I'll try to develop some notation and explanations that might make the Friedman equations more intuitive to some beginners. They might look a bit odd in their new get-up--if that bothers you please simply ignore this thread. The scale-factor function a(t) keeps track of the size of a generic distance, as it grows over time. It is a real-number-valued function of time, normalized to equal one at present: a(now) = 1. The derivative or slope a'(t) at a particular time t is the amount a(t) grows per unit time. For example it could be a "per second" or "per year" quantity. You need to specify a timescale. The fractional growth rate H(t) = a'(t)/a(t) is what fraction or percentage of itself a(t) grows by, per unit time. Again you have to specify a timescale. For example the present distance growth rate is 1/144% per million years, or 1/14400 per million years. A generic distance (between objects that are at rest relative to the ancient light Background) grows by 1/14400 of itself per million years. According to standard cosmic model, this fractional growth rate of distances is expected to continue declining (more and more slowly) and level out at around H∞ = 1/17300 per million years. Or if you like to think in percentages think of the eventual steady growth rate as H∞ = 1/173% per million years. You can see these numbers (in reciprocal form) in the R column of Jorrie's Lightcone tables, and get an idea of how they change over time. Just click on the Lightcone link that some of us have in our PF signatures at the end of our posts. You will get a table, where you can increase the number of rows or steps in the table (and, if you wish, shorten the span of time covered) to get finer resolution. Obviously if you have a way of finding the history of what H(t) the fractional growth rate of a(t) has been and will be, from early times on into the future, then you have a way to plot the history of a(t) itself. Remember that a(now) = 1. So you or your computer can start at the present value and (using the growth rate) work forwards and backwards in time, constructing the entire history. What I want to do in this thread is introduce the Friedman equations which give cosmologists a handle on the distance growth rate H(t) and the scalefactor a(t).