That’s not a realistic outcome. The accessible volume grows as t^3, at most, while population can grow exponentially with a fairly short doubling period. An exponential will always outrun a polynomial.
I could mention other reasons, but this one will do.
Why? If humans are spreading out through the universe faster than the population is growing, then everyone can stay just ahead of the Malthusian trap.
That’s not a realistic outcome. The accessible volume grows as t^3, at most, while population can grow exponentially with a fairly short doubling period. An exponential will always outrun a polynomial.
I could mention other reasons, but this one will do.