Observations of clusters gives a total matter density $\Omega_{\mathrm{M}} \simeq 0.3$. At the same time observation of the microwave background indicates a flat Universe $\Omega_0=1$. There must, therefore, be another source of mass energy that does not cluster to make up the difference, the dark energy $\Omega_{\mathrm{d}}$ (now) $\simeq 0.7$. This mass density cannot have been dominant in the past otherwise it would have prevented the formation of structure, so it must grow with time relative to the matter density. Use local energy conservation and the equation of state $P=w \rho c^2$ to show that, at time $t$,
$$
\rho_{\mathrm{d}} R^{3(\beta+1)}=\text { constant, }
$$
where $\beta=P /\left(\rho c^2\right)$. By considering the ratio $\Omega_{\mathrm{d}} / \Omega_{\mathrm{m}}$ deduce that the pressure of the dark energy must be negative. (This makes the vacuum energy or quintessence a candidate for the dark energy.)