Let's back track a little... what is the argument for "inflation"? What is the data we are interpreting as "inflation"? Is there any other possible explanation?
The strongest argument for inflation is the so-called horizon problem. Without inflation, there would be a''(t)<0. Then, there would be only a finite, quite short time for causal influences between different places. And even if the distance between far away places goes to zero in the limit, the region which can be causally influenced by some single event remains finite and is, in fact, quite small.
The image of the BB singularity as a single point is quite misleading. Take some event after the BB singularity. Compute the light cone which can be causally influenced. This is a finite region. Now move the event into the past, closer to the big bang singularity. The light cone, of course, increases. But this increase is limited, it cannot become greater than some limiting light cone, if a''(t)<0. This limit is the horizon. It appears quite small, much less than the part of our universe we can observed as the CMBR.
But what we observe as the CMBR is, first of all, very homogeneous. But if one wants to explain this homogenity as some thermal equilibrium, one has to presuppose some causal contact - without causally influencing each other, one cannot create an equilibrium. This problem could be solved by postulating, without further explanation, a homogeneous distribution as the initial value. But, unfortunately, what we observe is not an exact equilibrium. It has some small (in amplitude) but large (in size) inhomogeneities. For an explanation based on thermal equilibrium this would be unproblematic, this would simply be fluctuations. But fluctuations require causal contact even more. And the excuse with an exactly homogeneous initial value does not work for the fluctuations. The size of these fluctuations is much larger than the horizon, thus, cannot be created after the BB. One would have to assume a quite complex configuration as the initial value, almost homogeneous but with large (in their size) inhomogeneities - very implausible as an initial value, not as horrible but comparable with assuming the world we observe, with all the dinosaurier remains, as an initial value 5000 years ago.
So, what we need to use the much more natural explanation of the CMBR picture we see as some thermal equilibrium with fluctuations? We need much more time before the start of the CMBR, enough time to obtain a thermal equilibrium in all the visible universe. The a(t) at CMBR time is much too large for this. Thus, a(t) should have been much smaller before. This is what can be obtained only with a''(t)>0. This is named "inflation".
To name it inflation is very misleading, because what is named "inflation" in common sense is high inflation rate, thus, big a'(t), while this "inflation" means increasing inflation rates. So, we have the strange situation that in the initial part of "inflation" the inflation rate has been much lower than the inflation rate after the end of "inflation".
But to summarize: All the empirical evidence one needs is the picture of the CMBR, with its almost ideal homogenity, but also with its small inhomogenities, which have nonetheless a large size. This picture is incompatible with a theory which assumes a''(t)<0 all the time, simply because in this case no causal mechanism exists which could explain the homogenity as well as the fluctuations on such a large scale.
