Standard cosmology is a numerical science (math models instead of verbal concepts and analogies). So a way to get understanding and test your grasp is to try and see what you can calculate on your own. We can get a lot of basic cosmo stuff from just two numbers (71 and 0.73) and two short equations. The simple beginnings of this were introduced in post#39 of the basic cosmo Sticky thread: https://www.physicsforums.com/showthread.php?p=1957688#post1957688 In that post we saw how easily to calculate that distances are increasing at a rate which is 7.3 percent per billion years. Also that that presentday total energy density is about 0.85 nJ per cubic meter and the pressure (due to dark energy) is - 0.62 in the same units. I am going to try sometimes using purple-colored words to name physical quantities, instead of conventional symbols such as Greek letters, which may put some folks off. To proceed then a bit further along the same lines, let's bring in the two Friedmann equations. In barebones form (space being nearly flat, we restrict to the case of zero spatial curvature) the two Friedmanns are: First Friedmann (a'(t)/a(t))^2 = (8 pi G/(3c^2)) density Second Friedmann a"(t)/a(t) = - (4 pi G/(3c^2)) ( density + 3 pressure) Note that a(t) is usually normalized to equal one at t=now, the present era. So when calculating presentday numbers we can ignore the a(t) in the denominators. Recalling that in the Sticky an easy way was shown to calculate that presentday density= 0.85 nJ per m3 and presentday pressure= - 0.62 nJ per m3 (or equivalently, if you like, 0.62 nanopascal), the reason why the pressure's quoted here in nJ per m3 is that to add density-and-pressure we need them both expressed in the same units. Obviously density + 3 pressure = 0.85 - 1.86 = -1.01 nJ per m^3 So Second Friedmann says that for t=now: a"(t)/a(t) = (4 pi G/(3c^2)) ( 1.01 nJ per m^3) That is something we can type into the Google calculator pretty much verbatim, and it will give us the current acceleration of expansion. The only downside is it gives it in terms of per second per second. And then the rate is a tiny tiny number. To get it on a per billion years per billion years basis, so it is more reasonable sized, multiply the above thing by (10^9 year)^2, put this verbatim into Google, and press return. (10^9 year)^2 (4 pi G/(3c^2)) ( 1.01 nJ per m^3) When I put the blue thing into Google searchbox, what it calculates is 0.00313 That means 0.00313 (or if you like, 0.313 percentage points) per billion years per billion years. To illustrate. the present rate that distances are increasing is 7.3 percent per billion years (see calculation in Sticky thread) and we now know that percentage rate of increase is itself growing by 0.313 per billion years. Heading toward 7.613 in other words. The rough linear approximation is meant to give a feel for what the second time derivative a"(t) of the scale factor signifies and how it affects things.