PIRSA:14100076 Scientific Series Cosmology

DOI10.48660/14100076

Series: Cosmology and Gravitation

Uhlemann, C. (2014). Schroedinger method as field theoretical model to describe structure formation. Perimeter Institute. https://pirsa.org/14100076

Uhlemann, Cora. Schroedinger method as field theoretical model to describe structure formation. Perimeter Institute, Oct. 21, 2014, https://pirsa.org/14100076 

@misc{ pirsa_PIRSA:14100076,
  doi = {10.48660/14100076},
  url = {https://pirsa.org/14100076},
  author = {Uhlemann, Cora},
  keywords = {Cosmology},
  language = {en},
  title = {Schroedinger method as field theoretical model to describe structure formation},
  publisher = {Perimeter Institute},
  year = {2014},
  month = {oct},
  note = {PIRSA:14100076 see, \url{https://pirsa.org}}
}

Abstract

We investigate large-scale structure formation of collisionless dark matter in the phase space description based on the Vlasov equation whose nonlinearity is induced solely by gravitational interaction according to the Poisson equation. Determining the time-evolution of density and velocity demands solving the full Vlasov hierarchy for the cumulants of the distribution function. In the presence of long-range interaction no consistent truncation is known apart from the dust model which is incapable of describing the formation of bound structures due to the inability to generate higher cumulants like velocity dispersion. Our goal is to find a simple ansatz for the phase space distribution function that approximates the full Vlasov distribution function and can serve as theoretical N-body double to replace the dust model. We present the Schroedinger method which is based on the coarse-grained Wigner probability distribution obtained from a wave function fulfilling the Schroedinger-Poisson equation as sought-after model. We show that its evolution equation approximates the Vlasov equation in a controlled way, cures the shell-crossing singularities of the dust model and is able to describe multi-streaming which is crucial for halo formation. This feature has already been employed in cosmological simulations of large-scale structure formation by Widrow & Kaiser (1993). We explain how the coarse-grained Wigner ansatz allows to calculate higher cumulants like velocity dispersion analytically from density and velocity in a self-consistent manner. On this basis we show that instead of solving the Vlasov-Poisson system one can use the Schrödinger method and solve the Schrödinger-Poission equation to directly determine density and velocity and all higher cumulants. As a first application we study the coarse-grained dust model, which is a limiting case of the Schrödinger method, within Eulerian and Lagrangian perturbation theory.

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