In a 2014 paper on the topic, my colleagues and I rebutted many of these claims. In particular, we argued that one should not conflate the population growth in a single settlement with that of all settlements. There is no reason to suppose that population growth, resource depletion, or overcrowding drives the creation of new settlements, or that a small, sustainable settlement would never launch a new settlement ship. One can easily imagine a rapidly expanding network of small sustainable settlements (indeed, the first human migrations across the globe likely looked a lot like this). [...]
The results are pretty neat. When we let the settlements behave independently, Hart’s argument looks pretty good, even when the settlement fronts are slow. Even if all the ships have a very limited range (only able to reach, say, the nearest stars to Earth) and even if they are no faster than our own interstellar ships today (like Voyager 2), we find they can still settle the entire Milky Way in less than its lifetime, supporting Hart’s version of the Fermi Paradox. [...]
On the other hand, there are a lot of assumptions in Hart’s arguments that might not hold. In particular, his assertion that if the sun has ever been in range of a settled system then it would have been settled and the settlers would still be here. Perhaps Earth life for some reason keeps the settlements at bay, either because “they” want to keep Earth life pristine or it’s just too resilient and pernicious for an alien settlement to survive. Perhaps Earth is Aurora?
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