As rockets burn propellant, the propellant tanks are emptied, and that space needs to be filled with some gas. This can actually be a significant problem.
Just having some gas in your propellant tanks to start out with is not practical. The pressure drops as the tank is emptied, so you need to start at a much higher pressure and waste a lot of tank space. It's better to have a separate high-pressure gas tank that is emptied into the propellant tank to maintain constant pressure. And then you need a heater system, because expanding gas makes it cool down.
The tank pressurization system is normally relatively small, but if tank pressure was much higher, then it would be that much larger. Some people have proposed pressure-fed rockets as an alternative to rockets using pumps, but they generally don't consider the tank pressurization system needed. Taking that into account, would pressure-fed rockets actually be cheaper in practice? Maybe. It depends on the design and manufacturing process. And anyway, "pump-fed" is a rather broad description for rockets.
Suppose you had a pressure-fed rocket, but you expand the pressurizing gas in a turbine that also pumps propellant. Now you're not wasting energy from the pressure difference. Seems easy, right? But is that cheaper than other turbopump systems for rockets? Maybe, maybe not. People aren't used to thinking about systems where fuel pumps are more expensive than the fuel.
Helium is the current standard for pressurizing rocket tanks. It's light and doesn't burn. But helium is rare, it tends to leak, and liquid helium is hard to use because it's so cold. Hydrogen, methane, ammonia, and nitrogen can also be used.
Gas density is generally proportional to pressure/temperature. Less dense propellants obviously require bigger tanks.
So, a given amount of tank pressure is a bigger problem for low-density cold propellants. A pressure-fed rocket using liquid oxygen pressurized by oxygen gas would have too much residual gas in the tank to be practical.
The amount of turbopump you need is also proportional to volume, so higher density is always good for rocket propellants. That's one reason why methane is not a very good fuel for rockets, the other reason being that methane has a small temperature range where it's liquid. But there are some people who are oddly enthusiastic about methane as a rocket fuel. This is a common pattern with novices in any field: they learn about something important like isp, they try to optimize that without considering the design as a whole, and then they often congratulate themselves on their cleverness.
If I was choosing a high-density rocket fuel, I would probably go with tetralin (hydrogenated naphthalene), cyclohexylbenzene (benzene alkylated with cyclohexene), and/or JP-10 (hydrogenated dicyclopentadiene).
Hydrogen peroxide is more dense than liquid oxygen, it doesn't need to be kept cold, and you use less (generally lower-density) fuel with it. It can also be decomposed with catalysts, which helps with ignition, propellant mixing, and maybe turbopump design. On the other hand, it gives lower isp, it's more expensive, and it slowly decomposes over time. Overall, H2O2 is probably better than liquid oxygen for first stage pressure-fed rockets.
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