The Friedmann equation expressed in natural units ($\hbar=c=1$) is given by $$\left(\frac{\dot a}{a}\right)^2 = \frac{l_P^2}{3}\rho(t) - \frac{k}{R^2}$$ where $t$ is the proper time measured by a comoving observer, $a(t)$ is the dimensionless scale factor, $l_P=\sqrt{8\pi G\hbar/c^3}$ is the reduced proper Planck length, $\rho(t)$ is the proper mass density, the curvature parameter $k=\{-1,0,1\}$ and $R=R_0a$ is the proper spatial radius of curvature.
Now each quantity in this equation has dimensions of powers of $[\hbox{proper length}]$ therefore it seems reasonable to refer to it as the proper Friedmann equation.
I wish to find the corresponding comoving Friedmann equation that is defined solely in terms of quantities with dimensions of powers of scale-free $[\hbox{comoving length}]$. I want to then solve this equation to find the constant comoving mass density $\rho_0$ from first principles.
In order to achieve this goal I define conformal or scale-free time $\eta$ using $$dt=a\ d\eta$$ so that \begin{eqnarray*} \frac{da}{dt}&=&\frac{da}{d\eta}\frac{d\eta}{dt},\\ \dot{a} &=& \frac{a'}{a}. \end{eqnarray*} The Friedmann equation then becomes $$\left(\frac{a'}{a}\right)^2 = \frac{l_P^2}{3}\rho(\eta) a^2 - \frac{k}{R_0^2}.$$ Now the LHS of the equation and the second term on the RHS have dimensions of $[\hbox{comoving length}]^{-2}$ as required. But the remaining term involving $l_P^2\rho$ still has dimensions of $[\hbox{proper length}]^{-2}$. In order that it has dimensions of $[\hbox{comoving length}]^{-2}$ we need to express the Planck length squared, $l_P^2$, in terms of $[\hbox{comoving length}]^2$ and the mass density in terms of $[\hbox{comoving length}]^{-4}$. In order to express the proper Planck length in terms of $[\hbox{comoving length}]$ we need to divide by the scale factor so that $l_P \rightarrow l_P/a$. Finally, we replace the proper mass density $\rho(\eta)$ with the comoving mass density $\rho_0(\eta)$ which has dimensions $[\hbox{comoving length}]^{-4}$.
Thus the complete comoving Friedmann equation is given by $$\left(\frac{a'}{a}\right)^2 = \frac{l_P^2}{3}\rho_0(\eta) - \frac{k}{R_0^2}.$$ Now each quantity in this equation has dimensions of powers of $[\hbox{comoving length}]$.
Does this dimensional reasoning make sense?
Now let us find a cosmological solution for a Universe with a constant comoving mass density $\rho_0(\eta)=\rho_0$. This Universe would obey the so-called perfect cosmological principle in that it is homogeneous and isotropic in both time and space provided one uses spacetime coordinates that are themselves independent of scale.
I can solve the comoving Friedmann equation for the constant comoving mass density $\rho_0$ by defining a constant time $t_0$ such that $$\left(\frac{a'}{a}\right)^2=\frac{1}{t_0^2}$$ which has the solution $$a(\eta)=e^{\eta/t_0}.$$ By substituting $a(\eta)$ back into the comoving Friedmann equation one finds that the comoving mass density $\rho_0$ is given by $$\rho_0=\frac{3}{l_P^2t_0^2}\left(1+\frac{k t_0^2}{R_0^2}\right).$$ By substituting the $a(\eta)$ expression into $dt=a\ d\eta$ and integrating I find that the scale factor as a function of proper time $t$ takes the simple linear form $$a(t)=\frac{t}{t_0}.$$ By substituting $a(t)$ back into the proper Friedmann equation one finds that the proper mass density $\rho$ is given by $$\rho=\frac{\rho_0}{a^2}.$$ Therefore the comoving Friedmann equation implies a unique functional form for the proper mass density that does not depend on assumptions about the constituents of the Universe.
