# Redshift at 11.9 and the Hubble Constant

1. Apr 2, 2013

### phtdegroot

Recently, astronomers from Caltech and Edinburgh University discovered galaxies with a redshift of 11.9. With the Hubble constant at 67.8 km/s/Mpc ( according to the most recent survey with the Planck Satellite ) this means that the galaxies are at a distance of 14.24 billion light years !
With the age of the Universe ate 13.79 billion years how is this possible ?
I used the following formulas to calculate the distance :

(z+1)^2 = (c+v) / (c-v) which results in v= 2.96 x 10^8 m/s

d = v / H resulting in d = 14.24 x 10^9 ly

What did I do wrong ?,

Anybody willing to help out ?

Thanks

2. Apr 2, 2013

### Bandersnatch

Hello! Welcome to PF.

Here you go:

tl;dr;
The universe is larger than it's age*c due to expansion of space.
Iirc, the observable universe is some 92 billion ly across.

You might find Ned Wright's cosmology tutorial useful:
http://www.astro.ucla.edu/~wright/cosmo_01.htm
(part two covers distances in cosmology)

Last edited by a moderator: May 6, 2017
3. Apr 2, 2013

### P.Bo

Something wrong with your math somewhere... perhaps a rounding error... as z approaches infinity, the distance d approached the Hubble length c/H0 which is roughly 13.8 Gly. So you should not get 14.24Gly for an answer

FYI, has nothing to do with the "observable size" of the Universe.

4. Apr 2, 2013

### marcus

Urgently advise you learn to use one of the online calculators. With Jorrie's since the factor 1+z comes up all the time we use S=1+z instead of z itself. So the light wavelengths are 12.9 times original size.

Plug that into the calculator as the upper limit, and put S=1 (the present day) for lower limit.
you can make the number of steps zero if you like to get a one-line table. I will put it at 10 steps.
http://www.einsteins-theory-of-relativity-4engineers.com/TabCosmo7.html
$${\begin{array}{|c|c|c|c|c|c|c|}\hline Y_{now} (Gy) & Y_{inf} (Gy) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14&16.5&3280&69.86&0.72&0.28\\ \hline\end{array}}$$ $${\begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&T_{Hub}(Gy)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)\\ \hline12.900&0.077519&0.378564&0.569593&32.820&2.544&3.770&1.074\\ \hline9.989&0.100108&0.556085&0.835690&30.816&3.085&4.667&1.587\\ \hline7.735&0.129279&0.816425&1.225009&28.540&3.690&5.733&2.344\\ \hline5.990&0.166950&1.197747&1.792530&25.958&4.334&6.973&3.458\\ \hline4.638&0.215599&1.754744&2.613391&23.038&4.967&8.375&5.095\\ \hline3.592&0.278423&2.563957&3.781500&19.752&5.499&9.900&7.495\\ \hline2.781&0.359554&3.726417&5.389206&16.095&5.787&11.471&10.994\\ \hline2.154&0.464326&5.360491&7.463489&12.113&5.624&12.964&16.046\\ \hline1.668&0.599628&7.571179&9.852215&7.937&4.759&14.238&23.226\\ \hline1.291&0.774357&10.393552&12.170853&3.802&2.944&15.185&33.196\\ \hline1.000&1.000000&13.753303&13.999929&0.000&0.000&15.793&46.686\\ \hline\end{array}}$$
Time now (at S=1) or present age in billion years:13.753301
'T' in billion years (Gy) and 'D' in billion light years (Gly)
===========================

So that's what the calculator gives you, for that galaxy. If anyone is interested in cosmology they will want to learn how to interpret the top row of the table, what Hubble time means etc.

In this example, the galaxy emitted the light in year 378 million.
And it was then at a distance of 2.54 billion lightyears from "us" i.e. from our matter.
Distances and wavelengths have been expanded by a factor of 12.9 since then. So obviously the distance from us now is 32.8 billion lightyears. (Has to be 12.9 times 2.54)

You can find the Hubble law recession speed of the galaxy at the time it emitted the light (not speed of motion thru space but the speed the distance was increasing.) You just have to divide the distance 2.544 by the Hubble time 0.5696. that give the Hubble law speed as a multiple of c.

2.544/0.5696 = 4.466 c
So when it emitted the light we are now receiving the distance to it was increasing at a rate which was about 4.47 times the speed of light.

Last edited: Apr 2, 2013
5. Apr 2, 2013

### phyzguy

With the new value of the Hubble constant (67.8), the Hubble length is now about 14.4 Gly. As the earlier posts point out, there is no contradiction here with an age of 13.8 Gy, since the universe is not expanding at a constant rate.

6. Apr 2, 2013

### marcus

If you want to use the new Planck mission numbers we just got in March then we can do the same calculation as before with the now and eventual Hubbletimes 14.56 and 17.6 Gy.
$${\begin{array}{|c|c|c|c|c|c|c|}\hline Y_{now} (Gy) & Y_{inf} (Gy) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline14.56&17.6&3400&67.17&0.684&0.316\\ \hline\end{array}}$$ $${\begin{array}{|r|r|r|r|r|r|r|} \hline S=z+1&a=1/S&T (Gy)&T_{Hub}(Gy)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)\\ \hline12.900&0.077519&0.370952&0.558110&32.681&2.533&3.831&1.053\\ \hline9.989&0.100108&0.544901&0.818918&30.717&3.075&4.751&1.557\\ \hline7.735&0.129279&0.800043&1.200686&28.486&3.683&5.847&2.299\\ \hline5.990&0.166950&1.173878&1.757787&25.955&4.333&7.129&3.391\\ \hline4.638&0.215599&1.720345&2.565392&23.090&4.978&8.588&4.997\\ \hline3.592&0.278423&2.515502&3.720049&19.862&5.530&10.192&7.352\\ \hline2.781&0.359554&3.661402&5.324395&16.258&5.845&11.866&10.790\\ \hline2.154&0.464326&5.281940&7.431824&12.309&5.715&13.490&15.767\\ \hline1.668&0.599628&7.496781&9.933097&8.126&4.873&14.913&22.870\\ \hline1.291&0.774357&10.365286&12.465965&3.925&3.039&16.006&32.787\\ \hline1.000&1.000000&13.834293&14.559933&0.000&0.000&16.730&46.281\\ \hline\end{array}}$$Time now (at S=1) or present age in billion years: 13.834286
'T' in billion years (Gy) and 'D' in billion light years (Gly)
===============
So with Planck mission numbers and the same 12.9 stretch factor, we get that
distance then=2.533 Gly
distance now = 32.68 Gly
recession speed at emission = 2.533/0.5581 = 4.54 c
recession speed now = 32.68/14.56 = 2.24 c

Last edited: Apr 2, 2013
7. Apr 2, 2013

### marcus

You did not calculate the present day recession speed right. Hubble law uses universe time (time as measured by CMB stationary observers) and the corresponding distance.
SR is not readily applicable. Think of the Hubble law distance as what you would measure at this given moment if you could stop the expansion process, freeze it as the given moment, so as to have the opportunity to measure it without having it change. That distance is proportional to the rate that distance is changing, at that moment in universe time.

In the case of the S=12.9 galaxy, the present distance to it is 32.8 Gly. You got something else.
And the present-day recession speed, by Hubble law, is 2.24 times the speed of light. You got something else.

I think the basic reason you went wrong is that you were using Special Relativity, which is only valid locally in a small neighborhood. The space of SR does not expand so, unless you do a lot of manipulation e.g. with a long chain of many successive local approximations, it is of limited value---so better to use the calculator:
http://www.einsteins-theory-of-relativity-4engineers.com/TabCosmo7.html

Last edited: Apr 2, 2013
8. Apr 2, 2013

### P.Bo

Am I reading this right, recession speed was faster 14 Gy ago and has slowed down? By a factor of about 2?

I'm sorry, the most updated data hasn't filtered down to the masses (the ones who make text books) and they're still stating the universe's expansion is speeding up.

9. Apr 2, 2013

### marcus

Until about year 7 billion the recession speed of a generic (distant) galaxy was slowing down.

Then after year 7 billion the gradual acceleration began. Here is a curve of the scale factor a(t).

It is the distance to a generic object normalized to equal 1 at present. Think 1 billion ly at present for example.
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure14.jpg
This is from an article by Charles Lineweaver back in 2003. Recently Jorrie generated a similar curve using up-to-date parameters with his tabular calculator. Qualitatively the same. Inflection point (changeover from slowing to speeding up) comes around year 7 billion in either case.

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10. Apr 2, 2013

### marcus

I think with the new Planck parameters, just to take an example (67, .32, .68) then any galaxy with z=2 or larger would have that property, or if you prefer (68, .32, .68) it doesn't make much difference.

With the 2010 numbers from WMAP it is pretty much the same. Certainly I would say z=3 or larger should suffice. This is based on a rough quick check.

But the majority of the galaxies we can see are beyond redshift 2 or 3.
So they would have the property that their recession speed at light emission was greater than when we receive the light. This is not in contradiction with the very slight acceleration occurring now.

11. Apr 2, 2013

### Chronos

Yes, present day acceleration is very feeble. The surprising thing was any acceleration whatsoever existed. The expectation was a monotonous decrease in the expansion rate over the history of the universe. But, Perlmutter and Reiss turned the world of cosmology on its head. Actually, their results make a lot of sense now that we have WMAP and Planck results.

12. Apr 3, 2013

### phtdegroot

Thanks guys !