Accelerating expansion of the universe in numbers

Well, the current value of $H$, $H_0$ is approx $67\ \mathrm{km/(s\ Mpsec)}$, and @hdhondt's answer is on that is accurate. This is to add a few facts and information, and calculate the acceleration in $\mathrm{km/s^2}$. It's instructive to get the units to that, one almost never sees it calculated. On the other hand, it's not hard, just carefully keep track. The acceleration is so small, compared to $g$ or the nuclear or electromagnetic accelerations in bound material, it just has no effect on earth, and still little at the edge of the observable universe (now)

The Hubble constant is a measure of the velocity of the universe. The per Mpsec indicates that it is not a unique velocity, but rather increases as one goes further away. $H$ is the rate of change of the scale factor, divided by the scale factor, so its units are $\mathrm s^{-1}$. @hdhondt's answer explains. The galaxies further away from us go away from us faster because of the universe expansion. At some distance (a few Gpsecs) they are actually expanding away from us faster than $c$.

The acceleration you asked for is not the velocity, but rather its change with time, if you wanted something equivalent to $g$.

A good description of the expansion and what it depends on is in one of the answers referenced in a comment by @Druv. The first question and (yes) answer by @John Rennie plots the scale factor, $a/a_0$, such that it is normalized to 1 now, as function of time. t=1 is now (normalized to the current Hubble time). You can see the scale factor slope, ie its change with time, ie equivalent to the expansion velocity when multiplied by distance, decreases early on in the universe, and after some time it starts increasing. The Hubble parameter, $H$ is shown decreasing in the third figure, plotting $H/H_0$ as function of time, where you can see it decreasing, with the value normalized to 1 now (it's this way because $a$ increases more rapidly than $\dot{a}$). The plot for $a$ shows (in the slope) the velocity decreasing early and later (actually about 5 billion years ago) increasing again – so acceleration now is positive. The Hubble parameter, in the current $\Lambda CDM$ model (the standard model in Cosmology, with current parameters from Planck) is decreasing slowly, and approaches a constant value at infinite time of $H \approx 0.67H_0$.

The reason it approaches a constant is that future times (and we're getting there) $H$ is dominated by the dark energy density which is constant (from what we've measured so far), and the expansion of the scale factor approaches an exponential expansion with $H$ the doubling rate.

From https://en.wikipedia.org/wiki/Deceleration_parameter

where q is the dimensionless deceleration parameter. In the $\Lambda CDM$ model you can see in the same wiki reference that with $w = -1$, $q = -1$, and we have an equation for the dimensionless acceleration $-q$ (it's minus because the thought was that acceleration was negative before the discovery of dark energy). Note that I approximated above also where $q = -1$ is true strictly at late times when dark energy is totally dominant. Before that $q > -1$. We can then solve for

$\mathrm d/\mathrm dt(\dot{a})/a = H^2$, and in numbers with $H = 67\ \mathrm{km/(s/Mpsec)}$ (where I approximate $H$ as $H \approx H_0$)

acceleration $\approx {5000}\ \mathrm{(km/Mpsec)^2/s^2}$ ie acceleration $\approx {5000}\ \mathrm{(km/Mpsec)(10^{-22}/s^2)}$ ie acceleration $\approx {10^{-18}}\ \mathrm{(km/s^2)/Mpsec}$

So, the acceleration due to the universe's expansion and acceleration, which we measure as acceleration of other galaxies with respect to us, is extremely small per Mpsec. At $1\ \mathrm{Mpsec}$ it is $10^{-18}\ \mathrm{km/s^2}$ (if my algebra didn't fail me), or $10^{-15}\ \mathrm{m/s^2}$

At the edge of the observable universe now (about $47\ \mathrm{Gly}$) it is only about $10^{-11}\ \mathrm{m/s^2}$. (again, with simple multiplies I hope I got right)

We only have been able to measure acceleration with very fine measurements for which a Nobel Prize was awarded. See for instance https://www.scientificamerican.com/article/the-2011-nobel-prize-in-prize-physics/

The fact that it is so small compared with $g$ is one reason that gravitationally bound bodies, and even more so nuclear or electromagnetically bound atoms and molecules and human-sized objects, and mountains and the earth, stay bound with the universe expansion – the cosmic acceleration is just minuscule and unmeasurable for material on the earth. The other reason is that $H$ has no effect also for small distances, you have to go Mpsecs away to make any difference due to just expansion velocity.

The parameter $g$, which is indeed approximately $9.8m/s^2$ is the acceleration caused by the earth's gravity - at the surface of the earth. It reduces as you get further from the earth's centre, and increases as you get closer. As a result, it is lower on a high mountain, and higher at the poles (the earth is a kind of flattened sphere). At the equator, the real value is about $9.78 m/s^2$ and at the poles $9.83 m/s^2$. At 400 km, the altitude of the ISS, its value is only about $8.7 m/s^2$.

If we ignore Dark Energy, then the universe is not accelerating. What we are seeing is that the further a galaxy is from us, the faster it is flying away from us. This speed is what is measured by the Hubble Constant ($H_0$. The value obtained by the Planck spacecraft in 2015 is $67.74km/s$ per $Mpc$. That means that galaxies at a distance of $1Mpc$ (3.26 million lightyears) will fly away from us at galaxies $67.74km/s$. Those that are 10 times further move 10 times faster, etc.

So yes, $H_0$ has dimensions that are different from those of $g$, but that is because it is not an acceleration; it is simply a measure of the expected recession speed of remote galaxies. While all objects on earth are equally subject to $g$, $H_0$ is only approximate. Individual galaxies can have slightly different speeds. For example, some of the galaxies in our local cluster (e.g. Andromeda) are actually moving towards us, rather than away from us.

Finally, while we expected the expansion of the universe to slow down over time, due to the gravitational interaction of the galaxies, it has been found that, over the last 5 billion years or so, it has been speeding up. The cause of this is still very much a mystery, but it has been given the name Dark Energy. Dark Energy means that the Hubble Constant has been slowly increasing in value. What will happen to it in the distant future may become clear when we understand what Dark Energy is. It may continue to increase, or it may reach a maximum value and start decreasing again.