Cosmic Challenge

At even close to the speed of light the light output would be so red shifted that it would be below the spectrum of visibility. Unless it was coming at us, then it is blue shifted. So it depends.

If, an object is ejected at the speed of light at the big bang it would be the age of the universe multiplied by the fraction of the speed of light (which would be 1) to get the number of light years that galaxy would be from galactic center.

Ah, so here is the puzzle! If that galaxy as well as ourselves are moving away from galactic center (albeit at different velocities), just how far apart would we be now?

The best we can do is give a range. Why? Because it depends on which direction the galaxy is going and which direction we are going. The minimum would be if we are both moving in the same direction, but we are far behind in speed, The maximum would be if we exited galactic center at direct opposite directions. More likely, we would have exited somewhere in between those two extremes.

Also, your question does not bound the time frame. Did you mean from T=0 until present time?

Quote: ".... Unless it was coming at us, then it is blue shifted. So it depends." The puzzle only talks about recession speed, so no bluse-shift.
Quote: "... Also, your question does not bind the time frame. Did you mean from T=0 until present time?" The question is distance, not time. If the recession speed equals c, how far must the galaxy be?

Some simple algebra leads to the conclusion that at a distance of 4.16 Gpc, stars are receding from us at the speed of light. This is equivalent to about 13.5 billion light years.

Seeing as Hubble's constant prescribes a linear relationship between distance and speed, it is logical to conclude that objects further away from us that 13.5 billion light years are moving away from us faster than the speed of light. Light emmited beyond this distance is limitted to travel at the speed of light, and therefore can never "catch up" to our solar system. It would seem, therefore, that it would be impossible to ever see, contact, or travel to any point beyond this minimum distance.

Something to take into consideration, however, is a point or planet halfway between us and this minumum distance. Lets call earth A, the planet 4.16 Gpc away C, and the planet halfway between B. A can never see C, yet B can see both A and C. Being half the minimum distance from each, A and C both travel away from B at a rate of half the speed of light. It seems possible, then, that an electromagnetic signal originating at A could reach B, be retransmitted, and subsequently reach C. Likewise, if people could manage to travel at a speed of just over c/2, we could eventually reach planet B, and subsequently travel to planet C.

All of this would of course take quite a long time, as this minimum distance is 13 billion light years, which curiously is roughly equivalent to the age of the universe. So even if there was someone you wanted to talk to on planet C, you would have had to send your message right after the Big Bang, and you'd be waiting another 14 billion years to hear back. Hope the news wasnt too pressing!

After some more thinking on the topic, I realized that travelling from A to B is more complicated than it seems. B is travelling away from A, even as we are travelling to it. If B was standing still in relation to A, it would take 7 billion years to get to at the speed of light. It is not standing still, however, and neither is planet C. I don't have time to do out all the math, but it seems like by the time a traveller or ray of light got from planet A to planet B, planet C would have moved beyond the minimum distance from B, making it once again unreachable. It seems that planet C really is impossible to reach.

Your answers to the puzzle are spot-on. Regarding using the planet at the halfway point as a 'relay': here things get a bit messy. In the original Einstein-de Sitter model of the universe, where the expansion is slowing down, one must just wait long enough and galaxies further than 13.5 billion ly. will come into view, relay or no relay. In the modern accelerating expansion models, galaxies that are now on the limit of the observable range will eventually move out of range. Galaxies further than 13.5 billion ly. will then never come into view, relay or no relay.

Of course the debate about the accelerating universe rages on. One theory claims taht the universe undergoes changes to the rate of expansion and that the universe eventually collapses again every 1 trillion years and starts over again.

If true, the "visible" horizon of the universe chanegs with age almost like a receding hairline, but it comes back. Yeah, we wish it (our hair) would come back. ;-)

Yeah, for the hair! The dominant view for the universe is the same though---the 'hairline' might just recede forever... There is some form of 'dark energy' out of the vacuum driving it. Somewhat similar to the hairline---if the roots of the hair find no more gray stuff, they fall out. If the distant galaxies find no more 'dark stuff' they disappear!

The best answer as to why a relay will not help you to "see" beyond the observable horizon: light has not have time to reach us yet. Galaxies beyond the horizon are further in light years than the age of the universe in years. In an accelerating expansion, that will remain so. In a decelerating expansion, the distance of some, yet undetectable galaxies will eventually become less than the light-distance to them.

I agree that the visible horizon is further away in light years than the age of the universe. Yet doesn't this mean that there cannot be any matter beyond this horizon? Matter cannot travel faster than the speed of light, and while the Earth is not at the center of the universe (we think) it seems impossible for any matter to be farther away from it than light can travel in the entire span of the universe's existance.

Standard cosmology theory says that it is space that expands---matter does not need to move through space to recede from us. Einstein's theory of relativity does not restrict the speed at which space can expand. Further, it does not restrict the distance that the furthest galaxies can be from us---they might be at infinite distance! The PDF linked from my web page Cosmology Introduction sheds some light on this issue.