ϕ (k, τ) = - \frac{4 π G a ^{2} ρ ˉ _{m}}{k ^{2}} δ_{m} = - \frac{3 Ω _{m, 0} H _{0}^{2}}{2 a k ^{2}} δ_{m} .

δ_{m}

, and use the Rayleigh plane-wave expansion. Projection onto

Y_{l m}

a_{l m}^{ψ} = 12 π Ω_{m, 0} H_{0}^{2} i^{- l} \int_{0}^{χ_{e}} d χ \int \frac{d ^{3} k}{( 2 π ) ^{3}} \frac{1}{k ^{2}} \frac{χ _{e} - χ}{χ χ _{e}} \frac{δ _{m} ( k , τ )}{a ( τ )} j_{l} (k χ) Y_{l m}^{*} (k) .

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