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Wikipedia says that in new inflation, the slow-roll conditions must be satisfied for inflation to occur.

What is fast-roll or non-slow-roll inflation and why is slow-roll is prefered?

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Slow roll inflation is quantified in terms of a sequence of parameters which measure the local smoothness of the inflaton potential. The lowest-order parameter is $\epsilon \propto (V'/V)^2$, the next-lowest is $\eta \propto V''/V$, and so on. Higher-order parameters measure ever more localized kinks in the potential.

The values of $\epsilon$, $\eta$, etc are constrained by a few different things. For one, inflation needs to last long enough to solve the horizon and flatness problems. Inflation occurs as long as $\epsilon < 1$, and its rate of change is $$\frac{{\rm d}\epsilon}{{\rm d}N} = 2\epsilon (\eta - \epsilon)$$ where $N$ is a time variable denoting the number of e-folds before the end of inflation: current estimates require that $\Delta N$ be at least around 60. So if $\epsilon$ is too large (the potential has too steep a grade), then the field rolls down too quickly to drive sufficient inflation. Also, if $\eta$ is too large (the potential has too much curvature) then a similar fate awaits. So the first reason we want inflation to be "slow roll" is that we want to get enough inflation, and we need a reasonably flat potential to achieve that.

The other big reason that slow roll inflation is desired is due to the influence that inflationary dynamics have on the shape of the density perturbation spectrum, $P(k)$. Measurements of the cosmic microwave background and large scale structure surveys have revealed that $P(k)$ is very nearly scale invariant, $P(k) \sim k^{n-1}$ with $n \approx 1$. A lengthy and involved (though fun and rewarding) calculation reveals that we can connect the spectral parameters directly to the inflationary dynamics in terms of the slow roll parameters. In particular, $$n - 1= 4\epsilon -2\eta,$$ revealing that slow roll inflation leads to a nearly scale invariant power spectrum.

None of this is to say that there cannot be non-slow roll behavior during inflation, just that it needs to be the exception rather than the rule. There are many models involving localized, transient kinks in the potential that create interesting features in the spectrum on specific scales. People have also looked at "fast roll" inflation in which, though the potential may be flat, the field velocity is high compared to the expansion rate. This is also non-slow roll behavior and one finds that the resulting power spectrum deviates from a power law on the affected scales (fast rolling fields tend to slow down even as they roll down hill during inflation, due to Hubble drag, and so slow roll inflation can be achieved even with initially fast-rolling fields).

Lastly, we only have observations across a limited range of scales (maybe the first dozen or so e-folds of inflation), and so the potential could do all sorts of non-slow roll tricks on these scales (both larger and smaller) as long as sufficient inflation is achieved and there aren't dramatic spikes in the power spectrum (that might, for example, generate unacceptable amounts of primordial black holes.)

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  • $\begingroup$ Can Hubble drag substantially slow down the rolling enough to turn fast-roll into slow-roll? $\endgroup$
    – parker
    Aug 15, 2018 at 21:26
  • $\begingroup$ Yes. And the faster the rolling, the greater the drag. In fact, it’s generally difficult to maintain fast roll long enough to have a substantial effect across more than only a few decades of $k$. $\endgroup$
    – bapowell
    Aug 16, 2018 at 0:58
  • $\begingroup$ If the potential is too steep, fast-roll occurs and slow-roll cannot be achieved. But if the potential is rather flat and kinetic energy(field velocity) is high, fast-roll initially occurs but then slow-roll can be achieved because of Hubble drag. Have I got it right? $\endgroup$
    – parker
    Aug 16, 2018 at 17:47
  • $\begingroup$ Yes, that's right. I should qualify my comment above that I'm referring to the case of high field velocity on a relatively flat potential. Of course, if the potential is too steep, you'll never achieve slow roll. $\endgroup$
    – bapowell
    Aug 16, 2018 at 18:39
  • $\begingroup$ “revealing that slow roll inflation leads to a nearly scale invariant power spectrum.” Does that mean non-slow roll inflation will not lead to a nearly scale invariance? $\endgroup$
    – parker
    Nov 3, 2021 at 19:09
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The "non-slow-roll" inflation is the original model developed by Alan Guth in 1981 usually referred to as the old inflation (with slow-roll inflationary model being the new inflation)

According to the old inflation scenario, inflation is an exponential expansion of the universe in a supercooled false vacuum state. This expansion makes the universe very big and very flat. The transition to the non-expanding state occurs through bubble nucleation and subsequent reheating happens when walls from different bubbles collide and decay, with the energy stored in the walls converting into hot matter.

Unfortunately it was soon realized that this picture of inflation has problems. Bubbles grow at about the speed of light and so each bubble forms independently of one another. If the probability of bubble formation is large, bubbles are formed close to each other, inflation is too short and the bubble wall collisions make the universe extremely inhomogeneous. If the probability of bubble formation is small and inflation is long then there are too few collisions between bubbles with expansion continuing indefinitely between bubbles and each of the bubbles would represent a separate and almost empty universe. Both of these options are unacceptable, thus the conclusion is that this (old inflation) scenario does not work.

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  • $\begingroup$ Why do scientists think that the energy from false vacuum decay is stored in the wall, not inside the bubble? $\endgroup$
    – parker
    Aug 10, 2018 at 9:52
  • $\begingroup$ @parker: By solving field equations for inflaton field. Though the initial 'seed' of a bubble is created by a quantum process, its subsequent evolution is well described by classical equations. In the simplest model it is Klein–Gordon equation with a potential $V(\phi)$ like the one here. Solution is quite simple and well understood due to high degree of symmetry. Inside the bubble $\phi$ corresponds to true vacuum while outside is false vacuum, and all the potential energy freed goes into the energy of the wall. $\endgroup$
    – A.V.S.
    Aug 10, 2018 at 16:28

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