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Given the rate of expansion of the universe and the speed with which galaxies separate: Can a rough estimate be given, at which time T (in its proper time) a civilization that lives on a planet which is X light years away from Earth today would have had to send off a light signal or spaceship that can reach Earth in our days?

For which distances X would T be much smaller than the 13.80 - 4.28 = 9.52 billion years that it took life to appear on earth (only 120 million years after the first appearance of liquid water). Especially so small, that not even liquid water could have appeared on that planet? This might restrict the distance from which we could receive signals significantly below the event horizon, resp. radius of the observable universe.

But maybe I'm completely off the track.

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What you are looking for is the size of the past lightcone of our present moment, trying to find the time $T$ when it is $X$ lightyears in radius. Measured in co-moving coordinates the distance is $$\chi(T)=c\int_T^{t_{now}} \frac{dt}{a(t)}$$ where $a(t)$ is the scale factor of the universe. Note that by now the expansion has turned the distance into a (proper) distance $X=\chi(T)/a(T)$ (where we have used that $a(t_{now})\equiv 1$).

Annoyingly, the actual scale factor does not have a closed form expression. During the matter dominated era it grew as $t^{2/3}$ which might be enough for a rough approximation in this case. But you can for example use Edward Wright's cosmological calculator to find the co-moving distance for a given light travel time. That calculator gives $\chi(9.52\cdot 10^9)=14.758$ Gly. Unfortunately it does not give the scale factor, but using for example this calculator and a bit of search I get $a(T)=0.723$, so the current distance to where the civilisation that sent the signal/spacecraft was is 20.41 Gly.

(Note that this distance is somewhat bigger than our current event horizon, $\approx 5$ Gpc $\approx 16.3$ Gly, so we can never reach that spot ourselves in the future.)

Generally, the further back in time a civilisation could have been possible, the tougher the Fermi paradox becomes since the volume of sites where civilisations could have originated grows a lot. It does not get infinite, though. It approaches a finite co-moving volume with radius $\chi\approx 45$ Gly as $T$ approaches 0.

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If I understand your question right, you want to know the horizon where an Alien civilization might observe the Earth, see life on the Earth, and send us a message that we would receive by now.

So images leave Earth showing a planet with the some primitive life, perhaps 4 billion years ago, these images reach an advanced race in a galaxy far far away, and they send a signal back to us "To serve man" so to speak.

The upper limit would be 2 billion light-years without expansion, 2 billion there, 2 billion back. When you apply dark energy expansion, I figure, about 1.8 billion years for the light to reach them, 2.2 billion years for the light to return, so how a 2 way message over 4 billion years can be received from 2.2 billion light years away might seem counter intuitive but the universe was smaller when the initial signal left Earth. (I can't do the math, so 2.2 billion light years is an estimate).

The more realistic limitation is, say there is an intelligent alien species out there. Would they be looking for Earth's 1 or 2 billion light years away? I don't want to calculate how many galaxies and planets that would be, or if they could even see the Earth through the body of the milky-way. The theoretical limit of about 2.2 billion light years might be technologically unfeasible.

Now consider, they send a message to us "are you there" and we send a message back "yes we are" and they send a message back to us "how's the weather over there?" and you get into 3 transits of communication or 4 to begin to exchange technology - the limit for 3 transits or 4 grows even smaller, not to mention the 4 plus billion year wait between "yes we are here" and their return message. That's a pretty big reason why communication that far might be disappointing or, just one way. We could broadcast Ali McBeal to them for thousands of years and they might be entertained, but we wouldn't know they liked it until a message form them back to us reached us 2 billion years later and each back and forth would take longer and longer.

There are other theoretical horizons, like, how large a telescope can be built before it's gravity causes interference (I suppose that could be re-calculated using math, but there's still practical limits on how far telescopes can see details on a distant planet though I suppose a string of 100 telescopes or 1,000 would expand any theoretical limits, but there's still only so far that stuff can realistically observed. Communication over 2 billion light years might be close to impossible. Even seeing planets that far away and trying to see any signs of technology or life might be impossible.

And space travel. If the maximum practical space travel velocity is .5 or .75c, that makes the theoretical travel horizon quite a bit smaller than any 2-way communication horizon.

Did I miss-read your question or is this the gist of it?

Here's a good summary of cosmic horizons but they don't discuss theoretical 2-way communication horizon, which I think is what your question is about.

In a practical sense - looking and communicating that far is silly. We should focus any searches in our back hard. A 5 million year for a back and forth message to Andromeda is quite long enough.

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  • $\begingroup$ Thanks for the interesting answer. No, you didn't misread my question, at least not basically. What you don't take into account is the time it takes life to appear, but probably this isn't really an issue. $\endgroup$ Commented Nov 7, 2017 at 9:21

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