133. Multi-Field Inflation on the Landscape

This is the guest blog post by my friend Thorsten Battefeld from Princeton. Dmitry.

Dmitry asked me to write a guest post about a recent paper written by Diana and myself on “Multi-field Inflation on the Landscape” (a followup to “Staggered Multi-Field Inflation”) where we ask a simple question: assuming that inflaton is driven by many (say of the order of ) scalar fields, what are the observable consequences of fields decaying or stabilizing one after the other during inflation?

Before addressing this question, let me say a few words about multi-field inflation in general: first of all, why do we bother to work with more than one field? After all, single field inflation is still in good agreement with observations, and it seems that one can implement such models in string theory (the KKLMMT proposal is one example). Studying string theory, one quickly realizes that many new degrees of freedom are present, such as shape and size moduli, fluxes, branes etc. A popular approach of constructing inflationary models consists of stabilizing all but one (or two) degrees of freedom and tuning the ingredients to yield a flat potential for the remaining field. This means that the era of moduli stabilization (see i.e. Kofman et. al.) is relegated to an unobservable time-frame before the last sixty e-folds of inflation, yet one field (the inflaton) happened to be dynamical much longer. In a sense, the inflaton has to be quite special. Further, since one has to make sure that corrections do not spoil the flatness of the potential once the field traverses a super-Planckian stretch in field space (the -problem, see i.e. the series of papers by Baumann et. al.), the construction becomes fine-tuned.

It appears to me that a generic type of inflation ought to employ several dynamic fields, since there is no a priory reason why fields should be fixed early on. A benefit of using many dynamic fields is that the -problem can be alleviated, since no field needs to evolve much during the last sixty e-folds of inflation. Over the last few years there have been several proposals of this type of assisted inflation, such as inflation from M5-branes as proposed by Becker, Becker and Krause, inflation from axions such as N-flation by Dimopoulos, Kachru, McGreevy and Wacker, or inflation from tachyons by Majumdar and Davis).

At this point, one may argue that even if many fields are dynamical, one can always make a field re-definition and take the length along the path in field-space as an effective inflaton. Thus, one would be back at a standard single-field slow roll model. But, is there a problem with this line of reasoning? One difference concerns the end of inflation: in a simple single-field model, inflation comes to an end once slow roll is violated; this is a sudden event. For multiple fields, the end is more gradual as some fields can violate slow roll early while the other fields still drive inflation. To clarify things, let’s look at a concrete example: in the proposal by Becker, Becker and Krause (BBK), the M5-branes are initially distributed in a narrow stack within an orbifold, and the inflatons correspond to the distances between the branes. Inflation takes place while the branes spread out over the quasi static orbifold. Whenever a brane comes close to a boundary brane, it dissolves and converts some of its energy into other degrees of freedom (such as radiation), while the inflaton corresponding to the just dissolved brane simply drops out of the model; one could say that this inflaton decays. Geometrically, it is impossible for the inflatons to decay all at once. How much time this process takes is model dependent (it happens to be a few e-folds in the BBK setup, so an effective single field mode is indeed satisfactory here). However, the just mentioned single-field description becomes unreliable if the decay takes place over the last sixty e-folds; then, the energy driving inflation does not only decrease due to slow roll, but also because fields drop out, which has observable consequences in the CMB power-spectrum. For some reason, this observation has been largely ignored in the literature.

We dubbed this effect staggered inflation, which is evidently not a new type of inflation but a feature of certain existing models of multi-field inflation. In our articles, we derive an analytic formalism to compute some observational consequences due to staggered inflation, apply it to a couple of concrete models (such as inflation from tachyons and models motivated by the landscape picture) and compare predictions with CMB data.

The main approximation we make to describe this effect analytically consists of a coarse graining: we promote the number of fields to a smooth, time dependent function so that we can introduce a smooth decay rate describing how fast fields decay or stabilize. This rate is model dependent and usually not a free parameter once the setup is specified. Due to , energy is continually infused from the inflationary sector to an additional component of the energy momentum tensor, i.e. radiation (this is somewhat similar to warm inflation, but since the energy is not infused via an additional friction term, many problems of warm inflation are avoided). The ratio needs to be small, since inflation would shut off otherwise. At the perturbed level, it is this new small parameter which appears alongside the usual slow roll parameters. One word of caution: one needs to be careful to include also perturbations in in addition to the fluctuations of the inflaton fields.

By searching for generic situations arising on the landscape, we arrive at scenarios involving linear or hilltop potentials and evenly spread out initial field values. Following their slow roll evolution, fields decay/stabilize once they encounter a sharp drop in their potentials. Focusing on cases where the new contributions in the expressions for the scalar spectral index and the tensor to scalar ratio dominate over the slow roll ones, we find that linear potentials are in good agreement with observations, while hilltop potentials are already borderline ruled out at the -level by WMAP5 data. Thus, shallow linear potentials (terminated by sharp drops after ) provide a new class of inflationary models on the landscape that would be considered as ruled out, were it not for the effect of staggered inflation. Such a model is not severely fine tuned and was the most simple and generic one that we could think of (albeit, there are some ambiguities related to initial conditions). The scenario is similar in spirit to chain inflation which is supposed to operate via successive, rapid tunneling between different vacua on the landscape. However, instead of a rapid succession of first order phase transitions, which is difficult to achieve and can cause fatal problems in chain inflation related to bubble nucleation (see Ashoorioon, Freese, Liu), fields follow a smooth path through the landscape.

I would like to mention some limitations of our approach: due to the coarse graining, we cannot recover any features directly caused by the sudden evolution of individual fields once they decay, such as a ringing in the power spectrum. In this light, our work should be seen as a first step to compute observables in staggered inflation, not the final answer. Indeed, there are several expected unique signatures of staggered inflation caused by decaying/stabilizing fields, such as Non-Gaussianities and additional gravitational waves. These signatures would help to discern staggered multi-field inflation from single-field models. To compute these effects one needs to understand better the nature of the fields’ decays and one has to go beyond the analytic approximation. Regarding our implementation of inflation into the landscape, it is heuristic primarily due to our current inability of assigning a canonical measure an the space of initial conditions, and there is clearly room for improvement.

There are more reasons why an effective single field description is usually insufficient: for instance, (p)reheating differs considerably from the one in single-field scenarios if more then one field contributes, since resonances can be shut off due to dephasing (see Battefeld and Kawai); there are still possibilities to reheat efficiently, i.e. via tachyonic resonance, which should be less influenced by dephasing (J. Braden pointed this out at COSMO08). Another potential problem lies in the danger of reheating primarily hidden sectors if couplings are not fine tuned (see i.e. Green). Further, single-field models are generically insufficient for computing isocurvature perturbations and non-Gaussianities.

Since this is already a pretty long post, I will not go into any more details here, but I hope I could convey at least some of the reasons why I think that multi-field models are natural and promising candidates for inflation, which deserve to be studied in more detail.

Cheers,
Thorsten Battefeld