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.

---

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.

---

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.

---

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.

---

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.