## Thursday, 15 August 2019

### Expanding Glome to Special Relativity

In Big Fat Coincidence and Problems that Don’t (Seem to) Exist, I laid out the FUGE model works.  What I didn’t do, because I didn’t think about it at the time, was explain how one reaches an explanation of (Special) Relativity from an expanding glome.  So here goes …

The equation for a glome is:

Δx2 + Δy2 + Δx2 + (cΔτ)2 = r2 = (cΔt)2

where x, y and z are spatial units, τ is a temporal unit and r is the radius, which is given by the change in time, t, times c, which is a constant required to mediate the units.

We can use this equation to consider a change in spatial location on the glome in the period Δt:

v2 = Δx2/Δt2 + Δy2/Δt2 + Δz2/Δt2

so:

v2.Δt2 + (cΔτ)2 = (cΔt)2

and then, rearranging:

c2Δτ2 = c2Δt2 - v2.Δt2

Δτ2 = Δt2 - (v2/c2).Δt2

Δτ = √(1 - (v2/c2)).Δt

Which is the equation for temporal dilation where Δt is by convention expressed as t' and Δτ as t.  Note that “(a)fter compensating for varying signal delays due to the changing distance between an observer and a moving clock (i.e. Doppler effect), the observer will measure the moving clock as ticking slower than a clock that is at rest in the observer's own reference frame”.  If we are counting ticks, we are actually measuring a frequency (at a rate of one tick per second) and this is why the time dilation equation will usually appear as something like this:

t' = t / √(1 - (v2/c2))

To get length contraction, one simply multiplies through by c:

Δτ.c = √(1 - (v2/c2)).Δt.c

ΔL = √(1 - (v2/c2)).ΔLo

Alternatively, given that the surface volume of the expanding glome is flat in the FUGE model, one could merely use the approach described in Galilean to Special in One Page.

As for mass-energy, the total energy of a mass is given by:

Etotal = m.vspacetime2 = m.v2 + m.c2Δτ2 = mc2 mo.c2 + ½mo.v2

See On Time where I explain why m.v2 + m.c2Δτ2 mo.c2 + ½mo.v2.

## Monday, 12 August 2019

### Big Fat Coincidences and Problems that Don't (Seem to) Exist

If our universe were to be undergoing flat expansion (Flat Universal Granular Expansion, or FUGE) then, I suggest, a lot of what I have recently identified as “Big Fat Coincidences” would not be coincidences at all but would rather be the natural consequences of the process of that FUGE.  In addition, in the past few posts, I have identified a couple of problems that, given FUGE, aren’t actually problems – namely the Flatness Problem and the Cosmological Constant Problem.  Please note carefully, I am not suggesting that I have solved the problem, I am merely saying that if FUGE is true, then these are not problems.

I am going to summarise how I envisage that FUGE works and try to hit as many big fat coincidences and problems that don’t (seem to) exist as I can on the way.  If I miss any, or it’s just too awkward to address them at the time as they come up, I’ll list them at the end.

---

First and foremost, we know that the universe expands.  We can see evidence of that as we look at distant galaxies that are receding from us at a rate proportional to their distance from us.  This rate is what we all the Hubble parameter (H) or, more often, the Hubble constant (Ho) which merely is the value of the Hubble parameter today (meaning that it isn’t really a constant, since it changes with time).

Secondly, our measurements of the curvature of the universe imply that it is flat.  This flatness of the universe (which must be more extreme as we go back in time, per the Flatness Problem) tells us that the density of the universe is critical and that, per the Friedmann equations, is ρc = 3H2/8πG.

As can be seen above curvature of the universe and the Hubble parameter are linked.  In the FUGE model, the universe is expanding and it remains flat throughout that expansion.

The final element of the FUGE model is its granularity.  This granularity is at the Planck scale, which is the scale “below which (or beyond which) the predictions of the Standard Modelquantum field theory and general relativity are no longer reconcilable”.  To be more precisely, in the EFGE model, the universe is granular at the Planck length and Planck time.

To expand, the radius of the universe must increase.  Note however that this is the radius of a 4D shape, specifically a glome.  The 3D universe does not have a radius in the same way that a sphere has a radius, but the volume of the 3D universe is linked to the radius of the glome (r) thus:

Vsurface = 2π2r3

It is this radius that increases.  Clearly the universe, to expand, must expand at a rate.  In the FUGE model, the expansion is given by:

Δr/Δt=c

At the granular level, this is to that the radius of the universe (as glome) expands by one Planck increment each Planck time.  I say one Planck increment because it could be said that the expansion is time, so the expansion rate is one Planck time per Planck time however given the interchangeability of space and time, this is equivalent to one Planck length per Planck time, which is the value of r that can be used to determine the surface volume of the universe – r = ct, where t is the age of the universe.

As the universe expands, it is filled with Plank atoms – where a Planck atom is the 4D equivalent of Planck volume, so lpl3.tpl.  The surface volume of the universe is the current layer of Planck atoms, which appear to us at the macro level as 3D space.

Note that as the surface volume of the glome expands, room will become available for more Planck atoms.  In other words, gaps will open up.  These gaps will open up everywhere with a random distribution at a rate proportional to ct.  The consequence of this is that, within the surface volume, for a sphere of rLH = ct (that is a sphere defined by a radius equal to the light horizon, which is the distance that light could have travelled in the age of the universe, t, to reach an observer in the centre of that sphere), the rate of expansion would be c.  The recession of any object at distance D, as observed from the centre of that light horizon sphere, would be given by:

v = (c/rLH).D = (1/t).D = H.D

This accounts for one big fat coincidence, namely that the value of the Hubble parameter today, Ho, is the inverse of the age of the universe.  In the FUGE model, the value of the Hubble parameter is always the inverse of the age of the universe.  It also accounts for the fact that the Hubble length (lH = c/H) is the same as the light horizon.

Note that our observations are based on the light horizon in which we exist.  The volume of that light horizon is VLH = 4πr3/3, where r=c.t.

As mentioned above, is the universe is flat, then ρc = 3H2/8πG.  Given that H = 1/t  and ρc = M/V and Δr/Δt = c and VLH = 4πr3/3, consider the change in mass (ΔM) over a period of Δt:

ΔM/(4π(cΔt)3/3) = 3/8πGΔt2

ΔM/Δt = (c3/G)/2 = (Mpl / tpl)/2

This is telling us that mass (and thus mass-energy) is increasing within the light horizon at a rate of one half Planck mass per Planck time.  This resolves the flatness problem since, if this is true, the universe will be maintained at precisely critical density forever – and will be flat forever.  Note that this applies to whichever light horizon we choose, from either here in our current spacetime location, or from any other event location.  This implies that the entire universe is increasing in mass at rate that is greater by a factor of 3π/2.

Note that this rate of mass increase is not time constrained, so it would apply even for very small values of t.  This means that there is no singularity problem.  For sure, the universe would have been significantly denser at the beginning, when Hubble parameter values would have been very high, but that density would not have been infinite.

In the FUGE model, the universe does have a size because, while it’s not bounded (in 3D), it’s not infinite.  The volume of the universe is given by 2(ct)3 = 4.39x1079 m3.  It should be noted that the volume inside a sphere defined by a radius equal to the “comoving distance to the edge of the observable universe” is greater than this.
The radiation that is received by us today as cosmic microwave background radiation has travelled across an expanse of space that now has a comoving distance of 46 billion light years despite the fact that the radiation has travelled at precisely the speed of light for only 13.8 billion years.  Comprehending this difference is a little mind-bending but, in short, it is incorrect to think of a sphere painted with something like the image below receding from us at Ho.cd (where Ho is the current value of the Hubble parameter and cd is the comoving distance to the origin of the cosmic microwave background radiation (not to the “the edge” of the observable universe)).

Instead, think of an expanding sphere, on which there is a “flat” (unbounded but not infinite) 2D surface area and imagine that 2D information travels along the surface in a straight line (or rather a geodesic or a “straight arc”).  Imagine further that the sphere is expanding in such a way that information can’t quite circumnavigate the sphere:

The value rp is what could be called the “pseudo-radius” in that is the apparent radius of the large circle that the observer (denoted by the star) perceives herself to be the centre of.  Note that there is overlap, in that the observer can spin around to face the other direction and “see” the same expanse again so, to express the area of the apparent surface, we must use only half of the pseudo-radius, rp/2 – even if the observer would think she is looking at rp.

As said above, the surface volume of a glome of radius ct is given by 2(ct)3.  This can be equated to a sphere with a radius of half the pseudo-radius, or:

(4π/3).(rp/2)3 = 2π2(ct)3

rp3 = 12π.(ct)3

rp = 3√(12π).ct

When ct=13.8Gly, this gives a value of rp = 46.3Gly, which is (give or take a little) equal to the comoving distance to the “edge” of the observable universe.  Another big fat coincidence.

As discussed above, mass-energy enters the universe at a rate of one half Planck mass per Planck time within the light horizon (where the light horizon can be taken from any location within the surface volume of the universe.  Given that M = ((c3/G)/2).t, and V = (4π/3).(ct) 3, this gives us a mass-energy density of:

E/V = M.c2/V = ((c5/G)/2).t / ((4π/3).(ct) 3) = 3c2/(8πG.t 2)

At the current age of the universe at 13.8 Gy or 4.35x1017s, that gives us a mass-energy density of 8.48x10-10 J/m3.  Noting that baryonic and dark matter make up 32% of the mass-energy of the universe, that means 68% is dark energy.  Specifically, that is 68% of 8.48x10-10 J/m3 which is 5.76x10-10 J/m3.  According the WMAP Survey the “positive energy density (is) about 6 × 10-10 joules per cubic meter”.  Another big fat coincidence.

Above I arrived at ΔM/Δt = (c3/G)/2 = (Mpl / tpl)/2.  It can be seen that his represents the lower bound of the Heisenberg Uncertainty Principle, because:

(ΔE/Δt) = (ΔM/Δt).c2 = (c5/G)/2 = (ℏ/2)/tpl2

Noting that the Planck time is the smallest division of time (equated with the lower limit of Δt), we can multiply through by tpl2 to find:

ΔE.tpl = /2

Given that Δt ≥ tpl, then we have ΔE.Δt/2, which is the Energy-Time variant of the Heisenberg Uncertainty Principle equation.  Another big fat coincidence.

Finally, there are some aspects to the fine-tuned universe argument that are addressed by the FUGE model.  I only want to go into two here, as an example; the value of Ω and the value of Λ as argued by Martin Rees.

Omega (Ω) is the density parameter and, he argued, its value is very close to 1.  If the value of Ω deviated from 1 by any significant margin then gravity would be either too strong or too weak – too strong and the universe would collapse, too weak and stars would not have been able to form.  In the FUGE model, the value of Ω in not “close to 1”, it is precisely 1 as argued above in regard to flatness.

Lambda (Λ) is the cosmological constant and, Rees argued, its value is very, very small.  Making some assumptions, including that the dark energy density is constant, he arrived at 10-122 as its approximate value which as a very small, but not zero value is curious.  In the FUGE model, however, one of the assumptions made by Rees does not hold.  Dark energy density is not a constant.  Overall mass-density is inversely proportional to time, so density of dark energy is must be decreasing albeit at a slower rate than the density of baryonic and dark matter.

---

I’ve not progressed much further than this.  I don’t have anything to say, at this time, about how mass-energy coalesces into baryonic or dark matter.  I have some inkling about how the probability of a Planck atom appearing at any location is inversely proportional to the concentration of mass-energy in that location and the increased likeliness of space to appear where there is already space leads to localised curvature which manifests as gravity.  For the moment, these can remain as projects for the future.

## Tuesday, 6 August 2019

### 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:

rs = 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 lpl / tpl = 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 . G/c3 (noting that Δr2 = lpl2)

=> Δp . Δx = /2

This is the lower limit of Heisenberg’s Uncertainty principle (Δp . Δx ≥ /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 ≥ /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.

---

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.

---

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).

## Sunday, 4 August 2019

### Vacuum Energy, Dark Energy and the Units of the Planck Parameter

So, I’ve been asking some questions and getting answers which indicate that the questions are somehow annoying (but which don’t actually address the questions asked).  In the process, the topic of vacuum energy came up which is something that I had not even considered.

I sat at my desk for a while pondering how I would work out the amount of energy entering the universe at a given time and then get the average amount per cubic metre.  Then I intended to compare that value to the value given for vacuum energy, which I naïvely thought I’d just look up (it’s never that simple).

But as I sat there pondering, I thought: I already have a value that I could work with.  I concluded in Is the Universe Getting More Massive?  (Flatness, not Fatness) that mass-energy enters the universe at a rate of one Planck mass per Planck time.  I worked out that the density of the universe, if flat, after 13.8 billion years of this process would be the critical density at that time, which is approximately 10-26 kg/m3.  Now we know that E=mc2 (it’s really Eo=moc2, since we need to consider rest mass but I’m sure we can get past that).  Given that I already say mass-energy, I don’t have any problem expressing a mass in terms of its energy equivalent and in this case that is approximately 9x10-10 J/m3.  According to current estimates, 32% of the universe is either baryonic matter or dark matter, so … if the rest is just dark energy burbling away in “empty space”, then that would 68% of 9x10-10 J/m3, or about 6x10-10 J/m3.

At the Wikipedia article on vacuum energy, the first value given for the vacuum energy of free space is 10-9 J/m3.  This is the value estimated “using the upper limit of the cosmological constant”, and Sean Carroll is cited as the source (via C-SPAN’s Cosmology at Yearly Kos Science Panelbroadcast, Part 1).  The same value is quoted by John Baez, and he goes on to write:

One can know something is very close to zero without knowing whether it is positive, negative or zero. For a long time that's how it was with the cosmological constant. But, recent measurements by the Wilkinson Microwave Anisotropy Probe and many other experiments seem to be converging on a positive cosmological constant, equal to roughly 7 × 10-27 kilograms per cubic meter. This corresponds to a positive energy density of about 6 × 10-10 joules per cubic meter.

Interesting, huh?  Another big fat coincidence.

---

In a parallel discussion in which I was accused of saying that there’s a speed (distance/time) associated with the expansion of the universe despite having carefully written, in reference to a hypothetical universe:

The universe expands such that the radius increases by 1 Planck increment every 1 Planck time (possibly with smaller increments depending on at what point the granularity kicks in).

There is a lack of clarity with respect to that statement but I am not saying that the universe expands at any specific rate, I am just saying 1) the universe expands and 2) due to that expansion the radius increases at a rate that looks like it could be a speed.  In reality, I think the universe expands at a rate of 1 Planck time per Planck time, and that’s not a rate at all, it’s dimensionless.  Note that I am not currently thinking of the universe as a simple sphere, but even as a glome, the surface volume of universe would still expand in direct proportion to the radius of the glome.  Anyway …

I pointed out to my interlocutor that the Hubble parameter (today) is cited as ~70 because it’s expressed in km/s/Mpc, I assume because these are convenient figures in cosmology.  However, if you express this figure in Hubble lengths (where HL = c/H = 13.8 billion light years) and meters, rather than megaparsecs and kilometres, you get a value of 300,000,000 m/s/HL.  And, to more significant figures than is strictly necessary, this is the speed of light.  So, the expansion of the universe is associated with a very important speed, a speed when expressed in Planck units is 1.  But the expansion itself is not a speed, by its dimensions it’s more of a frequency – once every Planck time.

And the question that arises when thinking of the expansion of the universe as being a frequency is … a frequency of what?  It implies, strongly to me at least, that something is happening to the universe every unit of Planck time.  And for me, the answer is obvious, it’s expanding by an increment (be that a unit of Planck time, or a unit of Planck time multiplied by c, or a Planck length, or however you prefer to think of it).

---

Finally, when looking up the amount of dark energy in the universe, I found the NASA webpage on the issue.  On that page is the following text (for the purposes of transparency I should advise that it is followed immediately by a section of text that I am still a little dubious about although I plan to give it some more thought and I should highlight that, even though from NASA, they are only talking speculatively):

One explanation for dark energy is that it is a property of space. Albert Einstein was the first person to realize that empty space is not nothing. Space has amazing properties, many of which are just beginning to be understood. The first property that Einstein discovered is that it is possible for more space to come into existence. Then one version of Einstein's gravity theory, the version that contains a cosmological constant, makes a second prediction: "empty space" can possess its own energy. Because this energy is a property of space itself, it would not be diluted as space expands. As more space comes into existence, more of this energy-of-space would appear.

I recall reading that, in terms of the FLRW metric, dark energy increases but I can’t find it again.  However, the Wikipedia article on dark energy quite clearly indicates that dark energy increases:

when the volume of the universe doubles, the density of dark matter is halved, but the density of dark energy is nearly unchanged (it is exactly constant in the case of a cosmological constant)

This is entirely consistent with my model – at least now that I have got a better handle on how dark energy might fit in (ie all the energy that is entering the universe today is in the form of dark energy).

Oh, and by the way, I do understand that I am implying that dark energy and vacuum energy might be the same thing.  It’s clearly not outside the realm of possibility though, since actual scientists in the field have made similar claims.

## Tuesday, 30 July 2019

### 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.

---

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.

---

Take two facts from the universe, as we observe today (in cosmological terms).  First, the Hubble parameter at this point in time, Ho, is such that Ho = 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
V = 4πr3/3
ρ = M/V = 3c2/8πGr2 = 3H2/8πG = ρc

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

ρc = 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 mpl / lpl = 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 mpl / tpl = 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 Epl / tpl = 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.

---

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 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.

## Tuesday, 23 July 2019

### 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?

---

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.
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?

---

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.

---

However … this still 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 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?

---

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.

---

Please note the tag below "cynicism".  This is, in part, referring to the paranthetical "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.