Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 310 / 3 / d

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- Solution d

Solution

Define the radial transfer integral

F_{l} (k) = \int_{0}^{χ_{e}} d χ \frac{χ _{e} - χ}{χ χ _{e}} \frac{D _{i} ( τ _{0} - χ )}{a ( τ _{0} - χ )} j_{l} (k χ),

where

δ_{m} (k, τ) = D_{i} (τ) δ_{m} (k, τ_{i})

and

D_{i} (τ_{i}) = 1

. With

⟨ δ_{m} (k, τ_{i}) δ_{m}^{*} (k^{'}, τ_{i})⟩ = (2 π)^{3} δ^{(3)} (k - k^{'}) P_{i} (k),

the angular and radial integrations give

C_{l}^{ψ} = \frac{18}{π} Ω_{m, 0}^{2} H_{0}^{4} \int_{0}^{\infty} \frac{d k}{k ^{2}} P_{i} (k) F_{l}^{2} (k) .

This expression makes the lensing signal a weighted projection of the evolving matter power spectrum.

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