Birth information is public record. If you know where the child was born, mother's name and date of birth you can contact the local courthouse where child was born and they'll give you that information. They'll even tell you who is listed as the father. Hiring a PI is a good idea to check on her background and follow her for a few days. I wouldn't pay the PI to get the birth information when you can get it for free.
Good luck!
Good luck!