Birth Certificate/last names???

Ha
Ok, so I'm due with my first child Feb. 9th 
Recently I found out (about 2 weeks ago) that my boyfriend of 6 1/2 years has been cheating on me, and might leave me for this other female. I'm letting him stay to help me for now because I am on maternity leave, no money, and no place to go. If I stay at my current apartment, my rent would be more then my entire paycheck every month....
So, this brings me to my problem. He's going to be there when baby is born, we are actually trying to work things out. But if I give baby my last name, does he still have equal right to her because he's on the birth certificate? Or can he be on the birth certificate, baby has my last name, but his rights aren't the same because she has my name...make sense? This whole thing is confusing to me...also if I give baby his last name and he's on the birth certificate, does he have more rights? Or is it the same either way? 
I'm just worried he'll try to take her from me, or he's going to try for equal custody...she's a newborn, and I feel like she needs to be with me as much as possible. 

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