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? <picard clip> 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. <CLIPS> 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.
