### 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 σ^{2} = σ_{L}
^{2} + (-1.14 - n)σ_{L}
^{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 language | English |
---|---|

Journal | Astrophysical Journal |

Volume | 583 |

Issue number | 1 II |

DOIs | |

Publication status | Published - Jan 20 2003 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Space and Planetary Science
- Nuclear and High Energy Physics

### Cite this

*Astrophysical Journal*,

*583*(1 II). https://doi.org/10.1086/368013

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

Research output: Contribution to journal › Article

*Astrophysical Journal*, vol. 583, no. 1 II. https://doi.org/10.1086/368013

}

TY - JOUR

T1 - Cosmological perturbation theory using the Schrödinger equation

AU - Szapudi, I.

AU - Kaiser, Nick

PY - 2003/1/20

Y1 - 2003/1/20

N2 - 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.

AB - 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.

KW - Bcosmology: theory

KW - Cosmic microwave background

KW - Methods: statistical

UR - http://www.scopus.com/inward/record.url?scp=0142023393&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0142023393&partnerID=8YFLogxK

U2 - 10.1086/368013

DO - 10.1086/368013

M3 - Article

AN - SCOPUS:0142023393

VL - 583

JO - Astrophysical Journal

JF - Astrophysical Journal

SN - 0004-637X

IS - 1 II

ER -