Deriving the slow-roll parameter $\eta$ In inflationary theory, many papers start off by making the slow-roll approximation, on which many things depend. This approximation is usually presented by requiring that two 'slow-roll parameters' are small:
$$\epsilon_V\equiv\frac{1}{16\pi G}\left(\frac{V'}{V}\right)^2 \ll 1$$
$$|\eta_V|\equiv \frac{1}{8\pi G}\left(\frac{V''}{V}\right)\ll 1$$
We then have 
$$H^2=\frac{8\pi G}{3}V$$
$$3H\dot{\phi}=-V'$$
Now, the first of these two conditions is reasonably easily derived:
\begin{align*}
\frac{\ddot{a}}{a}&\gg 0\\
\dot{H}+H^2 &\gg 0\\
-\frac{\dot{H}}{H^2}&\ll 1\\
\frac{1}{16\pi G}\left(\frac{V'}{V}\right)^2=\epsilon_V&\ll 1 \hspace{1cm}\text{used slow roll approx.}
\end{align*}
However, I'm not sure how to find the second one (note that I am not asking for an intuitive explanation, I understand what the second parameter represents, I just wanna know how to derive it). Could someone tell me how to derive it (or what the original premise, analogous to $\ddot{a}/a\gg 0$ for $\epsilon_V$, is)?
 A: For inflation the potential energy of the field dominates the kinetic energy
$\dot{\phi} \ll V(\phi)$
This limit is referred as slow roll and under such conditions the universe expands quasi exponentially
$a(t) \propto \exp \left( H dt\right) = e^{-N} $
where we define the number of e-folds $N$ as:
$dN = -H dt$
so that $N$ is large in the far past and decreases as we go forward in time and as the scale factor $a$ increases.
With this we have:
$\epsilon = -\frac{\dot{H}}{H^{2}} = \frac{1}{H}\frac{dH}{dN}$
Accelerated expansion will only be sustained for a sufficiently long period of time if the second time derivative of $\phi$ is small enough:
$|\ddot{\phi}| \ll  |3H\dot{\phi}|, |V'(\phi)|$
So that the equation of motion for the scalar field is approximately:
$3H\dot{\phi} + V'(\phi) \simeq 0$
This condition can be expressed in terms of a second dimensionless parameter, defined as:
$\eta \cong -\frac{\ddot{\phi}}{H\dot{\phi}} \cong \epsilon + \frac{1}{2\epsilon}\frac{d\epsilon}{dN}$ 
then
$\eta \simeq \frac{1}{8\pi G} \left( \frac{V''(\phi)}{V(\phi)} \right)$
In the slow regime 
$\epsilon, |\eta|\ll 1$, where the last condition ensures that the change of $\epsilon$ per e-fold is small. notice that $\eta$ need not be small for inflation to take place. Inflation takes place when $\epsilon <1$, regardless of the value of $\eta$
