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All know that Inflation may solve some of the problems of the Standard Model of Cosmology and today we suppose that the field responsible for that is the inflaton. The properties of the potential $V(\varphi)$ allow via the momentum-energy tensor to define a barotropic fluid for which $$p=-\rho$$ and when the slow-roll conditions $\epsilon\ll1$, $|\eta|\ll1$ are satisfied, this fluid with $w=-1$ has some sort of negative pressure or repulsive gravity that allow the expansion of Universe. So a scalar field may act like a cosmological constant $\Lambda$ giving some energy to the vacuum. Now seems that the Universe undergoes an epoch of "second inflation" with $$\ddot{a}>0.$$ My question is: why there is no exponential inflation like the old one if today the dominant fraction of the Universe budget is again a sort of cosmological constant $\Lambda$? Maybe the two fluids are of completely different kind, or the behavior depends on the energy and temperature conditions or i am wrong something?

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The universe will expand exponentially once the cosmological constant dominates the expansion. At the moment it doesn't completely dominate, so the expansion isn't exponential, but as the expansion procedes and the universe tends towards a de Sitter universe the expansion will become exponential.

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