The Complexity of the Dark Matter Sheet

The Nature of Dark Matter

There seems to be overwhleming evidence that most of the Universe is made of invisible dark matter, but unfortunately we do not know much about the concrete nature of it. It seems likely that dark matter should have certain properties like a warmth (referring to the thermal velocity dispersion), a self-interaction strength and a quantum wavelength. All these properties could leave some observable traces in our universe. For example, if dark matter is warm, the density perturbations on small scales would be surpressed through the free-streaming of dark matter. In this case fewer small haloes and galaxies could have formed in our Universe than if dark matter was colder. In principle, we could detect such differences observationally. This requires precise and reliable theoretical predictions.

Sheet + Release Simulations

We can predict the formation of structures by running large cosmological simulations, for example the Milennium Simulation or the BACCO simulations. Such simulations have typically been run with the N-body simulation method. However, in universes with a surpression of small scale structure (like warm dark matter) the N-body simulation method produces large numbers of small numerical artefacts.

As an alternative, "sheet" simulations have been introduced by Hahn et al. (2013), Hahn & Angulo (2016) and Sousbie & Colombi (2016). In these simulations the dark matter distribution is reconstructed to almost perfect accuracy by interpolation of the dark matter sheet in phase space, thereby showing incredibly sharp density estimates and avoiding the artificial fragments of N-body simulations. However, this comes at a price: the dark matter sheet becomes incredibly complex inside of the dark matter haloes in the course of a simulation. Therefore, the reconstruction becomes practically impossible in haloes, leading to simulations that either have to be canceled early or that give biased density estimates inside haloes. We could summarize the situation in the following table:

Simulation Method Outside Haloes Inside Haloes
N-body Artificial Fragmentation OK
Sheet Perfect Too Complex

It seems clear that one could get a fully functional simulation scheme by combining the best from both worlds. Therefore, we have introduced in Stücker et al (2020) the "Sheet + Release" simulation method. In these simulations, initially all mass is traced through the dark matter sheet reconstruction. However, in regions where the sheet becomes too complex (mostly haloes) the method automatically "releases" mass elements into a set of mass carrying particles that behave like traditional N-body particles. Therefore, "sheet + release" simulations can reliably follow the evolution of the density field inside and outside of haloes. Here an example of how it looks in practice for a hot dark matter simulation:

Having an accurate density field outside of haloes is not only important for a reliable evolution of the simulation overall, but it might also be relevant for accurate predictions of gravitational lensing as investigated by Richardson et al. (2021).

The halo mass function of non-cold dark matter Universes

In Stücker et al (2021) we present the first set of reliable simulations of "non-cold" dark matter models -- run with the sheet + release scheme on the MareNostrum super computer. With "non-cold" dark matter we refer to any dark matter model that has a small scale surpression in the power spectrum. This includes the canonical "thermal relic" warm dark matter, but also other dark matter candidates like sterile neutrinos, ETHOS models, fuzzy dark matter and many more... Here you can see two of these simulations tiled together, one "warmer" simulation with a larger surpression scale ("1keV sharp" in the paper) and one "colder" simulation with a smaller surpression scale ("1.75keV sharp" in the paper):

You can see, how the colder case exhibits many more small scale objects than the warmer one. This can be quantified through the halo mass function. We have measured the halo mass function in dependence of two parameters which describe the supression of the initial power spectrum (one describing the slope and one the length scale of the cutoff) and we have come up with a parametric description of the halo and subhalo mass function that can be evaluated for almost any non-cold dark matter model! You can find a short python code for evaluating them here on github:
We hope that these simple, but robust predictions will facilitate constraining dark matter for future observational studies. Futher, we hope to motivate other researchers to use to use observations not only to constrain the fiducial thermal relic warm dark matter scenarios, but also other non-cold dark matter candidates.

Jens Stücker -- in collaboration with Raul Angulo, Oliver Hahn and Simon White


Links & References

  • Simulating the complexity of the dark matter sheet I: numerical algorithms
    Stücker J., Hahn O., Angulo R. E., White S. D. M., 2020, MNRAS, 495,4943
  • Simulating the complexity of the dark matter sheet II: the halo mass function of non-cold dark matter models
    Stücker J., Angulo R. E., Hahn O., White S. D. M., 2021, arxiv (*)
  • Non-Halo Structures and their Effects on Gravitational Lensing
    Richardson T. R. G., Stücker J., Angulo R. E., Hahn O., 2021, arXiv:2101.07806
(*) If you want to use any of the material on this page, please cite this paper.