The de Sitter horizon is a serious problem for the long-term chance of survival for life on Earth and, we can assume, other planets.
Edwin Hubble discovered redshift when examining distant galaxies with a spectrometer. He found that all the galaxies were receding and the further away they were, the faster they were receding. The universe was expanding. This was eventually hailed as evidence of the Big Bang.
When Einstein formulated the Theory of Relativity, he discovered that it implied the universe was expanding. He talked to cosmologists, and their rate of expansion did not match his, so he added what he called the 'cosmological constant' to balance it. Later he called this the biggest mistake of his life. But recent evidence shows that Einstein was right, except his constant wasn't large enough. The Universe is not only expanding, it is accelerating. The constant itself is still a very small, non-zero number.
Willem de Sitter (1872 - 1934) realised that universal expansion presented a problem. If distant galaxies were receding faster than nearby ones, then at some point out in space, the galaxies would be receding from us at the speed of light. This is called the de Sitter horizon. Any galaxy beyond the horizon is forever beyond our reach, as one would have to travel faster than light to get there, and special relativity prevents this.
The Universe will then fragment into a series of 'bubbles' separated by de Sitter horizons: a de Sitter space. This can be thought of as a series of small universes, separate from each other, like islands in an ocean. The universes are totally separate, as only faster-than-light travel could propel you between them. Because their edge is expanding at the speed of light, they can be thought of as having infinite size.
As all galaxies would eventually pass beyond the horizon, eventually there would be no natural resources left for intelligent beings to consume. Each bubble would eventually settle into a bland, lifeless uniformity.
The situation may even be worse than this, as there is evidence that, far from slowing down, the expansion seems to be accelerating.
The de Sitter horizon would also place limits on our knowledge, as we would be unable to look beyond it, or receive information from it.
This means that the longer the universe continues expanding, the less chance we have of observing the earlier parts of it, because they will pass beyond the horizon and will not be visible anymore.
This won't be a problem for the human race. It won't be anything to worry about for several billion years. We'll be long extinct by then.
Some physicists argue that even after all the universes have passed beyond the horizon, there will still be hope. Given enough time, the laws of thermodynamics would seem to indicate that the Universe would return to a state where life could flourish again. This may seem as likely as a drop of ink dispersed in a water from gathering into a single drop, but wait long enough and it will happen.
The reasoning behind this is the anthropic principle. It states that no matter how unlikely the Universe seems, the very fact that we are here to ask such questions resolves the problem of 'Why is the universe the way we see it?'. If things were different, life wouldn't exist and we wouldn't be around to ask the question1.
An Alternative Viewpoint
However, Leonard Susskind of Stanford University, California thinks otherwise. His team showed that the Anthropic Principle doesn't solve the problem, because a vast number of universes would permit life and still be quite different from this one. All of these habitable universes would result from highly unlikely statistical events. But there are so many of them that they would vastly overwhelm a cosmos like ours. Even if 'something' had set the peculiar initial conditions of our universe, this would only apply for its first run. Subsequent recurrences would produce a quite different universe.
If that were the case, then we are in the first expansion of this universe. Cosmologists say this all seems too much like a coincidence.
So either there is no cosmological constant after all - in which case, why is the universe accelerating? - or we're missing something fundamental.
The Simple Version
The cosmological constant was an initial condition of the pre-big bang set-up for the universe. It is only a valid physical force if this universe is the first expansion of the universe.
If, however, the universe expanded before and this is the second or third or three billionth expansion, the constant is no longer valid, as each expansion-de Sitter-coalescence and re-expansion would produce a universe based on different conditions.
In other words, the initial conditions of the universe for the first expansion are meaningless for all other expansions.
So, the antigravitational force we observe is either:
The cosmological constant of the first expansion of the universe.
Or (if we are in a subsequent expansion) it's something else and nobody seems to knows what. Cosmologists hate knowing that they don't know something.