# Determining H value from scalefactor

• I
I’m probably missing something obvious here but I can’t figure out what.

Consider a galaxy, starting at an initial distance of ##D_0##, recessing with a constant velocity. After a time Δt, we’d measure a larger distance ##D_{t}##. From this scenario I’d conclude that the Hubble value based on this object is calculated by:
$$H = \frac{D_t – D_0}{Δt} \cdot \frac{1}{D_0}$$
That is, the velocity, which is obtained by the first fraction, divided by the initial distance ##D_0## of the galaxy gives the Hubble value when the galaxy was at distance ##D_0##

Now, from what I understand, the scale factor ##a## is how large the reached distance ##D_t## is relative to the initial distance ##D_0##. Thus; ##a = \frac{D_t}{D_0}##
Furthermore, the rate of change of the scale factor ##\dot a## is how much the distance has increased per unit of time, relative to the initial distance ##D_0##. So ##\dot a## could be calculated by adding the velocity (a distance per unit time) to the initial distance ##D_0## and dividing that by ##D_0##:
$$\dot a = (\frac{D_t – D_0}{Δt} + D_0) \cdot \frac{1}{D_0}$$
When combining these formulations for ##H##, ##a## and ##\dot a## I’d get: ##H+1 = \dot a##.
This is obviously not true since it should be ##H = \frac{\dot a}{a}##.

What am I missing?

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PeterDonis
Mentor
2020 Award
from what I understand, the scale factor aa is how large the reached distance ##D_t## is relative to the initial distance ##D_0##.

Not with the definition of ##H## you are using. If ##H = \dot{a} / a##, then ##a## has units of distance, which means ##a## at time ##t## is just ##D_t##, and ##a## at time ##t_0## is just ##D_0##.

So ##\dot a## could be calculated by adding the velocity (a distance per unit time) to the initial distance ##D_0##

No. The derivative ##\dot{a}## is just the change in distance over the change in time (in the limit as the change in time goes to zero), so it would be

$$\dot{a} = \lim_{\Delta t \rightarrow 0} \frac{D_t - D_0}{\Delta t}$$

So ##H = \dot{a} / a## at time ##t_0## would be

$$\frac{\dot{a}}{a} = \frac{1}{D_0} \lim_{\Delta t \rightarrow 0} \frac{D_t - D_0}{\Delta t}$$

So ˙ a a could be calculated by adding the velocity (a distance per unit time) to the initial distance D 0 D 0 and dividing that by D 0 D 0 :
Your resulting equation for a-dot makes no sense dimensionally.

Not with the definition of HHH you are using. If H=˙a/aH=a˙/aH = \dot{a} / a, then aaa has units of distance, which means aaa at time ttt is just DtDtD_t, and aaa at time t0t0t_0 is just D0D0D_0.

Thanks for the reply. Could you please explain a bit more why my defintion of ##H## says that ##a = D_0## or ##a = D_t## at ##t_0## and ##t## respectively? I thought a scale factor is the ratio between the new ##D_t## and the original distance ##D_0##.

PeterDonis
Mentor
2020 Award
I thought a scale factor is the ratio between the new ##D_t## and the original distance ##D_0##.

It isn't. It's just the distance. More precisely, it's just the distance if you are defining ##H## as ##\dot{a} / a##. That should be evident from the fact that ##H## is the fractional rate of change of distances, i.e., the ratio of recession speed to distance, which is the ratio of distance per unit time to distance. So if ##H = \dot{a} / a##, then ##\dot{a}## must be recession speed--distance per unit time--and ##a## must be distance.

It isn't. It's just the distance. More precisely, it's just the distance if you are defining ##H## as ##\dot{a} / a##. That should be evident from the fact that ##H## is the fractional rate of change of distances, i.e., the ratio of recession speed to distance, which is the ratio of distance per unit time to distance. So if ##H = \dot{a} / a##, then ##\dot{a}## must be recession speed--distance per unit time--and ##a## must be distance.

I'm surprised, because this wiki does use ##D_t = a \cdot D_0## and ##H = \frac{\dot a}{a}## simultaneously. From the article it does look like they distinguish the distance from the scale factor.

PeterDonis
Mentor
2020 Award
From the article it does look like they distinguish the distance from the scale factor.

What they are doing amounts to measuring distances in units of ##D_0## instead of in units of, say, light-years. So ##a = 1## just means the distance is ##D_0## in units of light-years (or whatever ordinary units you want to use).

Mathematically, if you really want to interpret ##D_0## this way, as a distance unit, then you would write

$$H = \frac{\dot{a}}{a} = \frac{D_0}{D_t} \frac{d}{dt} \frac{D_t}{D_0}$$

Since ##D_0## doesn't change with time, the ##D_0## factors just cancel out and you have the same thing I wrote before. In terms of the limits, you would have

$$\dot{a} = \lim_{\Delta t \rightarrow 0} \frac{a_t - a_0}{\Delta t} = \frac{1}{D_0} \lim_{\Delta t \rightarrow 0} \frac{D_t - D_0}{\Delta t}$$

And the ##D_0## will just cancel out, once again, when you evaluate ##\dot{a} / a##.

Explanation

Thanks a lot for the detailed explanation. It makes sense now.

I just noticed my error myself as well. The Wiki is considering ##D_t## as a distance in the past while ##D_0## being the current distance at this moment. I was considering them to be the other way around whch led me to erroneously think that ##a## should be >1.

After correcting for this, when one writes the distance in units of ##D_0##, the ##\dot{a}(t)## is the speed at distance ##D_t## relative to the distance ##D_0## just like you wrote it.
If the speed has been constant all throughout history, then ##\dot{a}(t)## would equal the current Hubble value ##H_0##. If the speed hasn't been constant, then that means one would have determine ##\dot{a}(t)## at time ##t## and then use...
$$\dot{a}(t) \cdot D_0 = v(t)$$
...to calculate the speed at time ##t##. This calculated speed divided by the distance ##D_t## at that time would give the Hubble value at that time, ##H(t)##.
$$\frac{\dot{a}(t) \cdot D_0}{D_t} = H(t)$$
Since ##D_t = a(t) \cdot D_0## (a(t) being <1) this means that:
$$\frac{\dot{a}(t) \cdot D_0}{a(t) \cdot D_0} = \frac{\dot{a}(t)}{a(t)} = H(t)$$

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