Part of the problem with the Big Bang Theory is that, in some sense, the "infinite density" region almost certainly never existed. Beyond a certain scale quantum effects will have prevented that point from occurring. Infinite density regions are implied all over the place in Classical physics (here's a simple example: Newton's gravity, which among other things typically treats everything as a point-like mass ie infinitely dense. While you can get around that, the point is that in a picture that isn't the full story you will deal with infinitely-whatever things all the time).
I scanned over the paper a few days ago and can't say I followed it properly, but at any rate it's certainly too early in the research to be saying "ha! no Big Bang!". At any rate, more precisely they are saying no singularity at the beginning of the Universe, but this is an attempt to deal with the first 10^{-30} seconds or so of the early Universe; most of the actual physics of the Big Bang concerns what happened afterwards.
One thing I'll be interested to see later on is whether or not this "no beginning" idea turns out to be a relic of the coordinate system they were using, rather than an "actual" infinitely-old Universe. I expect they probably covered it already, but it's an interesting point anyway. General Relativity is very fuzzy on what counts as "time", it being relegated to just part of spacetime. As soon as you do that, how you measure or define time becomes flexible, with the effect that different measurements will give you different answers for how long something takes to happen.
The simplest example of this is when you measure how long something takes to reach the centre of a (Schwartzschild) Black Hole. In the most natural choice of coordinates, you never do (taking infinitely long to get there). In another choice you do in a finite amount of time. These things are very fiddly, anyway.