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The Lambda-CDM model combines a cosmological constant with cold dark matter and the standard baryonic and radiation components in an expanding Friedmann-Lemaitre-Robertson-Walker metric.

The Friedmann acceleration equation and $H=\dot{a}/a$ give the deceleration parameterAn Einstein-de Sitter universe contains only pressureless matter, so $q_0=1/2$ and cannot explain the observed negative value. For the stated Lambda-CDM model,so its cosmological constant produces the required accelerating universe.

The measured Hubble time would give the Einstein-de Sitter agewhich is less than the measured $12\mathrm{Gyr}$ age of stars that must themselves be younger than the universe. The measurements are therefore incompatible with an Einstein-de Sitter universe. For the stated spatially flat Lambda-CDM model, direct evaluation of the preceding integral givesand hence $t_0\simeq 13.5\mathrm{Gyr}$. The period of accelerating expansion caused by the cosmological constant therefore allows an age consistent with the stellar lower bound.

The shell feels only radial gravity and the radial force due to the cosmological constant, so its torque vanishes and its specific angular momentum $L=r^2\dot{\theta }$ is conserved. Multiplyingby $\dot{r}$ and integrating gives the conserved specific orbital energy

For a uniform sphere, assembling concentric shells gives its gravitational potential energyThe cosmological-constant potential per unit mass is $-\Lambda r^2/6$. Since $⟨r^2⟩=3r_{\mathrm{ta}}^2/5$ in a uniform sphere,

The scalar virial theorem weights a potential homogeneous of degree $n$ by $-nW$. Gravity has degree $-1$ and the $\Lambda$ potential degree $2$, so the final state obeysAt turnaround $T=0$, while the virial relation gives $E_f=W_{G,f}/2+2W_{\Lambda ,f}$. Conservation of energy, together with $M=4\pi {\rho }_{\mathrm{ta}}{r}_{\mathrm{ta}}^3/3$, then gives, for $x=r_f/r_{\mathrm{ta}}$ and $\eta =\Lambda /(4\pi G{\rho }_{\mathrm{ta}})$,The root connected continuously to the $\Lambda =0$ solution has $x=1/2-\eta /8+O({\eta }^2)$, equivalentlyto first order in $\eta$. With $\Lambda =0$, virialization occurs at half the turnaround radius. At fixed turnaround state, positive $\Lambda$ makes this equilibrium root slightly smaller because its repulsive quadratic potential enters both energy conservation and the virial relation; sufficiently strong repulsion instead prevents a bound virialized state. Negative $\Lambda$ shifts the root in the opposite direction.

In the orthonormal frame, the comoving perfect fluid in general relativity has $T_{00}=\rho$ and $T_{ij}=p{\delta }_{ij}$. The Einstein field equations with cosmological constant therefore giveThusChoosinggives $p=0$ and $\rho =1/(16\pi )>0$. The metric is then the Einstein static universe supported by pressureless matter and positive vacuum energy.

In a region with vanishing stress-energy tensor, the Einstein field equations reduce to $R_{\mu \nu }=0$ when the cosmological constant vanishes.