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 tra cks today, one has Q = m(P1) 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 potent ial deduced from the supergravity model has the form V(Q) = Lambda(4 + alph a)/Q(alpha)e(mu/2Q2). We show that despite the appearance of positive power s of the field, the coincidence problem is still solved. If alpha greater t han or equal to 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 fo r which it is difficult to go beyond omega(Q) approximate to -0.7. For Omeg a(m) approximate to 0.3, the model presented here predicts omega(Q) approxi mate to -0.82. Finally, we establish the Omega(m) - omega(Q) relation for t his model. (C) 1999 Published by Elsevier Science B.V. All rights reserved.