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In a situation like this, if there is no court order in place, then both parties have exactly equal rights to the child. When the father signed the birth certificate or affidavit of paternity, then he becomes the presumptive father and is put on equal legal ground with the mother.
What this means is that whoever has physical custody of the child has the right to continue that custody and the other parent can't force them to surrender physical custody without a court order. So if mother has the child, the presumptive father can't force her to allow him to have physical custody or even visitation.
However, the door swings both ways and if the father is in physical custody of the child, he could refuse to return the child to the mother until there is a formal court ordered custody/visitation schedule put in place.
So legally whoever is presently in physical custody of the child has no legal duty to allow the other parent to have custodial time if they feel that it is not in the child's best interests.