ok first of all why are you hiding things from your boyfriend and why is he snooping around your house...most people are very understanding if your straight up and honest. it seems that there is trust issues in this relationship however you do seem to really love eachother. if your babies dad really wants to know about your child then by all means go ahead with updating him however being that you are in a relationship with someone else I suggest you keep him informed of the contact and leave the contact as much as you possibly can just about your child. trust is something that has to be earned and when it is broken it is hard to get back...it can be done though...my advice is to completely forthright from this point on...he is still with you so he obviously he really loves you.
Did you tell him how you felt about your ex, that you don't love him? He has to realize that even if you have no feelings for your ex that he IS the father of your child, which means if he wants to know about his baby that he's going to be in your life. Let your ex read what you write to him, if you're ok with it to prove you are just keeping up with him because he is the father of your baby. If he sees that it's all just about the baby and you wanting to keep him updated and such, maybe he will understand. When you have a baby by someone else they are entitled to their rights as a parent, unless they give up those rights, in which case I would have nothing to do with the person, and wouldn't bother writing them. But if you honestly are just keeping in contact because of the baby then that should be understandable. Just make sure your ex knows there's no chance and that you are happy with your new man. Good luck, I hope your man starts to be more understanding.