What rights would the father have if I let him sign the birth certificate?

If he signs the affidavit of paternity, which will put his name on the birth certificate, there will be evidence of his paternity and either one of you is free to file a petition for a parenting plan and child support order. If he doesn't, he can file a parentage action, do a DNA test to prove he's the father and then get a parenting plan and child support order. If you are receiving any public benefits for the child, the state will file a parentage action. Until a parenting plan is signed by the court, he has no right to take the child and you should not give him the child unless you want him involved. Even if you do want him involved, there should be a parenting plan so that it is clear and enforceable who has the child when. There is no assurance if both parents are involved that one parent gets to make all the decisions. If you don't want him involved and there is time to go to CA now, do it and don't give him forwarding information. If you don't do it before legal documents are filed, you have to give him notice of the move and he can object. In other words, if you don't want him involved at all, then stop involving him. However, that means you have to be totally on your own financially, with no public help of any sort and if he's interested enough and can find you, he can always file to establish himself as the legal father. How much time he gets, depends on a long list of criteria, but if he has no major problems (drugs, alcohol etc) he would have exactly the same rights to the child as you do.