The optical absorption properties of hydrogenated amorphous silicon (a-Si:H) are important in solar applications and from the perspective of fundamental materials science. However, there has been a long-standing question from experiment of the dependence of the optical gap on the hydrogen content in a-Si:H. To reconcile this debate, we present density functional theory simulations of models of hydrogenated a-Si:H, with different hydrogen concentrations up to and including full hydrogen saturation. We discuss the dependence of the optical and mobility gaps in fully saturated and undersaturated a-Si:H. Oversaturation with hydrogen results in a dramatic change in the properties of a-Si:H and is beyond the scope of this paper. For undersaturated hydrogen contents, both gaps increase with increasing hydrogen concentration until hydrogen saturation is achieved. Our key finding is that at saturation the optical and mobility gaps converge to a value independent of the hydrogen content. Our analysis thus resolves the contradiction between experimental data examining the effect of hydrogen content up to saturation and interpretations based on conventional expectations regarding the hydrogen dependence of the optical and mobility gaps up to saturation, and it provides new insight into the materials properties of hydrogenated amorphous silicon that can be used for sample preparation.