birth certificate help

My son is a month old, and I just found out his dad cheated on me while I was 8 months pregnant. His dad hasn't helped out one bit, and since I left him he hasn't even tried to come see our son. 
So our son has his last name, but on the birth certificate only my name shows. His dad's name is not on there, so does he still have rights?? What rights does he have until this changes?? 
P.S. I have no problem if he wants to be part of his life legally, but he threatens to come to my house and take him if I don't take him back. I just worry as he is unstable and unfit 

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