The short answer is most likely a "yes". In California, there is a preference for joint legal and physical custody of a child. If a parent wishes to participate in a child's life, and they are a "healthy" parent (I will define healthy in a moment) then the court will likely order that you share joint custody. As to what I mean is healthy, if they are a typical parent, not addicted to drugs or alcohol, no violent tendencies and no problems with the law, then relatively speaking, they are a "healthy" parent.
If there is domestic violence, or a drug or alcohol problem, it may be reasonable for you to have been withholding visits with your son. However, if you are only withholding your son because you want the new relationship to have more of a bond with your son, the court would likely not approve.
The former boyfriend has been requesting to be part of this child's life. He is trying to step forward as a father. The court will appreciate that, and if he does file a motion, will grant an order for him to have a paternity test. If the results of the paternity test are positive, then he will have certain rights. You would be required to go to Family Court Services, for a Child Custody Recommending Counseling session. They Child Custody Recommending Counselor would then make a recommendation for a child sharing plan.
I would strongly suggest you consult with an attorney in your county to discuss your rights, options and obligations.
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Please be aware that any comments that I have made are preliminary and tentative and not based upon a thorough analysis of your case. I would need additional information and review the exact documentation to be sure of the above advice. The answer above does not create an attorney/client relationship and does not require me to answer any future questions.
There is a legal possibility if he files a Petition to establish paternity and the testing shows he is the natural father, it is statistically probable the court will award joint/legal custody with you having physical custody of the child. What usually keeps these guys away is the obligation of having to pay child support. The law goes in many directions. If he does not communicate with the child nor pay support for one year, you can also petition the court to terminate his parental rights. By and large, the court will not terminate his parental rights unless you also have a new acceptable parent to take your ex's place. If he really hasn't seen the child in 11 months, you would almost have to presume he is just making these noises because he wants to have contact with you.
I hope this is helpful.
John N. Kitta
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You have already received two great answers from two great attorneys. I do practice in Alameda County. To add on to what the other attorneys have said, I would like you to know that it is unlikely a he would get joint legal custody overnight. So far, you have been your child's primary care giver and have made all the most important decisions for your child. Since your baby's father has not been involved, most Judges or Commisioners will want to see the father slowly work his way back into the baby's life before granting joint legal custody. This is just the short answer to your issue. If you would like to schedule a consultation, please call me at (415) 457-4367.