We develop a phase space distribution function approach to redshift space distortions (RSD), in which the redshift space density can be written as a sum over velocity moments of the distribution function. These moments are density weighted and have well defined physical interpretation: their lowest orders are density, momentum density, and stress energy density. The series expansion is convergent if kμu/aH < 1, where k is the wavevector, H the Hubble parameter, u the typical gravitational velocity and μ = cos θ, with θ being the angle between the Fourier mode and the line of sight. We perform an expansion of these velocity moments into helicity modes, which are eigenmodes under rotation around the axis of Fourier mode direction, generalizing the scalar, vector, tensor decomposition of perturbations to an arbitrary order. We show that only equal helicity moments correlate and derive the angular dependence of the individual contributions to the redshift space power spectrum. We show that the dominant term of μ2 dependence on large scales is the cross-correlation between the density and scalar part of momentum density, which can be related to the time derivative of the matter power spectrum. Additional terms contributing to μ2 and dominating on small scales are the vector part of momentum density-momentum density correlations, the energy density-density correlations, and the scalar part of anisotropic stress density-density correlations. The second term is what is usually associated with the small scale Fingers-of-God damping and always suppresses power, but the first term comes with the opposite sign and always adds power. Similarly, we identify 7 terms contributing to μ4 dependence. Some of the advantages of the distribution function approach are that the series expansion converges on large scales and remains valid in multi-stream situations. We finish with a brief discussion of implications for RSD in galaxies relative to dark matter, highlighting the issue of scale dependent bias of velocity moments correlators.

Seljak, U; McDonald, P (2011). *Distribution function approach to redshift space distortions.* Journal of Cosmology and Astroparticle Physics, (11):39.

## Abstract

We develop a phase space distribution function approach to redshift space distortions (RSD), in which the redshift space density can be written as a sum over velocity moments of the distribution function. These moments are density weighted and have well defined physical interpretation: their lowest orders are density, momentum density, and stress energy density. The series expansion is convergent if kμu/aH < 1, where k is the wavevector, H the Hubble parameter, u the typical gravitational velocity and μ = cos θ, with θ being the angle between the Fourier mode and the line of sight. We perform an expansion of these velocity moments into helicity modes, which are eigenmodes under rotation around the axis of Fourier mode direction, generalizing the scalar, vector, tensor decomposition of perturbations to an arbitrary order. We show that only equal helicity moments correlate and derive the angular dependence of the individual contributions to the redshift space power spectrum. We show that the dominant term of μ2 dependence on large scales is the cross-correlation between the density and scalar part of momentum density, which can be related to the time derivative of the matter power spectrum. Additional terms contributing to μ2 and dominating on small scales are the vector part of momentum density-momentum density correlations, the energy density-density correlations, and the scalar part of anisotropic stress density-density correlations. The second term is what is usually associated with the small scale Fingers-of-God damping and always suppresses power, but the first term comes with the opposite sign and always adds power. Similarly, we identify 7 terms contributing to μ4 dependence. Some of the advantages of the distribution function approach are that the series expansion converges on large scales and remains valid in multi-stream situations. We finish with a brief discussion of implications for RSD in galaxies relative to dark matter, highlighting the issue of scale dependent bias of velocity moments correlators.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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Communities & Collections: | 07 Faculty of Science > Institute for Computational Science |

Dewey Decimal Classification: | 530 Physics |

Language: | English |

Date: | November 2011 |

Deposited On: | 19 Feb 2012 09:07 |

Last Modified: | 05 Apr 2016 15:20 |

Publisher: | Institute of Physics Publishing |

ISSN: | 1475-7516 |

Additional Information: | This is an author-created, un-copyedited version of an article accepted for publication in Journal of Cosmology and Astroparticle Physics. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher authenticated version is available online at 10.1088/1475-7516/2011/11/039 |

Publisher DOI: | 10.1088/1475-7516/2011/11/039 |

Related URLs: | http://arxiv.org/abs/1109.1888 |

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