# Why is the Hubble parameter constant for an accelerating universe?

The Hubble parameter is thought to become constant and to attain a value of about 60 km/s per parsec. So every second a parsec of space grows with 60km. Is this consistent with an accelerated expansion of space? Wiki says it's a common misconception but that confuses me.

Say that a parsec of space grows with a parsec per second. Then after a second there are 2 parsecs. Then after two seconds there 4 parsecs. Four gives 8, etc. Is this what is meant with accelerated expansion? Or is this a constant expansion? If I look at a distant mass at rest wrt comoving coordinates, it seems to move away at increasing speed. But the Hubble parameter is constant.

About 5 billion years ago the universe started to accelerate. Does that mean the Hubble constant was smaller then? I know it was huge during inflation. Then it decreased suddenly and a slower expansion took over. But now expansion appears to be accelerating again.

I just don't see why this implies a constant Hubble parameter in the future of 60km/s per parsec.

– pela
Sep 19, 2022 at 10:35
• @pela I think so, yes. Thank you! The second graph shows what I mean, I think. Doe it say the Hubble parameter grows? Or is that aH? Is the unit of H km/h per parsec? Sep 19, 2022 at 11:37
• @pela Does an accelerated expansion mean that instead of, say, 1,2,4,8,16, etc, it goes like say, 1,2, 6, 11, 20, etc.? Sep 19, 2022 at 11:39
• The second plot shows not H but a×H, which from H's definition is the same as da/dt. In other words it shows how much (the "da" part) the Universe expands in a given time (the "dt" part). You see that 1) it is always positive, meaning that the Universe has always expanded, 2) it first decreases, meaning that the expansion happened slower and slower, although the Universe still did expand, and 3) at some point it started increasing, meaning that expansion sped up again.
– pela
Sep 19, 2022 at 12:00
• So, as long as the curve in plot #2 is positive, we have expansion. "Accelerated expansion" just means any way you can imagine the the curve rises. One particular example of accelerated expansion is exponentially increasing expansion ("1, 2, 4, 8, 16, …"). And yes, the units of H is "speed per distance", though usually that distance is given not in parsec, but in mega-parsec (Mpc). Currently, H = H₀ ~ 70 km/s/Mpc, meaning that a galaxy 1 Mpc recedes at 70 km/s.
– pela
Sep 19, 2022 at 12:07

The Hubble parameter is the constant of proportionality between the rate of change of separation and the separation. Clearly, if the Hubble parameter is constant, but the separation is increasing with time, then the rate of change of that separation must also be increasing with time. i.e. an acceleration. $$\frac{da}{dt} = a(t) H(t)$$