Level 1 parallel worlds arise if - because - space is infinite. Not so far back we told you it is finite, because the Big Bang happened a finite time ago so it's not had time to expand to an infinite extent. [1] Apparently, though, data on the cosmic microwave background do not support a finite universe. Even though a very large finite one would generate the same data.
`Is there a copy of you reading this article?' Tegmark asks. Assuming the universe is infinite, he tells us that `even the most unlikely events must take place somewhere'. A copy of you is likelier than many, so it must happen. Where? A straightforward calculation indicates that `you have a twin in a galaxy about 10 to the power 10^21 metres from here'. Not 10^21 metres, which is already 25 times the size of the currently observable universe, but 1 followed by 1028 zeros. Not only that: a complete copy of (the observable part of) our universe should exist about 10 to the power 10^118 metres away. And beyond that ...
We need a good way to talk about very big numbers. Symbols like 10^118 are too formal. Writing out all the zeros is pointless, and usually impossible. The universe is big, and the multiverse is substantially bigger. Putting numbers to how big is not entirely straightforward, and finding something that can also be typeset is even harder.
[1] Curiously, it could expand to infinity in a finite time if it accelerated sufficiently rapidly. Expand by one light-year after one minute, by another light-year after half a minute, by another after a quarter of a minute ... do a Zeno, and after two minutes, you have an infinite universe. But it's not expanding that fast, and no one thinks it did so in the past, either.
Fortunately, we've already solved that problem with our earlier convention: if `umpty' is any number, then `umptyplex' will mean 10^umpty, which is 1 followed by umpty zeros.
When umpty = 118 we get 118plex, which is roughly the number of protons in the universe. When umpty is 118plex we get 118plexplex, which is the number that Tegmark is asking us to think about, 10 to the power 10 to the power 118. Those numbers arise because a `Hubble volume' of space - one the size of the observable universe - has a large but finite number of possible quantum states.
The quantum world is grainy, with a lower limit to how far space and time can be divided. So a sufficiently large region of space will contain such a vast number of Hubble volumes that every one of those quantum states can be accommodated. Specifically, a Hubble volume contains 118plex protons. Each has two possible quantum states. That means there are 2 to the power 118plex possible configurations of quantum states of protons. One of the useful rules in this type of mega-arithmetic is that the `lowest' number in the plexified stack - here 2 - can be changed to something more convenient, such as 10, without greatly affecting the top number. So, in round numbers, a region 118plexplex metres across can contain one copy of each Hubble volume.
Level 2 worlds arise on the assumption that spacetime is a kind of foam, in which each bubble constitutes a universe. The main reason for believing this is `inflation', a theory that explains why our universe is relativistically flat. In a period of inflation, space rapidly stretches, and it can stretch so far that the two ends of the stretched bit become independent of each other because light can't get from one to the other fast enough to connect them causally. So spacetime ends up as a foam, and each bubble probably has its own variant of the laws of physics - with the same basic mathematical form, but different constants.
Level 3 parallel worlds are those that appear in the many-worlds interpretation of quantum mechanics, which we've already tackled.
Everything described so far pales into insignificance when we come to level 4. Here, the various universes involved can have radically different laws of physics from each other. All conceivable mathematical structures, Tegmark tells us, exist here: How about a universe that obeys the laws of classical physics, with no quantum effects? How about time that comes in discrete steps, as for computers, instead of being continuous? How about a universe that is simply an empty dodecahedron? In the level N multiverse, all these alternative realities actually exist.
But do they?
In science, you get evidence from observations or from experiments.
Direct observational tests of Tegmark's hypothesis are completely out of the question, at least until some remarkable spacefaring technology comes into being. The observable universe extends no more than 27plex metres from the Earth. An object (even the size of our visible universe) that is 118plexplex metres away cannot be observed now, and no conceivable improvement on technology can get round that. It would be easier for a bacterium to observe the entire known universe than for a human to observe an object 118plexplex metres away.
We are sympathetic to the argument that the impossibility of direct experimental tests does not make the theory unscientific. There is no direct way to test the previous existence of dinosaurs, or the timing (or occurrence) of the Big Bang. We infer these things from indirect evidence. So what indirect evidence is there for infinite space and distant copies of our own world?
Space is infinite, Tegmark says, because the cosmic microwave background tells us so. If space were finite, then traces of that finitude would show up in the statistical properties of the cosmic background and the various frequencies of radiation that make it up.
This is a curious argument. Only a year or so ago, some mathematicians used certain statistical features of the cosmic microwave background to deduce that not only is the universe finite, but that it is shaped a bit like a football.* There is a paucity of very long-wavelength radiation, and the best reason for not finding it is that the universe is too small to accommodate such wavelengths. Just as a guitar string a metre long cannot support a vibration with a wavelength of 100 metres - there isn't room to fit the wave into the available space.
The main other item of evidence is of a very different nature - not an observation as such, but an observation about how we interpret observations. Cosmologists who analyse the microwave background to work out the shape and size of the universe habitually report their findings in the form `there is a probability of one in a thousand that such and such a shape and size could be consistent with the data'. Meaning that with 99.9 per cent probability we rule out that size and shape. Tegmark tells us that one way to interpret this is that at most one Hubble volume in a thousand, of that size and shape, would exhibit the observed data. `The lesson is that the multiverse theory can be tested and falsified even when we cannot see the other universes. The key is to predict what the ensemble of parallel universes is and to specify a probability distribution over that ensemble.'
This is a remarkable argument. Fatally, it confuses actual Hubble volumes with potential ones. For example, if the size and shape under consideration is `a football about 27plex metres across [1] - a fair guess for our own Hubble volume - then the `one in a thousand' probability is a calculation based on a potential array of one thousand footballs of that size. These are not part of a single infinite universe: they are distinct conceptual `points' in a phase space of big
[1] Actually a more sophisticated gadget called the Poincare dodecahedral space, a slightly weird shape invented more than a century ago to show that topology is not as simple as we'd like it to be. But people understand 'football'.
footballs. If you lived in such a football and made such observations, then you'd expect to get the observed data on about one occasion in a thousand.