Journal Article

Extending the domain of validity of the Lagrangian approximation

Sharvari Nadkarni-Ghosh and David F. Chernoff

in Monthly Notices of the Royal Astronomical Society

Published on behalf of The Royal Astronomical Society

Volume 410, issue 3, pages 1454-1488
Published in print January 2011 | ISSN: 0035-8711
Published online January 2011 | e-ISSN: 1365-2966 | DOI: http://dx.doi.org/10.1111/j.1365-2966.2010.17529.x
Extending the domain of validity of the Lagrangian approximation

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We investigate convergence of Lagrangian perturbation theory (LPT) by analysing the model problem of a spherical homogeneous top hat in an Einstein–de Sitter background cosmology. We derive the formal structure of the LPT series expansion, working to arbitrary order in the initial perturbation amplitude. The factors that regulate LPT convergence are identified by studying the exact, analytic solution expanded according to this formal structure. The key methodology is to complexify the exact solution, demonstrate that it is analytic and apply well-known convergence criteria for power series expansions of analytic functions. The ‘radius of convergence’ and the ‘time of validity’ for the LPT expansion are of great practical interest. The former describes the range of initial perturbation amplitudes which converge over some fixed, future time interval. The latter describes the extent in time for convergence of a given initial amplitude. We determine the radius of convergence and time of validity for a full sampling of initial density and velocity perturbations.

This analysis fully explains the previously reported observation that LPT fails to predict the evolution of an underdense, open region beyond a certain time. It also implies the existence of other examples, including overdense, closed regions, for which LPT predictions should also fail. We show that this is indeed the case by numerically computing the LPT expansion in these problematic cases.

The formal limitations to the validity of LPT expansion are considerably more complicated than simply the first occurrence of orbit crossings as is often assumed. Evolution to a future time generically requires re-expanding the solution in overlapping domains that ultimately link the initial and final times, each domain subject to its own time of validity criterion. We demonstrate that it is possible to handle all the problematic cases by taking multiple steps (LPT re-expansion).

A relatively small number (∼10) of re-expansion steps suffices to satisfy the time of validity constraints for calculating the evolution of a non-collapsed, recombination-era perturbation up to the current epoch. If it were possible to work to infinite Lagrangian order then the result would be exact. Instead, a finite expansion has finite errors. We characterize how the leading order numerical error for a solution generated by LPT re-expansion varies with the choice of Lagrangian order and of time-step size. Convergence occurs when the Lagrangian order increases and/or the time-step size decreases in a simple, well-defined manner. We develop a recipe for time-step control for LPT re-expansion based on these results.

Keywords: cosmology: theory; large-scale structure of Universe

Journal Article.  18979 words.  Illustrated.

Subjects: Astronomy and Astrophysics

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