Jorrie, nice article, but I've got a basic question - hope you don't mind my woolly-headedness, because I think you have explained this before!

According to Hubble's law, a specific redshift should be equivalent to a specific distance - it is km/s per Mpc, which is equivalent to redshift per million light years, is it not?

I can see the 6 and 8 billion light years on your diagram, but I can't understand how those values relate to Hubble's law.

SL

Hi SL, you asked about Hubble's law: "I can see the 6 and 8 billion light years on your diagram, but I can't understand how those values relate to Hubble's law."

Hubble's law only holds well for small redshift (up to z=0.1, or so), where we can use the approximations z ≈ v/c and look-back distance s ≈ 978 z / H₀. For higher redshift, H₀ is just a constant in a more complex equation. The only accurate way that I know of is to numerically integrate the full 'equation of motion' of cosmic expansion and then extract the look-back distance from the data. This is what the cosmological calculator does for us.

There is however a reasonable Doppler approximation, as described in the original full .pdf, equations 16.1, 16.2¹ and the calculations that follow after them.

-J

Note 1.

The complexity shown here arises from extracting apparent recession velocity from redshift in the relativistic case.

H₀=71 km/s/Mpc has clearly been used, not the 65 mentioned in the original text. That's what the balloon says in the eBook...

Jorrie, it makes some sense, thanks.

What still has me puzzled is the velocity that I calculate from your equations 16.1, 16.2. For z=1 it comes out as 0.6c. However, when I use your cosmic calculator for z=1, it gives two recession speeds, one for 'now' (0.79c) and one for 'then' (0.66c). I suppose it means the recession speed was 0.66c when the light left the galaxy and the recession speed of that galaxy is now 0.79c due to the accelerated expansion? But, the 0.6c from the equation does not fall in this range, so what the heck is it supposed to represent?

I understand that 16.2 is an approximate (empirical?) equation, but the form looks like Doppler shift and for Doppler shift the velocity must mean something.

Another puzzle is why the two different models that you pictured, using the same Hubble constant, give such widely different ages for the cosmos (10 and 14 b.yr). I thought Hubble time (1/H_0) is Hubble time, irrespective of model.

SL

Hi SL, you wrote: "But, the 0.6c from the equation does not fall in this range, so what the heck is it supposed to represent?"

In short: it is the relative (opening) speed in free, non-expanding space that will give a Doppler shift ratio of z = Δλ/λo = 1 (a 100% Doppler shift). The cause of the cosmological redshift is not quite a Doppler shift, IMO, although there are cosmologists who prefer to describe it as an accumulation of a large number of tiny Doppler shifts, all along the path of the photons. What's more, they do get the right answer, so in a way it is just a different interpretation of the observables. That's fine with me.

But, equation 16.2 is indeed empirical for the cosmological application and it only works closely for the present values of the ΛCDM model. As you can easily check, it will not work for the flat, matter-only case (like in the very distant past) and it will also not work in the very distant future, when the cosmological constant presumably completely overshadows all other energies. We just happen to live in an epoch where equation 16.2 works reasonably well, it seems.

Hubble time (t_H = 978/H₀ Gyr) actually has little bearing on the age of the universe, because the age also depends on all the energy densities, which changed over the history of the universe. It is (perhaps) a coincidence that today the predicted age of the cosmos is very close to t_H = 978/71 = 13.8 Gyr. It has to do with the nearly linear top part of expansion curve (ii) (pasted again below). Curve (i) is parabolic all the way, so the argument does not work there.

Hope this clears some issues.

-J

Tx again.

When I have more time, I'll look into this fascinating stuff again :)

SL

This was an elegant answer in my opinion for is said there are two ways to skin a cat, and the cat may well be either a kitten, or an old cat.

Then there may be many cats.

We are not in agreement far as multiple universes, and I can't read equations, but if I could read Absolam Absolam and understand it, I can read this stuff, and understand it.

Thing I really like about this Jorrie post is that it is very pragmatic, concise, and you need to read it closely to get the sense of it.

It also shows an openness to new ideas, I was wondering about. Now I see that Jorrie where Jorrie is coming from a bit better.