## Zel’dovich Pancake Day!

Today it’s Shrove Tuesday but unfortunately I forgot to buy shroves yesterday so will have to make do with pancakes instead, but not the usual kind. I’ve blogged before about the Zel’dovich Approximation (published in Zeldovich, Ya.B. 1970, A&A, 5, 84) but there’s no harm in describing this classic again. Here’s the first page of the original paper:

In a nutshell, this daringly simple approximation considers the evolution of particles in an expanding Universe from an early near-uniform state into the non-linear regime as a sort of ballistic, or kinematic, process. Imagine the matter particles are initial placed on a uniform grid, where they are labelled by Lagrangian coordinates $vec{q}$. Their (Eulerian) positions at some later time $t$ are taken to be

$vec{r}(vec(q),t) = a(t) vec{x}(vec{q},t) = a(t) left[ vec{q} + b(t) vec{s}(vec{q},t) right].$

Here the $vec{x}$ coordinates are comoving, i.e. scaled with the expansion of the Universe using the scale factor $a(t)$. The displacement $vec{s}(vec{q},t)$ between initial and final positions in comoving coordinates is taken to have the form

$vec{s}(vec{q},t)= vec{nabla} Phi_0 (vec{q})$

where $Phi_0$ is a kind of velocity potential (which is also in linear Newtonian theory proportional to the gravitational potential).If we’ve got the theory right then the gravitational potential field defined over the initial positions is a Gaussian random field. The function $b(t)$ is the growing mode of density perturbations in the linear theory of gravitational instability.

This all means that the particles just get a small initial kick from the uniform Lagrangian grid and their subsequent motion carries on in the same direction. The approximation predicts the formation of caustics in the final density field when particles from two or more different initial locations arrive at the same final location, a condition known as shell-crossing. The caustics are identified with the main elements we find in large-scale structure. Because the initial collapse is usually along one direction the dominant structures are known as pancakes (or, as Zel’dovich himself might have called them, blini…).

Here’s a picture of a simulation showing these structures from the classic paper of Davis, Efstathiou, Frenk & White (1985):

Despite its simplicity this approximation is known to perform extremely well at reproducing the morphology of the cosmic web, although it breaks down after shell-crossing has occurred. In reality, bound structures are formed whereas the Zel’dovich approximation simply predicts that particles sail straight through the caustic which consequently evaporates.