Do we know the rate of acceleration of the expanding universe? Question is in the title - I am aware that we believe the expansion rate of the universe is accelerating. Has anyone been able to measure it's rate of acceleration? Is it even possible to do that?
 A: This question has two possible answers depending on how we choose to define the "rate of acceleration". In either case, we can find a numerical value at present, but since anyone can plug numbers into an equation, I'm going to only go as far as to describe how to find it. Also, the value changes as the universe ages. There's not much point in my giving you the numerical value now if it's just going to be different in a hundred million years or so anyway.
The rate of expansion is agreed to be represented by $\frac{\dot a}{a}$, where $a$ is the scale factor and $\dot a$ its first time derivative. We could have used just $\dot a$ as the rate of expansion, but that would make it represent the amount the scale factor increases per unit time relative to what it is today. Using $\frac{\dot a}{a}$ means it represents the amount the scale factor increases per unit time relative to what it is at that same point in time.
Let me illustrate with an example: At some point in the past, the scale factor was $a=0.2$ - the universe was $20\%$ the size it is now. If, at that time, the universe were to double in size over the next second, $\dot a=0.2$ and $\frac{\dot a}{a}=1$. What would make more sense; writing the rate of expansion as $100\%$ larger per second, or writing it as larger by $20\%$ of the present day's size per second? Most people would agree that doubling every second should be represented as $100\%$ per second. So that explains how we chose to express the rate of expansion.
Building off that, there's a couple ways to define the rate of acceleration of the expansion. One way is by following the same train of thought and calling $\frac{\ddot a}{a}$ the acceleration. This is a fairly common way and has its own name; the Friedmann acceleration. The other way of defining the acceleration rate comes from the line of thinking that if $\frac{\dot a}{a}$ is the expansion rate, the acceleration should be simply the time derivative of this. Both definitions have merit and I'd call both valid. It all depends on your preference and purposes.
In the first case, the second Friedmann equation directly gives us the acceleration of expansion. If we work under the $\Lambda CDM$ model, you get something that looks like this:
$$\frac{\ddot a}{a}=-\frac{4\pi G}{3}\left(\frac{\rho_{0M}}{a^3}+2\frac{\rho_{0R}}{a^4}\right)+\frac{\Lambda c^2}{3}$$
where $G$ and $c$ are the gravitational and speed of light constants respectively, $\Lambda$ is the cosmological constant (representing the dark energy contribution), and the present day densities of matter and radiation respectively are $\rho_{0M}$ and $\rho_{0R}$. You can easily look up values for the present day densities and the cosmological constant and find this acceleration rate for any given scale factor. Notice that as the scale factor grows and the universe expands, this definition has the acceleration of expansion approaching a constant.
The second definition I mentioned was the time derivative of $\frac{\dot a}{a}$. Since we define the Hubble parameter, $H=\frac{\dot a}{a}$, the value we are looking for is $\dot H$. This ends up looking like this:
$$\dot H=-\frac{4\pi G}{3}\left(3\frac{\rho_{0M}}{a^3}+4\frac{\rho_{0R}}{a^4}\right)$$
You might be tempted to look at this and say "wait a moment, this definition of acceleration has the rate of expansion approaching a constant value as time goes on. Looks like it stops accelerating". That isn't quite right. $\dot H$ represents the rate of change of the numerical value of the expansion rate; however, the expansion rate we defined as being always relative to the instantaneous scale factor. On the other hand, $\frac{\ddot a}{a}$ represents the amount the rate of expansion will increase of the next moment of time relative to the scale factor of the current moment.
Let's give an example, because even I was confused by that last sentence. If the universe doubles in size every second, then $\dot H$ would be zero because at any given point in time the rate of expansion we defined will be $100\%$ per second; it's not changing. But if it doubles in size every second, then we know that one second from now it's going to be expanding twice as fast as it is right now (seems like it's accelerating to me), which corresponds to $\frac{\ddot a}{a}$ being $100\%$ per second per second.
This is why the best definition of the "rate of acceleration" of expansion depends on your needs. I'd say both are good. If you want to show the acceleration in recessional velocity for a fixed proper distance away from you, use $\dot H$. However, if you want to find the acceleration in recessional velocity between two comoving coordinates, use the Friedmann acceleration.
Now, as I said in the beginning, you can easily look up the proper values for densities from the internet. If our theories are correct, then this should give you the acceleration of expansion. We can't directly measure this kind of acceleration (there's no universal accelerometer or anything. The universe's smart phone doesn't have that feature). The best we could do in theory is measure the rate of expansion at multiple times and compare the values. However, the problem with this is that the relevant values for such a measurement aren't noticeably different from one year to the next. Understandably, universe-sized changes take universe-sized times. We simply don't have precise enough instruments to notice a difference between now and, say, 10 years from now. Until we can find a way to make this measurement, we'll have to rely on more indirect and theory-based measurements.
