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Axion monodromy inflation

Guest post by Prof Eva Silverstein, string theorist and cosmologist at Stanford & SLAC

Let us assume that the BICEP2 result is confirmed as cosmological, and indicates primordial gravitational waves generated during inflation. Within the context of inflationary theory, this groundbreaking discovery has important implications for quantum gravity, for which string theory is our leading candidate.

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}\frac{dt}{a}=\int\frac{HM_P}{\dot\phi}\frac{d\phi}{M_P}=\frac{8}{r^{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” (David Lyth, 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 \(\Delta \phi > M_{\rm Planck}\). 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 KKLMMT 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 AST 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_{\text{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 = \left|dC_p+ dC_{p-2}\wedge B+dC_{p-4}\wedge B\wedge B+\dots\right|^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({\rm moduli}) N_1^2 (b\!+\!Q_2)^2\!\!+ f_2({\rm 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. Banks et al. 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 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 arXiv:0907.2916 by Flauger et al as well interesting as field-theoretic treatments in e.g. Kaloper et al arXiv:1101.0026 and 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 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 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 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 arXiv:1011.4521 as well as other works. These include a useful paper by soon-to-be Stanford postdoc Guy Gur-Ari 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. arXiv:1303.2616, arXiv:1308.3736, arXiv:1308.3705, 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 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 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!

By Eva Silverstein, March 27th, 2014

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snail feedback (21) :

reader Luboš Motl said...

Dear Eva, thank you for this wonderful guest post. It's a good observation that theorists are better prepared than for the nonzero C.C. - to some extent, the tension between the positive C.C. and different theoretical expectations caused quite some years of frustration, I think.

Of course, I am the easiest to be convinced that the monodromy mechanism is natural and, let's say it, beautiful. The way how the inflaton range is effectively extended by a large integer factor is fully analogous to the long strings in matrix string theory which may also be called "the fundamental way" how string theory compresses information about long objects. It's a loophole that shows that the "no-go theorems" for the large-field inflation in string theory must be just order-of-magnitude estimates of a sort.

(This point should also have an analogy in the related "weak gravity conjecture" which is mathematically similar to the inequalities banning simple large-field inflation in string theory. As far as I could say, we never found "the" right universal inequality expressing the weak gravity conjecture for a general vacuum with a general spectrum, and things like complicated monodromies or otherwise complicated configuration spaces might potentially render the weak gravity inequality vacuous. Of course, the near-extremal black holes should still be able to radiate away, for QG consistency, but there might be a monodromy-like trick to guarantee this condition without constraining the spectrum too much, too.)

I obviously think that something like that - monodromy inflation, N-flation etc. - is necessary to avoid the genuine problems with the Lyth bound etc. On the other hand, from a practical perspective, even people who are rather careful could suggest that the string-theory modelling of these things "adds" a lot of stuff that isn't bringing insights practically. It seems like Nature tolerates one's being a sloppy field theorist who just thinks that a large-field inflation is OK, the huge QG corrections may be ignored or assumed to do the right thing, one inflaton is enough, and so on. In string theory, one is much less sloppy so she views these things as problems, and to solve them, she has to consider many axions or complicated monodromies etc. which, from a bottom-up field theorist's viewpoint, make the thing more complicated.

These are two (top-down, bottom-up) increasingly different ways of thinking which descriptions are a priori more natural, simpler, and more likely. I think that you're on the same side as how I feel. But do you think there is a chance that the features of the "careful top-down approach" and string theory in particular will be experimentally shown "necessary" and that the "sloppy bottom-up" field-theoretical based approach will be identified as insufficient - or at least, more contrived even from the sloppy field theorist's viewpoint?

reader Eva Silverstein said...

Hi Lubos, Thanks to you for the invitation and the discussion.

I like the analogy to long strings in matrix theory and black holes, another example of how string theory exhibits intricate structures required by physics (in that case thought experiments). I may be biased, but I don't regard these structures as contrived (as you say they are discovered not invented), although I completely agree that the theory adds a lot of degrees of freedom and certain complications, including moduli stabilization. I think there is a lot more to do conceptually to understand more about how even those complications might be relevant for e.g. de Sitter entropy, which might make their role more clear. In fact almost every inflationary mechanism I have stumbled upon was discovered serendipitously in the process of trying to address that problem.

The empirical question you ask is a of course a very reasonable and important one. It will be very interesting to see whether the data ends up centering on the m^2 phi^2 predictions (which may appear as a special case in string theory but doesn't look preferred). There is still some chance of much more detailed signatures at the level of structure in the power spectrum (e.g. oscillations), or particular shapes of non-Gaussianities, which could in principle provide smoking guns for certain mechanisms. But it's hard to assess the probability of that happening -- I was much more confident that B modes would be detected than I am about any of the other interesting signatures, because they seem more model-dependent, although I think they remain very interesting possibilities. Another wonderful thing about the BICEP result is that with relatively large r, some additional features could become observationally accessible with forthcoming experiments, such as its tilt and even some constraints on its non-Gaussianity.

Finally, perhaps we should approach this with due modesty, since after all the level of precision we now have in cosmology seemed like a pipe dream 30 years ago when inflationary theory began. I hope/expect that people will find some new ways to probe this physics that we have not anticipated.

reader Dilaton said...

Oh yeah, looking forward to read this after work and see what I can get out of it :-)

Cheers !

reader David Brown said...

"... approach this with due modesty." Let us assume that my quantum theory of gravity is complete rubbish. Are there any publications in refereed physics journals that consider the possibility that string vibrations are confined to the Leech lattice (or several copies of the Leech lattice)?

reader Luboš Motl said...

David, LOL, Eva may give you a better answer but: did you invent the concept of "string theory on the Leech lattice" yourself and independently?

It would be impressive because the answer is a resounding Yes. String theory on the Leech lattice, more precisely on the 24-torus R^24/Lambda where Lambda is the Leech lattice, exists and is very important.

First, it is the string theory that explains the monstrous moonshine:

More recently, Witten provided evidence that a related (in some sense the same) string theory on the Leech lattice is the holographic dual of pure gravity in AdS3, the simplest gravitational theory in 3D hyperboloid space.

Start with the oldest blog posts if you want them.

reader Luboš Motl said...

By the way, Dilaton (and others), if you had time, there is a cool Harvard colloquium from this Tuesday now posted:

Melissa Franklin introduces it, the same colloquium I would attend every other week or so for six years. ;-) Willie Soon sent me the link.

It's two hours - John Kováč and Alan Guth...

reader PlatoHagel said...

Yes thank you for your contributions to the world mind on string theory together with Lubos, John Preskill(on inflaton rolls) and Liam. You all have become a formidable force to the advancement of information as we lay people are moved forward into the realities of these frameworks and the theoretics gained from the scientific views.

reader lukelea said...

A very nice, clear discussion from the layman's point of view. And well written -- that aspect I can judge with some assurance. thanks,

reader Uncle Al said...

You cannot get by without a trace-broken symmetry. You have one before your eyes. Test for it locally: 90 days in existing bench top apparatus.

BICEP2 sees vortices. One cannot go behind them. They are chiral (re chiral relativistic beta-rays that slow to mere helicity). Physics postulates achiral isotropic space, then suffers unending parity violations, symmetry breakings, chiral anomalies, Chern-Simons repair of Einstein-Hilbert action, toward fermionic matter (hadrons, quarks). Trace chiral anisotropic space heals quantum gravitation, SUSY, and dark matter.

Visually and chemically identical single crystal test masses in enantiomorphic space groups, opposite shoes in left-footed vacuum, free fall along non-identical minimum action paths. Left-handed versus right-handed alpha-quartz violates the Equivalence Principle. A geometric Eötvös experiment contrasts paired 20 gram test mass loadings, 6.68×10^22 pairs of opposite shoes, pairs of 9-atom enantiomorphic unit cells. Look.

reader David Brown said...

By "confined" I mean that nature is infinite and digital. I conjecture that there are 3 basic possibilities for nature: (1) infinite and non-digital, (2) infinite and digital, or (3) finite and digital. By "confined" I mean that each string vibration has a digital representation with respect to the Leech lattice. Is there a publication in which the string landscape is conjectured to be infinite and digital?

reader Shyamoli Plassmann said...

Like your summary, Eva, and the mechanism of studying the axion potentials with higher rank pform potentials coupled via Chern Simons terms, and resulting axion/inflaton potentials. The pform gauge-Yang Mills symmetries are a beautiful constraint which arises directly from string/M theory. It would be nice if more complete phenomenology could begin, as suggested for example in Ibanez et al's recent paper, so that the supersymmetry breaking scale and inflation are considered simultaneously, and a more realistic and complete brane configuration/compactification be considered. Yes, I know that is asking for a lot, but the recent impetus gives us hope we can begin on that stage, right? Nicely written!

reader Giotis said...

Thanks for this detailed analysis and your effort.

What about Heterotic M-theory inflation? Nobody mentions these models and I remember that they produce detectable gravitational waves.

Also can you have the required symmetry in Heterotic M-theory?

reader Алексей Яковец said...

reader Physics Junkie said...

It looks like string theory is finally being tested. From the little I can gather, certain space time configurations of the hidden dimensions are being whittled down to incorporate simple inflation. I wonder who the negative, small minded trolls are spinning this who think string theory is bogus because it was 30 years ahead of it's time.

reader Giotis said...

Something that popped into my mind:

Is there a logical contradiction in trying to enforce naturalness in an inflationary model taking into account that future eternal inflation supports the multiverse idea which is often used as a justification of fine tuning?

reader Luboš Motl said...

Eva's answer may be different. ;-)

But my answer is No, there is no contradiction. Inflation may be viewed as another part of the standard physics obeying the old-fashioned rules including naturalness - the desire to avoid fine-tuning.

Eternal inflation is also sometimes used as a starting point for the anthropic reasoning that allows or encourages one to abandon the usual concepts of naturalness altogether but the anthropic reasoning in no way "follows" from the mechanisms of inflation themselves - not even when the multiverse is created - so there's no demonstrable contradiction between naturalness demanded from some mechanisms in inflation on one side; and the speculative ability of inflation to justify the abolition of the naturalness reasoning.

Another question is what the people who actually abandon naturalness - because of anthropic considerations - do with the apparent and sometimes "damn too obvious" fine-tuning that may be present in some models of inflation. I think that they don't know because there's really no convincing specific enough "realization" of the anthropic reasoning, at least not one that would produce some correct predictions and no wrong predictions. At the end, any consistent picture must have "some" notion of naturalness. It may be "biased" in favor of the creation of humans but it must exist. It should also exist within a well-defined dynamical theory, like string theory, but string theory itself doesn't seem to directly imply any anthropically biased rules of naturalness.

reader kashyap vasavada said...

I would like to double check my layman's understanding of the developments so far with experts.Please tell me if this is right or wrong. My feeling is that if BICEP2 is correct, it proves that one can get by with classical GR and QFT until 10^16 GeV. So one would need ST to take us from there to Planck energy. Also I have read that model of cosmic strings has been eliminated. Is this right or misinformation?

reader kashyap vasavada said...

Perhaps I should have been clearer. It would take me quite some time to even remotely understand Eva's blog and Lumos' comments. I would like to understand role of a possible future theory of QG in understanding BICEP2 results. Looks like you are saying that the results imply some form of QG. is that right?

reader Giotis said...

Nope we are still in the low energy effective four dimensional world. We don’t have to excite any KK (well for this we need to see at which scale the compactification volume moduli is stabilized) or stringy modes to produce the model. Of course in any effective filed theory we never really decoupled from the UV regime. We always have irrelevant operators in the Langrangian (produced by integrated out the heavy degrees of freedom) but they are of zero importance in low energy scales. In the large field inflationary model this is not the case anymore and we need to find a way to suppress them because they mess the potential. A symmetry is a way to do that…

reader David Brown said...

Is there a "natural" or "optimal" way of discretizing the orbifold?

reader kashyap vasavada said...

@Giotis. Thanks. OK. I agree we are now at low energy, effectively 4 Dim world. The question is whether this was true at 10^16 GeV. Does any interpretation of BICEP2 say that?

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