In 1823, German astronomer Heinrich Olbers looked up at the night sky and saw darkness.
He wondered, if the universe were infinite and eternally static, surely the night sky
should shine with the light of infinite stars - a dazzling brilliant sky. Olbers’ paradox
was so compelling that many considered it proof of a finite universe, a cosmos that
at some point simply ends. It wasn’t until a century later, when Edwin Hubble observationally
established the reality of an expanding universe, that Olber’s paradox was firmly solved - Olber’s
second assumption was wrong, the universe is not static. But what does modern astronomy
have to say about the size of the universe though? Is it finite and if so does that mean
there’s an edge? Or, could be infinite? An endless ocean of space, with challenging
philosophical consequences if so.
How big is the universe? It’s the sort of question that a child often asks when they
first encounter the concept of space, and yet, it is one which continues to perplex
and haunt astronomers. It’s often said that studying the cosmos is a humbling experience,
for whilst early thinking considered the Earth Sun and Moon to be the de-facto universe,
our modern understanding establishes one that it is unimaginably larger. At each stage in
science’s journey of revelation, humanity has had to swallow another great demotion,
quietly ushered down to an ever lower seat in the great cosmic hierarchy.
These demotions began with in the 16th century with Copernicus and Kepler, challenging the
geocentric view of their time and revealing that Earth is just another planet. Next, in
1838, Freidrich Bessel pioneered the parallax method to measure the distance to the star
61 Cygni - a staggering distance of more than 10 light years, or approximately 100 trillion
km. In 1920, famed astronomers Harlow Shapley and Heber Curtis debated at the Smithsonian
Museum of Natural History as to whether distant nebulae were small clouds on the outset of
our galaxy or entirely separate galaxies far further away. Soon after Edwin Hubble established
that Curtis was right by proving that Andromeda was in fact far outside of the Milky Way galaxy,
at a distance of 2.5 million light years.
Equipped with far superior telescopes, modern astronomy has pushed our observations to distances
unimaginable even to these pioneering astronomers. For example, look at this image. It is one
of the most incredible photos ever taken. You are looking at the most distant galaxy
ever observed, GN-z11. This galaxy is so far away, that the light forming this image left
it 13.4 billion years ago. A photographic time machine. The Universe itself was just
a few hundred million years old at this time, and so GN-z11 is a glimpse at what the first
galaxies to ever form looked like. If we could see GN-z11 as it is now, it would probably
be unrecognizable, and in fact likely have merged with other galaxies along the way.
GN-z11 is estimated to be 32 billion light years away from Earth. At first, this seems
impossible, if the light left 13.4 billion years ago, surely it’s 13.4 billion light
years away. But that’s again falls into Olber’s fallacy, the universe is not static.
In fact, GN-z11 has been hurtling away from us ever since the light from this image left,
or more accurately I should say space has been expanding, and so a correct calculation
of its current distance needs to account for that fact.
Objects like GN-z11 can thus be thought of as establishing a minimum size of the Universe.
But, given that it’s 13.4 billion years old, and the Universe is 13.8 billion years
old, then surely it should be possible to image an object that’s even older and thus
further away and that would extend this minimum size yet more.
It’s at this point it’s useful to talk about two distinct concepts. The visible universe
and the observable universe. Neither of these really have anything to do with telescope
capabilities but rather true limits dictated by spacetime. The observable universe would
be the distance to an object that was present at the very dawn of the universe itself and
whose light was now just reaching us. Of course the universe didn’t have any stars or galaxies
at the very beginning so there are no objects to detect. Nevertheless, based on the expansion
rate of the universe, as measured using telescopes like ESA’ Planck mission, we estimate that
this distance would be 45 billion light years away. If something was further away than this,
there simply wouldn’t have been enough time yet in the universe for it’s light to reach
us. We often say such objects lie beyond our horizon, like a ship that has dipped below
the ocean’s horizon. There’s simply no way to see them whether they exist or not.
In fact, the horizon of the observable universe has a special name - the particle horizon.
The visible universe is actually subtly different from this. For the first few hundred thousand
years after the Big Bang, the universe was so hot it was an ionized plasma which essentially
acted like a thick fog. Light couldn’t traverse the universe without interacting with one
of these ions. It’s only when the universe cooled enough for atoms to form, clearing
the fog away in an epoch known as re-ionization, that light can freely travel. So the visible
universe is a little bit smaller than than observable one, by about a billion light years.
This is a truly mind bogglingly big number, no words or analogies really do justice to
just how preposterously large the universe is. But it
could be even bigger? Take the observable universe, corresponding to a distance 45 billion
light years away. Usually, at this point, video like this take that number and double
it and call it the diameter of the observable universe. We have to be a little careful about
that because what if we see another GN-z11 over in the opposite direction, not a galaxy
similar to GN-z11 but the actual GN-z11. For example, if the universe has a positive curvature
like a sphere, then it’s quite possible we would see the same galaxies and patterns
in different directions. If that were true, doubling this distance wouldn’t correspond
to the diameter but something more like the circumference of the universe. These repeating
patterns seem to offer a possible way, then, for the universe to be smaller than we might
naively have guessed.
To answer this, let’s return that moment of re-ionization in the early universe. This
is the oldest light we can observe and it comes from all directions, after all the entire
universe was filled with this ionized plasma. The light is known as the cosmic microwave
background, or CMB, and it encodes the temperature of the universe at this time. Any patch of
the CMB more than 2 degrees away from another patch is too far away to have had time for
light to travel between them, at least given the age of the universe at the time, just
380,000 years. So, if we see a particular pattern or ripple repeated in the CMB but
separated by several degrees or more, that could reveal the universe wraps around itself
- implying a smaller universe.
Detailed studies of the CMB reveal no such repeating structures though. The lack of such
structure can be used to put lower limits on the size of the universe, but not surprisingly
these limits essentially correspond to the approximately size of the observable universe.
So the full universe really does appear to be at least as large as the observable part.
Now let’s suppose someone lives in the galaxy GN-z11, they would be able to look back along
that line of sight and see the proto Milky Way, as a faint distant galaxy too. But what
if they looked in the opposite direction? Would they see an edge or some kind of physical
boundary?
Whilst we cannot truly know what they might see, a basic assumption in astronomy is the
so-called cosmological principle, which states that we do not live in a special part of the
universe. So what we see is typical and by extension what GN-z11 inhabitants see is likely
not much different from us, at least on the largest of scales. The cosmological principle
does not rigorously prove anything about the size of the universe, but rather forms a logical
argument. By it, inhabitants at the edge of the observable universe should themselves
be able to see another entire observable universe around them. Filling out across the edges,
this would give us a volume 180 billion light years across.
We have to tread carefully here though, because the size of the universe could still be smaller
than this if it somehow wraps around itself. Consider in this direction at the edge of
our observable universe we have Alice. Alice looks out in that same direction and can see
at the very edge of her observable universe someone else, let’s call him Carlos. Now
let’s go back to Earth and look in the opposite direction. Over at the particle horizon here
we see Bob. As before, Bob looks out in the same direction and sees someone else, just
like Alice did. However, Bob is in fact looking at Carlos again. To each individual observer,
there is no repetition, but Alice and Bob are in fact both able to see the same person,
Carlos. And since Alice and Bob live far beyond each other’s particles horizons, there’s
no way they can communicate this to each other. No-one is able to tell that the universe repeats
and thus are left non the wiser.
So we can’t really use the cosmological principle to argue for a much larger size
to the universe, all we can say is that we don’t see repetition within our horizon
and thus the universe must be at least 2 x 45 billion light years in extent, or 90 billion
light years.
Repeating patterns are not the only way to place limits on size though, already I’ve
mentioned the concept of curvature and here perhaps we might finally have a clue unto
it’s true size. Standing on the Earth, we can’t directly see its curvature because
the radius of curvature is just so vast. But the curvature can be detected with some geometry,
as was famously done so by Eratosthenes of Cyrene two millennia ago by comparing the
lengths of shadows of sticks in the midday Sun at two different locations. In an analogous
way, astronomers can use the CMB radiation and some geometry to measure the curvature
of the universe itself. A simple way to think about this on a positively curved surface,
like a sphere, two parallel lines will eventually meet. For example, if we both head straight
north, no matter where you live, I will eventually bump into you at the north pole. On a flat
surface the lines stay true and parallel forever, but on a negatively curved surface, like a
saddle, they diverge. So how curved is the universe? The curvature
parameter, known as Omega_k, has been refined ever more over the years thanks to missions
studying the CMB. The earliest estimate, done by the Boomerang experiment
in 2000, found it to be flat to within 6%. Eighteen years later, ESA’s Planck mission
published our current most precise estimate to date, which is again flat to within 0.19%.
It is truly an amazing technical feat by legions of astronomers involved that we can measure
the curvature of space at this level. But what does this really mean? Let’s consider
that the number is indeed exactly 0, a flat universe. It’s at this point, that many
often look at that flatness and interpret it to mean the universe must be infinite.
Look, it’s perfectly reasonable to suggest the universe is infinite as
a possible hypothesis, but it’s not true it’s
the only explanation. It’s a non-sequitor that flatness requires infinite space.
To cosmologists flatness doesn’t really mean what we might naively think and a flat
universe can be finite. In fact we’ve already discussed an example of such a universe. Let’s
go back to our friends Alice, Bob and Carlos and imagine the universe really is 2-dimensional,
like a sheet of paper depicted here. If we had a warp drive, we could fly from Earth
in the direction of Alice, and when we cross Carlos we would re-emerge on the other side
and be heading home again. This is probably familiar to any computer gamers, games like
Pac-Man and Asteroids would often feature this rule. Since the the left-side and right-side
really correspond to the same line, the universe wraps around itself here. If up and down go
on forever, then what we essentially have is an infinite cylinder. In fact, since I
made this cylinder from a 2-dimensional sheet, it’s really a 2-cylinder. The 2-cylinder
is really defined by left-goes-to-right Pac-Man rule operating on the original 2D sheet, what
is known as the fundamental domain. That’s what makes it a 2-cylinder. When we roll it
up in 3-dimensions, we’re embedding the 2-cylinder into 3-dimensional space, but that’s
just us embedding it into 3D. So even though it looks curved when embedded in 3D, that’s
just a product of the embedding, it’s actually still a flat geometry in it’s fundamental
domain.
Now this 2-cylinder doesn’t really save us from an infinite universe because it’s
length extends out to infinity. So let’s go back to our fundamental domain and look
at our rules one more time. What if we add another rule that going past the upper edge
takes you back to the lower edge. Now we truly have full Pac-Man rules. A fundamental domain
described by these two rules is no longer an infinite 2-cylinder, but a finite 2-torus.
We call it a torus because if you imagine re-rolling it back up, we first get the cylinder
as before, and then we have to connect these two ends together. In practice, if I try to
do this I’d have to stretch out the material, distorting the plane, but we can see that
it should form something like a donut shaped torus.
Now at this point it’s useful to discuss John Nash, who you probably know from the
biopic A Beautiful Mind. The fact you have to stretch and distort the original plane
to create the torus shape means that you failed to isometrically embed it. And that isometric
part is key because without it distances are distorted in a way we don’t see in the universe
around us. But John Nash proved the it was in fact possible to isometrically embed a
a 2-torus into three-dimensions, by adding waves and waves of tiny corrugations during
the embedding procedure. This was a beautiful proof but it didn’t actually reveal what
this 2-torus embedded into 3D would actually look like. In 2012, a French team used supercomputers
to finally calculate what the shape looked like. This stunning morphology is in fact
flat in the fundamental domain and adheres to the 2-torus Pac-Man style rules.
The French team went a step further and even uploaded a 3D print design up on their website.
Thanks to those corrugations, it’s a formidable 3D print to actually pull off though. I asked
around several New York printing agencies about this and they all basically just said
straight up no. If there was anyone that could pull this off, it was going to be Joel, also
known as the 3D Printing Nerd - an awesome YouTube channel that explores everything 3D
print related. He took this on and has even created an awesome video describing this feat,
please do check it out when you’re done here.
So here it is, a 3D embedded 2-torus. It’s even more beautiful to hold and touch than
the renderings. And what’s truly mind blowing is that I may actually be holding the universe
in my hands. A finite flat cosmos embedded in some higher dimensional space. Of course,
remember that this is a 2-torus, it’s made from a two-dimensional plane. Since our universe
has three spatial dimensions, the universe would need to be a 3-torus, so a kind of hyper-dimensional
extension of this incredible shape.
The 3-torus is in fact just one of ten different possible manifolds that could represent a
flat, finite universe. Six of these are called orientable and include the 3-torus. But four
are so-called non-orientable manifolds, and include this thing, the bizarre Klein bottle
- a mind bending shape off technically zero embedded volume. This orientability property
is best understood by looking at a simpler manifold, the Mobius strip. If travel along
the Mobius, after one circuit your left and right have switched over - you’ve become
a mirror version of yourself. If the universe is truly one of these non-orientable manifolds,
it would have some bizarre consequences for a hypothetical warp ship. Leave Earth and
travel in one direction long enough and you return home. Arriving back on Earth, you would
at first feel the familiarity and relief of being back home, but would quickly realize
something was off. Your family’s faces were mirror versions of before, their watches ticked
anti-clockwise. But from their perspective, it would you who was changed, and taking you
to the doctor for a medical examination they’d discover your heart was now on the right-hand
side.
All ten of these manifolds are perfectly consistent with our observations, and all are finite.
So any of these could be true, but there is another possibility. The universe could be
truly infinite, either an infinite flat fundamental domain in all directions, or a negatively
curved infinite saddle-shaped universe. Either way, we are forced to face an ontological
crisis. What if the universe is infinite? Infinities
are uncomfortable to physicists because they bring along some disturbing consequences.
Given infinite possibilities, everything that can happen must happen, an infinite number
of times.
In seeking a way out of this, one might consider that yes, perhaps space is indeed infinite,
but, at some point its nature radically changes. There is an end to the universe. Indeed, there
is some theoretical work that might at first seem to fit the bill. In the so-called eternal
inflation model, our local universe is a bubble-like region where the hyper-expansion phase known
as inflation subdued, but travel far enough and you’d enter a region where the inflationary
field has a different state, expanding space so much that each bubble universe is separated
by unimaginable distances. However, since the entire landscape is still infinite, there
would be an infinite number of these bubble universes. So, in truth this doesn’t really
solve our dilemma.
So let’s just stop hiding and face the possibility head on. An infinite cosmos. Infinities disrupt
probability theory, since all events have a probability of occurring an infinite number
of times, like infinite monkeys on infinite typewriters. It challenges the cosmological
principle, because what does it really mean to assume you’re typical if there’s an
infinite number of outcomes out there.
At a human level, this infinite universe opens the door to some unsettling consequences.
Because it means that there is an infinite number of you’s out there. Not just similar,
but yous with the exact same arrangement of molecules and atoms, right down to the quantum
states of your subatomic particles. That immediately provides a kind of way out of the no-cloning
theorem, a quantum theory proved by James Part that states that its impossible to ever
create an independent and identical copy of an arbitrary quantum state. If these clones
share every single quantum state to you, surely that is you? And indeed these doppelgängers
would sit inside an entire universe that was identical to our own right down to the last
atom. Each one watching this video right now, each one their jaw dropping, their sense of
self dissolving, as they challenge the very meaning of who they are.
Yet more, in the past, in the future, there are infinite doppelgängers of you slightly
offset in time. A version of you one day behind, two weeks ahead. You are everywhere. And you
never end.
Despite the existential crisis this carries, it perhaps provides some relief in other ways.
For our close loved ones that we’ve lost continue to live somewhere. Both offset in
time, but also through different choices. An infinite number of loved doppelgängers
who didn’t get in the car that day, who decided not to take those terrible drugs,
who got to hospital in time. Still out there, somewhere, just preposterously distant.
At the same time, an infinite universe can also, paradoxically, feel empty and nihilistic.
For many, we define our goals in life in terms of our impact on the world. But that impact
dissolves away in infinity. For example, if your mission is to reduce suffering amongst
humanity, it is challenged by the fact that in an infinite universe there is infinite
suffering. And whatever you subtract from infinity, it’s still infinity. And likewise
there is infinite happiness, so if your goal is to increase happiness, surely your actions
are utterly futile.
As we look further and further out, we perhaps finally turn back to inner space. We can’t
affect an infinite universe, or even realistically the infinitely smaller one we see around us.
Rather than trying to derive meaning from our impact on the cosmic scale, the ensemble
of space, meaning might actually be much closer to home. Because whilst there might be an
infinite number of yous, only you can control your actions today, how you choose to live
your life, how you treat others, and so even the smallest acts of kindness carry infinite
weight.
So until the next video, stay thoughtful and stay curious.