99. Eternal inflation with many light scalar fields

I am going to briefly discuss one result from the recent paper by Peter Adshead, Richard Easther and Eugene Lim.

One subject of the authors’ study is the interplay between stochastic and eternal N-flation. Let us recall what is N-flation (or assisted inflation). Suppose that we have a large number of scalar fields with equal potentials and relatively large vacuum expectation values. How large? We want to have the ordinary slow roll inflation in this setup. While one needs an expectation value of the inflaton to be superplanckian in single field inflationary models in order for slow roll conditions to hold, expectation values of any one among N scalar fields can be rather small and still the inflationary slow roll regime will not be spoiled. Indeed, we have

,

and is sufficiently large even for small . Now, we notice that generally during one Hubble time there are fluctuations of the field with the wavelength of the quasi de Sitter horizon and the characteristic amplitude of the order of . Again, we see that there is one more amplification effect in N-flation compared to the single field inflation: the quantum fluctuations of individual scalar fields in N-flation are generally stronger than in a single field inflation (the amplitude amplification is proportional to ). As a result, the regime of stochastic inflation (see my lectures on stochastic inflation) for an individual field is generally achieved earlier for N-flation than for a single field inflation.

All this is rather clear, but the authors indicate the following thing which does not yet belong to the general lore: for N-flation, stochastic inflation of an individual field is not the same as the regime of eternal inflation! Indeed, in this setup eternal inflation would mean self-reproduction regime such that the classical displacement of the “mean field”

during one Hubble time is of the same order as the characteristic amplitude of the quantum fluctuation . Clearly, this regime is established at much higher values of than the regime of stochastic inflation for a single individual field .
In other words, for a single field inflation the onset of stochastic regime is synonymous to the amplitude of density fluctuations exceeding unity. For N-flation, stochastic regime is turned on at much lower amplitudes of density fluctuations.

Note that for both cases amplitude of density fluctuations exceeding unity=eternal inflation. This is a very generic and model-independent but (unfortunately) not very well known statement.