When doing digital recording, if your analog input level is very low, it makes sense to increase it. But this increase in volume in the analog domain must also increase the noise level. Alternatively, I guess you could simply record the very low analog signal and then normalize the recording digitally to 0 db after the fact. Has anyone actually compared these two approaches scientifically? Does one method introduce less noise than the other?

If you record at too low of a level (close to the noise floor), as you increase gain, the noise floor will become audible.

There is probably a lot written about the topic that you could search out. The general term is gain staging. The process is, generally, to set the gain of each stage in the analogue chain in order to get the most gain with the least noise. The ultimate goal is the optimum input signal with the least noise.

In general, noise in the analogue domain increases with gain but it usually isn’t linear. With many amplifiers there is a point where the increase in noise starts going up faster for additional amplification than it did at lower levels. Of course, if you only have one variable in your analogue chain, you have only to balance its setting against noise in the final product. It is a game of trial and error to find the optimum point.