# Inflaton after chaotic inflation

1. Feb 18, 2013

Can anyone explain to me why after chaotic inflation with $V=\frac{1}{2}m^2 \phi^2$ that the inflaton behaves as;
$\phi=\frac{m_p}{m \sqrt{3\pi}t} \sin(mt)$.

because this doesn't satify the slow roll conditions or the eqn of motion for the inflaton. I have read that it asymptotically approaches ths for large mt but i don't really understand it.
thx

2. Feb 18, 2013

### Naty1

3. Feb 18, 2013

I think i have actually solved it anyway, although feedback would be good.

The wiki article is misleading eternal and chaotic inflation aren't the same things.

Chaotic inflation ends when Slow roll ends and it is at this point the inflaton oscillates around the minima of the potential and we are in the 'reheating' phase. I found that the period is matter dom in 'preheating' and rad dom in traditional 'reheating' so using that and the eqn:
$\ddot{\phi} + 3H\dot{\phi} + m^2 \phi$ one can sub in the sinusoidal value and that show that it is a solution for this preheating phase.

4. Feb 18, 2013

### skydivephil

5. Feb 19, 2013

### bapowell

Chaotic and eternal inflation are not necessarily exclusive terms -- they describe different aspects of the inflationary peroid. Saying that inflation is "chaotic" is to say something about the initial conditions of the inflaton field. In Linde's original chaotic inflation model based on the $m^2\phi^2$ potential, the field takes on a wide range of values throughout the universe. This is to be contrasted with 'new', or 'hilltop' inflation in which the field begins inflation fairly localized at the local maximum of the potential. Linde described this property as "chaotic".

Meanwhile, the term "eternal" describes the global properties of the inflationary era. Of course inflation was not eternal in our observable universe, because we stopped inflating. But it's entirely possible that inflation is ongoing somewhere else in the universe -- outside our Hubble sphere. Linde's chaotic model is in fact eternal, in the sense that one finds that there are regions of the universe that are always still inflating (depending on the field vev, $\langle \phi \rangle$, quantum fluctuations are large enough to work against the classical slow roll motion towards $V=0$, so that certain regions of the universe always have $\langle \phi \rangle > a\, few\, M_{Pl}$ so that inflation keeps going.)

But there are also regions of the universe where the $m^2\phi^2$ finds its way down close enough to $V=0$ to end inflation. At that time, the field oscillates about the minimum of V, giving the time dependence $\phi(t) \propto \sin(mt)$ quoted by the OP.

So chaotic inflation is eternal, but that's not inconsistent with the fact that we live in a region of the universe in which inflation ended.