Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 310 / 4

Taking the divergence of the linear Euler equation and defining the peculiar-velocity divergence $\theta =\nabla \cdot \mathbf{v}$ givesThe assumption of negligible vorticity ensures that this longitudinal variable captures the velocity perturbation relevant to scalar density growth.

Solved by gpt-5.6-sol high.

Differentiate the linearized cosmological continuity equation, $\dot{\delta }=-\theta /a$, and use $\dot{a}/a=H$:Substitution of part (a), followed by the Poisson equation ${\nabla }^2\Phi =4\pi G{a}^2\bar{\rho }\delta$, yields

Solved by gpt-5.6-sol high.

For a barotropic equation of state, $\delta P=c_s^2\delta \rho =c_s^2\bar{\rho }\delta$. A spatial Fourier mode of comoving wavenumber $k$ therefore satisfiesThe Jeans wavenumber is defined by equality of the pressure and self-gravity terms:For $k\gg k_J$, pressure dominates and produces acoustic oscillations whose amplitude is affected by Hubble friction. For $k\ll k_J$, self-gravity dominates and a growing mode develops: this is Jeans instability.

Solved by gpt-5.6-sol high.

Cold dark matter has $c_s=0$. During matter domination,The density equation becomesTrying the power law $\delta =t^p$ giveswhose roots are $p=2/3$ and $p=-1$. Thus the cosmic-time matter density modes areand the leading growing mode is $\delta \propto t^{2/3}\propto a(t)$.

Solved by gpt-5.6-sol high.

The constant-equation-of-state density scaling givesFor a spatially flat universe dominated by this component, the Friedmann equation gives $H^2\propto a^{-1}$. Integrating $\dot{a}/a\propto a^{-1/2}$ yieldsThis is accelerated expansion because $w=-2/3<-1/3$.

Solved by gpt-5.6-sol high.

The dominant component is smooth, so only the subdominant matter density clusters. The cold-matter perturbation equation is thereforeAt late times, $a\gg 1$, the source term is negligible compared with the expansion terms. Setting $\tau =t-t_{\ast }$ and using $H=2/\tau$ givesIt integrates toThe constant mode shows the suppression of matter growth by smooth accelerated expansion; the second mode decays.

Solved by gpt-5.6-sol high.