The Big Bang model describes early moments of time for the entire universe. The official Lambda-CDM model allows for two possibilities. (1) If the universe is closed (global space has a positive curvature), then the universe is expanding from smaller to larger. (2) If the universe is open (global space is flat or negatively curved), then the universe is infinite and has always been infinite. In this case, the universe was initially infinitely large, with the infinite mass, and infinite density everywhere.
Infinities in physics are a problem, as infinite solutions are generally considered non-physical. A well known example is how a concern with infinite solutions in classical physics has lead to the development of quantum mechanics that avoided such infinities. Accordingly, the hope is that quantum gravity will bring more light on the first moments of the Big Bang and what the universe looked like back then.