 Solution

The Friedmann acceleration equation gives the gravitational deceleration from a uniform pressureless matter density, plus the acceleration from a cosmological constant. Here $\rho$ is the matter mass density, $G$ is the gravitational constant, $H=\dot{a}/a$ is the Hubble parameter, and $k$ is the constant spatial-curvature coefficient in the chosen normalization of $a$. The Friedmann equation is the first integral relating expansion to density, curvature and vacuum energy. With $c=1$, the matter, curvature and cosmological-constant contributions to $H^2$ are $8\pi G\rho /3$, $-k/a^2$ and $\Lambda /3$. The cosmological constant has effective density $\Lambda /(8\pi G)$ and equation-of-state parameter $w=-1$; its negative pressure explains its positive acceleration contribution when $\Lambda >0$.

For the Newtonian derivation with $\Lambda =0$, follow a test mass on the edge of a uniform sphere of physical radius $R=a\chi$. Conservation of the enclosed matter gives $M=4\pi \rho R^3/3$ constant. The shell theorem removes the force of external spherical shells and givesIntegrating the force equation once gives the conserved specific mechanical energyDivide by $R^2/2=a^2{\chi }^2/2$ to obtainHomologous motion makes $\epsilon /{\chi }^2$ constant between shells. Newtonian mechanics supplies this energy integration constant; its interpretation as intrinsic spatial curvature belongs to the relativistic FRW metric.

An outward acceleration $\Lambda \mathbf{r}/3$ comes from the Newtonian potential per unit masssince $-\nabla {\Phi }_{\Lambda }=\Lambda \mathbf{r}/3$. With conventional dimensional $\Lambda$, restore $c^2$ in this potential and the corresponding acceleration and Friedmann terms.