= Linearized cosmological Hamiltonian constraint
{title2=$\Delta\Phi-H\kappa=4\pi G\delta\rho$}
For a spatially flat <Friedmann-Lemaitre-Robertson-Walker metric>, negative <extrinsic curvature>, and $K=-3H+\kappa$, the <Hamiltonian constraint> gives the displayed scalar perturbation equation. Indeed $K^2=9H^2-6H\kappa$ and $K^i{}_jK^j{}_i=3H^2-2H\kappa$ to first order. The spatial <Ricci scalar> is $4\Delta\Phi$; subtract the background equation $6H^2=16\pi G\bar\rho$ and divide by four.
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