My question involves paternity law for the State of: California

I am an unwed mother of a 1/2 year old baby. On his birth certificate, the father name is blank. However, the baby has his dad's last name. The dad of the baby has never established for paternity. Baby has been living with me. The dad has been visiting the baby from time to time.

Recently my current company relocate me to a foreign country for an assignment (there is no deadline on the assignment). I wonder if I can just move with the baby without his knowledge of consent.

Can he file to established paternity after I moved to a foreign country?
If so, does he have the right to take the baby back? (Note that I will try my best to maintain the quality of life for the baby.)
If so, what is going to be likely for the term of visitation?
Will I have to pay for his ticket to meet the baby? (I make more money).
What if I loose the job while I am in the country. Do I have to move back?

NOTE: we are both U.S. citizen.

You might have trouble going to the new country with your baby because for him to get a passport (which he will need) both parents have to sign. Yes the father can file in court at anytime including while you are away for his visitation. At this point you can move to another state with out a problem because his father has no visitation but I really don't think another country will be the same story. Why wouldn't you move back if you lost your job? Generally if you are in another country you have to have a reason to be there and if you loose your job (your reason to be there) you would have to leave anyway.

Hm... if the father hasn't filed for the paternity and we are never married, does it mean the baby can never go out of the country? No right? What if some people who just had a baby out of a one night stand? I mean, I don't think father's signature is required. Thanks!

Let me repeat it again, the father name section of baby's birth certificate is empty.