= Solution
The result above gives $K=-3H+\kappa$. Since the shear is trace-free and already first order, it cannot contribute linearly to the quadratic <tensor contraction> $K_{ij}K^{ij}=\operatorname{tr}(K^2)$. Therefore
$$
K^2=9H^2-6H\kappa,\qquad K_{ij}K^{ij}=3H^2-2H\kappa,
$$
up to second-order <scalar cosmological perturbations>. Combining these with the supplied spatial <Ricci scalar> gives the <Hamiltonian constraint>
$$
6H^2+4\Delta\Phi-4H\kappa=16\pi G(\bar\rho+\delta\rho).
$$
Its background part is the flat <Friedmann equation> $6H^2=16\pi G\bar\rho$. Subtracting that equation proves the <linearized cosmological Hamiltonian constraint>:
$$
\boxed{\Delta\Phi-H\kappa=4\pi G\delta\rho.}
$$
Here $\rho$ is the <energy density> measured by the hypersurface normal, as required by the normal-normal projection of the <Einstein field equations>.
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