WHERE ARE ALL THE OTHER UNIVERSES?

 

Inflation and Tegmark’s “Level I Multiverse”

 If somebody asked me where all the other parallel universes are, I would probably respond that I don’t accept the premise of the question.  The question presumes that they are somewhere, but, in the Plexus model, each block universe is a mathematical construction that is self-contained: it certainly doesn’t contain other universes.

However, in 2012, Anthony Aguirre and Max Tegmark published a paper in which they claimed exactly that – in effect, they said that all the parallel universes that produce quantum uncertainty occupy different parts of the same, vast space.  This page examines that claim and is a summary of a paper that I posted on arXiv in February 2017.

In order to understand the Aguirre-Tegmark model, we need to discuss inflation. The idea of inflation was first published in 1981 by American physicist Alan Guth in order to explain (1) why conditions like temperature of the universe seem so uniform, on average, across such widely separated regions; and (2) why space, which might well have been curved (so that the angles on a vast, cosmic triangle, for example, sum to more than 180°) in fact appears to be very flat (but see discussion later).  Guth proposed that, in the first instants of our universe, after the temperature had time to settle to the same value everywhere, space suddenly expanded so fast that parts of it were travelling faster than the speed of light from each other.  This doesn’t violate special relativity, because we’re not talking about things travelling faster than light through space – it’s the space itself that is expanding.  This sudden expansion is called inflation.  It stopped almost immediately, but during that moment, the universe grew to a billion billion trillion times its original size!

 What drives inflation is repulsive gravity.  “What!” you say.  “Surely gravity is always attractive?”  Well, yes, normally it is.  However, we’re talking about totally unfamiliar conditions, and, indeed, repulsive gravity could be predicted from Einstein’s equations of general relativity when he first published them in 1915.  The equations say that the acceleration due to gravity is proportional to the negative of the sum of two components: the density of any matter in it and the pressure of space (the sum is actually written as -(ρ+3p/c2)).  Now, the density of matter, ρ, is always going to be positive (including radiation and antimatter), and the pressure, p, (for instance, the pressure of radiation) is generally going to be positive, too.  So the acceleration due to gravity is normally negative, which means that, instead of a piece of matter being accelerated in a (+x) direction away from, say, the Earth, it is going to be pulled back in a (–x) direction, towards the origin, the centre of the Earth.

 However, space can be in an energy state that is sometimes called a “false vacuum”.  Unlike an ordinary (true) vacuum, a false vacuum has a negative pressure.  It turns out that, when you add the positive density to the negative pressure in the sum in brackets in the previous paragraph, the negative term, 3p/c2, dominates: the negative pressure of the false vacuum wins.  So the acceleration due to gravity is proportional to the negative of a negative number, which means that it is positive – the piece of matter would fall in the (+x) direction away from Earth!  This is what drives the exponential expansion of space that we call inflation.  Physicists have a name for the field that drives inflation – the inflaton field.  (No, I haven’t made a spelling mistake – the “i” at the end of the word is deliberately left out to make it consistent with “electron”, “meson”, “boson” and so on.)

Figure 1: The grey area represents space that is expanding exponentially. The expansion is due to the inflaton field. Parts of the expanding space are continually dropping out of the false vacuum state and forming “bubble universes”, shown in white, one of which is our own.

 So the picture is like that in Figure 1, where the inflaton field causes space, in grey, to expand exponentially with time (which is in the vertically upward direction).  In this figure I have also drawn white regions labelled “bubble universes”.  Along with its negative pressure, the false vacuum has energy.  The energy corresponds to a particular value of the inflaton field.  Quantum uncertainty – sometimes called quantum jitters in this context – can move the value of the field in such a way as to lower the energy in a kind of runaway process.  The end result is that a region of space “condenses out” of the false vacuum.  This is the bubble universe I referred to above.  As you can see, many bubble universes are generated in the false vacuum – indeed, many physicists in the field believe that the process is unending, leading to “eternal inflation”.  In our bubble universe, the moment of inflation lasted perhaps ten trillionths of a trillionth of a trillionth of a second, during which time it expanded by the stupendous amount I said above.  After that, the space in our universe continued to expand, but at a much more leisurely pace, which is what Edwin Hubble, an American astronomer, was seeing when he discovered in 1929 that distant galaxies are receding from us – the more remote the galaxy, the greater the speed of recession.

 Now we can go back to seeing where the Plexus and Tegmark’s models differ.  Tegmark defines what he calls a “Level I Multiverse”.  In order to understand what he means, we need to get our terms straight and, in particular, what each of us means when we talk about a “universe”.  When Tegmark refers to the Level I Multiverse, he is thinking of a multiverse of universes all within one bubble universe – see Figure 2.

Figure 2: This is intended to clarify the usage of the word “universe” and other terms. The background grey represents the inflaton field and the two white “U” shapes are bubble universes. Comoving coordinates are used in the horizontal direction and time is in the vertical direction.

 In Figure 2, the walls of the two bubble universes appear nearly vertical.  That is because I have drawn the space axis, in the x-direction, in what are known as comoving coordinates.  Comoving coordinates enable us to keep, on one diagram, features that would otherwise be far off the page to the left and right.  Imagine for a moment that being carried by expanding space is like being a floating insect on the surface of a flowing river.  To someone on the river bank, you are moving relatively fast downstream, but to you, on the river surface, your neighbouring insects are not moving, because they are being carried along with you.  So, if you crawl along a line on the water surface perpendicular to the bank, it seems to you that you are moving vertically, while it will appear to the observer on the bank that you are moving diagonally.

 So this is what comoving coordinates do – roughly speaking, they remove the horizontal component of motion from the bubble universes in Figure 1 so that diagonal lines now appear vertical, as in Figure 2.

 Look at the larger of the two bubbles in this figure.  (A bubble universe is also called a thermalized region as well as a “pocket universe” by Guth, who wanted to avoid the impression that the surfaces of these bubble universes were like smooth bubbles.)  Tegmark says that our bubble universe is populated with many parallel universes called Hubble volumes.  From the point of view of every observer in our universe, space expands away in all directions, and the further a point is from the observer, the faster it is carried away by the expanding space.  The distance at which this speed reaches the speed of light is called the Hubble radius.  A Hubble volume centred around an observer is the spherical boundary – the spherical radius – at which objects that are locally stationary in space are receding from the observer at the speed of light.

 (It may sound counter-intuitive, but our observable universe is greater than the Hubble volume.  This is because, during the time light takes to reach us from the Hubble boundary, the space from where the light was emitted has expanded beyond it.  So Tegmark could have chosen observable universes rather than Hubble volumes for his parallel universes.  However, it ultimately makes no difference to the discussion, and so we shall use Hubble volumes for Tegmark’s parallel universes.)

 

The Aguirre-Tegmark hypothesis

 The important point about Tegmark’s claim is that, since he suggests (along with many others) that the bubble universe is infinite in extent, then our own Hubble volume must be repeated identically throughout the bubble universe, along with nearly identical copies.  This is because there are only so many different ways in which you can arrange all the particles in our Hubble volume and so, given an infinite amount of space and particles, the Hubble volume must be replicated identically an infinite number of times!  Aguirre and Tegmark claim in their 2012 paper that these copies are the parallel universes that account for quantum uncertainty.  Their actual words are “the Level I Multiverse is the same as the Level III Multiverse”.

Just to be clear, the Level I Multiverse that Tegmark is talking about is the collection of all of the Hubble volumes that make up the bubble universe in Figure 2.  This bubble universe is what we called our block universe in The Mathematical Structure of the Plexus and appears at the bottom of Figure 7 of that page.  Tegmark’s Level III Multiverse refers to the parallel universes of Everett’s Many Worlds Interpretation (see Many Worlds).  So what Tegmark and Aguirre are saying is that these parallel universes, the Level III Multiverse, are really those Hubble volumes shown in Figure 2, which Tegmark calls the Level I Multiverse.

At first glance, this may seem outlandish, because why could you not just take a very long journey from our own Hubble volume to our neighbouring one – you’d then be in one of Tegmark’s parallel universes, wouldn’t you?  Actually, no, you wouldn’t, because distant objects outside of our Hubble volume that are being carried away by the expansion of space faster than the speed of light will be forever beyond your reach.  In any case, Aguirre and Tegmark aren’t claiming that our nearest neighbouring Hubble volume is necessarily a parallel universe to ours; only that identical, or nearly identical copies of our own Hubble volume exist somewhere in our bubble universe – and generally extremely far away.

Figure 3: Three quantum events, A, B and C, have outcomes |A1〉, |A2〉, |B1〉, |B2〉, |C1〉, |C2〉 and |C3〉 respectively, with corresponding probability amplitudes a, b and c. As I mentioned in Tree structure of the Many Worlds Interpretation, the absolute square of the probability amplitude is the relative branch thickness. Each branch, at every level, is in an eigenstate.

Nevertheless, I believe that first instinct is correct – that you won’t find, in our bubble universe, the parallel universes that produce quantum uncertainty (from not knowing which universe you’re in).  I shall explain why below, but, just so you know where I’m going with this, essentially I’m going to show you that our entire bubble universe is in a single quantum state called an eigenstate, which rules out having any alternative, parallel universe of a different eigenstate inside it.

We start with Figure 4 from the Tree Structure of the Many Worlds Interpretation.  This is copied in Figure 3 here, except that the branches and twigs have now been labelled slightly differently.

 The symbol |  〉” that you see in this figure stands for a quantum state, so that, for example, |A1〉 is the quantum state where the outcome of a quantum event was A1.  Before the event, or experiment, happens, we might describe the state as the sum of its possible outcomes, such as (|A1〉+|A2〉), but, once the event has taken place, it can be in only one of these outcomes.  If, for example, the outcome is an electron with a spin in the upwards direction, then it cannot have a spin in the downwards direction at the same time.  We call such a state, after the event has happened, an eigenstate.  After a quantum event, or experiment – that is, when a quantum state is no longer the sum of possible outcomes – it will be in an eigenstate, but it can’t be in more than one, any more than an electron can be measured to have both an up-spin and a down-spin at the same time.

Figure 4: Observers A and B measure the polarizations of pairs of entangled photons, and the correlations confirm that the quantum eigenstate extends throughout A’s Hubble sphere.

In Figure 3, there are eight possible universes, labelled (1)-(8), each with a different history.  The quantum state of each universe is made up of the particular outcomes of the events that led up to its current state.  So, for example, the state of universe (6) is the tensor product of |B2〉⊗|C1〉⊗|A2〉.  We used the tensor-product symbol, “⊗”, before, to multiply kernels in The Origin of Quantum Probability.  The important point is that each of the possible universes (1)-(8) – which are block universes, as we saw in Figure 5 of the Tree Structure of the Many Worlds Interpretation – is an eigenstate.

If the eigenstate extends in space no further than, say, the Hubble boundary, then it is perfectly possible for the model of Aguirre and Tegmark to work, where they envisage other Hubble volumes of different eigenstates far removed in space from our own Hubble volume.  Crucially, however, the eigenstate that is our Hubble volume extends not only beyond the Hubble radius – it extends across the whole bubble universe.  Here is the reasoning.

 

 The quantum state of our universe extends across our entire bubble universe

 In Figure 4, A and B each take part in a quantum entanglement experiment – this time, with photons rather than with the electrons that we used in quantum entanglement experiments in the Quantum Entanglement and the Wave Function Collapse pages.  Instead of detecting the spin of the electron (spin-up or spin-down), we detect the photon’s polarization .  Polarization can be in two directions, horizontal or vertical, and, if you detect two entangled photons with polarizer detectors that are inclined at an angle of 30° to each other, you find that there is agreement (both polarizations horizontal with respect to the horizontal of the detector or both vertical) in ¾ of the measurements.  In ¼ of the measurements, there is disagreement.

Figure 5: As in Figure 4, except that, here, we see that the quantum eigenstate extends across B’s Hubble sphere.

 In Figure 4, A is at the centre of her (red) Hubble volume and B is just inside the boundary.  They detect pairs of entangled photons from a source, O, half-way between them.  Since B is within A’s Hubble radius, B can send his results to A and she will eventually receive them, thereby verifying that the quantum entanglement state holds up to her Hubble radius.  Since she can do this in any direction, the whole sphere is coloured red in the diagram to indicate that it is in a single eigenstate.

 Equally, of course, we can do the same thing from B’s viewpoint.  Figure 5 shows the same arrangement as in Figure 4, except that the Hubble volume we are considering is the one centred around B.  So we see that B, like A, finds that his whole Hubble volume is in an eigenstate, and so we colour it green to emphasize that this is all one eigenstate.

Figure 6: Since A’s and B’s Hubble spheres overlap, they must share the same eigenstate. I have called the process of demonstrating this, eigenstate annexation.

 

 

 

Figure 7: Continuing the process of eigenstate annexation shows that points that are receding from each other at superluminal velocities (such as C and D here, because of the expansion of space between them) nevertheless share the same quantum eigenstate.

However, the Hubble volumes of A and B overlap, as shown in the top part of Figure 6, coloured in yellow.  This yellow-coloured volume is part of the eigenstate centred on A, but it is also part of the eigenstate centred on B.  The only way for this to happen is for both A and B to be part of the same eigenstate, as shown in the bottom part of the figure, where the total volume covered by A and B is coloured yellow.  I call this trick with entangled photons, eigenstate annexation, because you start with one eigenstate, say A’s Hubble volume, and you show that it can be extended over B’s Hubble volume, effectively annexing it into A’s territory.

You can continue playing this game effectively forever.  Figure 7 shows what happens when A and B annex the Hubble volumes centred on C and D respectively: the resulting volume of the eigenstate is indicated by the grey envelope.

Notice, though, that the distance between C and D is definitely greater than a Hubble radius; indeed, it is more than a whole Hubble diameter.  This means that C and D are receding from each other at a velocity that is much greater than that of light.  Again, I must emphasize that we are not contravening special relativity here: it is the space between C and D that is expanding so fast that leads to superluminal velocities.

Nevertheless, it may seem impossible that C and D could be part of the same quantum state when there is no hope of a message ever reaching C from D, or vice-versa.  However, it is quite easy to show that entanglement between objects that are receding from each other at greater than the speed of light is perfectly natural!

 

Entangling objects with mutual recession velocities > c

Imagine climbing to the top of a mountain at night-time and looking at the sky to the east, just above the horizon.  The most distant photons reaching your eye are coming from near the boundary of the Hubble volume – even if there are too few and of too low an energy for you to see them yourself. 

Figure 8: Two entangled photons are sent in opposite directions from a mountain top on Earth towards observers A and B who are at opposite sides of our Hubble volume, close to its spherical boundary. A and B are receding from each other faster than the speed of light because of the expansion of the space between them.

Suppose that there is an observer A at this boundary, who has a polarization detector to measure polarization in the vertical direction.  Now look in the opposite direction, to the west, again just above the horizon, and, once more, photons will be coming from about as far from you as they possibly can – although this time they will be coming from a point diametrically opposite to A.  Suppose there is another observer, called B, at this diametrically opposite point (see Figure 8), who has a polarizer inclined at 30° to A’s instrument.

If you, on top of the mountain, send two entangled photons in opposite directions, towards A and B respectively, then they will eventually reach their respective polarization instruments.  Since both A and B are within the Hubble radius (albeit in opposite directions), they can each send a message back to you with the result of their particular experiment (although you will be very old by the time the message reaches you).  If enough pairs of photons are sent out, it will eventually become clear that the outcomes are correlated, as, of course, we should expect with entangled photons.  Remember, of course, that when you send out each pair of photons, they are in no way correlated, as we can prove with a Bell inequality experiment (see Hidden Variables).  It is only once they are measured by A and B that any correlation exists.

Now, since A and B are both receding from you at nearly the speed of light in opposite directions, and, since their recession velocities are due to the expansion of space, we can simply add their velocities together (which you couldn’t do if this were a problem in special relativity), and we find that they are receding from each other at well above the speed of light (nearly twice the speed of light, in fact).

So we have demonstrated (or, at least, we could demonstrate if we lived long enough) that a single quantum entanglement state can span a whole Hubble diameter, meaning that it extends between points receding from each other at well above the speed of light.

Figure 9: Region 1 and Region 2 may be thought of as universes (1) and (6) of Figure 3. The boundary of Region 2 is extended to perform an eigenstate annexation of Region 1.

So we shouldn’t be too surprised, after all, that the envelope of the annexed eigenstate includes parts of space that are receding so fast from each other that they can never be in direct communication.  Clearly, we can continue eigenstate annexation forever, until it becomes clear that the entire bubble universe is in a single eigenstate.

The fact that the whole bubble universe is in a single eigenstate should be enough to highlight the difficulty of Aguirre’s and Tegmark’s idea of the bubble universe containing the many different Hubble volumes required for the multiverse.  These Hubble volumes would generally be in different eigenstates from that of our bubble universe, and, as I said before, a region of space can’t be in two different eigenstates at the same time.  It is worth taking the time, though, to see just what would happen if we tried to populate the bubble universe with different Hubble volumes.

In Figure 9, I have drawn two possible Hubble volumes, Region 1 and Region 2, as they might appear in the Aguirre-Tegmark model.  They are not touching, but we can always perform an eigenstate annexation by extending, say, Region 2.  When we extend it sufficiently to overlap Region 1, we notice that the overlap now contains point P, which is part of Region 1.

Since point P belongs to Region 1, it is part of the quantum eigenstate, or coherent mathematical structure, of Region 1.  However, point P is also encompassed by eigenstate annexation from Region 2 and so it must also be part of the quantum eigenstate, or coherent mathematical structure, of Region 2.  However, because of their different histories, the mathematical structures of the two regions are in general different: they are in different eigenstates.  So, if event P is a logical consequence of the mathematical structure of Region 1 (and, therefore, part of it), it will not in general be a logical consequence of the mathematical structure of Region 2, and vice versa.  Essentially, of course, what we are saying once again is that the bubble universe cannot be in two different eigenstates.

Figure 10: The eigenstate of Region 1 cannot be annexed into the eigenstate of Region 2 because that would involve annexing the two opposing outcomes of event A, namely |A1〉 and |A2〉.
Figure 11: If a Hubble volume is to be replicated exactly, then so must the surrounding environment, including galaxy G, because it can be seen by observer A from the edge of the Hubble volume.

To take a specific example, look at Figure 10.  The quantum state of Region 2, within the solid-line circle on the right, is eigenstate |B2〉⊗|C1〉⊗|A2〉, which corresponds to universe (6) in Figure 3.  That of Region 1, on the left, is |B1〉⊗|A1〉,  corresponding to universe (1) in Figure 3.  If Region 2 is extended in an eigenstate annexation, it overlaps Region 1 as it did in Figure 9.  If outcome |A1〉 now falls within the compass of Region 2 in the same way that point P did, then all of the outcomes |A1〉, |B2〉, |C1〉 and |A2〉 must be part of the mathematical structure of Region 2.  However, the two possible outcomes of event A, namely |A1〉 and |A2〉, cannot both be present in Region 2 because that would be like saying that the outcome of quantum event A had a result that was both spin-up and spin-down at the same time.  The same conclusion applies if it is, instead, |B1〉 that falls within the compass of Region 2.

The one case in the Aguirre-Tegmark model that we have not yet considered is where the only copies of our Hubble volume are identical copies.  Figure 11 shows what happens when the Hubble volume on the left is to be replicated.  There is an observer, A, at the edge of the Hubble volume who can see a galaxy, G, beyond the Hubble volume.  (Remember that, although we, at the centre of our Hubble volume, cannot see far beyond its boundary, an observer at the boundary has a different Hubble volume, and can see much further than we can.)  If we are to replicate the Hubble volume, we have to replicate the observer, along with her observations, and so we have to replicate the environment that she sees.  This is why an identical galaxy G has been drawn to the right of the replicated Hubble volume, along with the replicated observer A.

In other words, it is not just the Hubble volume that is replicated, but the space between the spherical volumes also.  In Figure 12, the attempt to represent replication in all three dimensions looks like a carpet with a repeating pattern of period of Δ.

Such a repeating pattern is characteristic of a finite universe.  For example, three-dimensional space could have a slight curvature to it, in the same way that the surface of a sphere is curved in two dimensions.  Although careful measurement of the cosmic microwave background suggests that space is flat, if the universe is large enough, then it could still be curved over distances much greater than our Hubble volume.  In that case, if you set out on a journey in a straight line, and if there were no upper limit to the speed with which you could travel, you would eventually come back to your starting point.  It might seem to you, though, that you had simply encountered a repeating part of the universe with which you were familiar, not realizing that you had actually circumnavigated your entire universe!

Equally, space could be exactly flat, but arranged in such a way that, when you reach an edge, you enter the space at the corresponding point in the opposite edge.  Mathematicians regard this space as a torus, like a donut with a hole in it.  An old-fashioned computer-games screen serves as a two-dimensional illustration for a 3-torus: when an object disappears from the top of the screen, it reappears at the bottom, and similarly for the sides of the screen.

Figure 12: When the Hubble volumes are replicated as in Figure 11, so too is the surrounding environment. This leads to a patterned-carpet structure repeated in every spatial direction with a period Δ.

So the only way to accommodate the Aguirre-Tegmark model is for our bubble universe to be finite: the replication of our Hubble volume is just a characteristic of such a universe.  Notice, by the way (as you can see in Figure 12), that the distance you go before things are repeated may well be greater than the Hubble volume.

So we are left with the picture that our universe is finite, although it probably extends well beyond the Hubble volume.  The quantum uncertainty that we experience results from the multiverse of other bubble universes.  These other bubble universes are not generated by the inflaton field like the second bubble in Figure 2: they are mathematical bubble universes in exactly the same sense that our own bubble universe is purely a mathematical structure.  So each bubble universe is a block universe in a single eigenstate along with a multiverse of separate block universes in the same or different eigenstates which produce quantum uncertainty.  This, of course, is the model that we had already come up with at the end of The Mathematical Structure of the Plexus.

 

 Nature of the mathematical structure

There is, perhaps, a difference between the Aguirre-Tegmark model and the Plexus hypothesis that is more significant than whether parallel universes are all in the same three-dimensional space (in the Aguirre-Tegmark model) or not (in the Plexus hypothesis).  It is that the mathematical structure of the Plexus is assumed to be sufficiently complex that it is incomplete, whereas Tegmark avoids such Gödelian self-referential knots by allowing only a basic system that is so straightforward that it cannot even express ideas such as the Gödel statement, “G” (see Gödel’s Incompleteness Theorems).

The Aguirre-Tegmark system would have to be so basic, though, that we would have to use such a simple kind of arithmetic that you may well be unable, for example, to do simple multiplication with it.  You might think that repeated addition would do, but how do you express the product of two numbers, one of which is zero?  However, the Plexus axioms must include Schrödinger-like relations, for which you need at least multiplication!

In effect, what this means is that inhabitants of Tegmark’s universe ultimately have the tools to explain it completely (because there are no true statements that cannot be proved) whereas those of universes in the Plexus may speculate but can never be sure (because Gödel’s First Incompleteness Theorem applies).

Much more important, though, than the differences between Tegmark’s structure and the Plexus is the very fact that they are, indeed, both mathematical structures.  Remember, as we saw in Self-Awareness in a Toy Universe, even a basic mathematical structure consisting of only three lines is powerful enough for a Universal Turing Machine to emerge within it, capable of detecting, modelling and so being aware of its environment including, in principle, its own existence.  To such an emergent substructure, its world is most certainly “real”, and this applies to any part of the overarching mathematical structure that is capable of supporting such self-awareness.  At the deepest level, it is the mathematical structure that is at the very heart of reality.

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