# Catching up with a galaxy

1. May 3, 2013

### marcus

I tried out the new version of Lightcone today
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone3/LightCone.html

It brought to mind, basically just with the default settings, an amazing trek. The one thing I did was open "setup" and get check Vnow and Vthen while X-ing out event horizon, and particle horizon distances. IOW I de-selected columns I didn't need to make more room for the two recession speeds. But that was all, no other changes---then I clicked calculate:

$${\scriptsize \begin{array}{|c|c|}\hline R_{0} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.92&0.693&0.307\\ \hline \end{array}}$$

$${\scriptsize \begin{array}{|c|c|} \hline S&a&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 1090.000&0.000917&0.000373&0.000628&45.332&0.042&3.148&66.182\\ \hline 339.773&0.002943&0.002496&0.003956&44.184&0.130&3.068&32.869\\ \hline 105.913&0.009442&0.015309&0.023478&42.012&0.397&2.918&16.895\\ \hline 33.015&0.030289&0.090158&0.136321&38.052&1.153&2.642&8.455\\ \hline 10.291&0.097168&0.522342&0.785104&30.918&3.004&2.147&3.827\\ \hline 3.208&0.311718&2.977691&4.373615&18.248&5.688&1.267&1.301\\ \hline 1.000&1.000000&13.787206&14.399932&0.000&0.000&0.000&0.000\\ \hline 0.312&3.208025&32.884943&17.184900&11.118&35.666&0.772&2.075\\ \hline 0.132&7.580159&47.725063&17.291127&14.219&107.786&0.987&6.234\\ \hline 0.056&17.910960&62.598053&17.299307&15.536&278.256&1.079&16.085\\ \hline 0.024&42.321343&77.473722&17.299802&16.093&681.061&1.118&39.368\\ \hline 0.010&100.000000&92.349407&17.299900&16.328&1632.838&1.134&94.384\\ \hline \end{array}}$$

The story the last line tells is that there is a galaxy which as of today is 16.3 billion ly from us and we send a message to it, a flash of light. The distance to that galaxy is increasing slightly faster than c, namely Vnow 1.13 c. At first it looks like the galaxy is going to get away, the gap is actually widening. But it is not discouraged (being only a simple photon after all) and it persists.

Eventually, when the galaxy is a hundred times farther from us, 1630 billion ly, it catches up!
It reaches the galaxy in year 92 billion, roughly 80 billion years from now, when the galaxy distance to the galaxy is increasing at 94 times the speed of light.

Last edited: May 3, 2013
2. May 3, 2013

### Mordred

Lol I fully enjoy the new additives. As far as your example above. To the photon just prior to arriving at its destination that galaxies recessive velocity is negligable.
After all recessive speeds is distance dependant.
As you often point out as well as others the expansion per cubic meter means nothing in terms of the speed of a photon per cubic meter.
taking that further and on the same units of measure (cubic meters) As long as the velocity (cubic meters) of travelling "x" is
greater than the the expansion rate at the same unit of measure per time slice. "x'" will always
arrive. Though it may take a VERY long time

Last edited: May 3, 2013
3. May 4, 2013

### Chronos

There is a cutoff beyond which a photon will never catch a receeding galaxy.

4. May 4, 2013

### marcus

Right! The CEH (cosmic event horizon) distance, here denoted Dhor. In the example given above the galaxy (at Dnow = 16.3 Gly) was still within the CEH range (Dhor = 17.3 Gly).

So since Dhor is relevant I will select it to be part of the table:

$${\scriptsize \begin{array}{|c|c|}\hline R_{0} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.92&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize \begin{array}{|c|c|} \hline S&a&T (Gy)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 1090.000&0.000917&0.000373&45.332&0.042&0.057&3.148&66.182\\ \hline 339.773&0.002943&0.002496&44.184&0.130&0.179&3.068&32.869\\ \hline 105.913&0.009442&0.015309&42.012&0.397&0.552&2.918&16.895\\ \hline 33.015&0.030289&0.090158&38.052&1.153&1.652&2.642&8.455\\ \hline 10.291&0.097168&0.522342&30.918&3.004&4.606&2.147&3.827\\ \hline 3.208&0.311718&2.977691&18.248&5.688&10.827&1.267&1.301\\ \hline 1.000&1.000000&13.787206&0.000&0.000&16.472&0.000&0.000\\ \hline 0.312&3.208025&32.884943&11.118&35.666&17.225&0.772&2.075\\ \hline 0.132&7.580159&47.725063&14.219&107.786&17.291&0.987&6.234\\ \hline 0.056&17.910960&62.598053&15.536&278.256&17.299&1.079&16.085\\ \hline 0.024&42.321343&77.473722&16.093&681.061&17.300&1.118&39.368\\ \hline 0.010&100.000000&92.349407&16.328&1632.838&17.300&1.134&94.384\\ \hline \end{array}}$$

Last edited: May 4, 2013
5. May 4, 2013

### Mordred

lol I forgot to mention the cutoff. I am curious however if the realization of the cut off has been creditted to a particular founder? If so I would be interested in studying related papers on it. More for my self study of historical development than lack of understanding the why the cutoff exists

6. May 4, 2013

### Mordred

Is the proper term of the "cut-off". The co-moving future visibility limit?

7. May 4, 2013

### Chronos

It is usually referred to as the cosmic event horizon. See the link in marcus sig to Davis - Lineweave.

8. May 5, 2013

### Mordred

Yeah I compared the distance and redshift value to the cosmic event horizon. The two are the same. I wanted to make sure I had the right FLRW metric describing it.

9. May 10, 2013

### marcus

What I should have said is that, as of today when we send them a message, the event horizon Dhor is 16.472 Gly, and the galaxy we want to get the message is WITHIN that horizon. It's distance today as you can see from the last row of the table is 16.328 Gly. So it is out near the horizon but still reachable.

In the example we want the message to reach target at the future epoch when distances are 100 times present value, denoted by scalefactor a=100. So Dthen the galaxy's distance when our message finally catches up with it will be 1632.8 Gly. I'll use the default table of the new version, LightCone6.

$${\small\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.92&0.693&0.307\\ \hline \end{array}}$$ $${\small\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.001&1090.000&0.0004&0.0006&45.332&0.042&0.057&3.15&66.18\\ \hline 0.003&339.773&0.0025&0.0040&44.184&0.130&0.179&3.07&32.87\\ \hline 0.009&105.913&0.0153&0.0235&42.012&0.397&0.552&2.92&16.90\\ \hline 0.030&33.015&0.0902&0.1363&38.052&1.153&1.652&2.64&8.45\\ \hline 0.097&10.291&0.5223&0.7851&30.918&3.004&4.606&2.15&3.83\\ \hline 0.312&3.208&2.9777&4.3736&18.248&5.688&10.827&1.27&1.30\\ \hline 1.000&1.000&13.7872&14.3999&0.000&0.000&16.472&0.00&0.00\\ \hline 3.208&0.312&32.8849&17.1849&11.118&35.666&17.225&0.77&2.08\\ \hline 7.580&0.132&47.7251&17.2911&14.219&107.786&17.291&0.99&6.23\\ \hline 17.911&0.056&62.5981&17.2993&15.536&278.256&17.299&1.08&16.08\\ \hline 42.321&0.024&77.4737&17.2998&16.093&681.061&17.300&1.12&39.37\\ \hline 100.000&0.010&92.3494&17.2999&16.328&1632.838&17.300&1.13&94.38\\ \hline \end{array}}$$

As you can see the message, flash of light, we send them today will get to them in year 92 billion, at which time the galaxy's distance from us will be increasing at 94 times speed of light.