The energy content of the space (i.e. radiation, normal matter, dark matter, and dark energy) determines the dynamics of the universe. AS you have mentioned, the universe is homogenous for large scales (>100 Mpc). It is only in this regime that we can use simple math models that apply to the whole universe and avoid doing simulations.
As I said before, the density of energy/matter determines the expansion rate. But expansion affects the density of different components of the universe differently. For instance, if you double the scale factor, the mass density shrinks by a factor of $2^3$ since matter is constant but space volume has octuple. In contrast, the energy density of dark energy remains the same. Freedman equation completely captures this dynamic:
$$\frac{(a'(t)/a(t))^2}{H_ 0^2}=\frac{\Omega _{\text{R0}}}{a(t)^4}+\frac{\Omega _{\text{M0}}}{a(t)^3}+\frac{\Omega _{\text{$\kappa $0}}}{a(t)^2}+\Omega _{\text{$\Lambda$0}}$$
where the Hubble constant $H_0 = 67.8 \frac{\text{km}/\text{s}}{\text{Mpc}} = 0.0693 /\text{Gyr}$, the present value of the radiation density $\Omega _{\text{R0}} = 0.0000905$, the present value of the matter density $\Omega _{\text{M0}} = 0.308$ (which mainly consist of dark matter) , the present value of the curvature density $\Omega _{\text{$\kappa $0}} = 1 - (\Omega _{\text{R0}} + \Omega _{\text{M0}} + \Omega _{\text{$\Lambda$0}}) = 1$, and the present value of the cosmological constant $\Omega _{\text{$\Lambda $0}} = 0.692$. These values are from the [Plank Collaboration 2015][1].