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enter image description here(a) represents New inflation ; (b) represents Chaotic inflation.

From what I understand, chaotic inflation means chaotic initial conditions, i.e. different initial inflaton field values in different patches of space.

Why does chaotic inflation have to have the shape of the potential like in (b)? Can chaotic inflation also have the shape of the potential like in (a) , but with chaotic initial conditions?

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This is kind of a historical issue. The initial inflationary models had potentials with features such as local minima and plateaus, which were required to make the model work. But if you assume chaotic initial conditions, then inflation can work even with a very simple, polynomial potential. In principle you can make the potential more complicated than a polynomial, but you don't have to.

This issue is complicated by the fact that some people use "chaotic inflation" as a synonym for "polynomial potential". Models these days assume chaotic initial conditions by default.

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  • $\begingroup$ I read that the lastest data disfavor chaotic inflation. Does it refer to only chaotic inflation models with polynomial potential, or all chaotic inflation models? $\endgroup$ – parker Sep 14 '18 at 16:30
  • $\begingroup$ @parker The Planck data indeed disfavor some models with polynomial potentials, though not all. But the other models implicitly have chaotic initial conditions, I believe. $\endgroup$ – knzhou Sep 14 '18 at 16:45

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