Solving the Bernoulli-Zel'dovich equation

Astronomy and Astrophysics – Astrophysics

Scientific paper

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Bernoulli Theorem, Cosmology, Gravitational Fields, Universe, Astronomical Models, Lagrange Coordinates, Reconstruction

Scientific paper

The Zel'dovich approximation is very efficient in reconstructing the initial conditions (in the form of a displacement field, a deformation tensor, or the corresponding potential) from observational data, usually proper velocities. Recently it has been shown that this problem reduces to a differential equation for the velocity potential, the Bernouilli-Zel'dovich equation, and numerical methods were proposed to solve it. Here I derive an analytical solution to this equation which allows reconstruction without any numerical integration (except a spatial integration of the present velocity field which is the first step of any procedure). This allows the reconstruction of the initial potential at Lagrangian positions. I also propose an approximation which makes this method even more direct, allowing the reconstruction of the initial potential (Lagrangian or Eulerian) at Eulerian positions.

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