Can accelerating universe avoid Big Bang? Expansion of our universe is accelerating. This means that it was expanding less rapidly in the past.
Is in accordance with the laws of physics a model of accelerating universe in which the rate of expansion would continue to decrease when going back to time, drop to zero and flip the sign, significantly before the scale reaches zero (i.e. before Big Bang)?
I've noticed that many graphs depicting various way the universe could expand (e.g. this, this and this) depict accelerating universe as initially decelerating. Is this initial deceleration a generic feature of such an model, or is it present in those graphs for no particular reason and can be avoided?
 A: 
... drop to zero and flip the sign, significantly before the scale reaches
  zero (i.e. before Big Bang)?

That would mean not more and not less than that there was no Big Bang singularity. The universe could have started with a volume which was small but not zero. As we are not able yet to apply physical laws to the very first period of the universe, both possibilities cannot be excluded (pointlike singularity or very small volume).

Is this initial deceleration a generic feature of such an model, or is
  it present in those graphs for no particular reason and can be
  avoided?

What you are calling initial deceleration is due to the triple energy in the universe: radiation, matter and dark energy. In the beginning the universe was dominated by radiation and matter. Dark energy is proportional to the volume of the universe and has an extremely low energy density.
After the Big Bang, radiation and matter lost more and more of their pressure and the space expansion slowed down. This exactly is the initial deceleration of the universe. But as space expanded, dark energy became more and more dominant. We presume that 4 billion years after the Big Bang the universe became dark energy dominated. In contrast to radiation and matter, dark energy is proportional to the expanding volume, and its pressure is increasing and accelerating the expansion of the universe.
