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.

Regarding the ergodicity, if the inflaton is mobile in the landscape, ergodicity is more likely.

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.

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.

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.

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.

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 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.

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.

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.

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.

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.

Cheers,
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.

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.