Realistically, if we get immortality then we'd want to cap our population. The mechanics are actually really simple: Get the average number of children per coup

Realistically, if we get immortality then we'd want to cap our population. The mechanics are actually really simple: Get the average number of children per couple significantly lower than 2. Like, if it's a 1 child per couple rate, then it caps out at double the population. If it's 1.4 per couple, then it caps out at somewhere less than 4x our population, I think. Which, realistically, means we'll probably never have more than 40 billion people, since without immortality our population will cap at around 10 billion. The "worst case" scenario is if immortality is widely administered before birth rates drop to 2/couple or below. I'm not sure that will happen, because I'd expect immortality to start off fairly expensive and a populace getting wealthier usually results in birth rates dropping massively, AND there will be a huge focus on minimising birth rates once immortality (and the asociated population problems it causes) becomes widely recognised as imminent by the general population.