On inflation and the multiverse

[Note: in the following, and in the title, I have used the word multiverse a lot. When I do I am exclusively referring to this type of multiverse, which has, for example, been used to try to explain why the cosmological constant it so small. If you have any questions then please do ask them.]

About a week ago, Peter Coles, another cosmology blogger (who also happens to be my boss' boss' boss - or something), wrote a post expressing confusion about the association of inflation with the multiverse. His post was a reaction to a copy of a set of lectures posted on the arXiv by Alan Guth, one of the inventors of inflation (and discoverer of the name). Guth's lectures claimed, in title and abstract, that there is a very obvious link between inflation and a multiverse. Peter had some strong comments to make about this, including the assertion that at some points he's inclined to believe that any association between inflation and a multiverse is no different to a thought pattern of: quantum physics ---> woo ---> a multiverse!

I have some sympathy for Peter's frustration when people over-sell their articles/papers, and I would agree that inflation does not require a multiverse to exist, nor does inflation itself necessarily make a multiverse seem particularly likely/obvious. However, it is also true that, in a certain context, inflation and a multiverse are related. Put simply, through "eternal inflation", inflation provides a mechanism to create many Big Bangs. To get the sort of multiverse this post is about, these different Big Bangs need to have different laws of physics, which is not generic. However it can occur if the laws of physics depend on how inflation ends, in a way which I will describe below.

As with Peter though, I am unaware of any complete inflationary model that will generate a multiverse. We could both have a blindspot on this, but my understanding is that the situation is that people expect (or hope?) that complete models of inflation derived from string theory are likely to generate a multiverse for reasons that I will describe below.

Before that, you're probably wondering what this inflation thing is...

Inflation

The inflationary epoch is a (proposed - although the evidence for it is reasonably convincing) period in the past where the energy density of the universe was almost exactly constant and homogeneous (i.e. the same everywhere) and the expansion of the universe was accelerating. After this inflationary epoch ended, the expansion was decelerating (which isn't surprising given that gravity is normally attractive) and the universe gradually became less and less homogeneous, until it looked like it does today. We like inflation for all sorts of reasons, but for the purpose of this post, the preceding two sentences are all you need to know.

This is the "potential energy density" stored by a hypothetical inflationary field, \(\phi\). The x-axis is the value of \(\phi\). The y-axis is the energy density. The hatched region is where the conditions for "eternal inflation" would be satisfied.

This whole paradigm is depicted in the figure above, which is showing the potential energy density stored by the inflationary field, \(\phi\), as a function of its value. When the field, \( \phi \), has a value less than \( \phi_e \) (i.e. its value is on the left in the figure) inflation is occurring, when \(\phi>\phi_e\) inflation will end. This occurs because the potential function is flat enough that this potential energy dominates kinetic energy, and therefore it also dominates the gravitational effects in the universe. And, when a constant potential energy dominates in the universe, the expansion of the universe accelerates. Then, when the field has a value greater than \(\phi_e\) the expansion stops accelerating, starts decelerating, and the universe begins to do the stuff we know of as the Hot Big Bang. The reason inflation is interesting is that \(\phi\) inevitably has small, quantum, fluctuations in its value. Thus inflation ends at slightly different times in different parts of the universe, and, also thus, the Hot Big Bang starts at slightly different times in different parts of the universe. As a result, there are very small fluctuations in the density of the post-inflationary universe - and it is these small fluctuations that then grow to become temperature anisotropies, galaxies, solar systems and bloggers. We can predict the statistical properties of these density perturbations because we can predict the statistical properties of the fluctuations in \(\phi\).

This is all fine and good and this inflationary paradigm now has a lot of observational weight behind it. I won't go into any discussion about whether it actually is the way our universe got started or not, except to mention that it definitely seems possible and is the leading paradigm amongst cosmologists today, even if there hasn't yet been a way to know conclusively if it's right or not.

Eternal inflation

There is a curious feature that can arise in these models. As I explained above, in the inflating universe, \(\phi\) has different values at different points in the universe. Inflation will end where \(\phi > \phi_e \) and continue where \(\phi < \phi_e \). As time proceeds more and more of the universe leaves inflation and enters the Big Bang. Because the field is always rolling down that potential energy function, one would ordinarily expect that, after a sufficient length of time, the entire universe has stopped "inflating" and entered the Big Bang. However, if the rate at which \(\phi \) increases, and thus takes some volume of the universe out of inflation, is sufficiently slow, compared to the rate of accelerated expansion, it can actually occur that the total volume of the universe that is still inflating also continues to increase. If \(\phi \) can get close enough to the top of the hill in the above figure (i.e. enter the hashed region), then this is exactly what happens. When it does, although some of the universe undergoes a Big Bang, in other regions, inflation continues eternally.

In detail, for this to occur eternally, the initial distribution of \(\phi\) needs to be wide enough that there is always some small part of it that started arbitrarily close to the very top of that hill. In reality, because of those intrinsic quantum fluctuations, \(\phi\) cannot get arbitrarily close to the hilltop. The question then becomes whether or not the quantum fluctuations are large enough that they will ever dominate over the tendency for the field to roll down the potential. The quantum fluctuations in the field relate to the total energy density of the universe (the height of the curve in the figure above), and the evolution downward of the field depends on this and the slope of the potential energy function. For any model that looks like the figure above, this eternal inflation condition will be satisfied near the top of the hill, because there the slope tends to zero. For models that arise in regions that aren't hill tops, whether inflation continues eternally, will depend on the nature of the model. So, this eternal inflation scenario is actually quite generic for inflationary models, even if not all pervasive.

This is meant to show, in two dimensions, what is happening during eternal inflation. The red area is still "inflating". The yellow/white areas have entered Big Bangs. These Big Bangs will never collide, because the red, inflating region between them will push them apart.

But, what does this mean? It means that, although some region of the universe will escape this eternal inflation scenario and \(\phi\) will evolve downwards to a value where inflation ends and a Big Bang occurs, in most of the rest of the universe \(\phi\) has a value such that inflation continues. This Big Bang that occurs is then essentially an isolated bubble, surrounded by more inflation. Then, as time proceeds, some other region of the inflating universe will eventually roll to where \(\phi>\phi_e\) and another Big Bang will occur there, but, again, most of the rest of the universe will continue inflating. And so, as time goes on, you always have most of the universe inflating, with bubble Big Bangs coming off of it. This is meant to be depicted in the figure immediately above. The two white blobs would be bubble Big Bangs, and the rest of the red area is still inflating. Note that the size of each bubble Big Bang will grow with time (faster than light); however the bubbles will never meet, because the volume of the inflating red area is increasing even faster (thus the growing bubbles are pushed apart).

So what does this mean? Is it interesting? Well, in this scenario, not particularly. Every single one of these Big Bang bubbles will leave the eternally inflating patch in exactly the same way, which means that they all end up looking the same. It is true that the precise fluctuations in \(\phi\), from point-to-point, in each bubble universe, will be different, so the precise locations and history of the galaxies, solar systems and bloggers, in each bubble will be different, but the statistical properties of those fluctuations will be the same, and crucially, so will all the laws of physics.

Where does the multiverse come from?

I haven't yet shown you a multiverse (at least not of the kind that I promised at the beginning). I've shown you a way that inflation, reasonably generically, will create lots and lots of Big Bangs, each of which is separated from all the others. However a multiverse would want those different Big Bangs to have different physical laws as well.

The beginnings of a multiverse. Eternal inflation could happen in both "a" and "b", and the resulting Big Bangs can occur in either A or B, meaning that the Big Bangs would have different vacuum energies and potentially even different laws of physics

To see how a multiverse can come from this needs a slightly different function for the potential energy density. Consider the potential function shown above. Now there are two minima, A and B. If I chose an initial value for the field, entirely at random, it could "roll" to either minimum/vacuum. Now, remember all the lessons about inflation from the previous sections of this post. So long as both of the regions "a" and "b", support eternal inflation, this potential will generate bubble Big Bangs that are sometimes occurring in vacuum A and sometimes in vacuum B.

And here we have the beginnings of a multiverse. Firstly, the way I've drawn the potential you can see that the two minima have different potential energies. It is precisely this "vacuum" energy that would be responsible, today, for the effects labelled as "dark energy". Therefore, the bubble Big Bangs that end up in vacuum A would measure a different dark energy density to those that ended up in vacuum B. Moreover, suppose that some aspect of fundamental physics depends on the value of the field. Note that this is not particularly far-fetched at all. For example, the masses of many of the fundamental particles in the universe almost certainly depend on the value of the Higgs field. If some fundamental parameters did depend on the value of the inflationary field then the very nature of chemistry, atoms, biology, galaxies, bloggers, etc, in the bubbles that land in A would be completely different from those which land in B.

We're almost there, but this is still not quite the "multiverse" people are trying to motivate nowadays in fundamental physics. In that multiverse there are not just two types of bubbles, there are many. So how can that happen?

Now there are two field dimensions and eight potential vacuum energies/laws of physics. The hypothetical multiverse we're in would have many more field dimensions and many more potential vacua/minima. I would like to thank Stephen Wolfram for personally making this image for me after I entered the instructions at this http URL 

To see that, we need more than one inflationary field. Suppose that instead we were descended from a universe with just two field dimensions. In this universe, the potential energy function could look like the figure above, where \(x\) and \(y\) represent the values of the two inflationary fields. Now, you can see many different possible minima/vacua. So, lets follow the logic of the earlier sections of this post. If this potential energy function is flat enough near the centre to support eternal inflation, then there would be an infinite number of Big Bang's that would bubble off from the inflating patch. When these Big Bang's bubble off, they could roll to any one of the eight minima/vacua and thus the vacuum energy and physical laws could take up to eight different values and behaviours. This particular toy-toy-model has only eight vacua, but even in two dimensions, it is easy to imagine as many vacua as one wants, just by adding additional ripples in the potential.

And this, generalised to many more fields, is what people imagine when talking about the "inflationary multiverse". That is, an eternally inflating patch, with Big Bangs bubbling off from it, and descending into one of many different possible minima/vacua, each of which has a different vacuum energy and set of fundamental constants/laws.

[There is a very small part two here...]

Twitter: @just_shaun