The universe is a damn big place. When we look at the night sky, almost everything that can be seen with the naked eye is part of our galaxy: a star, a cluster of stars, a nebula. Behind the stars of the Milky Way, for example, is the Triangle galaxy. We find these "island worlds" everywhere in the universe, wherever we look, even in the darkest and empty spots of space, if only we can collect enough light to look deep enough.
Most of these galaxies are so distant that even a photon traveling at the speed of light will take millions or billions of years to traverse intergalactic space. Once it was emitted from the surface of a distant star, and now it has finally reached us. And while the speed of 299,792,458 meters per second seems incredible, the fact that we have traveled only 13.8 billion years since the Big Bang means that the distance that light has traveled is still finite.
You probably think that the farthest galaxy from us should be no further than 13.8 billion light years from us, but that would be a mistake. You see, in addition to the fact that light travels at a finite speed through the universe, there is another, less obvious fact: the tissue of the universe itself expands over time.
General relativity solutions, which ruled out such a possibility altogether, appeared in 1920, but observations that came later - and showed that the distance between galaxies is increasing - allowed us not only to confirm the expansion of the universe, but even to measure the rate of expansion and how it changed over time. The galaxies we see today were much further from us when they first emitted the light we received today.
Galaxy EGS8p7 currently holds the record for distance. With a measured redshift of 8.63, our reconstruction of the universe tells us that it took 13.24 billion years for the light from this galaxy to reach us. With a little more math, we find ourselves seeing this object when the universe was only 573 million years old, just 4% of its current age.
But because the universe has been expanding all this time, this galaxy is not 13.24 billion light years away; in fact, it is already 30.35 billion light years away. And don't forget: if we could instantly send a signal from this galaxy to us, it would cover a distance of 30.35 billion light years. But if you instead send a photon from this galaxy towards us, then thanks to dark energy and the expansion of the fabric of space, it will never reach us. This galaxy is already gone. The only reason we can observe it with the Keck and Hubble telescopes is that the neutral gas blocking light in the direction of this galaxy was quite rare.
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Hubble Mirror vs. James Webb Mirror
But don't think that this galaxy is the most distant of the most distant galaxies that we will ever see. We see galaxies at such a distance as far as our equipment and the Universe allow us: the less neutral gas, the larger and brighter the galaxy, the more sensitive our instrument, the further we see. In a few years, the James Webb Space Telescope will be able to look even further, as it will be able to capture light of a longer wavelength (and therefore with a large redshift), will be able to see light that is not blocked by neutral gas, will be able to see fainter galaxies than our modern telescopes (Hubble, Spitzer, Keck).
In theory, the very first galaxies should appear with a redshift of 15-20.