 Solution

In the Newtonian dust calculation, use a comoving spherical shell of physical radius $R(t)=a(t)\chi$. Homogeneity gives its enclosed mass $M=4\pi \rho R^3/3$, and the shell theorem removes the gravitational force of exterior shells. Assume pressureless matter, conserved enclosed mass, no cosmological constant, and no shell crossing. Its specific conserved mechanical energy isHomogeneity permits the energy per ${\chi }^2$ to be a constant, written $-k{c}^2/2$. Dividing by $a^2$ givesThis is a Newtonian derivation for dust. For radiation the enclosed gravitating mass is not conserved, so this same mechanical-energy argument cannot simply be reused; general relativity supplies the same Friedmann constraint with the total energy density divided by $c^2$, together with pressure-dependent fluid conservation.