<p>Accelerating expansion means that the separation between two points in the universe increases at an increasing rate.</p>
<p>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.
<span class="math-container">$$ \frac{da}{dt} = a(t) H(t)$$</span></p>
<p>The Hubble parameter is currently decreasing and was larger in the past. The sign of the acceleration changed because the expansion rate is proportional to the Hubble parameter <em>multiplied</em> by the separation between two points, and the latter was smaller in the past.</p>
<p>The maths and plots are shown in Pela's answer to a similar question. <a href="https://astronomy.stackexchange.com/a/25986/2531">https://astronomy.stackexchange.com/a/25986/2531</a></p>

CC BY-SA 4.0
2022-09-19T13:47:40.553Z
So the change, 5 billion years ago, from a decelerated expansion to an accelerated expansion has nothing to do with H getting higher but  instead with aH getting higher? In other words, even if you ad a fixed number of km per Mpar the space expansion accelerates because the space gets bigger?
Felicia
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2022-09-19T13:54:08.477Z
And so 1,2,4,8,16 is an accelerated expansion?
Felicia
CC BY-SA 4.0
2022-09-19T14:32:51.243Z
@Felicia it is an exponentially accelerating expansion.
ProfRob
CC BY-SA 4.0
2022-09-19T13:01:12.010Z
2022-09-19T14:34:33.173Z
2022-09-19T14:34:33.173Z
ProfRob
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>I just don't see why this implies a constant Hubble parameter in the future of 60km/s per parsec.</p>

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2022-09-19T10:35:24.250Z
Hi Frekkie, do the plots in [this answer](https://astronomy.stackexchange.com/a/25986/5264) answer your question (especially plot #2)?
pela
CC BY-SA 4.0
2022-09-19T11:37:05.437Z
@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?
Felicia
CC BY-SA 4.0
2022-09-19T11:39:21.373Z
@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.?
Felicia
CC BY-SA 4.0
2022-09-19T12:00:44.150Z
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
CC BY-SA 4.0
2022-09-19T12:07:06.380Z
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
CC BY-SA 4.0
2022-09-19T12:42:55.797Z
@pela So 1,2,4,8,16,... has constant Hubble parameter an is an accelerated expansion? What, in the second graph, grows then?
Felicia
CC BY-SA 4.0
2022-09-19T12:51:07.923Z
@pela If the universal expansion accelerates due to dark energy, will H become constant? I mean, doesn't that mean that the number of km's per Mpar grows?
Felicia
CC BY-SA 4.0
2022-09-19T14:11:43.457Z
H tells you how fast a galaxy at a given distance recedes. A constant H of 56 km/s/Mpc means that a galaxy at 1 Mpc recedes at 56 km/s. If you then wait 10B years, and measure the recession velocity of a galaxy at 1 Mpc, you'll find the same speed. But those galaxy cannot be the _same_ galaxy, because galaxy #1 has moved farther away. If, say, it has moved to a distance of 2 Mpc, it will now recede at 112 km/s, 10B years later at 224 km/s, and so on.
pela
<expansion><hubble-constant>
Why is the Hubble parameter constant for an accelerating universe?
Felicia
# Why is the Hubble parameter constant for an accelerating universe?

(Asked by: username_0 on 2022-09-19)

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.

**Tags:** expansion, hubble-constant
> **Comments:**
> **username_1**: Hi Frekkie, do the plots in [this answer](https://astronomy.stackexchange.com/a/25986/5264) answer your question (especially plot #2)?  
> **username_0**: @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?  
> **username_0**: @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.?  
> **username_1**: 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.  
> **username_1**: 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.  
> **username_0**: @pela So 1,2,4,8,16,... has constant Hubble parameter an is an accelerated expansion? What, in the second graph, grows then?  
> **username_0**: @pela If the universal expansion accelerates due to dark energy, will H become constant? I mean, doesn't that mean that the number of km's per Mpar grows?  
> **username_1**: H tells you how fast a galaxy at a given distance recedes. A constant H of 56 km/s/Mpc means that a galaxy at 1 Mpc recedes at 56 km/s. If you then wait 10B years, and measure the recession velocity of a galaxy at 1 Mpc, you'll find the same speed. But those galaxy cannot be the _same_ galaxy, because galaxy #1 has moved farther away. If, say, it has moved to a distance of 2 Mpc, it will now recede at 112 km/s, 10B years later at 224 km/s, and so on.  

---

## 1 answers

### Answer by username_2 on 2022-09-19. Score: 1

Accelerating expansion means that the separation between two points in the universe increases at an increasing rate.

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.
$$ \frac{da}{dt} = a(t) H(t)$$

The Hubble parameter is currently decreasing and was larger in the past. The sign of the acceleration changed because the expansion rate is proportional to the Hubble parameter *multiplied* by the separation between two points, and the latter was smaller in the past.

The maths and plots are shown in Pela's answer to a similar question. <https://astronomy.stackexchange.com/a/25986/2531>

> **Comments:**
> **username_0**: So the change, 5 billion years ago, from a decelerated expansion to an accelerated expansion has nothing to do with H getting higher but  instead with aH getting higher? In other words, even if you ad a fixed number of km per Mpar the space expansion accelerates because the space gets bigger?  
> **username_0**: And so 1,2,4,8,16 is an accelerated expansion?  
> **username_2**: @Felicia it is an exponentially accelerating expansion.  

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