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Redshift-Space Distortions, Pairwise Velocities and Nonlinearities

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Roman Scoccimarro

Astrophysics · Submitted 12 Jul 2004 · 67 references

Pith's verdict: The exact streaming relation and the demolition of the dispersion model are the real contributions; the deferred B_sigma term is a completeness gap in a secondary claim, not a crack in the headline.

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arXiv IDastro-ph/0407214 (version 2)
Pith ID5VVE4EVA, Pith's permanent record of this paper and its review
CitationRoman Scoccimarro. "Redshift-Space Distortions, Pairwise Velocities and Nonlinearities." arXiv:astro-ph/0407214 (2004). https://pith.science/paper/5VVE4EVA
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The paper

The paper claims

The mapping from real space to redshift space is exactly a convolution with the pairwise velocity distribution, and that distribution is non-Gaussian at every scale, so the Kaiser formula fails well before the nonlinear scale.

The reading

The paper's aim is to write down, with no approximation beyond the plane-parallel limit, how two-point clustering statistics in real space become the distorted statistics we observe in redshift space. The answer is a streaming model: the redshift-space correlation function is the real-space one convolved with the probability distribution of line-of-sight pairwise velocities, and the power spectrum is the same statement in Fourier space. If that identity is right, then everything hinges on the shape of that velocity distribution, and the paper's central warning is that the distribution is strongly non-Gaussian at all separations, so any model that assumes otherwise is modelling the wrong object. The practical stakes are large: the standard Kaiser-plus-dispersion recipe used to extract cosmological parameters from galaxy surveys is shown to correspond to a pairwise velocity distribution that is singular at zero velocity and discontinuous, and the paper argues that the distribution itself cannot be inverted out of clustering data at all. What can be salvaged is a corrected large-scale expansion of the power spectrum and a multipole combination that still recovers the real-space power spectrum rather well.

What carries the argument

The load-bearing object is the pairwise velocity generating function $Z(lambda,r) = [1+xi(r)] M(lambda,r) = langle e^{lambda Delta u_z} (1+delta)(1+delta') rangle$, evaluated at $lambda = i f k_z$ for the power spectrum. Everything else is a consequence of it: the streaming-model integral for $xi_s$, the cumulant expansion that writes $Z$ in terms of connected density-velocity correlators, and the small-$lambda$ expansion that produces the modified Kaiser limit. The second piece of machinery is the split of the pairwise dispersion into a Gaussian part $langle Delta u_z^2 rangle$ and non-Gaussian pieces $langle Delta u_z^2 (delta+delta') rangle$ and $langle Delta u_z^2 delta delta' rangle_c$, the last of which survives as a constant at infinite separation and therefore guarantees that linear theory never describes the second moment of pairwise velocities.

Extended reading notes

Core claim

The paper establishes that the redshift-space two-point correlation function is exactly $1+xi_s(s_parallel,s_perp) = int dr_parallel [1+xi(r)] P(r_parallel - s_parallel, r)$, where $P$ is the pairwise velocity PDF at separation $r$, with the power spectrum given by the corresponding generating function; this holds including all nonlinearities of both the dynamics and the real-to-redshift mapping, with the plane-parallel approximation as the only assumption. Measuring $P$ in a large N-body simulation, the author finds exponential wings, non-zero skewness, and kurtosis above unity at all scales from $r sim 2$ to $100$ Mpc$/h$, and explains why no Gaussian limit is reached: the velocity difference between two points is insensitive to modes larger than their separation but is weighted only by $k^{-1}$ for smaller modes, so it never stops sampling nonlinearity. Two corollaries follow. First, the widely used dispersion model $P_s(k,mu) = P_g(k)(1+beta mu^2)^2 / (1 + k^2 mu^2 sigma_p^2/2)$ implies a pairwise PDF with a delta function at the origin and a jump discontinuity whose size is set by the mean infall, which the simulations do not resemble. Second, because the true PDF depends on both scale and angle, redshift-space clustering cannot be deconvolved to yield it; the ratio $P_s(k)/P(k)$ generates only even moments and returns a symmetric pseudo-PDF whose second and fourth moments are pair-weighted mixtures of dispersion, infall, and skewness. On the constructive side, expanding real space about redshift space gives a large-scale power spectrum that differs from Kaiser by $f^2 k_z^2 (sigma_v^2 + A_sigma)$ and a non-Gaussian term $B_sigma(k)$, so the Kaiser formula holds only for $k mu < 0.2$ $h$ Mpc$^{-1}$; one-loop perturbation theory shows the velocity divergence power spectrum is suppressed well below linear theory at large scales because of tidal fields; and the multipole combination $P_0 - P_2/2 + 3P_4/8$ recovers the real-space power spectrum to $k sim 0.2$-$0.3$ $h$ Mpc$^{-1}$ even when the Kaiser form does not apply.

Load-bearing premise

The specific correction terms in the large-scale power spectrum rest on expanding the pairwise velocity distribution about redshift space and integrating term by term, an expansion the author himself says does not converge uniformly, and on a non-Gaussian coefficient whose value is quoted from a companion paper rather than derived here.

Editorial extensions

If this is right

  • Fitting a single effective pairwise dispersion $sigma_p$ to a measured quadrupole-to-monopole ratio does not recover a physical velocity distribution, so the fitted value cannot be interpreted as a moment of the true pairwise velocities.
  • The Kaiser limit is valid only for modes with a small line-of-sight component, roughly $k mu < 0.2$ $h$ Mpc$^{-1}$; beyond that, velocity dispersion terms of both Gaussian and non-Gaussian origin must be carried explicitly.
  • Using the redshift-space monopole as a stand-in for the shape of the real-space power spectrum biases constraints on the spectral index, its running, and the neutrino mass, because the monopole is suppressed by about 10 percent already at $k = 0.1$ $h$ Mpc$^{-1}$.
  • The velocity divergence power spectrum is suppressed relative to linear theory at scales far larger than the nonlinear scale, because velocity fields respond more strongly to tidal gravitational fields than the density does.
  • The real-space power spectrum can still be recovered from the $ell = 0, 2, 4$ multipoles by the same linear combination used in the Kaiser limit, since that combination relies only on Legendre orthogonality.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the velocity-difference PDF has exponential tails even in the volume-weighted (unclustered) limit, then the same non-Gaussianity should appear in other tracers of large-scale flows, such as the pairwise velocity statistics of voids or of low-density regions, which would be a clean independent check.
  • Because the non-Gaussian corrections scale as different powers of $sigma_8$ and of the linear bias than the Kaiser terms do, combining monopole, quadrupole, and hexadecapole at several scales could in principle break the $beta$-$sigma_8$ degeneracy that a single-parameter dispersion fit cannot.
  • The failure of the deconvolution argument suggests that any claimed measurement of the pairwise velocity PDF from survey clustering should be reported as a pair-weighted, symmetric functional of the true moments, and that comparisons with simulation PDFs should be made through that same functional rather than directly.
  • The suppression of the velocity divergence spectrum at large scales implies that linear-theory-based estimates of $f sigma_8$ from redshift-space distortions are systematically low, and that the size of the bias should depend on the shape of the linear power spectrum through the tidal term.
Original abstract

We derive the exact relationship, including all non-linearities, between real-space and redshift-space two-point statistics through the pairwise velocity distribution function. We show using numerical simulations that the pairwise velocity PDF is strongly non-Gaussian at all scales, and explain why this is so. We caution that a commonly used ansatz to model the redshift-space power spectrum gives rise to an unphysical distribution of pairwise velocities, and show that it is in general impossible to derive the distribution from measurements of redshift-space clustering. Methods that claim to do this obtain instead something else, whose properties we derive. We provide a general derivation of the large-scale limit of the redshift-space power spectrum and show that it differs from the Kaiser formula by terms that depend on Gaussian and non-Gaussian contributions to the velocity dispersion of large-scale flows. We also show that the large-scale evolution of velocity fields is not well described by linear theory and discuss how this impacts the redshift-space power spectrum. Finally, we stress that using the monopole of the redshift-space power as an indicator of the real-space power spectrum shape can lead to systematic effects in the determination of cosmological parameters; nevertheless a simple procedure is able to recover the large-scale real-space power spectrum rather well.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,752 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.53 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.94 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We derive the exact relationship, including all non-linearities, between real-space and redshift-space two-point statistics through the pairwise velocity distribution function. We show using numerical simulations that the pairwise velocity PDF is strongly non-Gaussian at all scales, and explain why this is so. We caution that a commonly used ansatz to model the redshift-space power spectrum gives rise to an unphysical distribution of pairwise velocities, and show that it is in general impossible to derive the distribution from measurements of redshift-space clustering. Methods that claim to do this obtain instead something else, whose properties we derive. We provide a general derivation of the large-scale limit of the redshift-space power spectrum and show that it differs from the Kaiser formula by terms that depend on Gaussian and non-Gaussian contributions to the velocity dispersion of large-scale flows. We also show that the large-scale evolution of velocity fields is not well described by linear theory and discuss how this impacts the redshift-space power spectrum. Finally, we stress that using the monopole of the redshift-space power as an indicator of the real-space power spectrum shape can lead to systematic effects in the determination of cosmological parameters; nevertheless a simple procedure is able to recover the large-scale real-space power spectrum rather well.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We derive the exact relationship, including all non-linearities, between real-space and redshift-space two-point statistics through the pairwise velocity distribution function. contribution 0.89 0.96 0.24 asserted 1.00
s1 We show using numerical simulations that the pairwise velocity PDF is strongly non-Gaussian at all scales, and explain why this is so. result 0.94 0.96 0.16 asserted 1.00
s2 We caution that a commonly used ansatz to model the redshift-space power spectrum gives rise to an unphysical distribution of pairwise velocities, and show that it is in general impossible to derive the distribution from measurements of redshift-space clustering. result 0.86 0.84 0.35 asserted 1.00
s3 Methods that claim to do this obtain instead something else, whose properties we derive. result 0.65 0.80 0.09 asserted 0.99
s4 We provide a general derivation of the large-scale limit of the redshift-space power spectrum and show that it differs from the Kaiser formula by terms that depend on Gaussian and non-Gaussian contributions to the velocity dispersion of large-scale flows. result 0.66 0.96 0.17 asserted 1.00
s5 We also show that the large-scale evolution of velocity fields is not well described by linear theory and discuss how this impacts the redshift-space power spectrum. result 0.91 0.90 0.10 asserted 1.00
s6 Finally, we stress that using the monopole of the redshift-space power as an indicator of the real-space power spectrum shape can lead to systematic effects in the determination of cosmological parameters; nevertheless a simple procedure is able to recover the large-scale real-space power spectrum rather well. interpretation 0.67 0.85 0.25 hedged possibility 0.59

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.95.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1: The parallel to the line of sight pairwise velocity
Figure 2. FIG. 2: Same as Fig. 1 but for pairs perpendicular to the
Figure 3. FIG. 3: Moments of pairwise velocities parallel to the
Figure 4. shows the result for the redshift-space correlation function ξs(sk, s⊥) in the exact Gaussian case Eq. (46) (solid) and the Kaiser limit (dashed), Eq. (61) below. Notice that there are significant deviations even at large scales, predominantly at small s⊥, we explain why this is the case in sec￾tion IV below. Since the correlation amplitude is so much smaller in this region compared to small sk, when multipoles are calculated integrating along fixed s the results are close to their Kaiser limit values. It is apparent that the qualitative behav￾ior of the corrections are to make the…
Figure 5. FIG. 5: Same as Fig. 4 but at small scales. Note the
Figure 6. FIG. 6: Non-linear corrections to the density-density
Figure 7. FIG. 7: Contours of the redshift-space power spectrum

Showing 7 of the paper's 8 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:astro-ph/0407214 by the authors.

References
How later papers used it

211 later papers cite it

Evidence: substantial. 203 of the citing papers share no author with it, and 49 of those carry a confident label from Pith's citation classifier.

Pith read 21 of the 423 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Builds on it 1 of 3 papers with this label

  • Extended the nonlinear approach to RSD of this paper to the wide-angle regime, deriving a non-perturbative expression for the wide-angle redshift-space correlation function.

    Nonlinear Redshift-Space Distortions on the Full Sky (2023) · 1.00 · paraphrased: the licence does not allow quoting

Uses its method or tool 6 of 40 papers with this label · 1 by the same authors

By the same authors

How the literature uses it

122 arXiv papers cite this in 273 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 174 sentences without a confident label

Builds on it 3 papers
Uses its method or tool 21 papers
Compares with it 7 papers
Cites it as support 7 papers
Cites it as background 40 papers

Discussion

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