If push comes to shove, the biological father can petition the court for a whole list of things such as custody, visitation, child support, preventing you from moving out of state (removal), et. al.,
Talk to a family law attorney before you do something that gets you hauled into court. What you want can be done, but it is too important to try and do this yourself and mess it up.
Sign up to receive a 3-part series of useful information and advice about child custody law.