### Abstract

We explore linear redshift distortions in wide-angle surveys from the point of view of symmetries. We show that the redshift-space two-point correlation function can be expanded into tripolar spherical harmonics of zero total angular momentum S_{l1l2l3} (x̂_{1},x̂ _{2},x̂). The coefficients of the expansion B_{l1l2l3} are analogous to the C_{l} of the angular power spectrum and express the anisotropy of the redshift-space correlation function. Moreover, only a handful of B_{l1l2l3} are nonzero: the resulting formulae reveal a hidden simplicity comparable to the distant observer limit. The B_{l1l2l3} depend on spherical Bessel moments of the power spectrum and f = Ω ^{0.6}/b. In the plane-parallel limit, the results of Kaiser and Hamilton are recovered. The general formalism is used to derive useful new expressions. We present a particularly simple trigonometric polynomial expansion, which is arguably the most compact expression of wide-angle redshift distortions. These formulae are suitable for inversion because of the orthogonality of the basis functions. An alternative Legendre polynomial expansion was obtained as well. This can be shown to be equivalent to the results of Szalay and coworkers. The simplicity of the underlying theory will admit similar calculations for higher order statistics as well.

Original language | English |
---|---|

Pages (from-to) | 51-55 |

Number of pages | 5 |

Journal | Astrophysical Journal |

Volume | 614 |

Issue number | 1 I |

DOIs | |

Publication status | Published - Oct 10 2004 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Space and Planetary Science
- Nuclear and High Energy Physics

### Cite this

*Astrophysical Journal*,

*614*(1 I), 51-55. https://doi.org/10.1086/423168

**Wide-angle redshift distortions revisited.** / Szapudi, I.

Research output: Contribution to journal › Article

*Astrophysical Journal*, vol. 614, no. 1 I, pp. 51-55. https://doi.org/10.1086/423168

}

TY - JOUR

T1 - Wide-angle redshift distortions revisited

AU - Szapudi, I.

PY - 2004/10/10

Y1 - 2004/10/10

N2 - We explore linear redshift distortions in wide-angle surveys from the point of view of symmetries. We show that the redshift-space two-point correlation function can be expanded into tripolar spherical harmonics of zero total angular momentum Sl1l2l3 (x̂1,x̂ 2,x̂). The coefficients of the expansion Bl1l2l3 are analogous to the Cl of the angular power spectrum and express the anisotropy of the redshift-space correlation function. Moreover, only a handful of Bl1l2l3 are nonzero: the resulting formulae reveal a hidden simplicity comparable to the distant observer limit. The Bl1l2l3 depend on spherical Bessel moments of the power spectrum and f = Ω 0.6/b. In the plane-parallel limit, the results of Kaiser and Hamilton are recovered. The general formalism is used to derive useful new expressions. We present a particularly simple trigonometric polynomial expansion, which is arguably the most compact expression of wide-angle redshift distortions. These formulae are suitable for inversion because of the orthogonality of the basis functions. An alternative Legendre polynomial expansion was obtained as well. This can be shown to be equivalent to the results of Szalay and coworkers. The simplicity of the underlying theory will admit similar calculations for higher order statistics as well.

AB - We explore linear redshift distortions in wide-angle surveys from the point of view of symmetries. We show that the redshift-space two-point correlation function can be expanded into tripolar spherical harmonics of zero total angular momentum Sl1l2l3 (x̂1,x̂ 2,x̂). The coefficients of the expansion Bl1l2l3 are analogous to the Cl of the angular power spectrum and express the anisotropy of the redshift-space correlation function. Moreover, only a handful of Bl1l2l3 are nonzero: the resulting formulae reveal a hidden simplicity comparable to the distant observer limit. The Bl1l2l3 depend on spherical Bessel moments of the power spectrum and f = Ω 0.6/b. In the plane-parallel limit, the results of Kaiser and Hamilton are recovered. The general formalism is used to derive useful new expressions. We present a particularly simple trigonometric polynomial expansion, which is arguably the most compact expression of wide-angle redshift distortions. These formulae are suitable for inversion because of the orthogonality of the basis functions. An alternative Legendre polynomial expansion was obtained as well. This can be shown to be equivalent to the results of Szalay and coworkers. The simplicity of the underlying theory will admit similar calculations for higher order statistics as well.

KW - Cosmic microwave background

KW - Cosmology: theory

KW - Methods: statistical

UR - http://www.scopus.com/inward/record.url?scp=10844251017&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=10844251017&partnerID=8YFLogxK

U2 - 10.1086/423168

DO - 10.1086/423168

M3 - Article

AN - SCOPUS:10844251017

VL - 614

SP - 51

EP - 55

JO - Astrophysical Journal

JF - Astrophysical Journal

SN - 0004-637X

IS - 1 I

ER -