Legal rights of spouse but not biological father

My husband and I were married (but having marriage difficulties though we were not legally separated) when I became pregnant by someone else. My husband & I reconciled & I went on to have the baby. No one signed the father part of the birth cert. My husband knows he is not the bio father. The bio father is a violent alcoholic with a felony DWI and while he is knows he is the father & has seen the child a few times when he was a baby (he is now 4 years old) he is not legally, financially, or emotionally in the picture at all. It is highly unlikely he will ever attempt custody. My husband and I had been married 3 years when the child was born, and my husband (not the bio father) was in the delivery room, and has been there everyday, he has fully taken on the role of father to the child and we have another younger child as well. What are his rights as a non bio parent? is he assumed the father because we were married even though the birth cert. is blank on that line? Can he sign it?