[Parallel Universes] Another You, Very Far Away

Is another you reading this sentence right now, somewhere very far away?

Ordinary cosmology says yes, if space goes on forever.

Tegmark's Level I argument

Max Tegmark calls this kind of parallel universe Level I in his article Parallel Universes, and it rests on three claims:

1. Space is infinite, or at least far larger than the part we can see.
2. Matter is spread evenly on large scales, and the details of where it ended up were set by chance at the start.
3. A region the size of our observable universe can be arranged in only a finite number of ways.

If all three are true, the conclusion follows: every possible arrangement of a region like ours happens somewhere, and it happens again and again, without end. One of those arrangements has an Earth with a Chelsea, a software engineer typing this sentence, and you reading it.

Step one: is space infinite?

Nobody can prove that space is infinite. Astronomers can measure its shape, though.

Flat, closed, and open space

General relativity says space can bend. On the largest scales, a universe that looks the same everywhere can have one of three kinds of curvature:

• Flat: the geometry you learned in school works. Parallel lines stay parallel, and the angles of a triangle add up to 180 degrees.
• Closed, or positively curved: like the surface of a ball, but in three dimensions. Lines that start parallel eventually meet, and the angles of a triangle add up to more than 180 degrees.
• Open, or negatively curved: like a saddle, in three dimensions. Lines that start parallel spread apart, and the angles of a triangle add up to less than 180 degrees.

Try the closed case on a globe. Start at the equator, walk north to the North Pole, turn, walk back down to the equator, and walk along the equator back to your start. If you choose the turns right, each of the three corners is a right angle. That's a triangle with 270 degrees.

A closed universe is finite, like the surface of the globe: go far enough in one direction and you come back to where you started. A flat or open universe can be infinite.

Measuring the shape of space

You can't walk around a cosmic triangle, but the sky gives you one for free.

The oldest light in the sky is the cosmic microwave background, or CMB. It was released when the universe was about 380,000 years old, when it had cooled enough for atoms to form and light could travel freely for the first time. That light reaches us from every direction at a temperature of 2.725 degrees above absolute zero.

The CMB isn't perfectly smooth. It has tiny hot and cold spots, differences of about 1 part in 100,000. The COBE satellite found them in 1992. Physicists can calculate how large the typical spot was when the light left. Then they compare it with how large the spots look in our sky.

That gives you a triangle: one corner is us, and the other two are the two edges of a typical spot. The length of the far side is known, and the distance to it is known. If space is flat, the spots appear at one size. If space is closed, the light rays bend toward each other on the way, and the spots look bigger. If space is open, they look smaller.

What Planck measured

The Planck satellite measured the CMB with great precision. Its 2018 results, combined with maps of how galaxies are spread out, put the curvature at 0.0007, give or take 0.0019. A curvature of zero means flat space, so as far as anyone can measure, space is flat.

Flat isn't the same as infinite

Flat doesn't mean infinite, though.

Comparison: think of an old arcade game where your ship flies off the right edge of the screen and appears again on the left. A triangle drawn on that screen has 180 degrees, but the screen is still finite, because its edges connect.

Space could work the same way in three dimensions, a shape physicists call a three dimensional torus. In a universe like that, if it were small enough, you could see the same galaxy in two different directions, or matching circles of temperature in the CMB on opposite sides of the sky. The Planck team searched its maps for those patterns and found nothing, so if space wraps around like a cube shaped torus, the cube must be at least about as large as the distance to the oldest light we see.

So, space is flat as far as we can measure, and if it wraps around, it does so on a scale larger than what we can see. It might be infinite, or finite but enormous, and measurement alone can't tell which.

Inflation, the leading theory of the first fraction of a second, predicts that space was stretched far beyond the observable universe. Tegmark describes an infinite universe as a generic prediction of inflation.

Step two: matter is spread evenly, by chance

The second step has two parts.

The first part is that, on large scales, the universe looks about the same everywhere and in every direction. Astronomers call this the cosmological principle. The CMB supports it: the temperature of the oldest light is the same in every direction to about 1 part in 100,000. Maps of galaxies support it too. On small scales, galaxies form clumps, chains, and empty regions. On very large scales, every big box of space contains about the same amount of everything.

The second part is about chance. The tiny hot and cold spots in the CMB became the seeds of galaxies. Where the early universe was slightly denser, gravity pulled more matter together, and galaxies formed there. According to inflation, those first differences came from quantum fluctuations: small, random variations that quantum physics allows everywhere.

So a region very far away follows the same laws as ours, but it starts from its own random details, which gives it its own history.

Step three: only a finite number of arrangements

The pigeonhole principle

In Indonesia, a bank card PIN has six digits. Each digit can be 0 to 9, so the number of possible PINs is 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000. Give a PIN to 1,000,001 people and at least two of them must share one. It doesn't matter how the PINs were chosen: only 1,000,000 options exist, and you have more people than options.

Mathematicians call this the pigeonhole principle: if you put more pigeons than holes, at least one hole gets more than one pigeon.

In space, the people are regions the size of our observable universe, and the PINs are the ways each region can be arranged. Infinite space never runs out of regions, so the argument only needs the number of arrangements to be finite.

Why the number of arrangements is finite

In everyday physics, you might think a particle can sit at any position at all, with infinite precision. Then a region would have an infinite number of possible arrangements, because you could always move one atom by a smaller and smaller amount.

Quantum physics rules that out, because a particle can't have a perfectly sharp position and speed at the same time. Below a certain scale, two arrangements are so close that no measurement, even in principle, could tell them apart, so they count as one arrangement and the total stays finite.

Tegmark gives a deliberately generous upper limit. Our observable universe contains about 10⁸⁰ protons. At temperatures up to 100 million degrees, quantum rules would allow you to pack in about 10¹¹⁵ of them, no more. Each of those 10¹¹⁵ places is either filled or empty, so the number of possible arrangements is at most about 2¹⁰¹¹⁵.

That's an absurdly large number, but it's finite. Infinite space has infinitely many regions the size of ours, so the regions outnumber the possible arrangements without limit.

So every possible arrangement of a region like ours appears somewhere, and each one appears infinitely many times. Mesmerizing enough, yeah?

How far away is the other you?

The magazine version of Tegmark's article rounds these a little differently, to 10¹⁰²⁸, 10¹⁰⁹², and 10¹⁰¹¹⁸ meters.

10²⁹ is a 1 followed by 29 zeros. The observable universe is about 93 billion light years across, which is about 8.8 × 10²⁶ meters. So 10²⁹ meters is only about a hundred times the size of everything we can see.

10¹⁰²⁹ is a different kind of number. It's a 1 followed by 10²⁹ zeros. You couldn't write it down. Your body contains about 7 × 10²⁷ atoms. If you wrote one zero on each atom in your body, you would run out of atoms after about 7 percent of the zeros.

That's why the distance to your copy barely changes when you measure it in meters, light years, or the size of the observable universe. Dividing a 1 with 10²⁹ zeros by a 1 with 27 zeros leaves a 1 with almost exactly 10²⁹ zeros. Your copy is far beyond any distance that has a meaning in daily life, or even in astronomy.

Tegmark also points out that your nearest copy is probably much closer than that estimate, because the processes that made planets and people are at work everywhere, not only here. The estimate counts every possible arrangement, including the vast majority with no planets at all.

Will you ever meet your copy?

No. And not because the trip is long.

The universe isn't only expanding. Since 1998, astronomers have known that the expansion is speeding up, a discovery that won the 2011 Nobel Prize in Physics. One consequence is that some galaxies are already so far away that light we send now will never reach them, no matter how long it travels. The space between us grows faster than the light can cross it.

That boundary is called the cosmic event horizon. Gott and colleagues calculate that a light signal we send now can reach, at most, a region that's now about 15.5 billion light years away. Anything further is out of reach forever. Your copy sits far beyond it. No signal will ever travel between you and your copy, in either direction.

That's what makes Level I a parallel universe, and more than a far place. The region exists in the same space as ours, under the same laws, but it's cut off forever.

Is your copy you?

Your exact copy has the same atoms in the same arrangement as you, so it has your brain, your memories, and your thoughts. Right now it's reading this sentence too.

It stays exact only for a moment, because the region around your copy has its own random events. A cosmic ray hits a different cell, or a neighbor plays a different song, and the copy's life starts to drift away from yours.

And the argument gives you more than exact copies. If every possible arrangement appears, then every version of you that's possible under the laws of physics appears too:

• A version who studied something else and never wrote a line of code.
• A version who lives in a different city.
• A version who supports a different football club. I'd rather not think about that one, and if I ever met a version of me supporting another club, I'd curse him lololol.

Each of those is a region of space somewhere, very far away, following its own history. Level I doesn't need quantum branches or extra dimensions to give you "a version of me who chose differently". It only needs space to be big enough and the start to be random.

Whether any of those people is "you" is a question for philosophy, not for a telescope.

The weak points

Space might not be infinite. If space wraps around on a scale only a little larger than the observable universe, which Planck's measurements still allow, the number of regions could be too small for any copy to exist.

The start might not cover every possibility. The argument assumes that the random starting conditions explore every allowed arrangement, an assumption physicists call ergodic. It's reasonable, but nobody has shown that it holds for the whole universe.

Infinity breaks probability. If every outcome happens infinitely many times, it becomes hard to say which outcomes are more likely. "Infinitely many copies of you" and "infinitely many slightly different versions" are both infinite. Comparing them needs a rule for counting, and physicists haven't agreed on one, a gap they call the measure problem. Tegmark calls it severe, and it comes back in every kind of parallel universe.

You can never check. Even if every step is correct, no observation will ever show you a copy. You can test the shape of space, the evenness of matter, and quantum physics, but not the conclusion itself.

What I think

Of all the parallel universe ideas, Level I asks for the least. It takes three things that cosmology already accepts or strongly suspects, and follows them to the end. It needs no new particles, no extra dimensions, and no new reading of quantum physics. Tegmark calls Level I "rather uncontroversial" for that reason.

I think this kind exists. Its weakest step is the first one, the size of space. Future measurements of the shape of space could push that question one way or the other.

The argument starts with a pigeonhole and ends with a copy of me. In software, the same principle is the reason hash collisions can't be avoided: put enough items into a fixed number of slots and two of them must share one. Applied to the universe, it hands me a doppelganger.