K.I.
Custody fight can take lots of time can be years.
But you don't have to worry about it right now.
As a single mom you have the total custody of your child. He can't take him away, or get visits if you don't want it( unless he can prove your bad parent , I mean abusive or negligent. But he need a legal case for that too,)
I don't know your situation but you can't do anything if he doesn't demand it first.
Enjoy your baby and hope you get things right with him after all is the father and if you can get an agreement that will be the best for your child.
He has to legitimate the baby first and pay child support in order to get visitation rights.
Hope this help a little
I'm not a lawyer but I was on your same shoes once.
Don't worry now Enjoy your baby,