# Distances in km/s vs mega parsecs or light years

1. Aug 18, 2015

### BWV

I know it has something to do with the rate of expansion, but why are large intergalactic distances measured in km/s rather than units of pure distance like mega parsecs? What is the conversion?

2. Aug 18, 2015

### Bandersnatch

You're most likely thinking about the Hubble constant: $H_0=68 km/s/Mpc$
This value describes the rate of expansion in the following way:
Objects (galaxies) you see at 1 megaparsec (Mpc) will be receding with recession velocity of 68 km/s. At 2 Mpc twice as fast and so on. This is nothing else than the Hubble law: $V=dH_0$

So the distances are indeed expressed in megaparsecs here, and the km/s is the recession velocity at a given distance.

3. Aug 18, 2015

### BWV

Thanks

the Hubble constant is not known with a great degree of accuracy and is changing due to the acceleration of expansion, but people don't worry about it at those distances?

4. Aug 18, 2015

### AgentSmith

The change due to acceleration of expansion of the universe is tiny on human epochs, but the value of the "constant" has changed since the time of Hubble due to observational issues.

5. Aug 18, 2015

### Staff: Mentor

The Hubble constant does not change with distance, but with time.

6. Aug 18, 2015

### Chronos

The rate of change of the Hubble constant over time is, however, fairly constant.

7. Aug 19, 2015

### Bandersnatch

I'm not sure about the claim about poor accuracy. The PLANCK mission results have narrowed it down to less than +/- 1 km/s/Mpc.

From the cosmological perspective, where the time scale of human observations is negligible, the Hubble constant is not changing. It is set as the value of the Hubble parameter at the present epoch - hence the 0 subscript. It represents the recession rate everywhere in the universe at this particular time (if you could see everywhere as it is now).
When tracing the changes in the recession rate, you're no longer talking about Hubble constant, but Hubble parameter to distinguish it from the present value. How it changes is determined by the matter/radiation/dark energy densities in the universe.

What we observe is old light, though, so the rate of recession at the present epoch leads to approximate recession velocities of only the closest (on cosmological scales) observed objects.
So you're right that the changing rate of the H parameter means that the simplistic approach of $V=dH_0$ doesn't really match observations for anything but relatively short distances (and correspondingly short look-back times). These 'short' distances are still in the order of hundreds of megaparsecs.
However, the evolution of H is derivable from observables and a given model of the universe, and so is the expansion history of the universe. People certainly do 'worry' about how it's changing. Here's a graph from Jorrie's calculator:

As you can see, the graph is nearly flat for something like last 4 billion years. The deviation from $H_0$ at the 10 Gy is less than 20%.

I don't know. Doesn't look terribly constant.

8. Aug 19, 2015

### BWV

OK thanks for all the responses. Shouldn't the slope of graph of H/H_0 turn positive in recent time to reflect acceleration? H today is higher than, say, H 1Gy ago?

9. Aug 19, 2015

### Bandersnatch

No, H always has been, and always will be going down with time, asymptotically approaching a value not much different from today's (63 km/s/Mpc iirc).
The switch to accelerated expansion is not readily visible on the above graph.

The acceleration/deceleration of the expansion means the evolution of scale factor (i.e. something like average distances between galaxies, where 1 means 'like today'):

Where the switch from downward-sloping (deceleration) to upward-sloping (acceleration) happens somewhere around 7 Gy.

10. Aug 19, 2015

### marcus

Fortunately, since we have simple formulas for the Hubble constant (and its time-derivative, or slope) over time, we can make this more precise! We can actually plot the history. Bandersnatch already did show a plot of H(t), three posts back in post #7. Here's the same but in zeit units---1 = 17.3 Gy. Present time = 0.8.

Since Chronos mentioned it, I'll try plotting the "rate of change" of H(t).
The derivative, H'(t) = dH/dt = -1.5/sinh2(1.5t) is the red curve in this picture. This keeps track of the SLOPE of the H(t) curve over time. In zeit units the present is 0.8 so we can see for much of the first half of cosmic history (up, say, to time 0.4) the slope is steeply negative compared with what it is later.
It's worth mentioning that the words "rate" and "rate of change" are vague in English and can refer to several quite different measures. They can mean "speed", or "slope". But they can also refer to the FRACTIONAL OR PERCENTAGE rate of change. So I suppose we ought to take the trouble to make clear which we are talking about.

The fractional rate of change of the Hubble constant is H'(t)/H(t), the change per unit time as a fraction of the whole, at that instant.
$$\frac{H'(t)}{H(t)} = \frac{-1.5}{\sinh(1.5t)\cosh(1.5t)}$$ This is the blue curve in the picture. Numbers on the vertical axis can be interpreted as percent change per 0.01 zeit, in other words percent change per 173 million years.

So we can see that at time 0.6 zeit H(t) was declining at rate 1% per 0.01 zeit (173 million years)
Somewhat earlier, at time 0.4 zeit, the Hubble constant was declining at rate 2% per same period of time.
And now, at time 0.8 zeit, the Hubble constant is declining at a little over half a percent per same period.

Last edited: Aug 19, 2015