Observing galaxies that recess faster than c

[marcus]

Some great points being made here, and that graph is very clear, thanks.

A question: I'm using Hubble's law with v = c, to find the distance where an object should be to recess at the speed of light and I find approx. 14 x 10⁹ l.y. (depending on the value of H).

That's right! I usually use 14.4 billion Ly as the value of the Hubble distance, but it does depend on the value for H.
==quote==
But then I look at the cosmological redshift, z = v/c, and I know we observed quasars with z = 6, would it mean recessing at 6c? But then it doesn't fit with those objects at 14 x 10⁹ l.y. that recess with c.
==endquote==
There's no substitute for checking out Jorrie's "Lightcone" calculator. It makes tables of universe history (past and future) where you choose the limits and the number of steps. It gives recession speeds both back then (when light was emitted) and now (when light is received).
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html

Check it out (ask questions here about any columns in the table you don't understand).
you will see that there is no simple exact correspondence between z and any speed v over c. That z=v/c thing is only an approximation and only works for comparatively near thing. Distance expansion is different from motion thru space. cosmo redshift is not doppler due to motion at some given time. cosmo redshift indicates the total amount of stretching that happened while the light was in transit.

since the rate of stretching constantly changes thru out history, the amount of stretch (I use the letter s = z+1 for the enlargement factor) does not depend in any simple way on the speed of recession at one or another time.
Lightcone also uses the symbol S = z+1 for "stretch factor". If some light comes in and its wavelengths are 3 times what they were when it started out then the stretch S = 3
that corresponds to a redshift of z = S-1 = 2.
I find S more intuitive, easier to use, because it corresponds to the actual enlargement factor by which both distances and wavelengths have expanded while the light is in transit.
Using z instead of S is just an historical accident, a quirk custom that astronomers got into in the early days and became traditional.

 

[marcus]

If you go to the LightCone calculator
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html
and want to use it to print a table IN A POST, so we can discuss it here at PF, say.
Then look at the yellow stripe of output options and tick the button that says "Tex script"
and then press "calculate"

Then you will get a Latex version of the output, already highlighted, so you just say "copy" and paste it into your post.
When I say "Tex script" and then "calculate", and then say "Command-C" to copy it copies the code on my scratchpad
then I go to Physicsforums, to this post and I say "Command-V" to paste and this is what I get---just a sample, we could change the columns and the upper and lower limits and the number of rows etc. This is just the default you get if you don't do anything:
{\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}} {\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.001&1090.000&0.0004&0.0006&45.332&0.042&0.057&3.15&66.18\\ \hline 0.003&339.773&0.0025&0.0040&44.184&0.130&0.179&3.07&32.87\\ \hline 0.009&105.913&0.0153&0.0235&42.012&0.397&0.552&2.92&16.90\\ \hline 0.030&33.015&0.0902&0.1363&38.052&1.153&1.652&2.64&8.45\\ \hline 0.097&10.291&0.5223&0.7851&30.918&3.004&4.606&2.15&3.83\\ \hline 0.312&3.208&2.9777&4.3736&18.248&5.688&10.827&1.27&1.30\\ \hline 1.000&1.000&13.7872&14.3999&0.000&0.000&16.472&0.00&0.00\\ \hline 3.208&0.312&32.8849&17.1849&11.118&35.666&17.225&0.77&2.08\\ \hline 7.580&0.132&47.7251&17.2911&14.219&107.786&17.291&0.99&6.23\\ \hline 17.911&0.056&62.5981&17.2993&15.536&278.256&17.299&1.08&16.08\\ \hline 42.321&0.024&77.4737&17.2998&16.093&681.061&17.300&1.12&39.37\\ \hline 100.000&0.010&92.3494&17.2999&16.328&1632.838&17.300&1.13&94.38\\ \hline \end{array}}

 

[marcus]

If you want to see what the block of Latex code looks like that makes that table, that you need to copy and paste, just press "reply" at the preceding post with the table. You don't have to be able to read Latex code to use this feature, you only need to copy and paste the block of code. The PF system will automatically make the table for you.
Here is a sample of the code block, without the "tex" tags that make it get interpreted and turn into a table

...\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}...
...\hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.001&1090.000&0.0004&0.0006&45.332&0.042&0.057&3.15&66.18\\ \hline 0.003&339.773&0.0025&0.0040&44.184&0.130&0.179&3.07&32.87\\ \hline 0.009&105.913&0.0153&0.0235&42.012&0.397&0.552&2.92&16.90\\ \hline 0.030&33.015&0.0902&0.1363&38.052&1.153&1.652&2.64&8.45\\ \hline 0.097&10.291&0.5223&0.7851&30.918&3.004&4.606&2.15&3.83\\ \hline 0.312&3.208&2.9777&4.3736&18.248&5.688&10.827&1.27&1.30\\ \hline 1.000&1.000&13.7872&14.3999&0.000&0.000&16.472&0.00&0.00\\ \hline 3.208&0.312&32.8849&17.1849&11.118&35.666&17.225&0.77&2.08\\ \hline 7.580&0.132&47.7251&17.2911&14.219&107.786&17.291&0.99&6.23\\ \hline 17.911&0.056&62.5981&17.2993&15.536&278.256&17.299&1.08&16.08\\ \hline 42.321&0.024&77.4737&17.2998&16.093&681.061&17.300&1.12&39.37\\ \hline 100.000&0.010&92.3494&17.2999&16.328&1632.838&17.300&1.13&94.38\\ \hline

One of the many good things about LightCone calculator is that it knows LaTex so it has this copy paste option. If you don't do anything it just shows you the table on your computer screen. But if you tick the "Tex script" button and then say calculate it gives you a block of stuff that will create the same table in a post here.

So you can call attention to something and ask about it.

 

[virgil1612]

If you want to see what the block of Latex code looks like that makes that table, that you need to copy and paste, just press "reply" at the preceding post with the table. You don't have to be able to read Latex code to use this feature, you only need to copy and paste the block of code. The PF system will automatically make the table for you.
Here is a sample of the code block, without the "tex" tags that make it get interpreted and turn into a table

...\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}...
...\hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.001&1090.000&0.0004&0.0006&45.332&0.042&0.057&3.15&66.18\\ \hline 0.003&339.773&0.0025&0.0040&44.184&0.130&0.179&3.07&32.87\\ \hline 0.009&105.913&0.0153&0.0235&42.012&0.397&0.552&2.92&16.90\\ \hline 0.030&33.015&0.0902&0.1363&38.052&1.153&1.652&2.64&8.45\\ \hline 0.097&10.291&0.5223&0.7851&30.918&3.004&4.606&2.15&3.83\\ \hline 0.312&3.208&2.9777&4.3736&18.248&5.688&10.827&1.27&1.30\\ \hline 1.000&1.000&13.7872&14.3999&0.000&0.000&16.472&0.00&0.00\\ \hline 3.208&0.312&32.8849&17.1849&11.118&35.666&17.225&0.77&2.08\\ \hline 7.580&0.132&47.7251&17.2911&14.219&107.786&17.291&0.99&6.23\\ \hline 17.911&0.056&62.5981&17.2993&15.536&278.256&17.299&1.08&16.08\\ \hline 42.321&0.024&77.4737&17.2998&16.093&681.061&17.300&1.12&39.37\\ \hline 100.000&0.010&92.3494&17.2999&16.328&1632.838&17.300&1.13&94.38\\ \hline

One of the many good things about LightCone calculator is that it knows LaTex so it has this copy paste option. If you don't do anything it just shows you the table on your computer screen. But if you tick the "Tex script" button and then say calculate it gives you a block of stuff that will create the same table in a post here.

So you can call attention to something and ask about it.

Thanks a lot Marcus, definitely lots of things to look at. I will try that calculator and I will post questions.
Virgil.

 

[virgil1612]

The blue curve is the Hubble distance which grows and in a sense reaches out to photons so that some of them eventually make it even though they may have, at first, gotten farther away (because the distance to the space they were traveling thru was increasing.)
View attachment 82596

How did you get this graph? I looked at the calculator and didn't find any means of graphing something based on that data.

 

[marcus]

How did you get this graph? I looked at the calculator and didn't find any means of graphing something based on that data.

under the yellow bar with display options there is a white space with two things to click on
"set sample chart range" will get a convenient interval of time for drawing charts (i.e. graphs, curves...)

"column definition and selection" will let you eliminate most columns and just include Time (T) Hubble radius (R) and distance then (D_then)
The columns in your table are what turn into curves of the chart (i.e. the graph) when you select graph output.

there is one more thing to know, how to select Time to be the x-axis if it doesn't happen automatically
but for now take a look at the "column definition" and try to produce a table with only three columns

 

[marcus]

When you click "select sample chart range" it sets the stretch factor S limits at 40 and 0.4
S=40 is in the past (light comes to us stretched by that factor) and S=1 is the present (light is not stretched at all)
S=0.4 is in the future when distances will be 2.5 times present size.
To get rid of the future part and make the table (or the graph) stop at present, change 0.4 to 1 in the S_lower box.

Now if you have opened "column selection" menu and de-selected everything except T, R, and D_then

you can make that graph you mentioned. You simply have to tick "chart" in the yellow stripe of display options, and press calculate.

Did you try this? Any luck, Virgil?

 

[virgil1612]

When you click "select sample chart range" it sets the stretch factor S limits at 40 and 0.4
S=40 is in the past (light comes to us stretched by that factor) and S=1 is the present (light is not stretched at all)
S=0.4 is in the future when distances will be 2.5 times present size.
To get rid of the future part and make the table (or the graph) stop at present, change 0.4 to 1 in the S_lower box.

Now if you have opened "column selection" menu and de-selected everything except T, R, and D_then

you can make that graph you mentioned. You simply have to tick "chart" in the yellow stripe of display options, and press calculate.

Did you try this? Any luck, Virgil?

Thanks

 

[virgil1612]

Thanks. No, not yet.

 

[marcus]

Shucks, that's frustrating. Easier to start out making a table, then make a graph when that's easy.
Have you tried making a table with just 3 columns: T, R, and Dthen?

Have you tried setting S_lower to 1, so the table stops at the present and doesn't go on into future?

Any success with the "table" stage of learning?

Any problems? Don't be reluctant to ask. Several people around here use Lightcone and can reply helpfully

 

[virgil1612]

When you click "select sample chart range" it sets the stretch factor S limits at 40 and 0.4
S=40 is in the past (light comes to us stretched by that factor) and S=1 is the present (light is not stretched at all)
S=0.4 is in the future when distances will be 2.5 times present size.
To get rid of the future part and make the table (or the graph) stop at present, change 0.4 to 1 in the S_lower box.

Now if you have opened "column selection" menu and de-selected everything except T, R, and D_then

you can make that graph you mentioned. You simply have to tick "chart" in the yellow stripe of display options, and press calculate.

Did you try this? Any luck, Virgil?

Got it! I think I will play with this a lot.

 

[virgil1612]

Shucks, that's frustrating. Easier to start out making a table, then make a graph when that's easy.
Have you tried making a table with just 3 columns: T, R, and Dthen?
Have you tried setting S_lower to 1, so the table stops at the present and doesn't go on into future?

Hi,

I reproduced the graph.
- for the part where the photons were outside of the Hubble radius, does that mean they weren't visible?
- I added D_now to the graph. What is the difference between D_then and D_now?

Thanks

 

[Chalnoth]

Hi,

I reproduced the graph.
- for the part where the photons were outside of the Hubble radius, does that mean they weren't visible?
- I added D_now to the graph. What is the difference between D_then and D_now?

Thanks

No photon is visible until it reaches the observer.

 

[marcus]

Got it! I think I will play with this a lot.

Congratulations! Good work!

Hi,
I reproduced the graph.
- for the part where the photons were outside of the Hubble radius, does that mean they weren't visible?
- I added D_now to the graph. What is the difference between D_then and D_now?
Thanks

Let me try to explain, just the red curve, D_then. This tells us the histories of all the photons which we are receiving at our telescope today.
Today is year 13.8 billion. Some of the arriving photons were emitted fairly recently, comparatively near by. Like emitted in year 10 billion at a distance of around 3.3 (does the curve at 10 look to you about 3.3 high?) It approached us on an almost straight line timetable. Sliding down the red curve, so to speak. Getting nearer at an almost constant speed.

All the photons traveled along this curve, so to speak, to get to us.
One was emitted at year 4 billion, at a distance of nearly 6 (do you read it at 4 as a little less than 6 high?)
At first he made hardly any progress. The curve of his distance from us is nearly level. He doesn't get much nearer.
But after 6 billion years have passed it is year 10 billion and he has gotten within 3.3 of us. His distance from us follows the curve down. The curve shows his progress.

In year 10 this old photon could just be passing where the other photon I mentioned is emitted (in year 10 billion). They are both 3.3 from us, traveling together at the same speed, down the red curve, arriving here at the same day.

And then there is the photon who was emitted in year 1 billion, at distance 4.
He arrives on the same day as the other two.
He is aimed towards us but at first he is keeps getting farther because he is traveling thru space which is getting farther.
Eventually in year 4 billion he is almost at distance 6 from us, and he meets the photon which was emitted just then at that same distance, and the two travel together.

At first they make hardly any progress, as I said before, because they are traveling at speed c thru space with is getting farther from us at speed c, but finally by year 10 billion they are only distance 3.3 from us and they meet up with the third photon which was just emitted in year 10 billion. The three travel together and arrive the same day, today.

For the sake of narrative I assumed they were all coming from the same direction, just emitted at different times. they could have been coming from different directions and just arrive simultaneously. The curve only describes the time&distance relation of all the photons arriving today from all directions.
===============
D_then tells the distance to the galaxy that emitted the photon when it emitted it so it tells the distance the photon was from us when it started.
D_now tells what the distance to the galaxy is now today when the photon finally gets here.
If distances and wavelengths have expanded by a factor of 2 while the photon was en route then the distance to the galaxy now will be twice what it was then.

By convention we call a factor of 2 expansion by the name "redshift z = 1"
The redshift number is, by astronomers' tradition, always one less than the actual expansion factor.
If you use Lightcone to make a table and include the expansion or "stretch" factor S they you may notice that the Dnow is exactly S times the Dthen. In the row where S=2, the distance now is twice the distance back then when the light was emitted.

 

[George Jones]

By convention we call a factor of 2 expansion by the name "redshift z = 1"
The redshift number is, by astronomers' tradition, always one less than the actual expansion factor.

$z$ is the fractional redshift or the relative redshift or the fractional change in wavelength or ..., i.e.,

$$z = \frac{\lambda_{ob}}{\lambda_{em}} - 1 = \frac{\lambda_{ob} - \lambda_{em}}{\lambda_{em}} = \frac{\Delta \lambda}{\lambda_{em}} .$$

 

[marcus]

$z$ is the fractional redshift or the relative redshift or the fractional change in wavelength or ..., i.e.,

$$z = \frac{\lambda_{ob}}{\lambda_{em}} - 1 = \frac{\lambda_{ob} - \lambda_{em}}{\lambda_{em}} = \frac{\Delta \lambda}{\lambda_{em}} .$$

Exactly!
And the ratio of wavelengths (observed over emitted) is what some, including myself, have been denoting by small or capital S.
$$\frac{\lambda_{ob}}{\lambda_{em}} = S$$
S is also the ratio by which distances are enlarged while the light is in transit.

Virgil was asking about the relation between distance to source now, and back then when the light was emitted...
$$\frac{D_{now}}{D_{then}} = S$$

So it is written in the light itself, as it comes to us (since we know the wavelengths of light that atoms emit).

 

[virgil1612]

View attachment 82784
===============
D_then tells the distance to the galaxy that emitted the photon when it emitted it so it tells the distance the photon was from us when it started.
D_now tells what the distance to the galaxy is now today when the photon finally gets here.
If distances and wavelengths have expanded by a factor of 2 while the photon was en route then the distance to the galaxy now will be twice what it was then.

Please tell me if I got it correctly.

D_then is the actual distance from the photon to us, from the moment it was emitted until we saw it. It follows the photon and shows the distance taking into account the dimension of the Universe at that point. If I read D_then correctly, not one of the photons we see today, coming from anywhere, has ever been at more than 6 GLy from us.

D_now shows the distance between past positions of the photon and us, calculated now, not when the photon was there. It doesn't show at any point the real distance between a real photon and us.

Virgil.

 

[Chalnoth]

Please tell me if I got it correctly.

D_then is the actual distance from the photon to us, from the moment it was emitted until we saw it. It follows the photon and shows the distance taking into account the dimension of the Universe at that point. If I read D_then correctly, not one of the photons we see today, coming from anywhere, has ever been at more than 6 GLy from us.

D_now shows the distance between past positions of the photon and us, calculated now, not when the photon was there. It doesn't show at any point the real distance between a real photon and us.

Virgil.

No. The actual distance traveled by the photon is given by the light travel time.

D_then is the proper distance measured at the time of emission, which is the distance if you could freeze the expansion at that precise moment, and bounce some light rays back and forth.

D_now is the current distance to the object that emitted the photon measured in the same way.

 

[marcus]

Please tell me if I got it correctly.

D_then is the actual distance from the photon to us, from the moment it was emitted until we saw it. It follows the photon and shows the distance taking into account the dimension of the Universe at that point. If I read D_then correctly, not one of the photons we see today, coming from anywhere, has ever been at more than 6 GLy from us.

D_now shows the distance between past positions of the photon and us, calculated now, not when the photon was there. It doesn't show at any point the real distance between a real photon and us.

Virgil.

Yes! that is good insight! You noticed that not one photon we are seeing has ever been more than 6 Gly from us, at any time (proper distance measured at that time). Actually you can get it down more accurately to 5.8 Gly using Jorrie's calculator. He and I discovered that for ourselves a few years back. It is a nice thing to realize.

I think you got it right. The part about D_now too.
Virgil if I remember right you and I went over the idea of "proper distance" already, didn't we? Defined at some particular instant of universe time.
The distance you would measure with string or radar or any conventional means if you could pause the expansion process at that given moment long enough to make the measurement.