166. Multifield Inflation with a Random Potential (I)
Save This Post as PDFThis is a guest blog post by Jiajun Xu from Cornell. Dmitry.
This is a guest post by Dmitry’s invitation. Here I would like to talk about the inflation scenario proposed in our recent paper “Multifield Inflation with a Random Potential“, collaborated with Prof. Henry Tye and a fellow graduate student Yang Zhang.
I would like to start with our motivations for this scenario. In building inflation models from string theory, one typically encounters many light scalar fields after dimensional reduction. These light scalar fields correspond to the moduli of the compactification, which parametrize the geometry of the internal space. In Calabi-Yau compactifications, one has Kahler moduli, complex structure moduli, and the dilaton, with a total number 
of hundreds in typical cases. It is a formidable task to compute the full potential as a function of all the moduli and even if we do so, it is hard to search for a flat path in the dimensional field space. The usual strategy is to stabilized all the moduli 
(but one 
) with positive mass 
, so that all these massive moduli do not participate in inflation and the one left will be the inflaton. However, one usually finds that whatever mechanisms that lift the flat directions also lift the putative inflaton direction , so some fine tuning is inevitable.
Now if one takes an alternative point of view, the complicated interactions between the moduli which exhibit an obstacle to single field inflation models, actually turn out to help in the multifield scenario. In single field inflation models, one needs a flat potential to support inflation for 50~60 e-folds, which is the origin of fine-tuning. However, with a multifield potential, especially one that is strongly coupled and thus looks random (like the string landscape), we would not expect the inflaton to simply roll down a particular path. Instead, the inflaton will bounce around frequently, resulting in dimensional Brownian motion. At the same time, the overall motion of the inflaton could still be downhill, determined by the large scale tilt of the potential. The actual motion is Brownian motion with a drift. The potential on small scales can be quite bumpy and the Brownian motion speed 
can be quite fast. However, the energy density of the universe decreases according to the drift speed 
, which can be much smaller than the Brownian motion speed. In general, a small still requires that the large scale structure of the potential be relatively flat. However, the flatness constraint is substantially relaxed compared to that in the slow-roll case, since the inflaton executes Brownian motion and back scatterings tend to slow down its overall motion. For this reason, we consider multi-field inflation in the landscape to be quite natural. Of course, if the inflaton hits a region with a cliff or big downhill slope in the landscape, inflation will end.
A key ingredient behind the above scenario is that the inflaton does not get trapped in the landscape. This can happen for many reasons. If the potential has a tiny bump, the inflaton can hop over it by quantum fluctuation 
. If the inflaton falls into a sharp dip in the potential, the Hubble friction does not come into play if the dip is narrow enough, then the kinetic energy of the inflaton will help it to climb out without getting trapped. Furthermore, it is argued in a series of papers by Henry Tye and collaborators that in a high dimensional field space 
, the wave function of the inflaton cannot get trapped, but instead is mobile in the landscape.
Last I would like to point out that the random walk picture we have is different from the picture of stochastic inflation or eternal inflation, in which quantum fluctuations dominate over the classical motion. In our picuture, the background path of the inflaton is already random before quantum fluctuations are introduced. We are mainly considering classical scatterings in our paper, and it will certainly be interesting to include quantum diffusions into this picture. Our paper is only the first step to quantitatively study this scenario.
Given the above scenario, one naturally wants to know what are the observable features, e.g. the primordial power spectrum, the tensor mode perturbations, the non-Gaussianity, etc. A key feature is a large tensor mode, enhanced by the ratio of the Brownian motion speed versus the drift speed. This ratio can be large. Another feature is the extra red-tilt of the power spectrum due to the classical randomness of the potential. Our paper employs the 
formalism plus a stochastic approach to calculate the two-point and three-point correlation functions of the primordial curvature perturbations 
. Since this post has become quite lengthy already, I will defer the technical details to a second post.
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Dear Jiajun,
thanks for your post and I am looking forward to read the next one! I have a couple of questions if you don’t mind…
We are talking about a random landscape (random potential). What correlation properties of the random potential do you have in mind? If it’s white noise correlated potential, why do you choose white noise correlation properties and not something else?
Cheers,
Dmitry.
Hi Dmitry,
Currently, we are assuming that when the inflaton makes turns in the landscape, these turns are correlated like white noises. This in some sense, is assuming a Gaussian landscape, e.g. if one expand the potential locally around a point in the landscape, the expansion coefficient might be drawn from a Gaussian distribution.
A realistic landscape may or may not be Gaussian. We are only taking the first step here. Introducing correlation properties other than the white noise will certainly be interesting. However, one probably needs a model of landscape to do this.
Best wishes,
Jiajun.
In reality, the potential is not quite random, what you are dealing with is a one particular realization of the random potential. If you have some realization of the random potential, then the evolution of the inflaton is predetermined. Why do you talk about Brownian motion
then? You probably average over different realizations of the random potential, am I right? Why can you average over them and why the result you get has a physical meaning?
Cheers,
Dmitry.
We are not averaging over different landscape. There is only one landscape here, but it is complicated, as a result, the inflaton scatters around in the landscape, probing a patch of it. And we are averaging over such a patch.
The path is pre-determined, but still random, think about a classical billard ball bouncing in the landscape, or think about motion of gas molecules, they are classical motion, but random.
Best wishes,
Jiajun.
Thanks again!
If you consider a power spectrum from such inflation, then it
should have bumps in places where potential jumps/changes strongly. In practice, what we see is a flat power spectrum without any bumps. Does it mean that for the last 60 efolds evolution of the inflaton was smooth?
Cheers,
Dmitry.
If some oscillation happens frequent enough in the power spectrum, say 100 times per l (l being the CMB multiple), data may not be able to pick it out, when doing data analysis, data points are binned according to “l”, and may lose information about some features.
Second, the oscillation in the power spectrum is suppressed in our model. The power spectrum has two terms, one from the adiabatic perturbations, which may have oscillations, the other term from entropic modes is quite smooth and in most cases, dominate. So we have oscillation on top of a smooth curve, and the domination of the smooth part suppresses the fractional change of the power spectrum
Best wishes,
Jiajun.
Hi again Jiajun
Also, I think that whether you should consider white noise or other correlation properties actually depends on the questions you ask. For example, if you are interested to know late time behavior of correlation functions of
or 
, Gaussian approximation can be very well enough – see my recent papers on dynamical RG for the landscape.
No, I think, the analogy with gas molecules does not necessarily work. There, the randomness pattern is time dependent (molecules collide), here it is frozen-in. If you want, landscape is more like glass, where disorder is also frozen in. Yes, averaging over patch (or different realizations of random potential) is equivalent to averaging over time or random trajectories, but only if ergodicity holds. Whether it holds for inflation in random potential, is not quite clear. For example, suppose that potential is random Gaussian (for different values of
– i.e., 
). Does it necessarily mean that the time correlations of 
are also Gaussian?
The question is whether entropic perturbations dominate in reality? The answer as we know is negative (contribution of adiabatic pert. was definitely more than 50% for the last 60 efolds).
Cheers,
Dmitry.
When I say entropic perturnations in the power spectrum, I am referring to the growth of the adiabatic mode after horizon exit, due to the conversion from entropic modes into the adiabatic mode. At the end of inflation, the adiabatic pert. contains one term form the adiabatic pert. at horizon exit and one term from its superhorizon growth, and the latter is smooth and in most cases dominate. This power spectrum is purely adiabatic.
In terms of the data constraint on isocurvature pert., in our scenario, the ratio of entropy perturbation to adiabatic perturbations after inflaiton, is roughly 1/Ne, with Ne between 50 and 60, so well within the bound.
Yes, I agree. The analogy of gas molecule is not so accurate. It is more like defects in the solids.
Regarding the ergodicity, if the inflaton is mobile in the landscape, ergodicity is more likely.
Dear Jiajun
Ok, full agreement regarding entropy perturbations – if you convert all of them into adiabatic ones, the spectrum is smooth.
As for ergodicity, I think, the latter holds for long time behaviour of correlation functions (i.e., overall N >> characteristic N between two subsequent scatterings).
Thanks,
Dmitry.