If a wife has another man’s child, what are her husband’s legal rights in relation to the child?

I understand this is a most unusual question. Hopefully someone could answer it. We’re a throuple. That’s me, my wife and another man, her boyfriend/partner. My wife had a baby daughter a few weeks ago, but I’m not the father. The father of her baby is the other man. It sounds complicated but it’s not. This was planned, so it’s what we all wanted. Now, my question is; who goes as father on the birth certificate? Both me and my wife want the actual father to be on the birth certificate and be a legal parent/guardian as does he. But about me? Would I be a legal parent/guardian of the baby as well. Could I have custody as well? We’re in this together for the long haul. I’m only asking in case anything happens in the future because you never know what life throws at you. by the way, I have 2 children with my wife already. And they’re definitely mine because they look like me. Not that it matters.

If a wife has another man’s child, what are her husband’s legal rights in relation to the child?
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