δ (x, t) = \int \frac{d ^{3} k}{( 2 π ) ^{3}} δ_{k} (t) e^{i k \cdot x} .

The spatial Laplacian multiplies a mode by

- k^{2}

, while its comoving wavenumber

k = ∣ k ∣

is time-independent. Writing

H = \overset{a}{˙} / a

\ddot{δ}_{k} + 2 H \dot{δ}_{k} + (\frac{c _{s}^{2} k ^{2}}{a ^{2}} - 4 π G \overset{ρ}{ˉ}) δ_{k} = 0.

The proper wavenumber is

k^{(p)} = k / a

, since the proper wavelength is

2 πa / k

k_{J}^{(p)} = \frac{4 π G ρ ˉ}{c _{s}}, \ddot{δ}_{k} + 2 H \dot{δ}_{k} + c_{s}^{2} [(k^{(p)})^{2} - (k_{J}^{(p)})^{2}] δ_{k} = 0.

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