Distribution function approach to redshift space distortions

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 k�u/aH < 1, where k is the wavevector, H the Hubble parameter, u the typical gravitational velocity and � = cos �, with � 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 �2 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 �2 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 Fingersof-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 �4 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. DOI: https://doi.org/10.1088/1475-7516/2011/11/039 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-54409 Accepted Version Originally published at: Seljak, U; McDonald, P (2011). Distribution function approach to redshift space distortions. Journal of Cosmology and Astroparticle Physics, (11):39. DOI: https://doi.org/10.1088/1475-7516/2011/11/039 ar X iv :1 10 9. 18 88 v1 [ as tr oph .C O ] 9 S ep 2 01 1 Prepared for submission to JCAP Distribution function approach to redshift space distortions Uroš Seljak, Patrick McDonald, Institute of Theoretical Physics, University of Zurich, 8057 Zurich, Switzerland Lawrence Berkeley National Laboratory, University of California, Berkeley, California 94720, USA Department of Physics, Department of Astronomy, University of California, Berkeley, California 94720, USA Institute for the Early Universe, Ewha Womans University, Seoul 120-750, S. Korea Physics Dept., Brookhaven National Laboratory, Building 510A, Upton, NY 11973-5000, USA E-mail: [email protected], [email protected] 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 μ 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 μ 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 μ 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. 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 μ 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 μ 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 μ 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.