Paternity, birth certificate, oh my!

Is anyone familiar with circumstances of not putting the father on the birth certificate? We split up and honestly don't want him interfering he said he doesn't believe the baby is his so at this point I'm not putting him on the birth certificate nor giving her his last name but if he should want to be involved can i leave establishing paternity up to him and not worry about it? Not concerned about child support at this time I just want to spend time w my daughter unbothered. Can I do so until he establishes his rights? Since I'm the custodial parent and he will not be on any paperwork?