Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 9D (Cosmology)

Differentiate $\mathcal{G}={\sum }_i{\mathbf{p}}_i\cdot {\mathbf{r}}_i$:

Pair the $ij$ and $ji$ terms and use ${\mathbf{f}}_{ji}=-{\mathbf{f}}_{ij}$:For gravity, ${\mathbf{f}}_{ij}=-G{m}^2({\mathbf{r}}_i-{\mathbf{r}}_j)/|{\mathbf{r}}_i-{\mathbf{r}}_j{|}^3$, so each summand is $-G{m}^2/|{\mathbf{r}}_i-{\mathbf{r}}_j|$. Their sum is $V$.

For a confined system, $\mathcal{G}(t)$ stays bounded, so its long-time averaged derivative is zero. The preceding identities give the virial theorem

Estimate $T\sim M⟨v^2⟩/2$ and $V\sim -G{M}^2/⟨R⟩$. The virial theorem givesup to a profile-dependent factor. If this exceeds the stellar/gas luminous mass, the difference is attributed primarily to dark matter (with modelling uncertainties and non-equilibrium effects checked).