Past exam of the cosmology course of the University of Oxford / 2025 / 2 / b / Solution

Use the source's $+---$ metric signature and $x^0=\eta$, with conformal time derivatives denoted by primes. For this vector cosmological perturbation, the metric tensor and inverse metric through first order areSpatial indices on perturbations are raised with the background Kronecker delta. The Christoffel symbol formula isBecause $a=a(\eta )$ only, it givesFor the other requested Christoffel symbols, only the time derivative of the spatial metric tensor contributes:ThusFinally contract the first upper and first lower spatial indices, summing over $i$:Since $J_{ii}=2{\partial }_i{F}_i=0$ by the transverse vector field condition on $\mathbf{F}$,The useful Christoffel symbol traces are consequently ${\Gamma }_{\mu 0}^{\mu }=4\mathcal{H}$ and ${\Gamma }_{\mu j}^{\mu }=0$ to first order: the first uses $J_{ii}^{\prime }=0$ and the second uses the contraction just proved.