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

Let $\delta =(\rho -\bar{\rho })/\bar{\rho }$ be the density contrast, let $\mathbf{x}$ be a comoving coordinate, and use $a_0=1$. Define $C=3H_0^2{\Omega }_0/2$, so $4\pi G\bar{\rho }=C/a^3$ for the pressureless background. The perturbation Poisson equation isFor the comoving velocity field $\mathbf{w}=\dot{\mathbf{x}}$, local mass conservation givesOne obtains this by subtracting $\dot{\bar{\rho }}+3H\bar{\rho }=0$ from ${\partial }_t\rho +3H\rho +{\nabla }_x\cdot (\rho \mathbf{w})=0$. In the linear regime it reduces to ${\partial }_t\delta =-{\nabla }_x\cdot \mathbf{w}$, the linearized cosmological continuity equation. The spatial derivatives are with respect to $\mathbf{x}$, and the time derivative in the conservative continuity equation is at fixed $\mathbf{x}$. In contrast the acceleration of an individual fluid element is a material derivative. The physical peculiar velocity is $a\mathbf{w}$.

Use $\dot{\mathbf{x}}=\dot{b}\mathbf{u}$ and $\phi =Cb\psi /a$, where the growing linear growth factor obeys $\ddot{b}+2H\dot{b}=Cb/a^3$. The Poisson relation in part (a) immediately becomesAlong a fluid trajectory,Substitute this into the pressureless equation of motion to obtainThe growth equation cancels the second coefficient. By the definition of the logarithmic growth rate, $\dot{b}=Hbf$, while $C/a^3=(3/2)H^2\Omega (t)$. ThereforeThis is the printed $d\mathbf{u}/db$ when interpreted along the particle. Dividing the Eulerian continuity equation by $\dot{b}$ instead givesThe density derivative in this conservative equation must be Eulerian. If both derivatives were read as material derivatives, the density equation would double-count advection; its material form is $D_b\delta +(1+\delta ){\nabla }_x\cdot \mathbf{u}=0$. These distinctions make the growth-time form of cosmological dust equations valid beyond the first-order terms, up to the single-stream pressureless-fluid limit. The reparameterization assumes $\dot{b}\ne 0$.

For the growing density mode, write $\delta (\mathbf{x},t)=b(t){\delta }_{\ast }(\mathbf{x})$. The new Poisson equation gives ${\nabla }_x^2\psi ={\delta }_{\ast }$ with no time dependence. With periodic zero-mean or fixed decaying boundary conditions, this determines a time-independent $\psi$ up to a spatially constant, dynamically irrelevant function. ThusIn the linear continuity equation, ${\partial }_b\delta ={\delta }_{\ast }$, so ${\nabla }_x\cdot \mathbf{u}=-{\delta }_{\ast }=-{\nabla }_x^2\psi$. For the pure longitudinal growing mode, this gives $\mathbf{u}=-{\nabla }_x\psi$ and hence a time-independent scaled velocity. Equivalently, neglect the quadratic advection term in the motion equation and put $\mathbf{h}=\mathbf{u}+{\nabla }_x\psi$. It obeysThe second expression follows directly from $\ddot{b}+2H\dot{b}=Cb/a^3$: its logarithmic derivative with respect to $b$ is the negative coefficient in the first equation. Excluding the homogeneous decaying-velocity component, $\mathbf{C}=0$, yields the pure growing-mode scaled velocityThe same holds along a trajectory to first order, since $\mathbf{u}\cdot \nabla \mathbf{u}$ is second order.

The pure growing-velocity assumption matters. A growing density alone does not exclude a divergence-free decaying velocity. For example, in an Einstein-de Sitter universe with $b=a$, one may add ${\mathbf{u}}_h=a^{-3/2}\sin (kx){\mathbf{e}}_y$. Its divergence is zero, it solves the homogeneous linear motion equation, and it changes neither $\delta =b{\delta }_{\ast }$ nor $\psi$, yet it is not constant. Thus the printed constant-velocity statement is correct for the intended pure scalar growing mode; without that condition the displayed field is a counterexample. This extra mode is removed by the usual irrotational, no-decaying-mode initial conditions.

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.

For the Zeldovich approximation, form the Jacobian matrix of the Lagrangian map:Let ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3$ be the eigenvalues of this potential Hessian matrix, with the convention that positive $\lambda$ means compression. Before shell crossing, conservation of the mass in a Lagrangian volume givesIf ${\lambda }_1>0$ and is distinct from the other eigenvalues, the first singularity occurs at $b=1/{\lambda }_1$. One principal length collapses while the other two remain nonzero. The resulting cosmological caustic is locally a sheet, conventionally a pancake, rather than a simultaneous spherical point collapse. Degenerate equal eigenvalues can instead give line- or pointlike special cases.

The predicted density divergence signals failure of the single-stream description. If the straight Lagrangian displacements are simply continued, particles cross the sheet and keep moving along their original displacement directions. They are not slowed and trapped by the evolved self-gravitating multistream structure. The streams therefore separate again, producing excessively broad pancakes at late times. In an N-body simulation, the changing gravitational field pulls crossing streams back, produces repeated crossings and binds matter into more compact structures. This is the limitation described by Zeldovich pancake thickness after shell crossing. The initial-displacement extrapolation is useful through the onset of collapse, but does not supply the self-consistent post-crossing gravitational dynamics or virialization. After crossing, the physical density would require summing inverse absolute Jacobians over all streams, not using one negative determinant as a density.