Top-hat collapse model

The movie below is intended as a pedagogical introduction to the collapse of a top-hat overdensity, i.e., a sphere of higher density than the background, in an Einstein-deSitter universe. All shells are initially expanding with the Hubble flow, but due to the excess gravity, the top-hat perturbation decouples and starts to collapse. It would rapidly collapse to a point mass, at which point the movie stops. Time is then reversed and we see the shells moving out again to half the turn-around radius, where they can be thought to have virialized. The overdensity at that point, 178, motivates common definitions of the “virial” radius.

Top-hat collapse
High quality (20s)mp4 (11.7 MB)tophat collapse model

Secondary infall model

The movies below show the secondary infall of collisionless dark matter shells onto a point perturbation, following the equations of Bertschinger 1985 (see also Fillmore & Goldreich 1984). The beauty of this solution is that it is self-similar, meaning that all shells follow the same trajectory in units of the turn-around radius and logarithmic turn-around time.

In the movies, the turn-around radius is indicated as the gray, outermost shell. All shells at larger radii expand with the Hubble flow, and are not shown here. Time proceeds in units of physical time. We note that the shells do not have equal mass, but are rather equally spaced in time for a better visualization. There are two versions of the movie, namely one with radii in physical units (i.e., a turn-around radius that expands with time) and one with radii re-scaled to the turn-around radius.

The visualizations highlight the pile-up of shells at the apocenter of their first orbit. This “splashback radius” has recently been proposed as a physically motivated halo boundary that separates infalling from orbiting material, even in realistic dark matter halos (Diemer & Kravtsov 2014, Adhikari et al. 2014, and More et al. 2015).

Secondary infall model (physical radii)
Low quality, 30s
High quality, 30s
mp4 (9 MB)
mp4 (37 MB)
shell model
Secondary infall model (radii scaled to the turn-around radius)
Low quality, 20s
High quality, 20s
mp4 (5 MB)
mp4 (22 MB)
shell model

Gaussian Random Fields & N-body simulations

This set of slides visually shows how the CMB power spectrum translates into a field of over- and underdensities, and eventually into a grid of particles as used in simulations of structure formation.

Slides on Gaussian Random Fields
key (6 MB)
pdf (7 MB)
gaussian random field

The (pseudo-)evolution of halo density profiles

The density profiles of most dark matter halos evolve in a similar way: after a turbulent fast accretion period, their growth slows down and the profile barely evolves (e.g., Bullock et al. 2001, Wechsler et al. 2002, Zhao et al. 2009, Ludlow et al. 2013). This type of evolution is shown in the first movie below (for a more or less randomly selected halo).

However, even after the physical evolution of the density profile has slowed down, the outer radius, R200c, evolves because the critical density decreases due to the expansion of the universe. We call this growth in radius and mass due to the redefinition of the halo boundary “pseudo-evolution” (Diemer et al. 2013, More et al. 2015). As concentration is defined as the ratio of outer and scale radius, it grows in the pseudo-evolution regime (as shown in the first movie).

The second and third movie depict an idealized version of pseudo-evolution, where we assume that the halo density profile is an NFW profile that is fixed in physical coordinates. The characteristic evolution of the concentration during this phase can be used to model the concentration-mass relation semi-analytically (Diemer & Joyce 2019).

Evolution of profile (from simulation)
High quality, 25smp4 (0.5 MB)shell model
Evolution of profile and concentration (from simulation)
High quality, 25smp4 (0.4 MB)shell model
High quality, 10smp4 (0.1 MB)
shell model
Pseudo-evolution (with concentration)
High quality, 15smp4 (0.1 MB)
shell model