So after quite a bit of work I have finally determined the solution to the continuous version of population growth, i. e. I can calculate what a colony's population is after a number of years with an initial population, and the growth modifier from governers assigned to planets and sectors.
Pf = ( t / 15 * (1+ Gp) * (1 + Gs / 4) + cube root(P0) )^3 when P0 > 0 else Pf = 0
Pf = Final population
P0 = Initial population
t = number of years passed
Gp = Planet Governor population growth modifier
Gs = Sector Governor population growth modifier
As far as I know Aurora calculates this discreetly (if I'm wrong let me know) so the actual population depends upon how small your construction cycle is as well as how small your time increments are, so if you really want to increase your population in the shortest game time possible, using smaller time-steps is generally better, which tends to lead to longer play-times overall. Using this equation instead should eliminate differences in population caused by time-step choices (rounding numbers may still cause some noticeable differences but it should be much more minor).
