$b \rightarrow s\gamma$ Decay in Supergravity Grand Unification and Dark Matter

An analysis of the $b \rightarrow s\gamma$ is given in supergravity grand unification using the framework of the radiative breaking of the electro-weak symmetry under three separate sets of constraints:1) neutralino relic density does not overclose the universe,2)p-stability constraint, and 3) the combined constraints of p-stability and COBE data .For case 1 it is found that the CLEO II data on the branching ratio already imposes very strong further constraints on dark matter analyses. For case2 the branching ratio is found to lie in the range (1.5-6.3)x$10^{-4}$ and thus the data does not at present significantly limit the analyses with p-stability constraint. It is shown that improvements by a factor of 3 in the $p \rightarrow\bar\nu K^{+}$ lifetime will reduce SUSY effects to less than O(30$\%$) of the Standard Model value.For case3,the branching ratio lies in the interval (3.1-5.3)x$10^{-4}$, and the SUSY effects lie within (-10$\%$,+50$\%$) of the SM value. Thus as experimental bounds on $B(b \rightarrow s\gamma)$ improve, one would need in cases 2 and 3 the next-to-leading order QCD corrections to disentangle SUSY effects.