Non relativistic small perturbation theory

Perturbation theory in expanding Universe has a number of distinctive festures. Strictly speaking, this theory shloud be based within the framework of general relativity. However, if inhomogeneities are small one could neglect the effects of curvature and finite speed of interaction and use newtonian dynamics.

To describe the fluctuations of density in this approximation we need the continuity equation $$ \frac{\partial \rho} {\partial t} + \nabla \cdot \left(\rho \vec v\right) = 0 $$ and Euler equation $$ \frac{\partial \vec v} {\partial t} + \left( \vec v\nabla \right)\vec v + \frac{1} {\rho }\nabla P + \nabla \Phi = 0, $$ where Newtonian gravitational potential satisfies the Laplace equation $$ \Delta \Phi = 4\pi G\rho. $$}

Problem 1

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