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Publication# A Framework for Numerical Simulations of Structure Formation

Abstract

In the absence of a full analytical treatment of nonlinear structure formation in the universe, numerical simulations provide the critical link between the properties of the underlying model and the features of the observed structures. Currently N-body simulations are the main tool to study structure growth. We explore an alternative framework for numerical simulations of structure formation. The underlying idea is to replace the long-range gravitational force in the Vlasov-Poisson system by a purely local interaction. To this end we trade the classical phase space distribution for its quantum mechanical counterpart, the Wigner distribution function. Its dynamical equation is equivalent to the Schrödinger equation and reduces to the Vlasov equation in the formal semi-classical limit. The proposed framework allows in principle to simulate systems with arbitrary phase space distributions and could for instance be beneficial for simulations of warm dark matter, where the velocity dispersion is important. We discuss several methods to obtain a set of wavefunctions whose Wigner distribution is close to a given initial phase space distribution function. An auxiliary gauge field is introduced to mediate the gravitational interaction, thereby obtaining a local Schrödinger-Maxwell system. We also use the ideas of lattice gauge theories to obtain a fully gauge-invariant discretization of the equations of motion. Their iterative solution was implemented in a three-dimensional simulation code. We discuss its computational complexity and memory requirements. Several testbed simulations were performed with this method. We compared the gravitational collapse of a Gaussian wavefunction with an independent numerical solution of the spherically symmetric Schrödinger-Newton system. The results were found to be in good agreement. Finally, a simple example of the growth of cosmic perturbations is investigated within our framework. We conclude by outlining various possible directions to optimize and develop our method.

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Claude Becker, Alexey Boyarsky, Oleg Ruchayskiy, Matthieu Schaller

The diversity of structures in the Universe (from the smallest galaxies to the largest superclusters) has formed under the pull of gravity from the tiny primordial perturbations that we see imprinted in the cosmic microwave background. A quantitative description of this process would require description of motion of zillions of dark matter particles. This impossible task is usually circumvented by coarse graining the problem: one either considers a Newtonian dynamics of 'particles' with macroscopically large masses or approximates the dark matter distribution with a continuous density field. There is no closed system of equations for the evolution of the matter density field alone and instead it should still be discretized at each time step. In this work, we describe a method of solving the full six-dimensional Vlasov-Poisson equation via a system of auxiliary Schrodinger-like equations. The complexity of the problem gets shifted into the choice of the number and shape of the initial wavefunctions that should only be specified at the beginning of the computation (we stress that these wavefunctions have nothing to do with quantum nature of the actual dark matter particles). We discuss different prescriptions to generate the initial wavefunctions from the initial conditions and demonstrate the validity of the technique on two simple test cases. This new simulation algorithm can in principle be used on an arbitrary distribution function, enabling the simulation of warm and hot dark matter structure formation scenarios.

Alexey Boyarsky, Frédéric Courbin, David Richard Harvey, Amélie Tamone, Wessel Valkenburg

Flux ratio anomalies in strong gravitationally lensed quasars constitute a unique way to probe the abundance of non-luminous dark matter haloes, and hence the nature of dark matter. In this paper, we identify double-imaged quasars as a statistically efficient probe of dark matter, since they are 20 times more abundant than quadruply imaged quasars. Using N-body simulations that include realistic baryonic feedback, we measure the full distribution of flux ratios in doubly imaged quasars for cold (CDM) and warm dark matter (WDM) cosmologies. Through this method, we fold in two key systematics - quasar variability and line-of-sight structures. We find that WDM cosmologies predict a similar to 6 per cent difference in the cumulative distribution functions of flux ratios relative to CDM, with CDM predicting many more small ratios. Finally, we estimate that similar to 600 doubly imaged quasars will need to be observed in order to be able to unambiguously discern between CDM and the two WDM models studied here. Such sample sizes will be easily within reach of future large-scale surveys such as Euclid. In preparation for this survey data, we require discerning the scale of the uncertainties in modelling lens galaxies and their substructure in simulations, plus a strong understanding of the selection function of observed lensed quasars.

2019Hakim Atek, Mathilde Jauzac, Jean-Paul Richard Kneib, Cristina Popa, Johan Richard

We map the lensing-inferred substructure in the first three clusters observed by the Hubble Space Telescope Frontier Fields (HSTFF) Initiative: Abell 2744 (z = 0.308), MACSJ 0416, (z = 0.396) and MACSJ 1149 (z = 0.543). Statistically resolving dark matter subhaloes down to similar to 10(9.5) M-circle dot, we compare the derived subhalo mass functions (SHMFs) to theoretical predictions from analytical models and with numerical simulations in a Lambda cold dark matter (LCDM) cosmology. Mimicking our observational cluster member selection criteria in the HSTFF, we report excellent agreement in both amplitude and shape of the SHMF over four decades in subhalo mass (10(9-13) M-circle dot). Projection effects do not appear to introduce significant errors in the determination of SHMFs from simulations. We do not find evidence for a substructure crisis, analogous to the missing satellite problem in the Local Group, on cluster scales, but rather excellent agreement of the count-matched HSTFF SHMF down to M-subhalo/M-halo similar to 10(-5). However, we do find discrepancies in the radial distribution of subhaloes inferred from HSTFF cluster lenses compared to determinations from simulated clusters. This suggests that although the selected simulated clusters match the HSTFF sample in mass, they do not adequately capture the dynamical properties and complex merging morphologies of these observed cluster lenses. Therefore, HSTFF clusters are likely observed in a transient evolutionary stage that is presently insufficiently sampled in cosmological simulations. The abundance and mass function of dark matter substructure in cluster lenses continues to offer an important test of the LCDM paradigm, and at present we find no tension between model predictions and observations.