Upper bounds on supersymmetry breaking from gauge coupling unification

I derive conservative upper bounds on the supersymmetry breaking parameter $m_{1/2}$ as a function of the strong coupling in the Standard Supersymmetric Model (SSM) using gauge coupling unification. I find that over more than $99\%$ of the parameter space, $\alpha_3>0.120$ implies that $m_{1/2}$ is below $10\TeV$ and $\alpha_3>0.129$ implies that $m_{1/2}$ is below $1\TeV$. I express the variation of these bounds over the SSM parameter space with a numerical coefficient, $c$. I also find that in the SSM, a reasonable value of $50\GeV<m_{1/2}<1\TeV$ requires $\alpha_3>0.119$ over the whole parameter space. These bounds are particularly sensitive to the value of $\sin^2\theta_W=0.2317\pm0.0005$ used in the calculation. In more realistic models, heavy thresholds and gravitational effects will modify this result. Although these effects are theoretically calculable in specific models, more realistic models contain many unknown parameters in practice. I illustrate this point with minimal supersymmetric $SU(5)$ where the combined constraints of gauge coupling unification and proton decay require $\alpha_3>0.119$ for $m_{1/2}<1\TeV$ and the upper bound on the supersymmetry breaking scale is greatly relaxed.