Asymptotic bound on slow-roll parameter in stringy quintessence model

Understand

We study the late time behavior of the scalar part of the volume modulus and the dilaton in stringy quintessence model, focusing on their contributions to the Hubble slow-roll parameter $\epsilon$ which directly measures the deviation of the spacetime geometry from de Sitter space.

• When only one of the moduli is allowed to move, $\epsilon$ converges to the stable fixed point at late time.
• The fixed point value is larger than $1$, thus the slow-roll cannot be realized.
• Moreover, if the decay rate of the quintessence potential is larger than some critical value, the positivity of the potential imposes that the stable fixed point value is just given by $3$, independent of the details of the moduli dynamics.