Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 45 / 2 / d / Solution

The condition is that the velocity be the pure irrotational growing-mode velocity, with the homogeneous component in part (c) absent. It is enough to impose ${\mathbf{u}}_{\mathrm{in}}=-\nabla {\psi }_{\mathrm{in}}$ and exclude additional harmonic or transverse motions through the boundary conditions. The linear equation for $\mathbf{u}+\nabla \psi$ then keeps that difference zero. A time-independent velocity also forces the same equality in the linear motion equation when its coefficient is nonzero.

At first order the gradient can be evaluated at the initial position: shifting its argument by the first-order displacement changes the velocity only at second order. Integrate $d\mathbf{x}/db=-{\nabla }_{{\mathbf{x}}_{\mathrm{in}}}{\psi }_{\mathrm{in}}$ from $b_{\mathrm{in}}$ to $b$, givingThe trajectory is straight in comoving position as a function of growth time. The physical trajectory $a(t)\mathbf{x}(t)$ need not be straight or have constant physical peculiar velocity. This linear displacement relation is the basis of the Zeldovich approximation.