According to cosmic inflation models, there is an early period where $\Lambda$ dominates and the scale factor grows exponentially. Among other things, this helps to give a reason for large-scale homogeneity (i.e. 'solves' horizon problem), because the parts of the universe that are causally connected become much larger. So far so good.
It seems to me that the reason the inflation picture achieves this causal connection is because it asserts that the growth of the scale factor during this very early period is very much slower than it would be on another model such as Friedman model without $\Lambda$.
(Vertical scale on this diagram is off of course!) Because the growth is slow, there is time for light-speed-limited communication around larger parts of the universe, because they are not being carried away from each other. But when you read presentations of inflation, you very commonly see a phrase such as "inflation solves X because the universe went through a period of extremely fast exponential growth", with the emphasis on fast. Now I don't deny that these early processes were fast, but surely the whole point about inflation is that it makes them slower not faster?
I understand that this is a process that is very fast compared to everyday timescales, but as far as I can see, the reason it solves the horizon problem, if it does, is because it makes the early expansion of the universe extremely slow compared to what one might otherwise expect, and what was in fact thought.
My question is: is that right or am I misunderstanding something?
(On the widely-used illustration of cosmic history from NASA/WMAP Science Team (e.g. at https://en.wikipedia.org/wiki/Chronology_of_the_universe) there appears to have been an effort to show a sharper growth in the early part, marked inflation, whereas I think a correct graph would have a point of inflection and look more like the one I drew above.)
This is similar to an earlier (unanswered) question, but perhaps I may have asked it more clearly.