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Maybe I am all wrong, but if the first light from the stars was emitted around 13 billions years ago, it means it has been travelling this entire time through the expanding universe while being redshiffted. This light is still reaching it since we can see it with HST and JWST in the future.

My question is: will this infrared light from that age ever stop reaching us? Shouldn't we get "younger" light with each new second? Or does the expanding universe cancels out that effect?

I know I haven't been really clear but I hope you understand what I am asking! Thanks!

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The most distant galaxy yet observed is GN-z11. The light we see from that galaxy is redshifted by a factor $z=11.1$, and was emitted when the Universe was 400 million years (Myr) old, and GN-z11 itself was roughly 50 Myr old. If we observe GN-z11 in, say, 10 Gyr from now (when the Universe is 23.8 Gyr old), we will see it having aged. But because of the expansion of the Universe, we will not see it having aged by 10 Gyr, but only by roughly 1.5 Gyr and thus be ~2 Gyr old. That is, in the future we will see light that today is younger (namely only ~11.8 Gyr old today), but at that time will be ~21.8 Gyr old.

So, we don't "get younger light with each second".

In the future we will build more powerful telescopes and thus be able to see farther than today, i.e. see light that today is older. And, in the future light from more distant galaxies in will have had the time to reach us, and we will be able to see a larger and larger part of the Universe. That is, the observable Universe keeps expanding. Assuming that by that time we have telescopes powerful enough to see that far, we will be able to see light that was emitted at the same time as the light we see from GN-z11 today, but which today is simply too far away to see yet.

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Adding to @pela's answer above:

The question of the OP assumes that if the light is there, we can observe it. But in fact, light from the era of the first stars will become increasingly redshifted towards infinite wavelengths and zero energy and thus be very hard, if not impossible, to observe.

The limit for how far away we can see light emitted at $t=0$ expands forever as @pela states above, this is correct. But the time span from $t=0$ to the accelerating expansion of the Universe takes an object across the event horizon -- the time after which we can no longer receive information from the object -- is also getting shorter for more and more distant regions. That is, we can see larger and larger parts of Space, but we can see smaller parts of their life span as we go farther away. At some point, these regions will cross the event horizon before the firsts stars have formed, and we will never see light from the first stars there.

Of course, light from the era of first stars just inside this limit will keep coming to us forever, just increasingly redshifted towards infinity. At some point, its wavelengths will be so long and its energy so low that it becomes impossible to observe. But it will of course still be there.

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Because of the size of the universe we live in at this moment, the space is being created (pushing matter and energy apart from eachother) so fast, that one far point of the universe is "moving" away from the completely other point faster than the speed of light. This means that we are starting to see less and less of those early moments in the universe after atoms formed. We can build bigger observational tools, but some photons that are really far away from us Will never reach us for those tools to detect them.

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