Cosmological perturbation theory using the Schrödinger equation

I. Szapudi, Nick Kaiser

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

We introduce the theory of nonlinear cosmological perturbations using the correspondence limit of the Schrödinger equation. The resulting formalism is equivalent to using the collisionless Boltzmann (or Vlasov) equations, which remain valid during the whole evolution, even after shell crossing. Other formulations of perturbation theory explicitly break down at shell crossing, e.g., Eulerean perturbation theory, which describes gravitational collapse in the fluid limit. This Letter lays the groundwork by introducing the new formalism, calculating the perturbation theory kernels that form the basis of all subsequent calculations. We also establish the connection with conventional perturbation theories, by showing that third-order tree-level results, such as bispectrum, skewness, cumulant correlators, and three-point function, are exactly reproduced in the appropriate expansion of our results. We explicitly show that cumulants up to W = 5 predicted by Eulerian perturbation theory for the dark matter field δ are exactly recovered in the corresponding limit. A logarithmic mapping of the field naturally arises in the Schrödinger context, which means that tree-level perturbation heory translates into (possibly incomplete) loop corrections for the conventional perturbation theory. We show that the first loop correction for the variance is σ2 = σL 2 + (-1.14 - n)σL 4 for a field with spectral index n. This yields 1.86 and 0.86 for n = -3 and -2, respectively, to be compared with the exact loop order corrections 1.82 and 0.88. Thus, our tree-level theory recovers the dominant part of first-order loop corrections of the conventional theory, while including (partial) loop corrections to infinite order in terms of δ.

Original languageEnglish
JournalAstrophysical Journal
Volume583
Issue number1 II
DOIs
Publication statusPublished - Jan 20 2003

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perturbation theory
perturbation
formalism
vlasov equations
skewness
gravitational collapse
correlators
shell
dark matter
breakdown
formulations
expansion
fluids
fluid

Keywords

  • Bcosmology: theory
  • Cosmic microwave background
  • Methods: statistical

ASJC Scopus subject areas

  • Space and Planetary Science
  • Nuclear and High Energy Physics

Cite this

Cosmological perturbation theory using the Schrödinger equation. / Szapudi, I.; Kaiser, Nick.

In: Astrophysical Journal, Vol. 583, No. 1 II, 20.01.2003.

Research output: Contribution to journalArticle

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