VIEWS OF DARK ENERGY

Edward Witten

STScI

May 5, 2008

Regrettably, I don’t have any new concept of dark energy to explain today.

I just want to explain two not very novel points:

• Discovery of dark energy greatly changed how we think about the laws of Nature

• The nature of the change depends crucially on whether dark energy is a “cosmological constant”

For me, the discovery of cosmic acceleration/dark energy was the most dramatic finding in physics since perhaps the discovery of the particle in 1974.

Of course, dark energy has the somewhat unusual property that it was a major embarrassment before it was discovered.

The reason for the embarrassment was simply that any reasonable calculation of quantum zero-point energy

gives an answer that is too big by orders of magnitude – by a lot of orders of magnitude.

In this formula, for bosons is

the ground state energy of a harmonic oscillator mode, where here

And the negative energy for

fermions comes from filling the Dirac sea.

Bosons and fermions cancel if they have the

same masses … a special case of supersymmetry.

The formula

is really a one-loop approximation, but trying to do a more complete calculation doesn’t seem to help.

The “sum” over states is really

and so is highly divergent.

What answer we get depends on what we do to cut off the divergence. However, the potential error is well over 100 orders of magnitude.

Even with the help of supersymmetry, one is still off by about 60 orders of magnitude.

Though there were some dissenters, the dominant reaction to this was to expect that somehow it would go away

… that some very deep and unknown mechanism, maybe involving mysteries of quantum gravity, would one day make the vacuum energy vanish.

Those of us who thought that way usually expected that the mystery mechanism would make the vacuum energy zero (and not just extraordinarily small) because it seemed that a value consistent with observational bounds, but not quite zero, would be unreasonably small to emerge spontaneously from a fundamental calculation.

The discovery of cosmic acceleration seems to show that the problem is never going to go away in that sense. But it doesn’t really prove it, and that is one of the reasons for the present meeting:

We don’t really know for sure if observed dark energy is really the energy of the vacuum or is there because we are not quite living in the vacuum, as in the picture:

There is a certain sense, however, in which the vacuum energy or “cosmological constant” hypothesis is the minimal one.

This is the only interpretation of dark energy that is based entirely on General Relativity with no fields beyond the gravitational field. One only needs a new constant of nature:

Every other theory of dark energy needs new fields (or more exotic new ingredients of some kind) and a more elaborate explanation.

This doesn’t necessarily mean that the cosmological constant is the right theory, but it is a simple and definite one and doesn’t yet really have a compelling competitor; I’ll say more about this at the end.

To try to give a sense of how cosmic acceleration has influenced our thinking about the laws of nature, it is reasonable to start with the minimal theory, the interpretation in terms of vacuum energy.

Quantum theory makes it clear that the “vacuum” that we live in shouldn’t just be taken for granted as “empty space,” as pre-quantum physicists probably tended to do.

The vacuum is filled with “virtual particles,” and its structure, which results from solving sophisticated equations of Quantum Chromodynamics (among other things) is largely responsible for the way the world is.

Before the discovery of the dark energy, quantum physicists tended to assume that the “vacuum” we live in has some very deep meaning that reflects Nature’s deepest secrets.

The discovery of cosmic acceleration has called this into question since, for one thing, the period of “inflation” that we are now apparently entering has an obvious analogy to the inflation that may have occurred in the past.

Inflation in the past didn’t go on forever and maybe that is also the case for the present epoch of inflation:

We are here!

If this is the right picture, our epoch and the vacuum

we live in seem a little less fundamental!

There is actually something that makes this picture seem plausible to me.

In the last decade or so, we’ve learned (through work of Maldacena and others) that it is definitely possible to make a stable quantum gravity vacuum of negative vacuum energy. Supersymmetrically, zero is also possible.

But it is extremely unclear whether in the presence of quantum gravity it is possible to have a stable world of positive vacuum energy.

If not, the Universe we see must be unstable, headed for a calamity (possibly in the very remote future long after all kinds of more conventional astrophysical calamities like the collision of the Milky Way and Andromeda and the burning out of the stars).

This may seem a little drastic. But actually in a way it is pretty tame …

The discovery of cosmic acceleration has motivated, to some, a more radical reassessment of the role of our “vacuum”:

What I’ve drawn is usually understood as a one-dimensional slice of a complicated multi-dimensional picture!

Us?

The idea here is that we shouldn’t aim to explain why “the vacuum” has a very tiny energy. Rather, we should look for a theory that generates all kinds of “vacua” with different properties – with energy large or small, positive, negative, or (in the supersymmetric case) possibly zero.

Such “vacua” are realized in different times and places in the Universe, perhaps as a result of cosmic inflation.

According to this picture, we live in a “vacuum” in which the cosmic acceleration is small, in part because that is where we can live.

In this view, the problem isn’t to explain why our vacuum has a small energy but why the Universe has all kinds of vacua with different properties.

This conception has been dubbed the “multiverse.”

Several distinguished physicists – among them A. Linde, A. Vilenkin, S. Weinberg, M. Rees – have proposed or advocated this picture for years. The motivations were cosmic inflation, the problem of the cosmological constant, and curiosity about whether the Universe could be like that. String theory wasn’t a primary motivation at all.

However, taking what we know at face value, string theory does seem to lead to something like this:

This is actually the main traditional embarrassment of the subject.

Since I am one of the people who developed methods to describe approximate string theory vacua, I am tempted to say that it was the embarrassment of my youth.

Until about ten years ago, string theorists generally assumed that we were getting this sort of result because we didn’t understand the theory. Who needs that mess? There is just one world we live in.

But since the dark energy was discovered, R. Bousso, J. Polchinski, L. Susskind, M. Douglas and other prominent string theorists have advocated that this sort of picture is the correct interpretation of string theory and the Universe.

I don’t know whether this view will survive, but I do think that coming to grips with the cosmological constant – if that is the right interpretation of cosmic acceleration – will lead physicists to think of the “vacuum” very differently from the way we used to.

Now I want to go into a little more detail about the impact of the dark energy phenomenon on views of string theory.

I’ll summarize the impact in terms of good news and bad news.

Prior to the dark energy epoch, we assumed that what we understood was seriously flawed for two reasons:

One is that there was no explanation for the vanishing of the cosmological constant.

The other had to do with trying to understand particle physics.

Though there was no one insuperable difficulty, and many things even worked elegantly, it seemed that trying to understand in string theory the details of particle physics – the quarks and leptons and gauge particles, unified with each other and with gravity – led to a maze of conceivable choices.

To describe particle physics via string theory, what one needs is to describe the “vacuum” of the theory – the observed particles and forces are then expected to result from small oscillations around this vacuum.

There were far too many approximate vacuum states, and none seemed to solve the most basic problem of all – vanishing of the cosmological constant after supersymmetry breaking.

We assumed – or at least I assumed – that for the theory to be right, one day a miraculous new idea would have to solve these problems.

Making the cosmological constant vanish would be a key test of this new idea.

And until it was found, our approximate string theory “vacua” could only be crude approximations.

The good news then is that if we are really living in a “multiverse,” it may be that the theory as we know it is pretty close to the truth.

If the “Universe” is really a “Multiverse,” finding the vacuum state we observe should be like searching for a needle in a haystack.

But this comes with a hefty dose of bad news … If the vacuum of the real world is really a needle in a haystack, it is hard to see how we are supposed to be able to understand it.

In other words, if an unimaginably large number of approximate “vacuum” states are realized in different parts of the Universe, none of them with any special meaning, and with the details of particle physics depending on where one happens to live, then what sort of understanding of particle physics can we hope to get?

I don’t have an answer to this question, but in a way the important point for today is that it is a different question from the ones we’d ask if we didn’t know about the dark energy.

Now I move on to the last part of this talk…

A lot of what I’ve said is most natural if we assume the minimal theory of dark energy – that is, the cosmological constant.

If the dark energy is something more complicated, then all bets are off.

For example, dark energy might really represent the discovery of a new elementary field or particle.

Here is a picture in which “dark energy” is really a sign of a new elementary particle:

We are here

There is a stable nonaccelerating vacuum, but we haven’t reached it yet.

If this is the right interpretation, then all the new thinking I’ve mentioned that has been occasioned by the discovery of cosmic acceleration may be misleading.

There might be, after all, a unique stable vacuum, with vanishing vacuum energy.

Elementary particle physics might spring uniquely from the underlying laws of nature, with uniquely determined values of dimensionless numbers like or

All the old viewpoints may be correct.

I have to say that I would be happy if this turned out to be the case – since I think it is more interesting if the dark energy isn’t just a constant, and I wish we would have a chance to compute one day the dimensionless constants of nature.

But there definitely are some difficulties.

In a picture like the one shown here, there should be a “fifth force,” resulting from couplings to matter

We are here

of the new and necessarily very light scalar field

Although there is no clear problem in principle, if one tries to make a model like this in string theory, one soon finds that it is difficult to get a potential that is flat enough to account for why our vacuum seems to be stable, and to make the scalar interact weakly enough to make the fifth force weak enough – that is, to preserve the successes of General Relativity.

There are lots of relatively light scalar fields -- that is where the “landscape” came from

but generally they aren’t really light enough, and/or they couple too strongly to ordinary

matter.

To me, the version that is closest to working is that the spin zero field could be one of the “axions” of string theory, a spin zero field similar to one of the fields that may be responsible for solving the “strong CP problem’’ (smallness of neutron electric dipole moment).

The potential of such a field looks like

with constants and .

There are three things one needs to make the model viable:

The vacuum energy should be suitably small; the Universe should evolve very slowly; and the couplings of to ordinary particles should be very tiny.

For this, should be rather close to the Planch scale and should be incredibly tiny. It actually isn’t hard to satisfy either one of these conditions, but it is pretty hard to get them both to work at once.

I was actually quite surprised at how hard it is to get both of the conditions to work, so as to get a competitor to the cosmological constant interpretation of dark energy.

But at any rate, our reward, if we succeed, and it happens that the true vacuum energy is really zero, is that we would be back where we used to be, not understanding the vanishing of the cosmological constant.

I think I’ll conclude by restating the two admittedly not very original points that I have aimed to convey:

• Discovery of dark energy greatly changed how we think about the laws of Nature

• The nature of the change depends crucially on whether dark energy is a “cosmological constant”

A last remark is to make an analogy with General Relativity.

Because the “cosmological constant” interpretation is so central in our thinking, tests of it are also central, even though there is no compelling competing theory.

VIEWS OF DARK ENERGY

Edward Witten

STScI

May 5, 2008