Troubles with the minimal SU(5) supergravity model

We show that within the framework of the minimal $SU(5)$ supergravity model, radiatively-induced electroweak symmetry breaking and presently available experimental lower bounds on nucleon decay, impose severe constraints on the available parameter space of the model which correspond to fine-tuning of the model parameters of over two orders of magnitude. Furthermore, a straightforward calculation of the cosmic relic density of neutralinos ($\chi$) gives $\Omega_\chi h^2\gg1$ for most of the allowed parameter space in this model, although small regions may still be cosmologically acceptable. We finally discuss how the {\it no-scale flipped $SU(5)$ supergravity model} avoids naturally the above troubles and thus constitutes a good candidate for the low-energy effective supergravity model.