Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 38B / b

Table of contents
- Solution b

Solution

Write

H = \overset{a}{˙} / a

. The nonzero Christoffel symbols for the flat FLRW metric are

Γ_{ij}^{t} = a \overset{a}{˙} δ_{ij}, Γ_{t j}^{i} = Γ_{j t}^{i} = H δ_{j}^{i} .

The mixed Einstein tensor is

G_{t}^{t} = - 3 H^{2}, G_{i}^{j} = - (2 \frac{a ¨}{a} + H^{2}) δ_{i}^{j} .

For a mixed tensor,

\nabla_{ν} G_{μ}^{ν} = \partial_{ν} G_{μ}^{ν} + Γ_{ν λ}^{ν} G_{μ}^{λ} - Γ_{νμ}^{λ} G_{λ}^{ν} .

For

μ = t

, direct substitution gives

\nabla_{ν} G_{t}^{ν} = \frac{d}{d t} (- 3 H^{2}) + 3 H (- 3 H^{2}) - 3 H [- (2 \frac{a ¨}{a} + H^{2})] = - 6 H (\dot{H} + H^{2} - \frac{a ¨}{a}) = 0.

For each spatial

μ

, homogeneity makes the partial derivatives vanish and the remaining diagonal connection terms cancel pairwise. This verifies the flat FLRW Einstein-tensor divergence directly.

G_{μ}^{ν} + Λ δ_{μ}^{ν} = 0.

The

tt

H^{2} = \frac{Λ}{3} .

Choosing the positive root gives the expanding de Sitter scale factor in flat slicing

a (t) = a_{0} exp (\frac{Λ}{3} t),

and the spatial equation is then automatically satisfied.

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