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Why does Standard inflation say that energy density drop in all regions of space and inflation stops, while Chaotic/Eternal inflation say that energy density drop in some regions and some other regions continue to inflate?

Which picture is the correct one?

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It depends on the nature of the inflaton potential. Suppose the classical evolution of the inflaton in a Hubble time, $\Delta t = H^{-1}$, is $\phi = \phi_0 - \Delta \phi$ (assuming without loss of generality that $\dot{\phi} < 0$). During this time, a region of size $l \sim H^{-1}$ increases in size by an amount $a \approx e^{H\Delta t} = e$. Now, there are also quantum fluctuations of the field to consider, so really $\phi = \phi_0 - \Delta \phi + \delta \phi$, where $\delta \phi$ gives the quantum variation of the field in our inflating region. Notice that if $\delta \phi \gg \Delta \phi$, the field won't roll down, but back up the potential. Since our inflating region has grown by a factor of $e$, it contains $\mathcal{O}(e^3)$ regions of size $H^{-1}$; we expect that in about half of these the field will evolve upwards rather than downwards due to the large quantum fluctuation. These regions will continue to inflate at a higher energy and hence higher rate of expansion, and over the course of the next Hubble time, this process repeats again. And again and again: inflation begets inflation if the fluctuations are large enough. The physical volume of the inflating region grows exponentially in time: this is what is called eternal inflation

Whether this happens depends on the model: Linde's "chaotic inflation" scenarios, $V \sim \lambda\phi^n$, indeed exhibit such large fluctuations because here $\delta \phi \propto \phi^{n/2}$ and $\phi$ tends to be quite large (in Planck units) in these models (the term "chaotic" refers to the highly random initial field values across the universe, some of which are very high up on the potential with $\phi \sim \mathcal{O}(10)M_{\rm Pl}$).

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