Well, You still can play it without the sun, but in certain parts, you need the sun in order to proceed. Part two of the boss battles is one of those times.
However, it is an interesting game. I went outside the other day.. in the sun for some relaxation. then decided to turn on the game.. and I appeared to have 3 flashes bars next to a little sun icon... turns out how much sun you have when you play the game affects the in game world.