Reference

A simple two-dimensional cosmological N-body code.

BVH(estimator::PS_DTFE, depth)

The bounding volume hierarchy of the estimator's triangles, as used before the cell index; for debugging, for instance with findBox and findIntersections(p, tree, estimator.positions, estimator.corners).

The cosmological box, setting the volume of the box L^2 and the number of particles N^2.

Caustic skeleton

The cosmological background in which the N-body simulation runs with the current Hubble rate H0, the matter density OmegaM, dark energy density OmegaL and the curvature component OmegaK.

GridIndex

Uniform G×G cell index of the triangles over the box [0, L)², in compressed form: the triangles whose bounding box overlaps cell k are tris[offsets[k]:offsets[k + 1] - 1], with cell k = i + (j - 1) G for x in cell i and y in cell j.

Preallocated grids and FFT plans for the particle-mesh force on the force grid `box`.

PS_DTFE(state_initial, state_final, box; G = n)
PS_DTFE(state_initial, state_final, depth, box)

Phase-space Delaunay tessellation field estimator. The particles are connected by the regular triangulation of their Lagrangian lattice (lattice_triangles), every triangle is unwrapped across the periodic boundaries on its own (unwrap_triangle), and triangles crossing the edge of the box are stored together with their periodic copies.

Triangles are located with a uniform G×G cell index (GridIndex); by default G is the size n of the particle lattice. The second form, with a depth, is kept for compatibility: the depth is not used for the queries, and a bounding volume hierarchy for debugging can be built with BVH(estimator, depth).

Query points are taken modulo the box. A point lying exactly on an edge or vertex of the tessellation is counted in every triangle that contains it. Fields can be evaluated at arbitrary points (ps_density, numberOfStreams, ps_velocity, velocitySum) or, much faster, on regular grids (ps_density_grid, numberOfStreams_grid).

The state of the N-body simulation in phase-space.

Constrained Gaussian random field object

Gridcell with attributes

Cloud-in-Cell density estimator on an N×N grid with spacing res, written to delta.

Cloud-in-Cell density estimator.

Compute the caustics

In-place drift step, X += Drift(a, δa, P, cosmology).

A drift step.

Push the skeleton from Lagrangian to Eulerian space

Push the skeleton from Lagrangian to Eulerian space

Push the skeleton from Lagrangian to Eulerian space

Push the skeleton from Lagrangian to Eulerian space

Push the skeleton from Lagrangian to Eulerian space

Generate an unconstrained Gaussian random field.

Generate an unconstrained Gaussian random field.

Fourth-order finite difference gradient along dimension i (in grid units), multiplied by scale and written to out.

Finite diference gradient

Two dimensional interpolation at the rows of x.

Two dimensional interpolation.

In-place kick step, P -= Kick(a, δa, X, ws.box, ws.mass, cosmology).

A kick step.

Lagrangian grid

The Leap Frog integrator. The returned positions are at state.time = af, the momenta half a step later at af + δa/2.

The Leap Frog integrator, returning the state after every step (at ai + δa, ai + 2δa, ..., af).

A leap frog step moving the positions from a to a + δa and the momenta from a + δa/2 to a + 3δa/2.

Compute the cusp (A3), swallowtail (A4) and butterfly (A5) caustics

Compute the cusp (A3), swallowtail (A4) and butterfly (A5) caustics

Compute the fold (A2) caustics

Compute the fold (A2) caustics

Compute the umbilic (D4) caustics

Compute the umbilic (D4) caustics

Interpolate a vertex along a line

Evaluate the Zel'dovich approximation, x = q - D(a) ∇ϕ, with the positions at a_pos and the momenta p = a² da/dt dx/da at a_vel.

The derivative of the scale factor.

Triangles whose bounding box overlaps the cell of the point (x, y) in [0, L)².

Generate an unconstrained, constrained Gaussian random field pair.

Convolution.

Finite difference order dx^4

Finite difference order dx^6

Finite difference order dx^2

Evaluate the first order directional derivative v₁⋅∇λ₁

Evaluate the second order directional derivative v₁⋅∇(v₁⋅∇λ₁)

Evaluate first eigenvalue and eigenvector field

Evaluate first eigenvalue and eigenvector

Generalized moment of the power spectrum P(k), smoothed on the scale σ.

Gradient of a scalar field

The growing mode.

The derivative of the growing mode with respect to the scale factor.

The normalization of the growing mode.

Hessian of a scalar field

Grid indices and bilinear weights of the point (x1, x2) in grid units on a periodic N×N grid.

Cubic interpolation of a regular grid

Laplace operator.

lattice_triangles(N)

Triangulation of the periodic N×N Lagrangian lattice. Every cell is split along the same diagonal, into (i, j)–(i+1, j)–(i+1, j+1) and (i, j)–(i+1, j+1)–(i, j+1), so that every particle is a vertex of exactly six triangles. Particle (i, j) has index i + (j - 1) N, matching reshape(state.position, :, 2).

Offset (dx, dy) of (x, y) from the first corner of triangle t, and whether the point lies in the triangle.

Mean of an Array.

The mean field given the constraints c.

Measure the constraints.

Number of leap frog steps from ai to af.

Number of streams at the point p.

numberOfStreams_grid(xs, ys, estimator)

Number of streams on the regular grid xs × ys, as a matrix with rows y and columns x; equal to [numberOfStreams([x, y], estimator) for y in ys, x in xs], but computed by rasterising the triangles.

Mesh plot of the N-body simulation

Mesh plot of the N-body simulation

Plot Caustic Skeleton

Plot Caustic Skeleton

Density at the point p, summed over all streams.

ps_density_grid(xs, ys, estimator)

Density on the regular grid xs × ys, as a matrix with rows y and columns x; equal to [ps_density([x, y], estimator) for y in ys, x in xs], but computed by rasterising the triangles.

ps_velocity(p, estimator)

Momentum p = a² dx/dt of every stream at the point p, one row per stream; divide by the scale factor a for the peculiar velocity.

rasterise(estimator, xs, ys)

Density and number of streams on the regular grid xs × ys (rows y, columns x), by looping once over the triangles and over the grid points inside each of them. The grid may extend beyond the box: every triangle is placed at all its periodic images that overlap the grid, so the result equals the point queries at mod.(p, L).

Smooth an Array.

Density, linearly interpolated on triangle s at the point p.

Density of triangle t at the offset (dx, dy) from its first corner.

Momentum p = a² dx/dt, linearly interpolated on triangle s at the point p.

Triangles containing the point p (taken modulo the box).

unwrap_s(s, L)

Cosmological N-body simulations normally work with periodic boundary conditions. This function undoes the wrapping of particles around the box in the displacement field, assuming the particles did not travel more than half the size of the box.

unwrap_triangle(x, sim, L)

Eulerian corners of the triangle with particle indices sim (rows of x). The second and third corners are placed at the periodic image nearest to the first corner, and the first corner is shifted into [0, L)². This only assumes that the edges of the triangle are shorter than L/2, so particles may have wrapped around the box any number of times.

The variance field of the constraints.

Stream-density weighted momentum p = a² dx/dt at the point p.

Periodic index in 1:N, with a fast path for indices that are already in range or one period out.