So first off I will assume that the generation ship has a reasonable capacity of 1 billion and 45,000 people are the first generation on it. Also assuming an 8/1000 per year death rate(the average death rate around the world) and an annual birth count of around 6,979 babies per 45,000(this takes into consideration miscarriage, stillbirth, fertility rate(and the fact that infertile couples make a minority on the generation ship), the 2 years between the birth of 1 baby and the conception of another and thus 11 pregnancies on average per generation between 20 years(youngest ideal age for pregnancy) and 50(average age of menopause), and multiple births from twins to quads(which has the lowest prevalence, like only 5 per 30 years per 45,000 if you take the natural conception rates into consideration only(like I did)), but not infant mortality and preterm birth because of these 2 assumptions:
- Technology to keep baby in the uterus until full term(like artificial amniotic sacks and stopping labor entirely until full term)
- Infant mortality is so low, even in those with severe defects, due to medical technology, that it simply isn't significant enough to affect the annual surviving birth count by much
that a linear graph would be very close to the actual population curve until 1 billion is reached, at which point people will have to die before anyone else is born so realistically, the curve would zigzag with the maximum always being 1 billion at most.
Now if we take the population density of Bangladesh, this means that at most, there will need to be 798,722 sq km just to fit 1 billion people in the generation ship, not to mention the complex water treatment plants with 100% of water being recycled from sewage and sodium hydroxide being used because, well, thousands of pregnant women in 1 cycle = a lot of HCl getting into the water = water approaching 2 on the pH scale = a lot of sodium hydroxide needed to form water in the least toxic and least likely to explode way(the only byproduct is sodium chloride which is soluble and can easily be drawn out of the water via evaporation. Or the tons of plants and animals including bees that would be required for the least gross space diet(seriously, eating algae all the time, um no thanks, that sounds like an unstable source and it just sounds gross) That would require so much fuel, it would only work if it was built in interstellar space, otherwise, gravity would prevent the generation ship from going into interstellar space.
So let's take the population density that is the highest of any state in the world, that of Macau, a city that is technically part of China but also technically its own state under One China 2 Systems. This brings the area needed for the humans way down to 46,704 sq km, an order of magnitude smaller than with Bangladesh's population density. But there are cities with an even higher population density than Macau. The most densely populated city in the world is Manila in the Philippines.
This doesn't bring the area needed down by an order of magnitude but it does go down by more than 10,000 sq km. If 1 billion people were in a place with the population density of Manila, Philippines, then the needed area would be 24,087 sq km. But this is only if you are talking about "city proper". According to the UN's Habitat Data, the most densely populated city is Dhaka, Bangladesh at 44,500 people/sq km which is even more dense than the "city proper" of Manila, Philippines. This would bring the needed area down to 22,471 sq km.
You see a pattern here? Yeah, as the density increases by a fixed amount, the amount of area needed gets smaller at a slower rate.
22,471 sq km though is still a lot and I think that is way too big for a generation ship, even though this does leave more added room for water treatment, farming, etc.
But just how dense can it get? I mean, I have heard of the theoretical maximum density of 5 people per sq ft but that seems way, way too dense for anything really, especially considering that that is a theoretical maximum assuming that people can't move because they are all contained in 6 cb ft chambers(1 sq ft * average height of 6 feet). That just would be impractical on so many levels, that even thinking that it could be possible is an overstatement.
So how dense would the population have to be in order for the generation ship to be small enough that it can relatively easily get out into interstellar space, even if it takes years to get to the sun's escape velocity due to the high mass? I'm assuming some very strong metal is used because metals can more easily block gamma rays(the most concerning of all radiation, considering that neutrinos don't cause damage and other subatomic particles at average energy would easily be blocked at a much lower thickness than is needed for gamma ray protection and very high energy particles like the OMG particle are very rare) than any other everyday material and I don't know if carbon fibers, which are way stronger than even the strongest metal, could block gamma rays with a reasonable thickness, much less that less thick than any metal that would be used.