Quintessence and Supergravity

In the context of quintessence, the concept of tracking solutions allows to address the fine-tuning and coincidence problems. When the field is on tracks today, one has $Q\approx m_{\rm Pl}$ demonstrating that, generically, any realistic model of quintessence must be based on supergravity. We construct the most simple model for which the scalar potential is positive. The scalar potential deduced from the supergravity model has the form $V(Q)=\frac{\Lambda^{4+\alpha}}{Q^{\alpha}}e^{\frac{\kappa}{2}Q^2}$. We show that despite the appearence of positive powers of the field, the coincidence problem is still solved. If $\alpha \ge 11$, the fine-tuning problem can be overcome. Moreover, due to the presence of the exponential term, the value of the equation of state, $\omega_Q$, is pushed towards the value -1 in contrast to the usual case for which it is difficult to go beyond $\omega_Q\approx -0.7$. For $\Omega_{\rm m}\approx 0.3$, the model presented here predicts $\omega_Q\approx -0.82$. Finally, we establish the $\Omega_{\rm m}-\omega_Q$ relation for this model.