They don't live together and not together. Both of them signed the birth certificate.
Estate Planning Attorney
When a child is born to unmarried parents, the mother has sole legal and physical custody until a court issues an order stating otherwise. In order to have parenting time and/or share custody, the unmarried father must establish himself as the legal father of the child. Simply being named on a birth certificate is not sufficient to establish an unmarried man as the legal father of the child.
Paternity can be established in one of two ways.
(1) By having both parents sign a Recognition of Parentage or ROP. The ROP must filed and accepted by the Department of Health, Office of State Registrar. Once filed and accepted, the ROP gives the unmarried father the right to go to court to ask for custody or parenting time or both. The ROP does not automatically give the unmarried father the right to see the child. Filing an ROP also gives the County right to begin a court action to establish child support.
(2) By going to court and asking the court to issue a Paternity Order. The court may grant some custody and parenting time as well as establishing child support in the same action.
You should note that the court looks at a number of factors the first time it decides custody and that it is very difficult to change a court's decision about custody after the decision has been made.
Paternity, custody, and parenting time are complex issues. You should really talk with an attorney about the specifics of your situation.