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In synchronous gauge in cosmology, $h_{0\mu }=0$, so the explicit metric terms vanish. Symmetry of the covariant energy-momentum tensor and the background metric implyHence the two expansion terms in part (c) combine to $3H\delta T^0{}_j$. Substituting the scalar velocity potential and scalar anisotropic stress definitions gives

Solved by gpt-5.6-sol high.

Use the Fourier convention $f(\mathbf{x})={\int }_{\mathbf{k}}f(\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{x}}$ with ${\int }_{\mathbf{k}}=\int d^{3}k/(2\pi {)}^3$. Neglect anisotropic stress and assume the pressureless velocity is irrotational, soFourier transforming $\nabla \cdot (\delta \mathbf{v})$ in the nonlinear continuity equation giveswhere the alpha mode-coupling kernel and its symmetrization areTaking the divergence of the Euler equation giveswith the symmetric beta mode-coupling kernel

In the Einstein-de Sitter universe, $a^{\prime }=\mathcal{H}a$ and ${\mathcal{H}}^{\prime }=-{\mathcal{H}}^2/2$. At linear order the ansatz gives ${\tilde{\theta }}^{(1)}={\tilde{\delta }}^{(1)}$. At second order, the continuity and Euler equations becomewhere each integral includes the momentum-conserving measure above. Eliminating ${\tilde{\theta }}^{(2)}$ yieldswith the standard perturbation theory density kernel

Solved by gpt-5.6-sol high.