In my limited experience, the deer/antelope sized animals respond to "stopping" power so-called beginning with a 30 caliber expanded hole. It may not kill them any quicker in time, but it sure seems to put the whop'em to them more so than anything smaller.
That is also what JJHack has corroborated in his African experiences on herd animals. He prefers that someone hits them with a 30 caliber or bigger to recognize that they got whopped, and to allow for easier tracking.
Not very scientific, but primarily just anecdotal observations over the years.
