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.

The Universe is Flat (as in Not Flat)

I posed the following question on Reddit, based on the pondering expressed in My Universal (and Expanding) Struggle:

Recent observations tell us that the expansion of the universe is accelerating. Other observations tell us that the universe is flat. This seems to be in contradiction, if you follow this logic:

The critical density found via the first Friedmann equation is ρc=3H²/8πG. As Sean Carroll points out, if the universe is flat, then the density of the universe is equal that of the mass required to obtain a Schwarzschild radius of one Hubble length (the speed of light divided by the Hubble parameter) divided by the volume of a sphere with that radius. The implication is that that Hubble parameter is inversely proportional to the radius of the observable universe (note I said "proportional", which eliminates the question of whether that is the naive value [13.7 Mly] or the calculated value [46.6 Mly]) and consequently also inversely proportional to the age of the universe.

​How can this be squared with the observation that the rate of expansion of the universe is apparently increasing?

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I note that this issue is effectively mentioned at wikipedia where is it stated that:

The discovery in 1998 that q is apparently negative means that the universe could actually be older than 1/H. However, estimates of the age of the universe are very close to 1/H.

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but the issue is not taken up for discussion. The fact that the age of universe is strikingly close to 1/H seems like too much of a coincidence, particularly if the deceleration parameter, q, has varied during the life of the universe. It would put us in the middle of an era of the universe that would appear to contravene the Copernican principle. Or am I missing something?

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I got one response which was nice enough from u/nivlark:

The exact proportionality you describe only holds if q remains constant over the lifetime of the universe (and in particular, if it is equal to zero). In the absence of dark energy, we'd instead have a positive deceleration parameter and a universe younger than 1/H.

As to why the universe's age is very close to 1/H, we have the more complex situation of a time-varying deceleration parameter - dark energy only became dominant (i.e. expansion began to accelerate) relatively recently. Perhaps by coincidence, this means that 1/H has only recently 'caught up' to the age of the universe. The discrepancy between the two will widen in the future, eventually approaching some limiting value depending on the exact value of the deceleration parameter.

This led to me ask:

Does that mean that Sean Carroll is wrong when he writes "Note 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"?

Are you suggesting that the universe only appears to be flat (as per

the WMAP and Planck surveys)?

Edit: I've read that if the universe is flat then q=1/2 (precisely, not more, not less), there may be caveats involved with that though.

u/nivlark responded with:

No, that is correct. A universe which is exactly flat will always be so, but it's an unstable equilibrium: deviations from flatness must grow such that non-flat universes become more open or closed with time.

Measurements of curvature from Planck &c. are consistent with flatness, but with some observational error (I have the number 0.4% in memory for the size of this error, but that may be out of date). So we can say that either the universe is exactly flat, or that it has a small non-zero amount of curvature consistent with these bounds. Neither is wholly uncontroversial: zero curvature suggests very finely-tuned initial conditions, while nonzero but small curvature requires a process like inflation to be invoked to produce the exceedingly small initial value of the curvature.

q=1/2 indicates a flat universe, but specifically one that is dominated by matter. A cosmological constant-dominated flat universe would instead have q=-1.

I just responded with “Thanks”, in part because I didn’t have more to ask at the time and in part because I’ve made enough of a fool of myself with mathematical questions, I don’t want to get into similar problems with physics.

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However … this answer by u/nivlark still bothered me.  He seemed to be saying, “yes, a universe which is exactly flat will always be so” and then immediately saying that there will inevitably be deviations from flatness (“it's an unstable equilibrium: deviations from flatness must grow such that non-flat universes become more open or closed with time”).  Let’s say that the universe started off not quite flat, but really close to flat.  The implication here is that the deviation can really only tend to one direction, because if it’s a tiny bit open and tended towards being a tiny bit closed, the universe would pass through exactly flat and he also said that exactly flat is flat forever.

Now, you could argue that the issue is tied in with variations not only over time, but also across space – local space to one observer might appear entirely flat, but another observer a cosmically significant distance away would see it as slightly open, or slightly closed.  That is, to be entirely flat, the universe would have to be eternally and universally flat, all the time, everywhere.

That would mean however that we just happen to be, just at the time that we first have the ability to measure the (local) density of the universe, just in the right place to measure that density to be completely consistent with a flat universe, neither a tiny bit open nor a tiny bit closed.

Which contravenes the Copernican principle, doesn’t it?

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The thinking above led to the following exchange:

neopolitan-

Can you confirm that you are happy with the fact that the Copernican principle is being contravened. In your argument you seem to be saying that the universe is not (entirely?) flat, not exactly flat, but it deviates from flat. However, our readings of the data, just now, just when we are just beginning (in cosmic timescales) to measure the curvature (or lack thereof) of the universe, we happen to be in an era and/or a sector in which our measurements tell us that the universe is flat.

This would, in a sense, make our era and/or sector special, would it not?

nivlark-

That isn't what I wrote. I said that the measurements we have are consistent with flatness, but that there is an observational error associated with those measurements which means that the best we can say is that the curvature is no greater than the magnitude of that error.

I then went on to say that there are some as-yet unsolved theoretical difficulties with both perfectly-flat and slightly-curved universes, so theory cannot help us by eliminating one of the possibilities.

The Copernican principle only applies to our spatial location: we do appear to occupy a privileged position in time. Whether by coincidence or by appeal to the anthropic principle, we appear to exist at an era when the densities of matter and dark energy are comparable, which will not be the case for the vast majority of the universe's lifetime.

neopolitan-

I know it wasn't what you wrote, that's why I asked for confirmation of what I interpreted from what you wrote (ie what you seemed to be saying, from my perspective). I'm sorry that I didn't make that more clear.

I agree that there is potential for observational error and there is also potential for what could be called "assumption error", since the measurements are based on certain assumptions, all of which might be perfectly correct but might also be slightly wrong (or more so).

And I might be extending the notion of the Copernican principle too far by considering a temporal aspect as well, but ... I suspect that we could be running into a simultaneity issue if we suggest that our position i(s) privileged only in time. There's an implication in your statement that the universe changed from matter dominated to dark energy dominated everywhere at the same time. Alternatively, we are in a part of the universe in which dark energy and matter are comparable (and/or in which the effects of there being a balance of matter and dark energy have manifested), which makes our location privileged as well.

Note, I have in mind the concept that I think of as "evenness" together with curvature, by which I mean that the extent to which the universe is flat or not, if it fluctuates as you suggest, won't be precisely the same everywhere - so it'd be "uneven". The flat universe that Sean Carroll referred to would also be even - flat everywhere, all the time. It seems to me that deviations from flat would also lead to deviations from even.

nivlark-

“There's an implication in your statement that the universe changed from matter dominated to dark energy dominated everywhere at the same time.”

This is the case...

“Alternatively, we are in a part of the universe in which dark energy and matter are comparable”

...as is this, and it is also true everywhere. By construction, we model a universe that is homogeneous on large scales, because that's what observations indicate to be the case.

“Note, I have in mind the concept that I think of as "evenness" together with curvature”

These are different quantities. The curvature referred to when talking about the flatness of the universe is a global quantity which is an intrinsic property of spacetime, and there's no theoretical basis to suspect it varies with position. However, 'local' curvature is produced by every massive object - this is what we perceive as gravitational fields. As a result of this the geometry of spacetime is lumpy/uneven on small scales (where 'small' here means galaxy-sized), and this can be the case irrespective of what the global curvature is. Cosmological models are applicable on much larger scales than this though, and so the real universe is very well-approximated by models of a perfectly homogeneous one.

neopolitan-

You haven't addressed the simultaneity issue associated with the entire universe fluctuating, or do you mean to do that by saying that the curvature is "an intrinsic property of spacetime"? If that is the case, would we not still expect to see the consequences of fluctuations in the intrinsic property of spacetime rippling through the universe due to simultaneity/relativism issues? Or do you suggest that we might if it weren't for the lumpiness of space at the galaxy level?

nivlark-

I don't know what the "simultaneity issue" you're referring to is. The global properties of a homogeneous universe are perceived to evolve simultaneously by any comoving observer (i.e. any observer who has no proper motion and is carried freely by expansion). This does not contradict relativity or the cosmological principles.

As I said in my previous comment, the flatness of the universe is such a global property. It does not depend on position. Superimposed on that global curvature is a time- and position-dependent local curvature, which occurs due to the presence and movement of mass. This has local effects, which we call 'gravity', but these are negligible on the scales relevant for cosmology because the magnitude of the local curvature falls with distance from the source (in classical language: the gravitational force weakens with distance).

neopolitan-

> I don't know what the "simultaneity issue" you're referring to is.

I'll try to explain, please forgive me if I don't use the precise terminology that you favour. There is a "slice" of the universe that constitute the comoving coordinates. It's this set that is normally referred to when considering the "shape of the universe" or the curvature. There's a reason for taking this particular slice such that the coordinates are comoving, namely that you can't really talk about the universe as a whole at a single point in time - the comoving coordinates set is as close as you can get (I'm assuming that it's basically the circular cow of a universe you need without anything in it to mess up the calculations). The comoving coordinates constitute a set of coordinates that are not collocated, and therefore a change of curvature that manifests across the entire set that is happening simultaneously is problematic.

Now I can accept that there is a process going on such that at the end of that process, no matter which localised subset of the comoving coordinates you consider, the curvature will fluctuate in the same way and therefore you'd see the entire manifold fluctuate at the same time (within the comoving frame). But if that were possible, it seems that that would be an alternative solution to the homogeneity of the CMB and inflation would not be necessary (and from what I read, something like inflation is necessary).

The conversion seemed to have died at that point, although it could be that it was the weekend and u/nivlark has a life.  Being the weekend, I did have some time to ponder though.

It’s possible that what u/nivlark is saying is that there’s a tendency to expansion (dark energy) and a tendency to contraction (gravity due to matter).  When the universe is small(ish) and there is a certain amount of matter in that small(ish) volume, then contraction has more sway than when the universe is larger.  When the density reduces to a certain point, we can say that dark energy now dominates; it would not be a punctuated transition but rather just a point of interest on a smooth curve.  If this were the case, then simultaneity would not be an issue.

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Please note the tag below "cynicism".  This is, in part, referring to the parenthetical "as in Not Flat" in the title.  My position is that the reason that all the measurements tell us that the universe is flat (at this time) is because the universe is actually flat.

Another Teeny Tiny Struggle

I recently wrote of my My Universal (and Expanding) Struggles.  The title is very tongue-in-cheek, but it is mildly amusing to think that some deranged skinhead might stumble on my page and think to himself “what the heil?”  (Oh ok, I admit it, that too was a joke.)

Back to the matter at hand – physics and more specifically, cosmology … even more specifically another one of my struggles associated with it.

I had forgotten about this particular struggle, but I was reminded by something said in skydivephil’s latest video, The Story of Loop Quantum Gravity - From the Big Bounce to Black Holes.  I think of the universe, in one sense, as being the eventual consequence of a prior universe in which matter has collected into a black hole over a period of eternity within that prior universe.  Time that other universe is orthogonal to time in our universe and you can think of all the mass energy being deposited instantaneously, or rather within the first quanta of time.

What I don’t envisage is that all this mass-energy appears in a singularity.  Instead, I see it all being compacted into the minimum amount of space consistent with the universe’s mass-energy being at the Planck density.

However, people are always going on about the initial singularity or the big bang starting with everything condensed down to infinite density.  There’s a corresponding argument regarding the centre of a black hole.

Imagine my joy then to hear that, in June 2001, there was a short paper by Martin Bojowald that suggests that in quantum geometry the singularity is removed (video reference | original paper).  Furthermore, in the video, the presenter specifically refers to planck density shortly after this image is shown:

Note that my conceptualisation isn’t completely consistent with this image, since I would not see the universe on the other side of the big bang as being in line with our universe, but rather orthogonal (ie perpendicular) to it.  This concept is however hinted at slightly later in the video, when the Hartle-Hawking No Boundary model is discussed.  “There is a real and an imaginary time.”  A feature of “imaginary” time is that it is orthogonal to “real” time.  Note that imaginary time isn’t made up, it’s a reference to imaginary numbers which are extremely handy for various tasks in physics and engineering.

The key point in the image is that there is no singularity, the size of universe decreases significantly, but not infinitely, and then expands back out again.  In respect to the planck density, the presenter refers to another paper by Ashtekar, Pawlowski and Singh (video reference | paper) – note that one of the people being interviewed for the video is the first listed on this paper.  The narrow tube shown to sort of link one universe with the other is an (almost certainly extremely brief) era in which planck density is achieved.

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So, I did struggle with regard to the singularity, but it seems I didn’t need to.  If quantum loop theory is close to the money, there is no singularity.

My Universal (and Expanding) Struggles

As stated in Is the Universe (in) a Black Hole, I've struggled with the idea of inflation for quite some time and, more recently, I also struggle with the idea that the expansion of the universe is speeding up.  I threatened that I would try to explain, so here goes.

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In the Standard Model of Cosmology, the universe was initially in a hot dense state.  Space-time abruptly appeared (or manifested) about 13.7 billion years ago and the universe immediately began expanding.  A very short period later (10^-36s) there was a brief phase of inflation (about 10^-32s), and then standard expansion began (or resumed).  In the words of Wikipedia:

Although a specific "inflationary epoch" is highlighted at around 10⁻³² seconds, observations and theories both suggest that distances between objects in space have been increasing at all times since the moment of the Big Bang, and is still increasing today (with the exception of gravitationally bound objects such as galaxies and most clusters, once the rate of expansion had greatly slowed). The inflationary period marks a specific period when a very rapid change in scale occurred, but does not mean that it stayed the same at other times. More precisely, during inflation, the expansion accelerated; then, after inflation and for about 9.8 billion years, the expansion was much slower and became an even slower expansion over time (although it never reversed); and then since about 4 billion years ago it has been slightly speeding up again.

Now, it is generally understood that the Big Bang happened 13.7 billion years ago.  It can also be understood that the observable universe has a radius of 13.7 billion light years, but there is another radius (comoving distance to the edge of the universe) of 46.6 billion light years, although this is often referred to as a diameter of 93 billion light years.  This second radius (or diameter) is calculated using the fact that light that reaches us today from what is now about 13.7 billion light years away (ie the cosmic microwave background (CMB)) was emitted by particles that are now even further away because of the on-going expansion.  It’s a bit complicated, so rather than risk miswording it, I’ll just refer you to a page that tells you all about this little nugget, or you could watch a video.

The CMB was “created” about 380,000 years after the big bang.  At that time, what is now the observable universe is thought to have been 42 million light years across and the entire universe must have been bigger (otherwise the CMB, using the logic of the calculation, would be just about to turn off and we’d be in a relatively special time and place in the universe to have seen it at all).  Just how big it was is unclear.

Anyway, the points that I want to make here are that 1) the inflationary epoch of the universe was well and truly over by the time that the CMB formed and 2) there is an implication that the universe must be significantly bigger than not only the naïve 13.7 billion light year radius, but also than the 46.6 billion light year radius that is implied by the 42 million light year radius at 380,000 years after the big bang.

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Now the first thing that I struggle with and have struggled with for a long time is this inflation.  One of the reasons that I struggle with it is that if you consider the universe to be expanding at a rate of one Planck unit of length per Planck unit of time (ie the radius expands at that rate), then this resolves down to a expansion rate at this time of 70.75 km/s/MPc – the expansion rate we currently measure (within a couple of percent).

Another reason is that there’s the amazing coincidence that the density of the universe is equal to the mass that creates a Schwarzschild radius equal to 13.7 billion light years divided by the volume of a sphere with that radius.

Then there’s the fact that if the universe is a 4D hypersphere (with the 3D spatial universe mapped over the surface with time as the radius – known as a 3-sphere or glome), then you can arrive at the equations of special relativity.

It’s possible that the last of these can operate fine even with a period of inflation, while the first and second seem not to work – even though they all just fall out of the mathematics.  There is a problem with the second, in that there’s an implication of increasing mass in the (observable) universe and then there’s the fact that the surface volume of a glome is greater than the volume of a sphere with the same radius by a factor of 3π/2.

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And then there is inconsistent expansion.  Not just inflation, but expansion of some kind for a fraction of a second, then inflation, then slower and decelerating expansion and now accelerating expansion (as in the quoted text above).

The current expansion, today, can be expressed as the inverse of the age of the universe today.  This is an amazing, or should I say unbelievable, coincidence if the expansion of the universe has followed the pattern suggested.

It also presents a problem with regard to the flatness of the universe.  In response to Is the Universe (in) a Black Hole, at least one Redditor (and Sean Carroll) argued that the fact that (deep breath) the density of the universe is equal to the mass consistent with a Schwarzschild radius equal to the age of the universe times the speed of light divided by the volume within a sphere with the same radius (release breath) is consistent with a flat universe.  Sean Carroll goes on to say that “a spatially flat universe remains spatially flat forever”.

But hang on here a moment.  If the rate of expansion of the universe were accelerating, then (absent the addition of mass) the density of the universe would be decreasing at the cube of an accelerating rate (ie with increase of the radius).  However, for the universe to be flat, the density would decrease only with the square of the increase of the radius.  This implies that if the universe were flat then not only would mass in the universe need to be increasing, but it would also need to be increasing at an accelerating rate.

It should be noted that Sean Carroll does not directly reference the age of the universe.  He refers to the Hubble length and the Hubble parameter (aka the Hubble constant), where “the Hubble length … is the speed of light divided by the Hubble parameter” and goes on to say that when the universe is flat, the Hubble length equals the corresponding Schwarzschild radius.

This is indeed simple to show, using the Friedmann equation for a flat universe (where spatial curvature, k=0) which gives a density ρ_c of:

ρ_c=3H²/8πG

Given the Schwarzschild radius equation:

R_s=2GM_s/c²

Thus:

M_s=R_sc²/2G

And the volume of a sphere of radius R_s is:

V_s=4πR_s³/3

So the corresponding density, ρ_s, is:

ρ_s=M_s/V_s=R_sc²/2G/(4πR_s³/3)=3c²/8πGR_s²=3/8πG.(c/R_s) ²

and if ρ_c=ρ_s, then:

H=c/R_s, and

R_s=c/H

Note however that this equation implies that as R_s increases, noting that R_s is implied to be equal to the Hubble length, and that this is linked to the naïve radius of the observable universe (ꬱ.c, where ꬱ is the age of the universe), the Hubble parameter decreases.  This implies that the rate of expansion should be decelerating, not accelerating – if the universe is flat, which measurements taken by both WMAP and Planck confirm to a high level of accuracy.

Something seems to be wrong since we seem to have inconsistent observations.  It’s probably in my understanding (or misunderstanding) of something, hence my struggle.

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I guess I should include two other, more internal struggles.

The first, which I’ve already mentioned, involves two models I have lodged in my mind.  One being a glome which is expanding out from the past (thus implying that the universe is a 3D surface on that glome) and the other being an effective white hole encased within a Schwarzschild radius.  The volumes don’t match because the surface volume of a glome of radius R is 3π/2 times larger than that of a sphere with the same radius.  Equally, the way that each expands doesn’t match, and the Schwarzschild radius model initially seems to imply a curved 2D surface to the universe, which I don’t consider feasible.  I do have the glimmer of a solution, considering a 2D observer on the surface of a globe, who looks around to see a perfect (observable) circle around her.  The radius of that circle and the radius of the globe are the same.  If the globe is expanding, the radius of the circle and the globe will remain in lockstep.  The entire surface area of the globe will be significantly greater than the area of the (observable) circle and even the surface area of the semiglobe that is perceived to be a circle is greater than that of the circle (not by as much though).  Bumping this up by one dimension and replacing our 2D observer with a 3D observer (like ourselves), it’s possible to have what appears to be simple 3D geometry actually being the 3D surface of 4D geometry.  Maybe it works, maybe it doesn’t.  I’ll have to give it more thought.

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.