Is the Universe (in) a black hole?

In short, in answer to the titular question - "not as such" - or, to put it in a bit more detail, "not really, but sort of (keeping in mind that it's also sort of not) or perhaps maybe".  A lot comes down to the definition of a black hole and certain characteristics associated with a black hole.  And where the singularity is (or rather when).

Let’s go through the argument, and please pay attention to the caveats and definitions.

Say we have a mass, M, which has compacted into a non-rotating, chargeless black hole.  The event horizon for this black hole is defined by the Schwarzschild radius, which is given by the equation:

r_s = 2GM/c²

where G is the gravitational constant and c is the speed of light.

The volume of the black hole, using the event horizon as its edge and which I am going to call the Schwarzschild volume, V_s, is given by the equation:

V_s = 4πr_s³/3 = 32/3.πG³M³/c⁶

And the density of such a black hole (using the volume defined by the Schwarzschild radius), which I am going to call the Schwarzschild density, ρ_s, is:

ρ_s = M/V_s = M / (32/3.πG³M³/c⁶) = 3c⁶/ (32πG³M²)

This shows that the Schwarzschild density of a black hole is inversely proportional to the square of its mass - meaning that the heavier the black hole is, the lower the Schwarzschild density it has.

The question I have is this; how big would a non-rotating, chargeless black hole be if the Schwarzschild density associated with it was the same as the density of the universe.

The equation for this mass (after rearranging the above) is:

M = sqrt (3c⁶/ (32πG³ρ_s))

The density of the universe is 9.9x10^-30 g/cm³ (which, to use SI units, is the same as 9.9x10^-27 kg/m³).  The mass equivalent to this, if it were a non-rotating, chargeless black hole, is 8.6x10⁵² kg.  And the Schwarzschild radius of a non-rotating, chargeless black hole with this mass is 1.3x10²⁶ m.

When we get to such large distances though, we use a more convenient measure like the light year.  This means that the radius of a non-rotating, chargeless black hole with the density of the universe is 13.7 billion light years.

It just so happens that the age of the universe is 13.8 billion years (one of the most recent measurements is from the Planck Collaboration, which has the figure at 13.82 billion years).

Perhaps it’s merely coincidence (it's not by the way, but it's not strictly related to being (in) a black hole).

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There is however an immediate question that arises.  The radius of the observable universe, at the "age" discussed above is the radius of the observable universe today (by which I mean within a hundred million years or so).  And that is not believed to be 13.8 billion light years - it's actually thought to be in the order of 46 billion light years.  And then there is the actual universe which could be significantly larger.

To avoid pain here, I'm going to use the term "light horizon" defined as the distance light travels in the period equal to the age of the universe.  It would be an expanding sphere and it's possible that it doesn't have any physical significance.  It would, however, be the distance to the edge of the universe if the universe were expanding at the speed of light (and had always expanded at no more and no less than the speed of light).  Please note that physicists tell us that this is not the case.

At the time that the Earth formed, the light horizon would have been closer to 9 billion light years.  If the total mass of the universe were a constant, and the light horizon were all that there was then, at that time, the density of the universe would have been 3.3 times greater than today and calculations based on that density would give us a Schwarzschild radius of 7.4 billion light years.  It wouldn’t match up.

When the sun is scheduled to explode into a red giant, in about 5 billion years or so, the light horizon will be about 19 billion light years.  Making the same presumptions, that would give us a density about 1/3 of what it is today, leading to a Schwarzschild radius of 22 billion light years.  And, again, it wouldn't match up.

It might be difficult to accept that it is just a staggering coincidence that the density of the universe, right now, is equal to that of a non-rotating, chargeless black hole with a Schwarzschild radius equal to the light horizon, right now - remembering that the light horizon may have no physical significance.

The reason that it looks this way however is because the universe is flat.  A ramification of the universe being flat is that, as Sean Carroll puts it:

... some folks will stubbornly insist, there has to be something deep and interesting about the fact that the radius of the observable universe is comparable to the Schwarzschild radius of an equally-sized black hole. And there is! It means the universe is spatially flat.

You can figure this out by looking at the Friedmann equation, which relates the Hubble parameter to the energy density and the spatial curvature of the universe. The radius of our observable universe is basically the Hubble length, which is the speed of light divided by the Hubble parameter. It’s a straightforward exercise to calculate the amount of mass inside a sphere whose radius is the Hubble length ( M = 4π c3H-3/3), and then calculate the corresponding Schwarzschild radius (R = 2GM/c2). You will find that the radius equals the Hubble length, if the universe is spatially flat. Voila!

Note that what I am calling the light horizon, Carroll is describing as "a sphere whose radius is the Hubble length"… that the total mass in the universe is increasing in proportion with the rate of expansion.  If it’s the latter, then I am tempted to think that dark matter is appearing in the universe at a rate of 6.4x10⁵¹ kg every billion years.  I know that this sounds like quite a lot, but that’s 0.2 μg of black matter appearing in a volume of space equal to the Sun each year.

Is there another interpretation?

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Note that the density of the universe given above, 9.9x10^-30 g/m³, is based on the notion that the universe is flat.  Observations from both WMAP and the Planck Collaboration confirm that the universe is flat with a 0.4% margin of error.  Such a density is known as the critical density, meaning that universe will neither collapse nor expand forever without end.  Instead, it will eventually stop expanding – after an infinite time.  (At this site, an apparently different value for the density is presented, 10^-26 kg/m³.  The difference here is due only to different units being used and the figure being rounded up.)

The fact that the universe appears to have this density is astounding (or “somewhat surprising” according to the understated people at Swinburne).  However, something should be noted: This value is the critical density right now.  In the past, the critical density would have been different and in the future it will be different again.  The equation for the critical density is:

ρ_c = 3H²/8πG

where H is the Hubble Constant.  As noted in an earlier article, the Hubble Constant is the inverse of the age of the universe.  This means that as the universe ages, ρ_c decreases.  And if ρ_c = ρ_s, as is the case in a flat universe, then:

3H²/8πG = 3c⁶/ (32πG³M²)

After introducing ꬱ = 1/H and rearranging, we have:

M = ꬱ.c³/2G

This implies that the universe is getting heavier by 1 unit of Planck energy every two units of Planck time.  Of interest is the fact that, with the assumption that the universe is flat and that it is expanding at the speed of light, we can recall the equation for the Schwarzschild radius and say that:

r_s = ꬱ.c = 2GM/c²  =>  M = ꬱ.c³/2G

How much heavier would the universe get over a 1 billion years, using this calculation?  Plugging in the values, it should come as no surprise that we get 6.4x10⁵¹ kg, which is what I calculated a completely different way above.

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So what does all this mean?

Perhaps nothing.  The few people who have engaged with me from reddit.com seem to agree that the above does not have physical significance and one even accused me of numerology/astrology - a fair call when I had a few rookie errors in an earlier version including the wrong radius of the observable universe and a less carefully crafted comment about black holes.   The comment at the top of this article is still too controversial for one correspondent - he wants the answer to be a hard "no".  I think that if I meant what I think he thinks I mean, then the answer would be a hard no, but I don't think he knows what I mean.  Perhaps even if he knew what I mean, he'd still want a hard "no" and that's okay, I might be entirely wrong.

Since then, I've had some more time to think and some helpful interactions with some experts.  I've struggled with the idea of inflation for quite some time and I also struggle with the idea that the expansion of the universe is speeding up, although that's a little more recent.  I'll try to put together something on that topic in the next few days, including what might be a resolution of the first struggle, although potentially replaced by a different, albeit related one.

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Further corrections gratefully welcomed.