= 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 gives
$$
\ddot R=-\frac{GM}{R^2}=-\frac{4\pi G}{3}\rho R,
\qquad
\boxed{\frac{\ddot a}{a}=-\frac{4\pi G}{3}\rho.}
$$
Integrating the force equation once gives the conserved specific <mechanical energy>
$$
\varepsilon=\frac12\dot R^2-\frac{GM}{R}.
$$
Divide by $R^2/2=a^2\chi^2/2$ to obtain
$$
H^2=\frac{8\pi G}{3}\rho+\frac{2\varepsilon}{a^2\chi^2}
=\frac{8\pi G}{3}\rho-\frac{k}{a^2},\qquad k=-\frac{2\varepsilon}{\chi^2}.
$$
Homologous motion makes $\varepsilon/\chi^2$ 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 mass
$$
\boxed{\Phi_\Lambda(\mathbf r)=-\frac{\Lambda r^2}{6}+\text{constant},}
$$
since $-\nabla\Phi_\Lambda=\Lambda\mathbf r/3$. With conventional dimensional $\Lambda$, restore $c^2$ in this potential and the corresponding acceleration and Friedmann terms.
Back to article page
