Distribution function approach to redshift space distortions. Part IV: perturbation theory applied to dark matter

TL;DR

The paper develops a phase-space distribution function approach to redshift-space distortions (RSD) and evaluates a perturbative expansion of the redshift-space power spectrum into velocity-moment correlators. It demonstrates that standard and extended PT can model several terms (e.g., P_{00}, P_{01}, P_{11}) at one-loop, but small-scale velocity dispersion and FoG effects necessitate halo-model–inspired corrections for other terms (e.g., P_{02}, P_{12}, P_{22}, P_{03}, P_{13}); a FoG resummation framework and velocity-dispersion parameterization are proposed to capture these effects. Comparisons with N-body simulations reveal substantial progress in predicting μ^2 and μ^4 dependencies, with notable improvement when incorporating dispersion terms, though several terms remain challenging, particularly at higher k, requiring additional loops or phenomenological modeling. The work provides a physically transparent, term-by-term understanding of RSD and highlights the central role of velocity dispersion in shaping the redshift-space power spectrum, motivating extensions to galaxy biasing and practical application to real surveys.

Abstract

We develop a perturbative approach to redshift space distortions (RSD) using the phase space distribution function approach and apply it to the dark matter redshift space power spectrum and its moments. RSD can be written as a sum over density weighted velocity moments correlators, with the lowest order being density, momentum density and stress energy density. We use standard and extended perturbation theory (PT) to determine their auto and cross correlators, comparing them to N-body simulations. We show which of the terms can be modeled well with the standard PT and which need additional terms that include higher order corrections which cannot be modeled in PT. Most of these additional terms are related to the small scale velocity dispersion effects, the so called finger of god (FoG) effects, which affect some, but not all, of the terms in this expansion, and which can be approximately modeled using a simple physically motivated ansatz such as the halo model. We point out that there are several velocity dispersions that enter into the detailed RSD analysis with very different amplitudes, which can be approximately predicted by the halo model. In contrast to previous models our approach systematically includes all of the terms at a given order in PT and provides a physical interpretation for the small scale dispersion values. We investigate RSD power spectrum as a function of μ, the cosine of the angle between the Fourier mode and line of sight, focusing on the lowest order powers of μand multipole moments which dominate the observable RSD power spectrum. Overall we find considerable success in modeling many, but not all, of the terms in this expansion.

Figures (13)

• Figure 1: $P_{00}(k)$ power spectrum term is plotted at four redshifts $z=0.0,~0.5,~1.0$ and $2.0$. We show linear result (black, dotted), one loop PT (blue, solid), two loop closure (green, dashed), corrected Zel'dovich (red, long-dashed) of Tassev:2011ac , simple Zel'dovich (magenta, dot-dashed) and simulation measurements (black dots). The error bars show the variance among realizations in simulations. The power spectrum is divided by no-wiggle fitting formula from Eisenstein:1997ik, to reduce the dynamic range.
• Figure 2: $k$-dependence of $P^{ss}_{01}$ term of redshift power spectrum is plotted at four redshifts $z=0.0,~0.5,~1.0$ and $2.0$. This term has simple $\mu^2$ dependence in all nonlinear orders. Here we show linear Kaiser result (black, dotted), one loop PT (blue, solid), corrected Zel'dovich (red, dashed) model from Tassev:2011ac, simple Zel'dovich (magenta, dot-dashed), and simulation measurements (black dots). The error bars show the variance among realizations in simulations. The power spectra are divided by second, no-wiggle, term of Kaiser formula to reduce the dynamic range.
• Figure 3: $k$-dependence of the scalar part of $P^{ss}_{11}$ term. Power spectrum is plotted at four redshifts $z=0.0,~0.5,~1.0$ and $2.0$. This term has a simple $\mu^4$ dependence. Here we show linear Kaiser (black, dotted) and one loop PT (blue, solid) result, and compare it to simulation measurements (black dots). The error bars show the variance among realizations in simulations. The power spectra are divided by the no-wiggle linear term.
• Figure 4: $k$-dependence of scalar and vector part of $P^{ss}_{11}$ term of the redshift power spectrum is plotted at four redshifts $z=0.0,~0.5,~1.0$ and $2.0$, assuming $\mu=1$. Scalar part has simple $\mu^4$ angular dependence while the vector part has $~\mu^2(1-\mu^2)$ angular dependence at all (nonlinear) orders. We show linear/Kaiser result (black, dotted), one loop PT result for scalar part (blue, solid), one loop PT result for vector part (lighter red, dashed), relevant part of two loop PT for vector part (red, solid) and simulations for scalar (blue points) and vector (red points) part. We also show scalar contributions of $C_{11}$ term at one (lighter green, dashed) and two (green, solid) loop order.
• Figure 5: $k$-dependence of isotropic and anisotropic part of $P^{ss}_{02}$ term of redshift power spectrum is plotted at four redshifts $z=0.0,~0.5,~1.0$ and $2.0$. Isotropic part $P^{ss,I}_{02}$, computed in one loop PT (red, solid) is plotted, as well as using the model presented above (blue, dot-dashed). Isotropic part has simple $~\mu^2$ angular dependence while the anisotropic part $P^{ss,A}_{02}$ (green, dashed) has $~\mu^2(3\mu^2-1)/2$ angular dependence. Simulation measurements (dots) for the corresponding terms are also presented. The power spectra are divided by $k^2\sigma_v^2P^{\text{nw}}_L$ without the wiggles.