Quote by chronon
The gravitational force between the masses decreases as the square of the distance between them, while the force due to dark energy is proportional to the distance, so that their ratio goes as the cube of the distance.

this doesn't sound correct to me, after all...
in both cases (x^2 + x) ≠ x^3 and (x^2 x) ≠ x^3
so if the two forces are regarded as both positive or one positive and the other negative in neither case will they make x^3
again i am no mathematician, but i know addition and subtraction (of the dark energy over distance) doesn't multiply or divide the force or the distance of gravitational attraction.
Then again, maybe I'm just reading the equation and/or logic incorrectly... think on it and let me know what you come up with.