Deriving the slow-roll parameter $\eta$

Question

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)?

Answer 1

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 $\eta$