The dark matter sheet

Figure 4.2: N-body density estimate versus a density estimate inferred by interpolation of the dark matter sheet in phase space. The N-body density estimate shows regular lumps which will grow into fragmented artificial haloes. However, the continuum density estimate shows no such artefacts.

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Figure 4.3: Top: Illustration of the dark matter sheet in phase space. Dark matter oc-cupies a submanifold (a “sheet”) in phase space. This submanifold can be reconstructed by interpolation from a finite number of tracers. Note: in this image we purposely use only linear interpolation to emphasize the difference between the true dark matter sheet and the one reconstructed from interpolation. When a finite number of tracers is used, the interpolation gets worse with time, because the complexity of the sheet grows with time.

Bottom: Illustration of the release. Originally the mass (in red) in all Lagrangian volume elements was traced by the interpolated sheet. However, in the course of the simulation some mass elements have been flagged for release and are now represented by N-body particles instead.

Figure 4.4: The projection of a pseudo-2d sheet into Eulerian space (see text for details) as reconstructed with varying number of tracer particles. Top with 64² particles, center with 128² particles and bottom with 512² particles. Left: showing a large region with various different structure types. Inside low-density regions (which are the majority of the volume) the interpolation seems to be well converged already at moderate resolution levels – thereby getting close to the continuum limit. Right: zoom onto a halo. The reconstructed density field in and around haloes still seems to be resolution dependent even when most other regions are already converged at those resolutions. (This figure is not part of the actual paper draft.)

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representation of the dark matter sheet can then be obtained by using a higher interpola-tion order, or by an adaptive higher number of particles as has been explored by Hahn &

Angulo (2016) and Sousbie & Colombi (2016).

Let us have a short look at how the quality of the inferred density field changes with the number of available tracers. For this we select from an N = 512³ simulation of 1 keV warm dark matter (without refinement) a pseudo-2D sheet by selecting a slice with a fixed Lagrangian z coordinate q_z = const. In Figure 4.4 we show how its projection to Eulerian space changes with the resolution. We change the resolution that is used for reconstruction to 512², 128² and 64² particles by selecting every, every 4th and every 8th particle along each axis as support points of the interpolation. Inspection of Figure 4.4 already gives qualitatively the insight that we will test more quantitatively in the other sections of this paper: The density field is reconstructed very well in low-density regions like single-stream regions, pancakes and filaments. At finite resolution the sheet density estimate gets already extremely close to the continuum limit. However, in and around halos the density field still seems resolution dependent at relatively high resolution, because the dark matter sheet is quite complex here.

As already explained in section 2.4.4, the schemes of Hahn & Angulo (2016) and Sousbie

& Colombi (2016) use adaptive refinement approaches to increase the Lagrangian resolution where the sheet has a higher complexity and keep the resolution low where convergence can be achieved with a low number of tracers. By a refinement criterion, these schemes try to make sure that the interpolated sheet is consistent with the continuum limit at all times (up to some error tolerance).

With these schemes it is, in principle, possible to make a simulation that exactly traces the dark matter sheet. However, this turns out to be impossible in practice, since the dark matter sheet grows in complexity very rapidly. Tracing it requires an extraordinary amount of computational resources. Sousbie & Colombi (2016) managed to carry out a mX = 250 eV WDM simulation in a 28 Mpc box until a = 0.31 and found the number of simplexes required to scale with the twelfth power of time. Assuming that this scaling remains valid until a = 1, running that simulation until the present time would require roughly 10⁶ times as much memory and probably also computational time. However, it is already an optimistic assumption that this scaling can be extrapolated so easily. So soon after their formation, their haloes had probably had no mergers yet and so had maintained a relatively simple phase space structure. It cannot be excluded that chaotic orbits with an exponentially growing complexity arise from merging haloes. Further the simulation described in Sousbie & Colombi (2016) has a relatively low force-resolution of 28 Mpc/1024 ≈ 27 kpc. Therefore the central structure of haloes is resolved poorly.

The complexity of the sheet is expected to grow most rapidly in the centers of haloes.

Vogelsberger & White (2011) find that the number of streams at a single point in the center of the halo of a Milky-Way-type dark matter halo in a cold dark matter universe might already get as high as 10¹⁶. If that is true, a dark matter sheet plus refinement based simulation scheme for such a halo would require far more than 10¹⁶ resolution elements.

Even in that most optimistic case it seems unlikely that a simulation like that in Sousbie

& Colombi (2016) can be run until the present day, a = 1. We will demonstrate in this

Figure 4.5: Stream-Densities on an infinitesimally thin plane in Lagrangian Space. Left:

Exact stream densities from the GDE distortion tensor. Center: Finite differences ap-proximation, representing the derivatives of the interpolated sheet. Right: Morphology classification as described in section 2.6.4. The stream densities agree extraordinarily well in regions where they vary slowly with the Lagrangian coordinates, but get into complete disagreement in regions where they vary rapidly - that is in haloes. The sheet is too complex here for accurate reconstruction.

chapter how to deal with this with affordable computational costs. We propose a simulation scheme with a “release” mechanism that uses a sheet-interpolation scheme where it is well converged, and switches to a particle based N-body approach in regions where the sheet becomes too complex. This allows us to perform the first warm dark matter simulations that do not fragment in low density regions while remaining accurate in the inner regions of haloes. We illustrate qualitatively in the bottom panel of Figure 4.3 how the release could look in the phase space of a one dimensional world. Note that in the case of a three dimensional simulation the complexity in the released region would be much higher - for example it could have∼10¹⁶ foldings in the same region (Vogelsberger & White, 2011).

The next sections will explain how we identify regions where the interpolation scheme breaks down.