Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 58 / 2 / i

Table of contents
- Solution i

Solution

Write

H = a^{'} / a

, where primes denote conformal time. In the passive convention for a cosmological gauge transformation, retain terms linear in

h

or

ξ

and discard their products. The linear metric gauge-transformation law then gives

Δ h_{μν} = - η_{μα} \partial_{ν} ξ^{α} - η_{να} \partial_{μ} ξ^{α} - 2 H η_{μν} ξ^{0} .

For a single nonzero Fourier mode, suppress the common factor

e^{i k \cdot x}

. Its derivatives are

\partial_{0} ξ^{0} = T^{'}, \partial_{i} ξ^{0} = ik k_{i} T, \partial_{0} ξ^{i} = i k_{i} L^{'}, \partial_{j} ξ^{i} = - k k_{i} k_{j} L .

Using

η_{00} = - 1

and

η_{ij} = δ_{ij}

therefore yields

Δ h_{00} Δ h_{0 i} Δ h_{ij} = 2 (T^{'} + H T), = i k_{i} (k T - L^{'}), = 2 k L k_{i} k_{j} - 2 H T δ_{ij} .

The term proportional to

H T

comes from the time dependence of the background scale factor; omitting it would transform only the flat reference metric.

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