In the standard theory, t=0 is a "singularity." Energy density becomes infinite, for example, at t=0. What that really means is that t=0 is undefined in the same sense that 1/0 is undefined.

Consider the function 1/x. For all positive x, 1/x is defined. As you approach x=0 from the right, 1/x approaches infinity. Same thing for the density of the universe. But the point 1/0 itself is undefined, even though the limit is well defined from the right. Likewise, t=0 itself is undefined, even though the density is well defined for all times greater than t=0.

I wouldn't really consider this an "assumption." It is actually a consequence of assuming that the universe is expanding in accordance with the law of gravity and that energy is conserved. The only way to do it is to construct a "metric" ( http://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre... ) where a bunch of parameters are undefined at t=0, but well-defined thereafter.