I suggest reading these materials.
In California- if you are not married to the father of your child you can only put the father's birthdate and place of birth on the birth certificate.
In order to get the full name on the certificate there has to be a signed Acknowledgement of Paternity and request to amend birth certifcate filed.
Courts still need to rule on paternity for purposes of child support and visitation. In my case, even though the acknowledgement forms were filed, the father asked for a dna test, and the court ordered that one be done.
Then, unless paternity has been established, he is not currently the legal father. You have sole custody at the moment because of that but it is quite simple for him to seek paternity in association with the divorce or by itself.