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Wave-mechanics and the Zeldovich approximation

The equations of motion of a self-gravitating fluid of collisionless cold dark matter (CDM) particles can be written in the form of a Schrödinger equation coupled to a Poisson equation describing Newtonian gravity. In a spatially flat CDM-dominated universe the Schrödinger equation can be reduced to the free-particle Schrödinger equation and, in an appropriate limit, we show that the free-particle Schrödinger equation can be thought of as a wave-mechanical analogue of the Zeldovich approximation. We use the free-particle Schrödinger equation to formulate an alternative approximation method (the free-particle approximation) that is capable of following the gravitational collapse of density fluctuations into the quasi-linear regime. The free-particle approximation is tested by appealing to a cosmological N-body simulation and results are compared with the results obtained from two established approaches: linear perturbation theory and the Zeldovich-Bernoulli approximation. We find that the free-particle approximation comprehensively out-performs both of these approximation schemes in all tests carried out and thus provides another useful analytical tool for studying structure formation on cosmological scales.