Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 310 / 1 / a

Use units $c=1$ throughout. Set $\chi =|\mathbf{x}|$, so the spatial Euclidean metric is $d{\mathbf{x}}^2=d{\chi }^2+{\chi }^{2}d{\Omega }^2$. On either hemisphere of the embedded three-sphere, the constraint gives $u^2=R^2-{\chi }^2$ and henceIntroduce the dimensionless comoving coordinate $r=\chi /R$. ThenThis is the positively curved Friedmann-Lemaître-Robertson-Walker metric, with $a(t)$ the physical curvature radius. An equally valid convention retains $r=\chi$, takes $a=b$, and sets $K=R^{-2}$; only the coordinate normalization differs. The chart with $r\le 1$ describes a hemisphere, not the entire closed slice. Writing $r=\sin \psi$ gives $d{\ell }^2=R^2(d{\psi }^2+{\sin }^2\psi d{\Omega }^2)$, with $0\le \psi \le \pi$, displaying the full spherical spatial slices of a closed FLRW universe.

For separately conserved cosmological fluids, the prerequisite is absence of energy or momentum exchange between components. Each matter action then gives its own local conservation equation ${\nabla }_{\mu }{T}_i^{\mu \nu }=0$. In the homogeneous perfect fluid in general relativity, its time component isThis is the cosmological perfect-fluid continuity equation. Equivalently, a physical volume $V\propto a^3$ following the fluid satisfies $d({\rho }_{i}V)=-P_{i}dV$.

The Einstein field equations instead have the single shared metric on its left and ${\sum }_i{T}_i^{\mu \nu }$ on its right. The Friedmann equation and Friedmann acceleration equation therefore involve total density and total pressure; the same geometry cannot be sourced separately by each component's density alone.

The PDF's separate-conservation statement needs this noninteraction qualification. In an interacting system,where $Q_i$ is the energy-transfer rate into component $i$. Total conservation remains true, but the source-free equation does not hold individually during processes such as annihilation or particle decay.