For the so-called ‘benchmark universe’, matter and lambda, the scale factor, a(t), is proportional to (sinh(t))^(2/3).
Hi Adam. Thanks for taking this on and replying to my wacky ideas! To be honest I am trying to catch up with where you are with your understanding. I didn’t realize that in the matter dominated era in which we are now the scale factor a(t) for a lamda-CDM model could be so neatly expressed as a sinh(t) function. I found this paper (pdf) on the internet which I found helpful. I think their equation (5) is what you are referring to. What a beautifully elegant result! I always assumed that a model with lamda would be difficult to compute. So yes, as you say, this would be the benchmark to compare to. It has two nice asymptotic forms. When H0 t is << 1 the form is as in equation (6). The lamda term becomes negligible and a(t) is proportional to (t)^(2/3) which is like the Einstein-de Sitter model. When H0 t is >> 1 the form is as in equation (8) where the lamda term dominates and a(t) is proportional to exp(k t) where k is a constant which is like the de Sitter expansion model. I hope I have got this right!
I am going to split my replies due to issues of losing my posts when editing them.