# Would eternal inflation be really eternal since quantum fluctuations are usually very rare and very small?

I’ve read somewhere that quantum fluctuations in the inflaton field are usually very rare and very small.

If the effect of quantum fluctuations pushing the potential uphill is too rare and too small compared to the rate of the field rolling downhill, the inflating volume will not grow so it will not keep inflation going eternally.

What I am asking is that whether the effect of quantum fluctuations in the inflaton field is enough to keep inflation going eternally? Or does it depend on the model?

The "size" of quantum fluctuations is given in terms of the variance on the scale $k$, $$\langle | \delta \phi_k|^2 \rangle = \frac{H^2}{2k^3}.$$ Up to the $k$-dependence, this quantity depends on the inflationary dynamics through $H$. If the energy scale is large enough, then the variance can indeed be large. Meanwhile, the classical rolling of the inflaton covers ground $\Delta \phi \sim \dot{\phi}/H$ in a Hubble time. The variance of the quantum fluctuation in proportion to the classical field excursion is $H^4/2\dot{\phi}^2k^3$, and is particularly large for high-energy scale, slow roll inflation. Linde's original chaotic inflation model fits these requirements, with different regions of the universe having initial field values across a wide range of potential energies, some easily approaching the Planck scale.
• The energy scale of inflation is tuned by adjusting the height of the potential energy function. But, even if the energy is low in our observable universe, in a scenario like chaotic inflation the idea is that there are regions of the universe where it is high (the potential is $m^2\phi^2$), and so eternal inflation implies that there are always regions in the universe that are inflating. – bapowell Aug 8 '18 at 16:54