BICEP2 string-theoretical winner: N-flation

Last Tuesday, I discussed some of the winners and losers of the apparent discovery of the primordial gravitational waves.

The first winner that I mentioned was Andrei Linde's "chaotic inflation" with the potential \(V=\frac 12 m^2 \phi^2\). It is extremely simple and it matches all the observations at this point. However, it is just an effective field theory, not a full-fledged compactification of string theory.



What are the actual compactifications or scenarios in string theory that are winners? We will turn to this question later in this blog post.

One point that I haven't sufficiently emphasized is that the anti-physics activists, the sort of computer administrators, senile teaching assistants, and other assorted unfriendly individuals are the top losers – in the excitement following the discovery, we have almost completely forgotten that they exist. Imagine how you must feel if you have been preaching for decades that physics at dozens of \(\TeV\) is untestable, unfalsifiable, or any other adjective that was popular among similar cranks... and suddenly, a cheap $10 million experiment measures the value of a new parameter that describes some physics at the supersymmetric GUT scale, around \(10^{16}\GeV\), and it eliminates 90% of the theories that say something about the scale. 90% is quite a brutal case of falsification, isn't it? It would be appropriate if at least 90% of these physics haters disappeared now, too.

Yakov B. Zeldovich once said that the Universe is the poor man's accelerator. In the USSR, they had sufficiently many poor men which is why they had to look to the Universe and why a significant fraction of the pre-fathers and early co-fathers of inflation were Russians. And as another Slavic physicist and probably the first man who thought about the decomposition of CMB into E-modes and B-modes, Uroš Seljak, observed, "[the discovery] may force us in the direction of string theory; it also fits in with models of continuing inflation that produce multiple universes."




We are suddenly observing physics at time scales of \(10^{-35}\) seconds, length scales \(10^{-27}\) meters, and so on, i.e. billions of times shorter distances and times than those at the LHC, and even though BICEP2 has effectively measured 1-2 new nonzero parameters only, we may proceed with the spring cleaning which eliminates a large portion of the detailed inflationary (and an even higher fraction of alternative!) models. A majority of string compactifications may now be excluded, too.

So which string compactifications and scenarios are dead and which of them look really good right now? The dust hasn't settled yet and there's actually some disagreement. For example, even when we think about string-inspired scenarios in phenomenology, most experts that I communicated with think that the old large dimensions (ADD: Arkani-Hamed, Dimopoulos, Dvali) have been excluded and many of them believe that the warped extra dimensions (RS: Randall-Sundrum) share their fate. However, Lisa Randall thinks that RS is doing fine as the inflation "takes place" at the Planck brane and is unaffected. And Edward Witten was heard as saying that even ADD may actually be viable after BICEP2.




It would be helpful if these physicists revealed some details of their reasoning.

But I think that in general it's true that most "papers on stringy models of inflation" in existing literature focus on models that are now ruled out. While the cyclic and ekpyrotic models predict "virtually zero" B-modes, i.e. the value of the tensor-to-scalar ratio \(r=0\), the inflationary models predict a nonzero \(r\) but in most cases, the predicted value of \(r\) is just much lower than \(r=0.20\pm 0.05\) suggested by BICEP2.

You should understand that \(r=0.20\) is really near the "maximum" value of \(r\) that has been considered in any theoretical paper. Significantly higher values of \(r\) are really of "order one" which seemed "manifestly excluded" to the people which is the probable reason why that option hasn't even been considered. A value of \(r\) of order one would apparently mean that the polarization would have to be "visible to the naked eyes" but it had not been discovered earlier, which is why the people didn't consider it at all. I think that if people wanted, they may easily find models with still larger values of \(r\).

On the other hand, you should understand that values of \(r\) obeying \(r\leq 0.001\) or so are unobservable. It's because the value of \(r\) cannot be measured quite accurately. We measure it from the amplitudes of the spherical harmonics \(Y_{\ell m}\) for \(\ell\approx 100\), and there are just thousands of such particular coefficients. Each of them is inevitably random to some extent and even if you incorporate all of them into your calculation of the "average intensity" or your calculation of \(r\) in order to suppress the noise, some noise just cannot be eliminated. The unavoidable error/noise that results from observing a finite number of coefficients – or a finite number of "vortices in the sky" – is known as the cosmic variance. Cosmic variance is an enemy of the observational cosmologist (yes, the preposterous universe is an enemy, too) because it prevents her from measuring things like \(r\) too accurately. And values \(r\leq 0.001\) or so are simply indistinguishable from \(r=0\).

Many inflationary models – and stringy descriptions of inflation – have predicted much smaller values, \(r\ll 0.001\), so they predicted that the primordial gravitational waves would never be discovered. If BICEP2 is right, these models are excluded.

I also agree with Liam McAllister who thinks that the old large dimensions and similar models are ruled out:

We do learn one model-independent thing about string theory: because the inflationary Hubble scale is so large,\[

H\approx 10^{14}\GeV,

\] we can exclude a wide range of models in which quantum fluctuations at this scale would destabilize the compactification. In particular, if the Kaluza-Klein mass is below \(H\), the same fluctuations that give rise to the scalar and tensor perturbations of the CMB would give rise to perturbations of the extra dimensions. When fluctuations of this sort are large, a four-dimensional description ceases to make sense, because the whole compactification is dynamical. By this logic we can exclude models with very low Kaluza-Klein scales, i.e. models of large extra dimensions. (Perhaps there is a model-building trick that can make large compactification robust against quantum fluctuations during inflation, but I'm not aware of a compelling idea.)

Well, perhaps the whole dynamical compactification could still survive the era of inflation and having a workable four-dimensional effective field theory might be unnecessary but I do agree with him that the default assumption is that a compactification destabilized in this way is probably devastating. I think that this eliminates old large dimensions with radii \(R\gg L_{\rm Planck}\) but also warped geometries with curvature radii or proper lengths of the extra dimensions \(R\gg L_{\rm Planck}\).

Building on the pro-anthropic KKLT paper, six authors KKLMMT 2003 began to study inflation in string theory. I believe that the "D3-brane inflation" described by that paper is excluded by BICEP2 if that experiment is right. It doesn't matter that the paper is approaching 1,000 citations and it doesn't matter that it talks about a formidable number of vacua, like \(10^{500}\); all of them may still be shown wrong. That's the power of the empirical evidence in science! The reason is that the location of the D3-brane in a throat which plays the role of the inflaton has a problem with the Lyth bound that was discussed by Liam: the vev of the field just can't change by higher-than-Planckian values which is needed for significant tensor modes and string vacua without some extra structure ban these large variations of the fields.

We see that while the BICEP2 discovery supports the general concept of the multiverse by strengthening "chaotic inflation" which apparently implies "eternal inflation", it may rule out the single most popular incarnation of the anthropic reasoning within string theory as described in the KKLMMT paper. I have never liked the pro-anthropic KKLT/KKLMMT scenarios and especially the ideology behind them because they would prefer silly arguments based on "majority" and "typicality" and all this left-wing junk as opposed to a clever description of the observed patterns by unique enough physical constructs. So I may be a bit biased against KKLT/KKLMMT. But nevertheless, I do think that this line of reasoning is in trouble. The most unequivocal assertion that these models predict low \(r\) appeared in this Linde-Kallosh 2007 paper.

A string theorist would write me an e-mail considering F-theory to be the main stringy winner of the BICEP2 announcement. Well, some people consider KKLMMT to be a "generic" or "most typical" representation of inflation within F-theory, and it may be ruled out. So the message shouldn't be this clearly positive. I feel that the traditional (and for 20 years, my favorite) heterotic vacua with the Calabi-Yau scale close enough to the Planck scale – and producing effective 4D field theories beneath this high scale – might be doing better.

N-flation: a winner

However, there exists another line of stringy model building that seems to have strengthened substantially. It was started in 2005 and empirically, it is almost exactly equivalent to Andrei Linde's simple "chaotic inflation". The paper was

N-flation by Savas Dimopoulos, Shamit Kachru, John McGreevy, Jay Wacker

and it appeared in JCAP in 2008 (quite a delay). Like the paradigm of the stringy axiverse (which also includes Savas Dimopoulos among the co-authors), N-flation uses the observation that string theory vacua are likely to predict a high number of periodic scalars, the axions. The high number is the reason behind the letter \(N\) in the name – and the name, "enflation", sounds almost like "inflation". The authors must consider this linguistic unintelligibility to be an advantage. ;-)



String theory vacua often predict a whole plethora of different axions. Picture taken from the blog of the UCLA professor who makes sure that Sheldon Cooper's science is kosher.

Note that there is the \(B\)-field in perturbative string theory, \(B_{\mu\nu}\), that plays the role of the electromagnetic potential \(A_{\mu}\) for the "string winding charge". You may integrate it over various 2-cycles of the compactification, and there may be many of them, to produce scalar fields. There may be other differential forms that may be integrated over cycles of appropriate dimensions to yield additional scalars, and so on. All these scalar fields are analogous (and sometimes U-dual to) "angles" of rectangular (I mean paralellogram/general toroidal) compactifications which are periodic variables, too.

If you have many axion fields like that, the potential\[

V = \sum_{i=1}^N \gamma_i a_i^2

\] allows you to pick vevs of the fields, \[

\langle a_i\rangle = \frac{v}{\sqrt{N}},

\] which means that by the Pythagorean theorem, the "collective" overall scalar field \(a\) is of order one (not a decreasing power of \(N\)) even though the individual fields \(a_i\) have vevs much smaller than that, by a factor of \(\sqrt{N}\), so the conflict with the Lyth bound is avoided. So far, I have assumed that the coefficients \(\gamma_i\) were the same for all axions \(a_i\). The authors then refine the model by allowing the axion masses to be different – e.g. uniformly covering the log scale. If that's so, the cosmic inflation naturally has "many stages" in which the "most important" (fastest running) inflaton is gradually switching from the heaviest axion to the lightest axion.

They also discuss various radiative corrections and radiative stability, in the presence of SUSY and without the presence of SUSY, and the important role of the shift symmetry.

The most important point is that this scenario ends up being extremely similar to Linde's simple chaotic inflation – except that it could perhaps also predict some of the non-power-law running that seems to follow from the (so far minor) tension between Planck and BICEP2. In other words, the large and simple form of the tensor perturbations could be evidence for Linde's simplest one-quadratic-scalar model; but it could also be evidence for inflation's dependence on many scalars, i.e. evidence for lots of scalars (axions) in Nature! Somewhat less directly, it could be evidence in favor of a complicated enough compactification of extra dimensions in string theory!

This model of N-flation is a "simpler" and "more stringy" cousin of assisted inflation. Assisted inflation also had the point of showing that inflation may be slow-rolling if there are many scalar fields – even though the rolling would be fast with each scalar field separately. They seem to require a particular form of potential, an exponential one (I really mean \(V=A\exp(B\phi)\)...), which is probably unnecessary. At any rate, I do think that N-flation is smarter, simpler, and contains the main viable ideas that were coined in assisted inflation.

I highlighted N-flation in order to have a particular brand but there are several stringy and semi-stringy models of inflation that are equally capable of predicting large enough tensor-to-scalar ratios. For example, this 2011 paper by Barnaby and Peloso lists several such paradigms:

[...] Moreover, \(f\gt M_p\) does not seem possible in string theory [Banks et al. 2003]. More recently, several controlled realizations of axion inflation have been studied – including double-axion inflation [Kim et al. 2004], N-flation [Dimopoulos et al. 2005, Easther+McAllister 2005], axion monodromy [McAllister et al. 2008, see guest post by Eva Silverstein], and axion/4-form mixing [Kaloper+Sorbo 2009] – which have \(f\lt M_p\) but nevertheless behave effectively as large field inflaton models (\(\phi\gtrsim M_p\)). [...]

Whether these models come in intimidatingly large groups supporting the anthropic delusions or not, they may be picked as the winners by the experiments, and that's what matters more in physics than philosophical preconceptions. I guess that when the dust settles, people will focus much more attention on these scenarios and the experiments may provide us with several new pieces of nontrivial information, too.

Particle accelerators will remain important and complementary but a "new" donor of exciting empirical data, the poor man's accelerator named the Universe, may gain much more importance than what most of us used to expect.