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I am studying inflation reading this article http://lanl.arxiv.org/abs/hep-ph/0406191 and in section 3 it states:

This inflaton field may evolve slowly down its effective potential, or not. While an approximately constant energy density seems to be required, a slow-roll field is only a simplifying assumption. Non-slow-roll models of inflation exist and for the moment make predictions that are compatible with observations.

I don't know what a slow-roll field is, and I am not able to find any good definition.

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The most straightforward theories of inflation assume there exists some scalar field $\phi$ that permeates the Universe and drives inflation. Over time this scalar field changes, and the rate of change is given by $\dot{\phi}$. There is also some "potential energy" associated with the scalar field, which is given by some function $V(\phi)$. The specific functional form of $V(\phi)$ is given by whatever theory of inflation that a theorist has postulated.

The "slow roll approximation" simply states that $\dot{\phi} \ll V(\phi)$. In other words, the "kinetic energy" of the field is negligible compared to the "potential energy" of the field. Under this assumption, it is possible to show that the equation of motion of the expansion of the Universe leads to approximately exponential expansion, i.e., inflation.

The slow roll approximation simplifies the equations of motion and guarantees that inflation will occur. However, as the text you have quoted states, it is also possible to have inflation without resorting to the slow roll approximation.

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