling us anything about the universe now; it always has been true, and will remain always true.”

There is also what is known as “the flatness problem”.  Effectively what this is about is that if the universe is very close to flat today, then in the past it must have been even more flat.  The universe is (apparently) such that it cannot have deviated from flat in the past and just tended towards flat today.  If the universe does deviate from flat (within the wriggle room provided by our inability to measure curvature with absolute precision), then it will eventually no longer appear flat.  That would make our measurement of how flat it is today, just when our technological advancement is sufficient to permit that measurement, a big fat coincidence.

I'm not on board with coincidences, but it doesn't matter, we can just think in terms of the past and I can still make my point.

There’s a conflict.  The universe cannot just currently be spatially flat, if it is flat right now (and measurements say that it is with great accuracy), then it has always been flat, since the beginning (as per the flatness problem).  However, if the universe is flat then the equation

ρ_c = 3H²/8πG

applies and if that equation applies, then H=c/r and M = r.c²/2G – which means that the mass of the universe is proportional to the Hubble length, which increases with time which means that the mass of the universe has been (and is) increasing!

It's pretty easy to work out that the mass has been increasing at a rate of M / r = c²/2G, which just happens to be proportional to the relationship between the Planck mass and the Planck length (ie m_pl / l_pl = c²/G).  Another big fat coincidence?

Now, given that I have suggested elsewhere that the universe is expanding with the speed of light, so that r = c.t, that gives us M / t = c³/2G, which is half the relationship between the Planck mass and the Planck time (ie m_pl / t_pl = c³/G).  This no greater or lesser coincidence than above, it's just simple division of the same (apparent) big fat coincidence.

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.  Again, if you carry out the simple algebra to convert mass to energy, you get E / t = c⁵/2G, which is of course proportional to E_pl / t_pl = c⁵/G.

While this does seem rather strange, the simple algebra works out and the result pretty much looks like a perfect balance between the Steady State model and the Big Bang model – you have the big bang and finite history, but you also have this strange continuous creation.  Without it though, it doesn’t seem that a flat universe is possible, and all our measurements seem to be telling that the universe is most definitely flat.

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Note that it can be gleaned from the above that there are alternative conclusions:

The Hubble parameter is not always the inverse of the age of the universe, and the fact that it is today is merely a coincidence.

The universe is not flat, and the fact that it (still) looks entirely flat is merely a coincidence, because it does deviate from flat by a margin that is lower than the level of precision to which we can currently measure universal curvature.

Perhaps one or both of these coincidences is in play and we don't need a model which explains why they aren't coincidences.  But I'm not going to just stop there and assume that it's all sorted via the coincidence card when there is a model which seems to explain it.

In such a model, the universe is flat, has always been flat and always will be flat.  In such a model, Hubble parameter is the inverse of the age of the universe.

A model that satisfies both of these requirements is (as pointed towards in Is the Universe Expanding at the Speed of Light?) one in which the universe is a glome, with time as its radius, expanding at one Planck increment per Planck time.  To make that model one of a flat universe (and one that is eternally flat) all that needs to be introduced is the notion that units of Planck mass-energy enter at a rate consistent with maintaining the critical density.

While the appearance of mass-energy might be counter-intuitive, it should be noted that it is believed by some that energy is increasing in the universe in the form of dark energy - “Note that (the value of omega-lambda) changes over time: the critical density changes with cosmological time, but the energy density due to the cosmological constant remains unchanged throughout the history of the universe: the amount of dark energy increases as the universe grows, while the amount of matter does not.”

In addition, it does resolve another issue – namely the initial instantaneous appearance of the entirety of the universe’s mass-energy in a singularity.  Instead all we have at the beginning is the appearance of precisely one Planck glome (radius equivalent to one Planck time) containing one unit of Planck energy.  The universe, tiny as it was, would then have started expanding with energy entering only as quickly as it could – at a rate consistent with the universe remaining precisely flat, maintaining the critical density all the way.

This, I believe, would resolve the flatness problem.