Sharvari Nadkarni-Ghosh, David F. Chernoff
We present a new method to calculate formation of cosmological structure in the Newtonian limit. The method is based on Lagrangian perturbation theory (LPT) plus two key theoretical extensions. One advance involves fixing a previously ignored gauge-like degree of freedom present in the formal LPT. The traditional derivation of the perturbation expansion introduces this unwanted freedom which it is crucial to eliminate. In effect, we transform the usual results of a LPT calculation by a frame shift to give answers sought by a particular observer. A second extension is based on our previous work where we showed that, independent of orbit crossing, LPT expansions converge only over a limited time interval. We had introduced the idea of a multi-step method to extend the solution as far forward in time as possible. Here, we implement both the frame shift and the multi-step method to produce an algorithm capable of solving for the cosmological evolution of cold matter. Extensive `proof of principle' tests validate the method. The algorithm behaves satisfactorily in all these trials. The rate of convergence is exponential in the grid size, exponential in the Lagrangian order and polynomial in the step size. There are three main advantages of this new technique. First, it employs a smooth representation of all fields and the results are not limited by particle induced shot-noise errors. Second, the numerical error for any problem can be controlled by changing Lagrangian order and/or number of steps. In principle, arbitrarily small errors can be achieved prior to orbit crossing. Third, the initial data is completely generic, including cases where the initial velocity field has a rotational component. Together, these properties make the new technique well-suited to handle problems on quasi-linear scales where analytic methods and/or numerical simulations fail to provide accurate answers.
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http://arxiv.org/abs/1211.5777
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