> Only when the sphere’s thin outer shell had reached the earth—perhaps carrying radio signals from the extraterrestrials’ early history, before their rapid expansion started. By that point, though, the expanding sphere itself would be nearly upon us!
Wouldn't the problematic portion of the sphere be, by definition, millions of years away from us? Given that radio waves travel at the speed of light, and the alien civilization travels at slightly less than the speed of light, it seems like we should have at least however many million years it took for those aliens to get from producing radio waves to "maxing out" travel.