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I'm currently writing a review paper on the accelerated expansion of the universe (focussing on scalar field models like quintessence) and was wondering if there is a specific name for the approximation used in obtaining the condition for cosmic acceleration, given by

$$\dot{\phi}^2 < V(\phi).$$

When this is substituted into

$$w = \frac{P}{\rho} = \frac{\frac{1}{2}\dot{\phi}^2 - V(\phi)}{\frac{1}{2}\dot{\phi}^2 + V(\phi)},$$

we get $w < -1/3$, which is the condition for cosmic acceleration. It is very similar to the slow-roll approximation used for inflation: $\dot{\phi}^2 \ll V(\phi)$.

Is there a name for this approximation?

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Short answer: no.

I have never seen this condition given a special name. You are fine to refer to it as "the condition for accelerated expansion" or words to that effect. (Indeed, the equations are sufficient, especially if you can show why $w < \frac{1}{3}$ leads to acceleration.)

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