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Dear Duncan, yes, that is the correct equation. I find it fascinating that a two component matter/lambda universe can be modelled with such a simple expression. But this is not restricted to matter/lambda. Other two component universes can be similarly modelled.
Duncan, this is an interesting idea. For the so-called ‘benchmark universe’, matter and lambda, the scale factor, a(t), is proportional to (sinh(t))^(2/3). So let’s try and see what the Friedmann equations look like if we keep this evolution of the scale factor, but restrict it to one component, which I assume is matter. We then get H^2 is proportional to (coth(t))^2, which I think means that if the matter density is proportional to coth(t), instead of the normal a^-3, we get the same evolution of the scale factor as we do with the benchmark universe. And of course, for large t, (coth(t))^2 tends to 1, which means H tends to a constant, i.e a de Sitter universe. And this is how we expect our benchmark universe to evolve over the long term. Adam
