Dark Energy May Be Incompatible With String Theory,exactly when I wanted to write something. Well, first, I must say that I already wrote a text about this dispute, Vafa, quintessence vs Gross, Silverstein, in late June 2018. You may want to reread the text because the comments below may be considered "just an appendix" to that older text. Since that time, I exchanged some friendly e-mails with Cumrun Vafa. I am obviously more skeptical towards their ideas than they are but I think that I have encountered some excessive certainty of some of their main critics.

Wolchover's article sketches some basic points about this rather important disagreement about cosmology among string theorists. But there are some very unfortunate details. The first unfortunate detail appears in the title. Wolchover actually says that "dark energy might be incompatible with string theory". That's the statement she seems to attribute to Cumrun Vafa and co-authors.

But that misleading formulation is really invalid – it's not what Cumrun is saying. Here, the misunderstanding may be blamed on some sloppy "translation" of the technical terms that has become standard in the pop science press – and the excessively generalized usage of some jargon.

OK, what's going on? First of all, the Universe is expanding, isn't it? We're talking about cosmology, the big bang theory (which I don't capitalize – to make sure that I am not talking about the sitcom), and the expansion of the Universe was already seen in the 1920s although people only became confident about it some 50 years ago.

In the late 1990s, it was observed that the expansion wasn't slowing down, as widely expected, but speeding up. The accelerated expansion may be explained by dark energy. Dark energy is anything that is present everywhere in the vacuum and that tends to accelerate the expansion of the Universe. Dark energy, like dark matter, is invisible by optical telescopes (that's why both of them are called dark). But unlike dark matter which has (like all matter or dust) the pressure \(p=0\), the dark energy has nonzero pressure, namely \(p\lt 0\) or \(p\approx -\rho\) where \(\rho\) is the energy density. That's how dark energy and dark matter differ; dark energy's negative pressure is needed for its ability to accelerate the expansion of the Universe.

Dark energy is supposed to be a rather general, umbrella term that may be represented by several known, slightly different theoretical concepts described by equations of physics. So far, the by far most widespread and "canonical" or "minimalist" kind of dark energy was the cosmological constant. That's really a number that is independent of space and especially time (it's why it's called a constant) which Einstein added to the original Einstein's equations of the general theory of relativity. Einstein's original goal was to allow the size of the Universe to be stable in time – because his equations seemed to imply that the Universe's size should evolve, much like the height of a freely falling apple. It just can't sit at a constant value – just like the apple usually doesn't sit in the air in the middle of the room.

But the expansion of the Universe was discovered. Einstein could have

*predicted it*because it follows from the simplest form of Einstein's equations, as I said. That could have earned him another Nobel prize when the expansion was seen by Hubble. (Well, Einstein's stabilization by the cosmological constant term wouldn't really work even theoretically, anyway. The balance would be unstable, tending to turn to an expansion or the implosion, like a pencil standing on the tip. Any tiny perturbation would be enough for this instability to grow exponentially.)

That's probably the main reason why Einstein labeled the introduction of the cosmological constant term "the greatest blunder of his life". Well, it wasn't the greatest blunder of his life: the denial of quantum mechanics and state-of-the-art physics in general in the last 30 years of his life was almost certainly a greater blunder.

In the late 1990s, the Universe's expansion was seen to accelerate which is why it seemed obvious that Einstein's blunder wasn't a blunder at all, let alone the worst one: the cosmological constant term seems to be there and it's responsible for the acceleration of the Universe. Suddenly, Einstein's cosmological term (with a different numerical value than Einstein needed – but one that is of the same order) seemed like a perfect, minimalistic explanation of the accelerated expansions. Recall that Einstein's equations say\[

G_{\mu\nu} +\Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}.

\] Note that even in the complicated SI units, there is no \(\hbar\) here – Einstein's general relativity is a classical theory that doesn't depend on quantum mechanics at all. Here, \[

G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu}

\] is the Einstein curvature tensor, constructed from the Ricci tensor and the Ricci scalar \(R\). It's some function of the metric and its first and especially second partial derivatives in the spacetime. On the right hand side of Einstein's equations, \(T_{\mu\nu}\) is the stress-energy tensor that knows about the sources, the density of mass/energy and momentum and their flow.

The \(\Lambda g_{\mu\nu}\), a simple term that adds an additional mixture of the metric tensor to Einstein's equations, is the cosmological constant term. It naturally reappeared in the late 1990s. It's a rather efficient theory. The term doesn't have to be there but in some sense, it's even "simpler" than Einstein's tensor, so why should it be absent? And it seems to explain the accelerated expansion, so we need it.

The theory is really natural which is why the standard cosmological model was the \(\Lambda{CDM}\) model, i.e. a big bang theory with the cold dark matter (CDM) and the cosmological constant term \(\Lambda\).

What about string theory?

String theory really predicts gravity. You may derive Einstein's equations, including the equivalence principle, from the vibrating strings. Einstein's theory of gravity is

*a prediction of string theory*, which is still one of the main reasons to be confident that string theory is on the right track to find a deeper or final theory in physics, to say the least. Aside from gravitons and gravity (and Einstein's equations that may be derived from string theory for this force), string theory also predicts gauge fields and matter fields such as leptons and quarks. They have their (Dirac, Maxwell...) equations and their stress-energy tensors also enter as terms in \(T_{\mu\nu}\) on the right hand side of Einstein's equations.

String theory demonstrably predicts Einstein's equations as the low-energy limit for the massless, spin-two field (the graviton field) that unavoidably arises as a low-lying excitation of a vibrating string. To some extent, this appearance of Einstein's equations is guaranteed by consistency of the theory (or by the relevant gauge invariance, namely the diffeomorphisms) – and string theory is consistent (which is a highly unusual, and probably unprecedented, virtue of string theory among quantum mechanical theories dealing with massless spin-two fields).

Does string theory also predict the cosmological constant term, one that Einstein originally included in the equations? At this level, the answer is unquestionably Yes and Cumrun Vafa and pals surely agree. To say the least, string theory predicts lots of vacua with a negative value of the cosmological constant, the anti de Sitter (AdS) vacua. In fact, those are the vacua where the holographic principle of quantum gravity may be shown rather rigorously – holography takes the form of Maldacena's AdS/CFT correspondence.

There are lots of Minkowski, \(\Lambda=0\), vacua in string theory. And there are also lots of AdS, \(\Lambda\lt 0\), vacua in string theory. I think that the evidence is clear and no one who is considered a real string theorist by most string theorists disputes the statement that both groups of vacua, flat Minkowski vacua and AdS vacua, are predicted by string theory.

The real open question is whether string theory allows the existence of \(\Lambda \gt 0\) (de Sitter or dS) vacua. Those seem to be needed to describe the accelerated expansion of the Universe in terms of the cosmological constant. After 2000, the widespread view – if counted by the number of heads or number of papers – was that string theory allowed the positive cosmological constant. Even though I still find de Sitter vacua in string theory plausible, I believe that it's fair to say that the frantic efforts to spread this de Sitter view – and write papers about de Sitter in string theory – may be described as a sign of group think in the community.

There have always been reason to

*doubt*whether string theory allows de Sitter vacua at all. At the end of the last millennium, Maldacena and Nunez wrote a paper with a no-go theorem. It was mostly based on supergravity, a supersymmetric extension of Einstein's general relativity and a low-energy limit of superstring theories, but people generally believed that this approximation of string theory was valid in the context of the proof.

Sociologically, you may also want to know that in the 1990s, Edward Witten was "predicting" that the cosmological constant had to be exactly zero (and a symmetry-like principle would be found that implies the vanishing value). He was motivated by the experience with string theory. Even before Maldacena and Nunez and lots of similar work, it looked very hard to establish de Sitter, \(\Lambda \gt 0\) vacua in string theory. However, some of these problems could have been – and were – considered just technical difficulties. Why? Because if the cosmological constant is positive, you don't have any time-like Killing vectors and there can be no unbroken spacetime supersymmetry. Controlled stringy calculations only work when the spacetime supersymmetry is present (and guarantees lots of cancellations etc.) which is why people were willing to think that the difficulties in finding de Sitter vacua in string theory were only technical difficulties – caused by the hard calculations in the case of a broken supersymmetry.

However, aside from Maldacena-Nunez, we got additional reasons to think that string theory might prohibit de Sitter vacua in general. Cumrun Vafa's Swampland – the term for an extension of the (nice stringy) landscape that also includes effective field theories that string theory wouldn't touch, not even with a long stick – implies various general (sometimes qualitative, sometimes quantitative) predictions of string theory that hold in all the stringy vacua, despite their high number. Along with his friend Donald Trump, Cumrun Vafa has always wanted to drain the swamp. ;-)

The Swampland program has produced several, more or less established, general laws of string theory – that may also be considered consequences of a consistent theory of quantum gravity. Wolchover mentions that the most well-established example of a Swampland law is our "weak gravity conjecture". Gravity (among elementary particles) is much weaker than other forces in our Universe – and in fact, it probably has to be the case in all Universes that are consistent at all.

The Swampland business contains many other laws like that, some of them are more often challenged than the weak gravity conjecture. Cumrun Vafa and his co-authors have presented an incomplete sketch of a proof that de Sitter vacua could be banned in string theory for Swampland reasons – for similar general reasons that guarantee that gravity is the weakest force.

This assertion is unsurprisingly disputed by lots of people, especially people around Stanford, because Stanford University (with Linde, Kallosh, Susskind, Kachru, Silverstein, and many others) has been the hotbed of the "standard stringy cosmology" after 2000. They wrote lots of papers about cosmology, starting from the KKLT paper, and the most famous ones have thousands of citations. At some level, authors of such papers may be tempted to think that their papers just can't be wrong.

But even the main claims of papers with thousands of citations ultimately

*may be wrong*, of course. Sadly, I must say that some of this Stanford environment likes to use group think – and arguments about authorities and number of papers – that resembles the "consensus science" about the global warming. Sorry, ladies and gentlemen, but that's not how science works.

Doubts about the KKLT construction are reasonable because the KKLT and similar papers still build on certain assumptions and approximations. I am confident it is correct to say that the authors of some of the critical papers questioning the KKLT (especially the final, de Sitter "uplift" of some intermediate AdS vacua, an uplift that is achieved by the addition of some anti-D3-branes) are competent physicists – at least "basically indistinguishable" in competence from the Stanford folks. See e.g. Thomas Van Riet's TRF guest blog from November 2014 (time is fast, 1 year per year).

Cumrun Vafa et al. don't want to say that string theory has been ruled out. Instead, they say that in string theory, the observed dark energy is represented by quintessence which is just a form of dark energy (read the first sentence of the Wikipedia article I just linked to) – and that's why Wolchover's title that "dark energy is incompatible with string theory" is so misleading. I think that the previous sentence is enough for everyone to understand the main unfortunate terminological blunder in Wolchover's article. Cumrun and pals say that dark energy is described by quintessence, a form of dark energy, in string theory. They don't say that dark energy is impossible in string theory.

Wolchover's blunder may be blamed upon the habit to consider the phrase "dark energy" to be the pop science equivalent of the "cosmological constant". Well, they are not quite equivalent and to understand the proposals by Cumrun Vafa et al., the difference between the terms "dark energy" and "cosmological constant" is absolutely paramount.

Quintessence is a philosophically if not spiritually sounding word but in cosmology, it's just a fancy word for an ordinary time-dependent generalization of the cosmological constant – that results from the potential energy of a new, inflaton-like scalar field. String theory often predicts many scalar fields, some of them may play the role of the inflaton, others – similar ones – may be the quintessence that fills our Universe with the dark energy which is responsible for the accelerated expansion.

Now, the disagreement between "Team Vafa" and "Team Stanford" may be described as follows:

Team Stanford uses the seemingly simplest description, one using Einstein's old cosmological constant. It's really constant, string theory allows it, and elaborate – but not quite exact – constructions with antibranes exist in the literature. They use lots of sophisticated equations, do many details very accurately and technically, but the question whether these de Sitter vacua exist remains uncertain because approximations are still used. Team Stanford ignores the uncertainty and sometimes intimidates other people by sociology – by a large number of authors who have joined this direction. The cosmological constant may be positive, they believe, and there are very many, like the notorious number \(10^{500}\), ways to obtain de Sitter vacua in string theory. We may live in one of them. Because of the high number, the predictive power of string theory may be reduced and some form of the multiverse or even the anthropic principle may be relevant.These are just two very different sketches how dark energy is actually incorporated in string theory. They differ by some basic statements, by the expectation "how very technical certain adequate papers answering a question should be", and in many other respects. I think we can't be certain which of them, if any, is right – even though Team Stanford would be tempted to disagree. But their constructions simply aren't waterproof and they look arbitrary or contrived from many points of view. And yes, as you could have figured out, I do have some feeling that the way of argumentation by Team Stanford has always been similar to the "consensus science" behind the global warming hysteria. Occasional references to the "consensus" and a large number of papers and authors – and equations that seem complicated but if you think about their implications, they don't really settle the basic question (whether the de Sitter vacua – or the dangerous global warming – exist at all).

Team Vafa uses a next-to-simplest description of dark energy, quintessence, which is a scalar field. This scalar field evolves and the potential normally needs to be fine-tuned even more so than the cosmological constant. But Team Vafa says that due to some characteristically stringy relationships, the new, added fine-tuning is actually not independent from the old one, the tuning of the apparently tiny cosmological constant, so from this viewpoint, their picture might be actually as bad (or as good) as the normal cosmological constant. The very large hypothetical landscape may be an illusion – all these constructions may be inconsistent and therefore non-existent, due to subtle technical bugs overlooked by the approximations or, equivalently, due to very general Swampland-like principles that may be used to kill all these hypothetical vacua simultaneously. Team Vafa doesn't have too many fancy mathematical calculations of the potential energy and it doesn't have a very large landscape. So in this sense, Team Vafa looks less technical and more speculative than Team Stanford. But one may argue that Team Stanford's fancy equations are just a way to intimidate the readers and they don't really increase the probability that the stringy de Sitter vacua exist.

Team Vafa proposes a new possibility and I surely believe it deserves to be considered. It's "controversial" in the sense that Team Stanford is upset, especially some of the members such as E.S. But I dislike Wolchover's subtitle:

A controversial new paper argues that universes with dark energy profiles like ours do not exist in the “landscape” of universes allowed by string theory.What's the point of labeling it "controversial"? It may still be right. Strictly speaking, the KKLT paper and the KKLT-based constructions by Team Stanford are

*controversial as well*. These

*a priori*labels just don't belong to the science reporting, I think – they belong to the reporting about pseudosciences such as the global warming hysteria. Reasonable people just don't give a damn about these labels. They care about the evidence. Cumrun Vafa is a top physicist, he and pals have proposed some ideas and presented some evidence, and this evidence hasn't really been killed by solid counter-evidence as of now.

Incidentally, after less than two months, Team Vafa already has 23+19 citations. So it doesn't look like some self-evidently wrong crackpot papers, like papers claiming that the Standard Model is all about octonions.

I was also surprised by another adjective used by Wolchover:

In the meantime, string theorists, who normally form a united front, will disagree about the conjecture.Do they form a

*united front*? What is it supposed to mean and what's the evidence that the statement is correct whatever it means? Are all string theorists members of Marine Le Pen's National Front? Boris Pioline could be one but I think that even he is not. ;-) String theorists are theoretical physicists at the current cutting-edge of fundamental physics and they do the work as well as they can. So when something looks clearly proven by some papers, they agree about it. When something looks uncertain, they are individually uncertain – and/or they disagree about the open questions. When a possible new loophole is presented that challenges some older lore or no-go not-yet-theorems, people start to think about the new possibilities and usually have different views about it, at least for a while.

What is Wolchover's "front" supposed to be "united" for or against? String theorists are united in the sense that they take string theory seriously. Well, that's a tautology. They wouldn't be called string theorists otherwise. String theory also implies something so they of course take these implications – as far as they're clearly there – seriously. But is there any valid, non-tautological content in Wolchover's statement about the "united front"?

It's complete nonsense to say that string theories are "more united as a front" than folks in any other typical scientific discipline that does things properly. String theorists have disagreed about numerous things that didn't seem settled to some of them. I could list many technical examples but one recent example is very conceptual – the firewall by late Joe Polchinski and his team. There were sophisticated constructions and equations in the papers by Polchinski et al. but the existence of the firewalls obviously remained disputed, and I think that almost all string theorists think that firewalls don't exist in any useful operational sense. But they followed the papers by Polchinski et al. to some extent. Polchinski and others weren't excommunicated for a heresy in any sense – despite the fact that the statement "the black holes don't have any interior at all" would unquestionably be a radical change of the lore.

This disagreement about the representation of dark energy within string theory is comparably deep and far-reaching as the firewall wars.

Again, I still assign the probability above 50% to the basic picture of Team Stanford which leads to a cosmological constant from string theory. But I don't think it has been proven (a similar warning I have said about \(P\neq NP\) and other things). I have communicated with many apparently smart and technically powerful folks who had sensible arguments against the validity of the basic conclusions of the KKLT. I am extremely nervous about the apparent efforts of some Stanford folks to "ban any disagreement" about the KKLT-based constructions, a ban that would be "justified" by the existence of many papers and their mutual citations.

That's not how actual science may progress for a very long time. If folks like Vafa have doubts about de Sitter vacua in string theory and all related constructions, and they propose quintessence models that could be more natural than once believed (the simple reasons why quintessence would be dismissed by string theorists including myself just a few years ago), they must have the freedom – not just formally, but also in practice – to pursue these alternative scenarios, regardless of the number of papers in literature that take KKLT for granted! Only when the plausibility and attractiveness of these ideas really disappears according to the body of the experts, it could make sense to suggest that Vafa seems to be losing.

These two pictures offer very different sketches how the real world is realized within string theory. Indeed, the string phenomenological communities that would work on these two possibilities could easily evolve into "two separated species" that can't talk to each other usefully (although both of them would still be trained with the help of the same textbooks up to a basic textbook of string theory). But as long as we're uncertain, this splitting of the research to several different possibilities is simply the right thing that should happen. Putting eggs to one basket when we're not quite sure about the right basket would simply be wrong.

Wolchover also mentions the work of Dr Wrase. I haven't read that so I won't comment.

But I will comment on some remarks by Matt Kleban (trained at Team Stanford, now NYU) such as

Maybe string theory doesn’t describe the world. [Maybe] dark energy has falsified it.Well, that's nice. String theory is surely falsifiable and such things might happen which would be a big event. But I think it's obvious that Kleban isn't really taking the side of the string theory critics. Instead this statement – that dark energy may have falsified string theory – is a subtle demagogic attack against Team Vafa which is whom he

*actually*cares about (he doesn't care about Šm*its). Effectively, Matt is trying to compare Vafa et al. to Šmoits. If the dark energy in string theory doesn't work in the Stanford way, I will scream and cry, Matt says, and you will give it up. Matt knows that the real people whom he cares about wouldn't consider string theory ruled out for similar reasons so he's effectively saying that they shouldn't buy Team Vafa's claims, either.

Sorry, Matt, but that's a demagogy. Team Vafa doesn't really claim that they have falsified string theory. There is a genuine new possibility whether you like to admit it or not. Also, Matt expressed his attacks against Team Vafa using a different verbal construction:

He stresses that the new swampland conjecture is highly speculative and an example of “lamppost reasoning,"...Cute, Matt. I always love when people complain about lamppost reasoning. I've had funny discussions both with Brian Greene and Lisa Randall about this phrase before they published their popular books. Lisa felt very entertained when I said it was actually

*rational*to spend more time by looking under the lamppost. But it

*is*rational.

I must explain the proverb here. There exists some mathematical set of possibilities in theoretical physics or string theory but only some of them have been discovered or understood, OK? So we call those things that have been understood or studied (intensely enough) "the insights under the lamppost". Now, the "lamppost reasoning" is a criticism used by some people who accuse others from a specific kind of

*bias*. What is this

*sin or bias*supposed to be? Well, the sin is that these people only search for their lost keys under the lamppost.

Now, this is supposed to be funny and immediately mock the perpetrators of the "sin" and kill their arguments. If you lose your keys somewhere, it's a matter of luck whether the keys are located under a lamppost, where you could see them, or elsewhere, where you couldn't. So obviously, you should look for the keys everywhere, including places that aren't illumined by the lamp, Kleban and Randall say, among others.

But there's a problem with this recommendation. You can't find the keys in the dark too easily – because you don't see anything there. Perhaps if you sweep the whole surface by your fingers. But it's harder and the dark area may be very large. If you want to increase the probability that you find something, you should appreciate the superiority of vision and primarily look at the places where you can see something! You aren't guaranteed to find the keys but your probability to find them per unit time may be higher because

*you can see there*.

And there might even exist reasons why the keys are even more likely to

*be*under the lamppost. When you were losing them, you probably preferred to walk at places where you could see, too. You may have lost them while checking the content of your wallet, and you were more likely to do it under the lamppost. So that's why you were more likely under the lamppost at that time, too! Similarly, when God was creating the world, assuming her similar mathematical skills, She was likely to start with discovering things that were relatively easy for us to discover and clarify, too. So she was more likely to drop our Universe under the lamppost, too, and that's why it's right to focus our attention there, too.

For a researcher, it's damn reasonable to focus on things that are easier to be understood properly.

The two situations (keys, physics) aren't quite analogous but they're close enough. My claim is even clearer in the metaphorical "lamppost" of physics. If you want to settle a question, such as the existence of de Sitter vacua, you simply

*have to build*primarily on the concepts – both general principles and the particular constructions – that have been understood well enough. You can't build on the things that are completely unknown. And if you build on things that are only known vaguely or with a lot of uncertainty, you can be misled easily!

So in some sense, I

*am*saying that you

*should*look for your keys under the lamppost, and then increase the sensitivity of your retinas and increase your range that you have a control over. That's how knowledge normally grows – but there always exist regions in the space of ideas and facts that aren't understood yet. The suggestion that claims in physics may be supported by constructions that are either

*completely unknown*or

*badly understood*are just ludicrous. They may sound convincing to them because the keys

*may be anywhere*, the keys

*may be in the dark*. But in the dark of ignorance, science can't be applied and we must appreciate that all our scientific conclusions may only be based on the things that have been illuminated – all of our legitimate science

*is*built out of the insights about the vicinity of the lamppost.

Whoever claims to have knowledge derived from the dark is a charlatan – sorry but it's true, Lisa and Matt! In this particular case, it's totally sensible for Team Vafa to evaluate the experience with the

*known*constructions of the vacua and conclude that it seems rather convincing that no de Sitter vacua exist in string theory and the existing counterexamples are fishy and likely to be inconsistent. This evidence is circumstantial because it builds on the "set of constructions" that have been studied or illuminated – constructions under the lamppost – but that's still vastly better than if you make up your facts and make far-reaching claims about the "world in the dark" that we have no real evidence of!

You surely expect comparisons to politics as well. I can't avoid the feeling that the Team Stanford claim that de Sitter vacua simply

*have to*exist is just another example of some egalitarianism or non-discrimination. Like men and women, anti de Sitter and de Sitter vacua must be treated as equal. But sorry to say, like men and women, de Sitter and anti de Sitter vacua are simply

*not equal*. The constructions of these two classes within string theory look very different and unlike the anti de Sitter vacua, it's plausible and at least marginally compatible with the evidence that the de Sitter vacua don't exist at all. A Palo Alto leftist could prefer a non-discrimination policy but the known facts, evidence, and constructions surely

*do discriminate*between de Sitter and anti de Sitter spaces – and Team Vafa, like any honest scientist who actually cares about the evidence, assigns

*some importance*to this highly asymmetric observation!

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