String theory contains a rather simple mathematical structure -- monodromy -- which naturally generates a significant tensor signal. In this guest post, I'll describe that mechanism, and discuss its range of applicability as we currently understand it. (String theory also contains multiple axion fields, which in itself gives an interesting realization of assisted inflation, N-flation, covered nicely in an earlier blog post. It was later realized that along each such direction the monodromy effect operates; in general, one may consider a combination of these two mechanisms.)
Before getting to inflation in string theory, it is important to understand the motivation for combining these subjects. Using the chain rule, one can relate the number of e-foldings of inflation to the field range, assuming no strong variations in the slow roll parameters during the process.
\[
N_e = \int \frac{da}{a}=\int\frac{da}{dt}dt=\int\frac{HM_P}{\dot\phi}\frac{d\phi}{M_P}=8r^{-1/2}\frac{\Delta\phi}{M_P}
\]
where in the last step we used the slow-roll inflation result for the tensor/scalar ratio \(r\) and we assumed that \(\frac{HM_P}{\dot\phi}\) varies slowly, as in simple slow-roll inflationary models. This relation, the so-called Lyth ``bound" http://arxiv.org/abs/hep-ph/9606387, combined with the BICEP2 result \(r\gg .01\), implies a super-Planckian field range for the inflaton field \(\phi\) during the process.
Inflation requires a slowly decreasing source of potential energy \(V(\phi)\) over this range \(\Delta \phi > M_{Planck}\). Variation of the potential \(V(\phi)\) over ranges in \(\phi\) at (or below) the Planck mass scale is strongly constrained by the requirement that \(V\) generate enough e-foldings of inflation, and by CMB data on the power spectrum. Turning this around, the process of inflation and the observed perturbation spectrum are sensitive to an infinite number of corrections to the inflaton potential which are suppressed by the Planck mass scale. We call this situation UV Sensitivity (although of course we're talking about much higher energies than the ultraviolet electromagnetic spectrum!), and it's a tremendous opportunity for getting a window into quantum gravity.
Said differently, if we were to parameterize our ignorance of such effects from the point of view of low energy effective field theory, we would have to take into account the possibility that as the field \(\phi\) rolls along its field space, corrections to the potential arise via couplings to whatever degrees of freedom UV complete gravity. Over a large range of field, the conditions could change dramatically, and it would seem miraculous to obtain the pristine conditions (slowly varying V) required for inflation. As we will see, the structure of monodromy along axion directions in string theory produces a large field range, with an underlying softly broken discrete shift symmetry maintaining similar conditions all along the super-Planckian trajectory. That is, the theory will naturally address this puzzle in a way that is tied to the structure of its gauge symmetries.
In general, an approximate symmetry under shifts of the field \(\phi\) can address this puzzle, even from the low energy field theory point of view. As such, traditional large-field models of inflation, such as Chaotic Inflation and Natural Inflation are internally consistent and `natural' from the Wilsonian point of view -- the potential is protected from problematic quantum corrections.
However, for the reason just discussed, such models make a strong assumption about the UV completion of gravity -- that its quantum (and classical) contributions to the potential not only respect a symmetry, but produce precisely the potential postulated in the field theory model.
Although the inflationary paradigm is compelling as it stands, and now well-tested, many theorists are not completely satisfied with purely ``bottom up" models (although needless to say some of these have been important and pioneering contributions). Those of us in this category regard large-field inflation and its associated tensor signal as requiring, or at least strongly motivating, a treatment which accounts for quantum gravity effects. Since string theory is a well-motivated candidate for quantum gravity (already passing many thought-experimental and mathematical consistency checks), it makes sense to analyze this question in that framework.
I will focus on one rather broad mechanism -- which has been realized by specific string theoretic models as proofs of principle. Before continuing, let me address an issue that sometimes arises. Various people have made comments along the lines that `most' string theory models are ruled out. Certainly small-field models predicting tiny values of the tensor to scalar ratio are now falsified, a healthy part of science. Those works, particularly http://arxiv.org/abs/hep-th/0308055, played a crucial role in establishing a standard of theoretical control in the field, emphasizing the effect of Planck-suppressed operators; others such as http://arxiv.org/abs/hep-th/0404084 helped stimulate a more systematic, model-independent understanding of inflation, leading to a more complete analysis of non-Gaussianity in the CMB. Although they played a useful role, these and many other models, at least in their original form, are dead given primordial B-modes from inflation. However, there is no credible argument that string theoretic inflation is generically small-field; in fact to me (even before the BICEP2 announcement), it has always seemed quite possible that it goes the other way because of the plethora of axion fields in the low energy spectrum arising from string theory. In any case, there are many works on both cases (large and small field); and needless to say the statistics of papers is a very different thing from the statistical distribution of string theory solutions.
String theory contains many axion-like fields, descending from higher dimensional analogues of the electromagnetic potential field \(A_\mu\). These include the 2-form potential field \(B_{MN}\) sourced by the fundamental string, and more general p-form potentials \(C^{(p)}_{M_1\dots M_p}\) sourced by the various branes of string theory. Axion-like scalar fields in four dimensions arise from integrating these potential fields over the extra dimensions, for example
\[
b(x)=\int_{2-cycle} B
\]
and its generalizations, some of which are related by string theory dualities. There is a beautiful structure of inter-related gauge symmetries which are respected by the effective action, including terms of the form
\[
|\tilde F|^2 = |dC_p+ dC_{p-2}\wedge B+dC_{p-4}\wedge B\wedge B+\dots |^2
\]
generalizing a Stueckelberg type coupling \[(d\theta + A)^2\] familiar from symmetry breaking in quantum field theory. In the latter case, the gauge symmetry \(A\to A+d\Lambda, \theta\to \theta -\Lambda\) is respected since it includes the transformation of \(\theta\). Similarly, in the string theory effective action, the gauge symmetry under which \(B\to B+d\Lambda_1\) goes along with compensating shifts in the $C_n$, leaving the \(|\tilde F|^2\) term invariant.
The next step is to note that the fluxes obtained by integrating the field strengths \(dC_n\) over internal cycles in the extra dimensions are quantized, taking integer values (appropriately normalized). Also, the size and shape of the internal dimensions are dynamical `moduli', descending from higher-dimensional Einstein gravity (along with other fields, like the string coupling) plus string-theoretic corrections. Altogether when we dimensionally reduce from higher dimensions to four dimensions, the potential is schematically of the form
\[
f_1(moduli) N_1^2 (b+Q_2)^2+ f_2(moduli) N_2^2(b+Q_2)^4 +\dots
\]
(and similarly for other types of axions).
We can read off several important features from this structure.
First, the theory as a whole has a periodicity: if we move from some value of the field \(b\), say \(b=b_0\), to \(b_0+1\), this is equivalent to shifting the flux quantum numbers \(Q_1\) and \(Q_2\). However, with a given choice of flux quantum numbers -- i.e. on a given branch of the potential -- the field range of \(b\) is unbounded. In particular, each branch of the potential is not periodic in \(b\), in the presence of generic fluxes. This is a relative of the Witten effect in gauge theory.
Another key point is that when we normalize the scalar field canonically, rescaling to form the inflaton field
\[
\phi = f b,
\]
the periodicity \(f\) in \(\phi\) is sub-Planckian, by a factor of \(1/L^2\), where \(L\) is the size of the internal dimensions in units of the string tension (see e.g. http://arxiv.org/abs/hep-th/0303252 ). This is an example of the fact that despite the many solutions of string theory (the `landscape'), the theory has a lot of structure. Not anything goes, even though it is true that the theory has many solutions. In any case, despite this sub-Planckian underlying period, the fluxes generically unwrap the axion potential, leading to a super-Planckian field range.
Because of the underlying periodicity much of the physics remains similar along the whole super-Planckian excursion of the field. This is in a nutshell how this mechanism in string theory addresses the original question raised by effective field theory above.
The same features arise in the presence of generic branes in string theory, something we can understand both directly, and using the AdS/CFT duality to relate the flux and brane descriptions. On of our original realizations of this mechanism in a string compactification http://arxiv.org/abs/arXiv:0808.0706 (with McAllister and Westphal) arises in this way, with a linear potential built up via the direct coupling of axions to branes. (See also http://arxiv.org/abs/arXiv:0907.2916 by Flauger et al as well interesting as field-theoretic treatments in e.g. Kaloper et al http://arxiv.org/abs/arXiv:1101.0026 and http://arxiv.org/abs/arXiv:1105.3740 and many other references.)
More simply, the potential is like a windup toy. This has recently been generalized, with an interesting mechanism to start inflation, in http://arxiv.org/abs/arXiv:1211.4589.
 |
| A schematic picture of one example of axion monodromy in string theory (the direction around the circle being related by a local string duality to an axion direction, and with potential energy built up by the stretched `D4-brane'). |
Secondly, working for simplicity on the branch \(Q_1=Q_2=0\), the potential is analytic in \(b\) near the origin, but at large values (as are relevant for inflation), the other degrees of freedom such as the `moduli' (and also the internal configurations of fluxes etc.) can adjust in response to the built up potential energy. As a result, we find a potential which near the origin is a simple power law, e.g. quadratic (or even quartic if \(N_2\) dominates -- something we are currently studying in light of the high BICEP2 central value for \(r\)).
But at large field values, the potential is typically flatter than the original integer power-law, as the additional degrees of freedom adjust. This is a simple way of understanding the lower-than-quadratic power, \(V(\phi)\propto\phi\) that we obtained explicitly in the original example (as explained in http://arxiv.org/abs/arXiv:1011.4521 on this `flattening' effect).
Finally, although each branch of the potential goes out to large field range, the underlying periodicity leads to a residual sinusoidal modulation of the potential. The amplitude and period of this modulation depends on the values of the moduli fields, and because those are dynamical they can themselves vary in time during the process.
Because of the moduli fields, the construction of complete string theory models realizing this (or any) mechanism for inflation is quite involved. The top-down construction http://arxiv.org/abs/arXiv:0808.0706 provides a proof-of-principle, and has become a benchmark example in CMB studies. But it is clear that the mechanism is much more general, as was emphasized in http://arxiv.org/abs/arXiv:1011.4521 as well as other works. These include a useful paper by soon-to-be Stanford postdoc Guy Gur-Ari http://arxiv.org/abs/arXiv:1310.6787 which lays out some possible realizations on twisted tori, while pointing out an error in my original attempt to stabilize string theory on nilmanifolds (happily, this flaw was not uncovered until after the twisted tori suggested the monodromy mechanism, which transcends that particular compactification...).
Monodromy inflation is falsifiable on the basis of its gravity wave signature, and so given the BICEP2 result of nonzero \(r\) at high statistical significance (assuming it is indeed primordial), large-field inflation in general and monodromy inflation in particular has passed a significant observational test. There are opportunities at a more model-dependent level for more detailed signatures, involving the residual modulation of the potential, see e.g. http://arxiv.org/abs/arXiv:1303.2616, http://arxiv.org/abs/arXiv:1308.3736, http://arxiv.org/abs/arXiv:1308.3705, http://arxiv.org/abs/arXiv:1308.3704 as well as the Planck 2013 release for interesting analyses putting limits on this possibility. As mentioned previously, the dynamical nature of the amplitude and oscillation period make this a subtle analysis and we should try to develop a more systematic theoretical understanding.
In any case, given at least the successful prediction for gravity waves, we are now very interested in the range of possible values of the tensor to scalar ratio \(r\) (and other observables) that this mechanism (and related ideas such as N-flation http://arxiv.org/abs/hep-th/0507205) covers. As a concrete first step, in ongoing work we have constructed additional examples of the `flattening' mechanism, but starting from quartic, \(|F\wedge B\wedge B|^2\) terms which can generate larger values of \(r\). The next step is to incorporate these into string compactifications with fully stabilized moduli, which would provide a proof-of-principle for relatively large values of the tensor signal. Beyond that, we would like to understand the range of \(r\) values in theory space (UV complete) as systematically as possible.
Incidentally, another mechanism for large values of \(r\) that we had previously considered (with Senatore and Zaldarriaga http://arxiv.org/abs/arXiv:1109.0542 ) involves another aspect of the quasi-periodic structure. This structure raises the possibility of periodically repeated and hence approximately scale-invariant production of particles or strings, which can themselves emit gravity waves. That still requires inflation, but can easily enhance the tensor signal by up to a few orders of magnitude (and easily by an order one factor). This mechanism is likely distinguishable by further data on the B modes (which will constrain their power spectrum and non-Gaussianity).
Altogether, I view monodromy inflation as more than just some random model but less than a complete theory. It is tied to the structure of string theory and its symmetries in a way that seems pretty robust. But at our current level of understanding, it is far from a complete theory -- if it were, we could write a computer program to generate the `discretuum' of values of observables like \(r\) that it UV completes. We are far from that systematic an understanding, and even if we had it the data will not allow us to `invert' the problem and deduce a very particular model, even given smaller error bars to come with future data on the tensor to scalar ratio, its tilt, and higher correlators.
Fortunately, some of the most important distinctions -- such as large versus small field -- are rather directly probed by the observations. This breakthrough is somewhat analogous to the cosmological constant discovery 15 years ago -- even a single number can be extremely significant. In the case of the B-modes, we theorists are a little better prepared than in the case of the cosmological constant, but there is much that remains to do. We have entered an era of genuinely data-driven string theory research. Exciting times!
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