Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 64 / 5 / iii

Table of contents
- Solution iii

Solution

The mixed stress tensor has the convenient exact form

T^{0}_{i} = \partial^{0} ϕ \partial_{i} ϕ - δ^{0}_{i} [\frac{1}{2} (\partial ϕ)^{2} + V] = \partial^{0} ϕ \partial_{i} ϕ .

Since the background scalar has no spatial gradient, retaining first-order terms gives

\partial^{0} ϕ = - \overset{ˉ}{ϕ}^{'} / a^{2}

and

\partial_{i} ϕ = \partial_{i} δ ϕ

in the product. Therefore

δ T^{0}_{i} = - \frac{ϕ ˉ ^{'}}{a ^{2}} \partial_{i} δ ϕ .

It is a gradient momentum density, hence a scalar velocity perturbation. With a rolling background, uniform-field slices also have vanishing scalar momentum density and provide the field's comoving slicing.

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