You're question is technically a bit more complicated than it appears on the surface. While a 40# bow is a 40# bow is a 40# bow, not all 40# bows perform alike. There are countless variables in laminate bows much less selfbows that affect speed and penetration, but here is what I would tell you.
I saw a study that said it takes 7# of pressure to penetrate the body cavity of a whitetail deer so if you took a sharp blade and put it between ribs then applied 7# of pressure then you would penetrate the body. So now you have to account for ribs, shoulder, organs and do you want an exit wound? I know that a lot of deer have been killed with 30 - 35# bows. If you hit them right it will do the job, but what I do not know the answer to is how many ran off wounded. Truthfully, our community has a hard enough time just getting the arrow in the vitals much less using a bow that requires precise arrow placement. Now, I hope a bunch of folks don't get offended by that statement, but I'm being honest. I'm simply a realist and I've shot with a lot of people. So to answer the question how light? Well....very light...if everything is perfect, but there are compromises.
Speed also reduces deer reaction time. If you keep the shots at 10 yards then it's not a big deal but a deer at 15 - 25 yards can move significantly if they hear the bow. A 35# selfbow with a light arrow might get you in the 140s (FPS). The light arrow with then reduce your penetration. A slow fast arrow hits like a hammer and drives deep, but you have to keep the shots close. Yes, I've seen the studies that show fast light arrows penetrating deep. This is an endless debate but I will tell you that the deer I've shot with moderate to heavy arrows get penetrated deeper. I think 9 - 11 gpp is optimal for speed and weight.
Now, performance depends on a lot of things. I have a 38# Osage bow and a 40# hackberry bow. The 38# osage is lighter in gross weight at a given draw length but shoots faster. The speed is in the woods cast or speed at which the limbs return from anchor to brace. This is where high performance strings can help you a bit if the wood can handle it, but keep in mind that the minimal stretch string will perform better on better quality wood. You get more energy into the arrow. So if you have a good sinew backed bow it could increase cast but it will increase weight. Sinew gives you the elasticity which in effect pulls the limbs back forward but it creates resistance which makes the bow higher in weight. A key advantage to sinew is that it can effectively eliminate the stress to the wood.
When you draw a bow you compress the belly and stretch the back. If this is a true statement, and we know it is, then there is a point in the wood that is neutral. It is neither being compressed nor stretched. When you put sinew on the bow you move that neutral point out towards the back of the bow. Ultimately, you can take the stress off the wood and put it on the sinew which is tremendously strong. That is perhaps the best benefit of the sinew.
Okay, I've rambled on and on. The bottom line is that in archery everything is a compromise. You never optimize everything. You trade speed for weight and cast for comfort. The key is finding what optimizes the most for your particular situation. I know I didn't directly answer your question but as I said in the beginning - it's complicated.
